C-PROJECTIVE GEOMETRY

DAVID M.J. CALDERBANK, MICHAEL G. EASTWOOD, VLADIMIR S. MATVEEV, AND KATHARINA NEUSSER

Abstract. We develop in detail the theory of c-projective geometry, a natural analogue of projective differential geometry adapted to complex manifolds. We realise it as a type of parabolic geometry and describe the associated Cartan or tractor connection. A Kähler manifold gives rise to a c-projective structure and this is one of the primary motivations for its study. The existence of two or more Kähler metrics underlying a given c-projective structure has many ramifications, which we explore in depth. As a consequence of this analysis, we prove the Yano–Obata Conjecture for complete Kähler manifolds: if such a manifold admits a one parameter group of c-projective transformations that are not affine, then it is complex projective space, equipped with a multiple of the Fubini–Study metric.

Introduction

C-projective geometry is a natural analogue of real projective differential geometry for complex manifolds. Like projective geometry, it has many facets, which have been discovered and explored independently and repeatedly over the past sixty years. Our aim in this work is to develop in detail a unified theory of c-projective geometry, which highlights its relation with real projective geometry as well as its differences.

Projective geometry is a classical subject concerned with the behaviour of straight lines, or more generally, (unparametrised) geodesic curves of a metric or affine connection. It has been known for some time [66, 99] that two non-proportional metrics can have the same geodesic curves: central projection takes great circles on the n-sphere, namely the geodesics for the round metric, to straight lines in Euclidean n-space, namely geodesics for the flat metric. The quotient of the round n-sphere under the antipodal identification may be identified with the flat model for n-dimensional projective geometry: the real projective n-space ℝℙ^{n}, viewed as a homogeneous space under the group PSL(n+1, ℝ) of projective transformations, which preserve the family of (linearly embedded) projective lines ℝℙ^{1} ↪ ℝℙ^{n}. More generally, a projective structure on a smooth n-manifold (for n ≥ 2) is an equivalence class of torsion-free affine connections having the same geodesic curves. In this setting, it is a nontrivial and interesting question whether these curves are the geodesic curves of a (pseudo-)Riemannian metric, i.e. whether any connection in the projective equivalence class preserves a nondegenerate metric, possibly of indefinite signature. Such projective structures are called metrisable and the corresponding metrics compatible. Rather surprisingly, the partial differential equations controlling the metrisability of a given projective structure can be set up as a linear system [13, 40, 70, 94]. More precisely, there is a projectively invariant linear differential operator acting on symmetric contravariant 2-tensors such that the nondegenerate elements of its kernel correspond to compatible metrics.

This article was initiated when its authors participated in a workshop at the Kioloa campus of the Australian National University in March 2013. We would like to thank the The Edith and Joy London Foundation for providing the excellent facilities at Kioloa. We would also like to thank the Group of Eight, Deutscher Akademischer Austausch Dienst, Australia-Germany Joint Research Cooperation Scheme for financially supporting the workshop in 2013 and a subsequent Kioloa workshop in 2014; for the latter, we thank, in addition, FSU Jena and the Deutsche Forschungsgemeinschaft (GK 1523/2)

for their financial support. The fourth author was also supported during part of this project by the Eduard Čech Institute. We would also like to thank the latter for supporting a meeting of the first, second and fourth author at the Mathematical Institute at Charles University in July 2014.

In modern language, a projective structure determines a canonical Cartan connection [39] modelled on ℝℙ^{n}, and hence projective geometry is a parabolic geometry [36, 42]. In these terms, the metrisability operator is a first BGG (Bernstein– Gelfand–Gelfand) operator, which is a differential operator of finite type [43]. Its solutions correspond to parallel sections of a bundle with connection, which is, up to curvature corrections, a linear representation of the Cartan connection. The kernel is thus finite-dimensional; it is zero for generic projective structures, with the maximal dimension attained on the flat model ℝℙ^{n}. The parabolic viewpoint on projective geometry has proven to be very useful, for example in understanding projective compactifications of Einstein metrics [31, 34], the geometry of holonomy reductions of projective structures [6], and (solving problems posed by Sophus Lie in 1882) projective vector fields on surfaces [22, 75].

Projective geometry has been linked to the theory of finite dimensional integrable systems with great success: the equation for symmetric Killing tensors is projectively invariant [42], and (consequently) the existence of two projectively equivalent metrics on a manifold implies the existence of nontrivial integrals for the geodesic flows of both metrics. This method has been effectively employed when the manifold is closed or complete (see e.g. [72, 74]). Moreover, the integrability of many classically studied integrable geodesic flows (e.g., on ellipsoids) is closely related to the existence of a projectively compatible metric, and many geometric structures that lead to such integrable geodesic flows have been directly related to the existence of a projectively compatible metric, see e.g. [9, 13].

C-projective geometry arises when one retells this story, mutatis mutandis, for complex or, indeed, almost complex manifolds, i.e. smooth manifolds equipped with an almost complex structure J, which is a smooth endomorphism of the tangent bundle such that J^{2} = − Id. On such a manifold M, the relevant (pseudo-)Riemannian metrics are Hermitian with respect to J, i.e. J-invariant, and the relevant affine connections are those which preserve J, called complex connections. Such connections cannot be torsion-free unless the almost complex structure is integrable, i.e. its Nijenhuis tensor vanishes identically [89]. This holds in particular if the Levi-Civita connection of a Hermitian metric g preserves J, in which case g is called a (pseudo-)Kähler metric.

In 1947, Bochner [12, Theorem 2] observed that any two metrics that are Kähler with respect to the same complex structure cannot be projectively equivalent (i.e. have the same geodesic curves) unless they are affinely equivalent (i.e. have the same Levi-Civita connection). This led Otsuki and Tashiro [90] to introduce a broader class of curves, which they called “holomorphically flat”, and which depend on both the connection and the (almost) complex structure. We refer to these curves as J-planar: whereas a geodesic curve for an affine connection ∇ is a curve c whose acceleration ∇ċċ is proportional to its velocity ċ, a J-planar curve is one whose acceleration is in the linear span of ċ and Jċ. On a Riemann surface, therefore, all curves are J-planar. The other common manifold where it is possible to see all J-planar curves without computation is complex projective space with its Fubini–Study connection. The point here is that the linearly embedded complex lines ℂℙ^{1} ↪ ℂℙ^{n} are totally geodesic. Therefore, the J-planar curves on ℂℙ^{n} are precisely the smooth curves lying within such complex lines. Viewed in a standard affine coordinate patch ℂ^{n} ↪ ℂℙ^{n}, the J-planar curves are again the smooth curves lying inside an arbitrary complex line {az +b} ⊂ ℂ^{n} but otherwise unconstrained. Evidently, these are the intrinsic J-planar curves for the flat connection on ℂ^{n}.

The J-planar curves provide a nontrivial notion of projective equivalence in complex differential geometry, due to Otsuki and Tashiro [90] in the Kähler setting, and Tashiro [98] for almost complex manifolds in general. Two complex connections on an almost complex manifold (M, J) are c-projectively equivalent if they have the same torsion and the same J-planar curves. An almost c-projective manifold is a complex manifold (M, J) equipped with a c-projective equivalence class of such connections. If J is integrable, we follow the usual convention and drop the word “almost” to arrive at the notion of a c-projective manifold. We caution the reader that Otsuki and Tashiro [90], and many later researchers, refer to “holomorphically projective correspondences”, rather than c-projective equivalences, and many authors use the terminology “h-projective” or “holomorphic(ally) projective” instead of “c-projective”. We avoid their terminology because the connections in a c-projective class are typically not holomorphic, even if the complex structure is integrable; similarly, we avoid the term “complex projective structure”, which is often used for the holomorphic analogue of a real projective structure, or related concepts.

During the decades following Otsuki and Tashiro’s 1954 paper, c-projective structures provided a prominent research direction in Japanese and Soviet differential geometry. Many of the researchers involved had some background in projective geometry, and the dominant line of investigation sought to generalise methods and results from projective geometry to the c-projective setting. This was a very productive direction, with more than 300 publications appearing in the relatively short period from 1960 to 1990. One can compare, for example, the surveys by Mikeš [82, 83], or the papers of Hiramatu [53, 54], to see how successfully c-projective analogues of results in projective geometry were found. In particular, the linear system for c-projectively equivalent Kähler metrics was obtained by Domashev and Mikeš [41], and its finite type prolongation to a connection was given by Mikeš [81].

Relatively recently, the linear system for c-projectively equivalent Kähler metrics has been rediscovered, under different names and with different motivations. On a fixed complex manifold, a compatible (pseudo-)Kähler metric is determined uniquely by its Kähler form (a compatible symplectic form), and under this correspondence, c-projectively equivalent Kähler metrics are essentially the same as Hamiltonian 2forms defined and investigated in Apostolov et al. [2, 3, 4, 5]: the defining equation [2, (12)] for a Hamiltonian 2-form is actually algebraically equivalent to the metrisability equation (125). In dimension ≥ 6, c-projectively equivalent metrics are also essentially the same as conformal Killing (or twistor) (1, 1)-forms studied in [86, 92, 93], see [2, Appendix A] or [78, §1.3] for details.

The work of [2, 3] provides, a postiori, local and global classification results for c-projectively equivalent Kähler metrics, although the authors were unaware of this interpretation, nor the pre-existing literature. Instead, as explained in [2, 3] and [26], the notion and study of Hamiltonian 2-forms was motivated by their natural appearance in many interesting problems in Kähler geometry, and the unifying role they play in the construction of explicit Kähler metrics on projective bundles. In subsequent papers, e.g. [4, 5], Hamiltonian 2-form methods were used to construct many new examples of Kähler manifolds and orbifolds with interesting properties.

Another independent line of research closely related to c-projectively equivalent metrics (and perhaps underpinning the utility of Hamiltonian 2-forms) appeared within the theory of finitely dimensional integrable systems. C-projectively equivalent metrics are closely related (see e.g. [61]) to the so-called Kähler–Liouville integrable systems of type A introduced and studied by Kiyohara in [59]. In fact, Topalov [100] (see also [60]) shows that generic c-projectively equivalent Kähler metrics have integrable geodesic flows, cf. [101] for the analogous result in the projective case. On the one hand, integrability provides, as in projective geometry, a number of new methods that can be used in c-projective geometry. On the other hand, examples from c-projective geometry turn out to be interesting for the theory of integrable systems, since there are only a few known examples of Kähler metrics with integrable geodesic flows.

Despite the many analogies between results in projective and c-projective geometry, there seem to be very few attempts in the literature to explain why these two subjects are so similar. In 1978, it was noted by Yoshimatsu [103] that c-projective manifolds admit canonical Cartan connections, and this was generalised to almost c-projective manifolds by Hrdina [55] in 2009. Thus c-projective geometry, like projective geometry, is a parabolic geometry; its flat model is ℂℙ^{n}, viewed as a homogeneous space under the group PSL(n + 1, ℂ) of projective transformations, which preserve the J-planar curves described above. Despite this, c-projective structures have received very little attention in the parabolic geometry literature: apart from the work of Hrdina, and some work in dimension 4 [28, 80], they have only been studied in [6], where they appear as holonomy reductions of projective geometries. A possible explanation for this oversight is that PSL(n+1, ℂ) appears in c-projective geometry as a real Lie group and, as such, its complexification is semisimple, but not simple. This is related to the subtle point that most interesting c-projective structures are not holomorphic.

The development of c-projective geometry, as described above, has been rather nonlinear until relatively recently, when a number of independent threads have converged on a coherent set of ideas. However, a thorough description of almost c-projective manifolds in the framework of parabolic geometries is lacking in the literature. We therefore believe it is timely to lay down the fundamentals of such a theory.

The article is organised as follows. In Section 1, we survey the background on almost complex manifolds and complex connections. As we review in Section 1.2, the torsion of any complex connection on an almost complex manifold, of real dimension 2n ≥ 4, naturally decomposes into five irreducible pieces, one of which is invariantly defined and can be identified as the Nijenhuis tensor. All other pieces can be eliminated by a suitable choice of complex connection, which we call minimal. In first four sections of the article we carry along the Nijenhuis tensor in almost all calculations and discussions.

Section 2 begins with the classical viewpoint on almost c-projective structures, based on J-planar curves and equivalence classes of minimal complex connections [90]. We then recall the notion of parabolic geometries and establish, in Theorem 2.8, an equivalence of categories between almost c-projective manifolds and parabolic geometries with a normal Cartan connection, modelled on ℂℙ^{n}.

As a consequence of this parabolic viewpoint, we can associate a fundamental local invariant to an almost c-projective manifold, namely the curvature κ of its normal Cartan connection; furthermore, κ ≡ 0 if and only if the almost c-projective manifold is locally isomorphic to ℂℙ^{n} equipped with its standard c-projective structure. Since the Cartan connection is normal (for this we need the complex connections to be minimal), its curvature is a 2-cycle for Lie algebra homology, and is uniquely determined by its homology class, also known as the harmonic curvature. We construct and discuss this curvature in section 2.7. For almost c-projective structures there are three irreducible parts to the harmonic curvature. One of the pieces is the Nijenhuis tensor, which is precisely the obstruction to the underlying almost complex manifold actually being complex. One of the other two parts is precisely the obstruction to there being a holomorphic connection in the c-projective class. When it vanishes we end up with holomorphic projective geometry, i.e. ordinary projective differential geometry but in the holomorphic category. The remaining piece can then be identified with the classical projective Weyl curvature (for n ≥ 3) or Liouville curvature (for n = 2).

Another consequence of the parabolic perspective is that representation theory is brought to the fore, both as the appropriate language for discussing natural bundles on almost c-projective manifolds, and also as the correct tool for understanding invariant differential operators on the flat model, and their curved analogues. The various BGG complexes on ℂℙ^{n} and their curved analogues are systematically introduced and discussed in Section 3.

In particular, there is a BGG operator that controls the metrisability of a c-projective structure just as happens in the projective setting. A large part of this article is devoted to the metrisability equation, which we introduce in Section 4, where we also obtain its prolongation to a connection, not only for compatible (pseudo-)Kähler metric, but also in the non-integrable case of quasi-Kähler or (2,1)-symplectic structures. For the remainder of the article, we suppose that the Nijenhuis tensor vanishes, in other words that we are starting with a complex manifold. In this case, a compatible metric is exactly a (pseudo-)Kähler metric (and a normal solution of the metrisability equation corresponds to a (pseudo-)Kähler–Einstein metric). We shall also restrict our attention to metric c-projective structures, i.e. to the metrisable case where the c-projective structure arises from a (pseudo-)Kähler metric. Borrowing terminology from the projective case, we refer to the dimension of the solution space of the metrisability equation as the (degree of ) mobility of the metric c-projective structure (or of any compatible (pseudo-)Kähler metric). We are mainly interested in understanding when the metric c-projective structure has mobility at least two, and the consequences this has for the geometry and topology of the manifold.

In Section 5, we develop the consequences of mobility for integrability, by showing that a pencil (two dimensional family) of solutions to the metrisability equation generates a family of holomorphic Killing vector fields and Hermitian symmetric Killing tensors, which together provide commuting linear and quadratic integrals for the geodesic flow of any metric in the pencil. In Section 6, we study an important, but somewhat mysterious, phenomenon in which tractor bundles for metric c-projective geometries are naturally equipped with congenial connections, which are neither induced by the normal Cartan connection nor equal to the prolongation connection, but which have the property that their covariant constant sections nevertheless correspond to solutions of the corresponding first BGG operator.

We bring these tools together in Section 7, where we establish the Yano–Obata Conjecture for complete Kähler manifolds, namely that the identity component of the group of c-projective transformations of the manifold consists of affine transformations unless the manifold is complex projective space equipped with a multiple of the Fubini– Study metric. This result is an analogue of the the Projective Lichnerowicz Conjecture obtained in [73, 74], but the proof given there does not generalise directly to the cprojective situation. Our proof also differs from the proof for closed manifolds given in [77], and makes use of many preliminary results obtained by the methods of parabolic geometry, which also apply in the projective case.

Here, and throughout the article, we see that not only results from projective geometry, but also methods and proofs, can be generalised to the c-projective case, and we explain why and how. We hope that this article will set the scene for what promises to be an interesting series of further developments in c-projective geometry. In fact, several such developments already appeared during our work on this article, which we discuss in Section 8.

Contents

Introduction 1 1. Almost complex manifolds 7 1.1. Real and complex viewpoints 7 1.2. Complex connections 10 2. Elements of c-projective geometry 11 2.1. Almost c-projective structures 11 2.2. Parabolic geometries 15 2.3. Almost c-projective manifolds as parabolic geometries 16 2.4. The curvature of the canonical Cartan connection 19 2.5. Curvature operators on c-projective densities 25 2.6. The curvature of complex projective space 26 2.7. The harmonic curvature 27 3. Tractor bundles and BGG sequences 28 3.1. Standard complex tractors 29 3.2. Cone description of almost c-projective structures 33 3.3. BGG sequences 34 3.4. Adjoint tractors and infinitesimal automorphisms 40 4. Metrisability of almost c-projective structures 43 4.1. Almost Hermitian manifolds 44 4.2. Other curvature decompositions 48 4.3. Metrisability of almost c-projective manifolds 50 4.4. The metrisability equation and mobility 53 4.5. Prolongation of the metrisability equation 56 4.6. The c-projective Hessian 59 4.7. Prolongation of the c-projective Hessian 61 5. Metrisability, conserved quantities and integrability 63 5.1. Conserved quantities for the geodesic flow 63 5.2. Holomorphic Killing fields 64 5.3. Hermitian symmetric Killing tensors 66 5.4. Metrisability pencils, Killing fields and Killing tensors 68 5.5. Conserved quantities on c-projective manifolds 71 5.6. The local complex torus action 75 5.7. Local classification 76 6. Metric c-projective structures and nullity 79 6.1. Metric projective geometry and projective nullity 79 6.2. C-projective nullity 83 6.3. Mobility, nullity, and the special tractor connection 86 6.4. The standard tractor bundle for metric c-projective structures 94 6.5. Special tractor connections and the complex cone 95 6.6. The c-projective Hessian, nullity, and the Tanno equation 97 7. Global results 99 7.1. Complete Kähler metrics with nullity 100 7.2. The Yano–Obata Conjecture for complete Kähler manifolds 103 7.3. The proof of the Yano–Obata Conjecture 104 8. Outlook 111 8.1. Metrisability and symmetry 111 8.2. Applications in Kähler geometry 112 8.3. Projective parabolic geometries 113 References 113

1. Almost complex manifolds

Recall that an almost complex structure on a smooth manifold M is a smooth endomorphism J of the tangent bundle T M of M that satisfies J^{2} = − Id. Equivalently, an almost complex structure makes T M into a complex vector bundle in which multiplication by i is decreed to be the real endomorphism J. In particular, the dimension of M is necessarily even, say 2n, and an almost complex structure is yet equivalently a reduction of structure group to GL(n, ℂ) ⊂ GL(2n, ℝ).

1.1. Real and complex viewpoints. If M is a complex manifold in the usual sense of being equipped with holomorphic transition functions, then T M is a complex vector bundle and multiplication by i defines a real endomorphism T M → T M, which we write as J. This is enough to define the holomorphic structure on M: holomorphic functions may be characterised amongst all smooth complex-valued functions f = u+iv as satisfying Xu = (JX)v for all vector fields X (the Cauchy–Riemann equations).

Thus, complex manifolds may be regarded as a subclass of almost complex manifolds and the celebrated Newlander–Nirenberg Theorem tells us how to recognise them:

Theorem 1.1 (Newlander–Nirenberg, [89]). An almost complex manifold (M, J) is a complex manifold if and only if the tensor

N^{J}(X, Y) := [X, Y] − [JX, JY] + J([JX, Y] + [X, JY])  (1)

vanishes for all vector fields X and Y on M, where [ , ] denotes the Lie bracket of vector fields.

Note that N^{J}: T M × T M → T M is a 2-form with values in T M, which satisfies N^{J}(JX, Y) = −JN^{J}(X, Y). It is called the Nijenhuis tensor of J. When N^{J} vanishes we say that the almost complex structure J is integrable. This viewpoint on complex manifolds, as even-dimensional smooth manifolds equipped with integrable almost complex structures, turns out to be very useful especially from the differential geometric viewpoint.

It is useful to complexify the tangent bundle of M and decompose the result into eigenbundles under the action of J. Specifically,

ℂT M = T^{1,0}M ⊕ T^{0,1}M = {X s.t. JX = iX} ⊕ {X s.t. JX = −iX}.  (2)

Notice that T^{0,1}M = \overline{T^{1,0}M}. There is a corresponding decomposition of the complexified cotangent bundle, which we write as ∧^{1}M or simply ∧^{1} if M is understood. Specifically,

∧^{1} = ∧^{0,1} ⊕ ∧^{1,0} = {ω s.t. Jω = −iω} ⊕ {ω s.t. Jω = iω},  (3)

where sections of ∧^{1,0} respectively of ∧^{0,1} are known as 1-forms of type (1, 0) respectively (0, 1), see e.g. [62]. Notice that the canonical complex linear pairing between ℂT M and ∧^{1}M induces natural isomorphisms ∧^{0,1} = (T^{0,1})^{∗} and ∧^{1,0} = (T^{1,0})^{∗} of complex vector bundles.

It is convenient to introduce abstract indices [91] for real or complex tensors on M and also for sections of the bundles T^{1,0}M, ∧^{0,1}, and so on. Let us write X^{α} for real or complex fields and ω_{α} for real or complex 1-forms on M. In local coordinates α would range over 1, 2, …, 2n where 2n is the dimension of M. Let us denote by X^{a} a section of T^{1,0}M. In any frame, the index a would then range over 1, 2, …, n. Similarly, let us write Xā for a section of T^{0,1}M and the conjugate linear isomorphism T^{0,1}M = \overline{T^{1,0}M} as X^{a} ↦ \overline{X^{a}} = \overline{X}ā. Accordingly, sections of ∧^{1,0} and ∧^{0,1} will be denoted by ω_{a} and ωā respectively, and the canonical pairings between T^{1,0}M and ∧^{1,0}, respectively T^{0,1}M and ∧^{0,1}, written as X^{a}ω_{a}, respectively Xāωā, an abstract index counterpart to the Einstein summation convention.

We shall need the complex linear homomorphism ℂT M → T^{1,0}M defined as projection onto the first summand in the decomposition (2) and given explicitly as X ↦ \frac{1}{2}(X − iJX). It is useful to write it in abstract indices as

X^{α} ↦ Π_{α}^{a}X^{α}.

It follows that the dual homomorphism ∧^{1,0} ↪ ∧^{1} is given in abstract indices by

ω_{a} ↦ Π_{α}^{a}ω_{a}

and also that the homomorphisms ℂT M → T^{0,1}M and ∧^{0,1} ↪ ∧^{1} are given by

X^{α} ↦ \overline{Π}ā ; αX^{α} and ωā ↦ \overline{Π}ā ; αωā,

respectively.

Let us denote by X^{a} ↦ Π_{a}^{α}X^{a}, the inclusion T^{1,0}M ↪ ℂT M, paying attention to the distinction in their indices between Π_{α}^{a} and Π_{a}^{α}. Various identities follow, such as Π_{a}^{α}Π_{α}^{b} = δ_{a}^{b}, where the Kronecker delta δ_{a}^{b} denotes the identity transformation on T^{1,0}M. The symbol Π_{a}^{α} also gives us access to the dual and conjugate homomorphisms. Thus,

ω_{α} ↦ \overline{Π}α ; āω_{α}

extracts the (0, 1)-part of a complex-valued 1-form ω_{α} on M. The following identities are immediate from the definitions

Π_{α}^{a}Π_{a}^{β} = \frac{1}{2}(δ_{α}^{β} − iJ_{α}^{β})  \overline{Π}ā ; α\overline{Π}β ; ā = \frac{1}{2}(δ_{α}^{β} + iJ_{α}^{β})
Π_{a}^{α}J_{α}^{β} = iΠ_{a}^{β}  \overline{Π}α ; āJ_{α}^{β} = −i\overline{Π}β ; ā  (4)
J_{α}^{β}Π_{β}^{a} = iΠ_{α}^{a}  J_{α}^{β}\overline{Π}ā ; β = −i\overline{Π}ā ; α.

They are indispensable for the calculations in the following sections. Further useful abstract index conventions are as follows. Quantities endowed with several indices denote sections of the tensor product of the corresponding vector bundles. Thus, a section of T M ⊗ T M would be denoted X^{αβ} whilst Φ_{α}^{β} is necessarily a section of T^{∗}M ⊗ T M or, equivalently, an endomorphism of T M, namely X^{β} ↦ Φ_{α}^{β}X^{α}, yet equivalently an endomorphism of T^{∗}M, namely ω_{α} ↦ Φ_{α}^{β}ω_{β}. We have already seen this notation for an almost complex structure J_{α}^{β}. But it is unnecessary notationally to distinguish between real- and complex-valued tensors. Thus, by ω_{α} we can mean a section of T^{∗}M or of ∧^{1}M := ℂT^{∗}M and if a distinction is warranted, then it can be made in words or by context. For example, an almost complex structure J_{α}^{β} is a real endomorphism whereas Π_{α}^{a} is necessarily complex.

Symmetry operations can also be written in abstract index notation. For example, the skew part of a covariant 2-tensor ϕ_{αβ} is \frac{1}{2}(ϕ_{αβ} − ϕ_{βα}), which we write as ϕ_{[αβ]}. Similarly, we write ϕ_{(αβ)} = \frac{1}{2}(ϕ_{αβ} + ϕ_{βα}) for the symmetric part and then ϕ_{αβ} = ϕ_{(αβ)} + ϕ_{[αβ]} realises the decomposition of vector bundles ∧^{1} ⊗ ∧^{1} = S^{2}∧^{1} ⊕ ∧^{2}. In general, round brackets symmetrise over the indices they enclose whilst square brackets take the skew part, e.g.

R_{αβ}^{γ}_{δ} ↦ R_{[αβ}^{γ}_{δ]} = \frac{1}{6}(R_{αβ}^{γ}_{δ} + R_{βδ}^{γ}_{α} + R_{δα}^{γ}_{β} − R_{βα}^{γ}_{δ} − R_{αδ}^{γ}_{β} − R_{δβ}^{γ}_{α}).

By (3) differential forms on almost complex manifolds can be naturally decomposed according to type (see e.g. [62]). We pause to examine the decomposition of 2-forms, especially from the abstract index point of view. From (3) it follows that the bundle ∧^{2} of complex-valued 2-forms decomposes into types according to

∧^{2} = ∧^{2}(∧^{0,1} ⊕ ∧^{1,0}) = ∧^{0,2} ⊕ ∧^{1,1} ⊕ ∧^{2,0}  (5)

and, as we shall make precise in Section 3.3, there is no finer decomposition available (it is a decomposition into irreducibles). Using the projectors Π_{α}^{a} and Π_{a}^{α}, we can explicitly execute this decomposition:

ω_{αβ} ↦ (\overline{Π}α ; ā\overline{Π}β ; b̄ω_{αβ}, Π_{a}^{α}\overline{Π}β ; b̄ω_{αβ}, Π_{a}^{α}Π_{b}^{β}ω_{αβ})
\overline{Π}ā ; α\overline{Π}b̄ ; βωāb̄ + 2Π_{[α}^{a}\overline{Π}b̄ ; β]ω_{ab̄} + Π_{α}^{a}Π_{β}^{b}ω_{ab} ← (ωāb̄, ω_{ab̄}, ω_{ab})

in accordance with (4). Notice that we made a choice here, namely to identify ∧^{1,1} as ∧^{1,0} ⊗ ∧^{0,1} in this order and, consequently, write forms of type (1, 1) as ω_{ab̄}. We could equally well choose the opposite convention or, indeed, use both conventions simultaneously representing a (1, 1) form as ω_{ab̄} and/or ω_{āb} but now subject to ω_{ab̄} = −ω_{b̄a}. Strictly speaking, this goes against the conventions of the abstract index notation [91] but we shall allow ourselves this extra leeway when it is useful. For example, the reconstructed form ω_{αβ} may then be written as

ω_{αβ} = Π_{α}^{a}\overline{Π}b̄ ; βω_{ab̄} + \overline{Π}ā ; αΠ_{β}^{b}ω_{āb}.

Two-forms of various types may be characterised as

ω_{αβ} is type (0, 2) ⟺ J_{α}^{γ}ω_{βγ} = iω_{αβ}
ω_{αγ} is type (1, 1) ⟺ J_{[α}^{γ}ω_{β]γ} = 0
ω_{αβ} is type (2, 0) ⟺ J_{α}^{γ}ω_{βγ} = −iω_{αβ}

(6)

but already this is a little awkward and becomes more so for higher forms and extremely awkward when attempting to decompose more general tensors as we shall have cause to do when considering torsion and curvature. Notice that forms of type (1, 1) in (6) are characterised by a real condition. Indeed, the complex bundle ∧^{1,1} is the complexification of a real irreducible bundle whose sections are the real 2-forms satisfying J_{[α}^{γ}ω_{β]γ} = 0. As for forms of types (0, 2) and (2, 0), there is a real bundle whose sections satisfy

J_{α}^{γ}J_{β}^{δ}ω_{γδ} = −ω_{αβ}

(as opposed to J_{α}^{γ}J_{β}^{δ}ω_{γδ} = ω_{αβ} for sections of ∧^{1,1}) and whose complexification is ∧^{0,2} ⊕∧^{2,0}. Thus, the real 2-forms split irreducibly into just two kinds but the complex 2-forms split into three types (5).

Notice that if E is a complex vector bundle on M, then we can decompose 2-forms with values in E into types by using the same formulae (6). In particular, we can do this on an almost complex manifold when E = T M, regarded as a complex bundle via the action of J. Writing this out explicitly, a real tensor T_{αβ}^{γ} = T_{[αβ]}^{γ} is said to be

of type (0, 2) ⟺ J_{α}^{γ}T_{βγ}^{δ} = T_{αβ}^{γ}J_{γ}^{δ}
of type (1, 1) ⟺ J_{[α}^{γ}T_{β]γ}^{δ} = 0
of type (2, 0) ⟺ J_{α}^{γ}T_{βγ}^{δ} = −T_{αβ}^{γ}J_{γ}^{δ}.

(7)

For example, as the Nijenhuis tensor (1) satisfies N^{J}(Y, JX) = JN^{J}(X, Y), it is of type (0, 2). Further to investigate this decomposition (7), it is useful to apply the projectors Π_{α}^{a} and Π_{a}^{α} to obtain

Tāb̄c̄ ≡ \overline{Π}α ; ā\overline{Π}β ; b̄\overline{Π}c̄ ; γT_{αβ}^{γ}  T_{ab̄}c̄ ≡ Π_{a}^{α}\overline{Π}β ; b̄\overline{Π}c̄ ; γT_{αβ}^{γ}  T_{ab}c̄ ≡ Π_{a}^{α}Π_{b}^{β}\overline{Π}c̄ ; γT_{αβ}^{γ}
Tāb̄^{c} ≡ \overline{Π}α ; ā\overline{Π}β ; b̄Π_{γ}^{c}T_{αβ}^{γ}  T_{ab̄}^{c} ≡ Π_{a}^{α}\overline{Π}β ; b̄Π_{γ}^{c}T_{αβ}^{γ}  T_{ab}^{c} ≡ Π_{a}^{α}Π_{b}^{β}Π_{γ}^{c}T_{αβ}^{γ}

satisfying

\begin{matrix}T_{ab}^{c} = T_{[ab]}^{c} & Tāb̄^{c} = T_{[āb̄]}^{c} & T_{ab}c̄ = T_{[ab]}c̄ & Tāb̄c̄ = T_{[āb̄]}c̄ \\ \overline{T_{ab}^{c}} = Tāb̄c̄ & \overline{T_{ab̄}^{c}} = −T_{bā}c̄ & \overline{T_{ab}c̄} = Tāb̄^{c} \end{matrix}

and from which we can recover T_{αβ}^{γ} according to

T_{αβ}^{γ} = \overline{Π}ā ; α\overline{Π}b̄ ; β\overline{Π}γ ; c̄Tāb̄c̄ + 2Π_{[α}^{a}\overline{Π}b̄ ; β]\overline{Π}γ ; c̄T_{ab̄}c̄ + Π_{α}^{a}Π_{β}^{b}\overline{Π}γ ; c̄T_{ab}c̄

+ \overline{Π}ā ; α\overline{Π}b̄ ; βΠ_{c}^{γ}Tāb̄^{c} + 2Π_{[α}^{a}\overline{Π}b̄ ; β]Π_{c}^{γ}T_{ab̄}^{c} + Π_{α}^{a}Π_{β}^{b}Π_{c}^{γ}T_{ab}^{c}.

From (4) and (7), the splitting of T_{αβ}^{γ} into types corresponds exactly to components

type (0, 2) ↔ (T_{ab}c̄, Tāb̄^{c})
type (1, 1) ↔ (T_{ab̄}^{c}, T_{ab̄}c̄)
type (2, 0) ↔ (T_{ab}^{c}, Tāb̄c̄).

(8)

Notice that, for each of types (1, 1) and (2, 0), a complex-valued 1-form can be invariantly extracted:

type (1, 1) :ϕ_{α} ≡ Π_{α}^{a}T_{ab̄}b̄ = \frac{1}{2}(T_{αβ}^{β} + iT_{αβ}^{γ}J_{γ}^{β})
type (2, 0) :ψ_{α} ≡ Π_{α}^{a}T_{ab}^{b} = \frac{1}{2}(T_{αβ}^{β} − iT_{αβ}^{γ}J_{γ}^{β}).

On the other hand, just from the index structure, tensors T_{αβ}^{γ} = T_{[αβ]}^{γ} of type (0, 2) seemingly cannot be further decomposed (and this is confirmed in Section 3.3). In any case, it follows easily from J_{α}^{γ}T_{βγ}^{δ} = T_{αβ}^{γ}J_{γ}^{δ} that T_{αβ}^{γ} of type (0, 2) satisfy T_{αβ}^{β} = 0 = T_{αβ}^{γ}J_{γ}^{β}.

1.2. Complex connections. The geometrically useful affine connections ∇ on an almost complex manifold (M, J) are those that preserve J_{α}^{β}, i.e. ∇_{α}J_{β}^{γ} = 0. We call them complex connections. The space of complex connections is an affine space over the vector space that consists of 1-forms with values in the complex endomorphisms 𝔤𝔩(T M, J) of T M. A complex connection ∇ naturally extends to a linear connection on ℂT M that preserves the decomposition into types (2). Indeed, preservation of type is also a sufficient condition for an affine connection to be complex.

Given a complex connection ∇, we denote by T_{αβ}^{γ} its torsion, which is a 2-form with values in T M. As such T_{αβ}^{γ} naturally splits according to type into a direct sum of three components as in (7). A straightforward computation shows that the (0, 2)- component of the torsion of any complex connection equals −\frac{1}{4}N^{J}. In particular, this component is an invariant of the almost complex structure and cannot be eliminated by a suitable choice of complex connection. However, all other components can be removed. To see this, suppose ˆ∇ is another complex connection. Then there is an element υ ∈ T^{∗}M ⊗ 𝔤𝔩(T M, J) such that ˆ∇ = ∇ + υ. It follows that their torsions are related by the formula T̂ = T + ∂υ, where ∂ is the composition

T^{∗}M ⊗ 𝔤𝔩(T M, J) ↪ T^{∗}M ⊗ T^{∗}M ⊗ T M → ∧^{2}T^{∗}M ⊗ T M
υ_{αβ}^{γ} ↦ 2υ_{[αβ]}^{γ}.

Notice that the image of ∂ is spanned by 2-forms of type (2, 0) and (1, 1). Consequently, its cokernel can be identified with forms of type (0, 2). Hence, any complex connection can be deformed in such a way that its torsion is of type (0, 2). We have shown the following classical result:

Proposition 1.2 ([62, 69]). On any almost complex manifold (M, J) there is a complex connection such that T = −\frac{1}{4}N^{J}.

Since ∂ is not injective such a complex connection is not unique. Complex connections ∇ with T = −\frac{1}{4}N^{J} form an affine space over

ker ∂ = (S^{2}T^{∗}M ⊗ T M) ∩ (T^{∗}M ⊗ 𝔤𝔩(T M, J))  (9)

and are called minimal connections.

From Proposition 1.2 and the above discussion one also deduces immediately:

Corollary 1.3. There exists a complex torsion-free connection on an almost complex manifold (M, J) if and only if N^{J} ≡ 0.

Remark 1.1. We have already noted that the cokernel of ∂ can be identified with tensors T_{αβ}^{γ} such that

T_{(αβ)}^{γ} = 0 J_{α}^{γ}T_{βγ}^{δ} = T_{αβ}^{γ}J_{γ}^{δ}.

Consequently, T_{αβ}^{β} = T_{αβ}^{γ}J_{γ}^{β} = 0. As we shall see in Section 3.3, such tensors are irreducible. More precisely, the natural vector bundles on an almost complex manifold (M, J) correspond to representations of GL(n, ℂ) and we shall see that coker ∂ corresponds to an irreducible representation of GL(n, ℂ). On the other hand, its kernel (9) decomposes into two irreducible components, namely a trace-free part and a trace part. We shall see in the next section that deforming a complex connection by an element from the latter space exactly corresponds to changing a connection c-projectively.

2. Elements of c-projective geometry

We now introduce almost c-projective structures, first from the classical perspective of J-planar curves and equivalence classes of complex affine connections [90], then from the modern viewpoint of parabolic geometries [36, 55, 103]. The (categorical) equivalence between these approaches is established in Theorem 2.8. This leads us to study the intrinsic curvature of an almost c-projective manifold, namely the harmonic curvature of its canonical normal Cartan connection.

2.1. Almost c-projective structures. Recall that affine connections ∇ and ˆ∇ on a manifold M are projectively equivalent if there is a 1-form Υ_{α} on M such that

ˆ∇_{α}X^{γ} = ∇_{α}X^{γ} + Υ_{α}X^{γ} + δ_{α}^{γ}Υ_{β}X^{β}.  (10)

Suppose now that (M, J) is an almost complex manifold. Then ∇ and ˆ∇ are called c-projectively equivalent, if there is a (real) 1-form Υ_{α} on M such that

ˆ∇_{α}X^{γ} = ∇_{α}X^{γ} + υ_{αβ}^{γ}X^{β},  (11)
where  υ_{αβ}^{γ} := \frac{1}{2}(Υ_{α}δ_{β}^{γ} + δ_{α}^{γ}Υ_{β} − J_{α}^{δ}Υ_{δ}J_{β}^{γ} − J_{α}^{γ}Υ_{δ}J_{β}^{δ}).

Note that υ_{αβ}^{γ}J_{γ}^{δ} = υ_{αγ}^{δ}J_{β}^{γ}. In other words υ_{αβ}^{γ} is a 1-form on M with values in 𝔤𝔩(T M, J), which implies that if ∇ is a complex connection, then so is ˆ∇. Moreover υ_{αβ}^{γ} = υ_{(αβ)}^{γ} and so c-projectively equivalent connections have the same torsion. In particular, if ∇ is minimal, then so is ˆ∇.

A smooth curve c: (a, b) → M is called a J-planar curve with respect to a complex connection ∇, if ∇ċċ lies in the span of ċ and Jċ. The notion of J-planar curves gives rise to the following geometric interpretation of a c-projective equivalence class of complex connections.

Proposition 2.1 ([55, 84, 90, 98]). Suppose (M, J) is an almost complex manifold and let ∇ and ˆ∇ be complex connections on M with the same torsion. Then ∇ and ˆ∇ are c-projectively equivalent if and only if they have the same J-planar curves.

Proof. Suppose ∇ and ˆ∇ are complex connections with the same torsion. If ∇ and ˆ∇ are c-projectively equivalent, then they clearly have the same J-planar curves. Conversely, assume that ∇ and ˆ∇ share the same J-planar curves and consider the difference tensor A_{αβ}^{γ}Y^{β} = ˆ∇_{α}Y^{γ} − ∇_{α}Y^{γ}. As both connections are complex and have the same torsion, the difference tensor satisfies A_{αβ}^{γ} = A_{(αβ)}^{γ} and A_{αβ}^{γ}J_{γ}^{δ} = A_{αγ}^{δ}J_{β}^{γ}. The fact that ˆ∇ and ∇ have the same J-planar curves and that any tangent vector can be realised as the derivative of such a curve implies that at any point x ∈ M and for any nonzero vector Y ∈ T_{x}M there exist uniquely defined real numbers γ(Y) and μ(Y) such that

A(Y, Y) = γ(Y)Y + μ(Y)JY.  (12)

Note that γ and μ give rise to well-defined smooth functions on T M \ 0. Extending γ and μ to functions on all of T M by setting γ(0) = μ(0) = 0, formula (12) becomes valid for any tangent vector, and by construction γ and μ are then clearly homogeneous of degree one. From A(Y, Y) = −A(JY, JY) we deduce that μ(X) = −γ(JX) whence

A(Y, Y) = γ(Y)Y − γ(JY)JY.

By polarisation we have for any tangent vectors X and Y

A(X, Y) = \frac{1}{2}(A(X + Y, X + Y) − A(X, X) − A(Y, Y))  (13)
= \frac{1}{2}((γ(X + Y) − γ(X))X + (γ(X + Y) − γ(Y))Y )
− \frac{1}{2}((γ(JX + JY) − γ(JX))JX − (γ(JX + JY) − γ(JY))JY ).

Suppose that X and Y are linearly independent and expand the identity A(X, tY) = tA(X, Y) for all t ∈ ℝ using (13). Then a comparison of coefficients shows that

γ(X + tY) − tγ(Y) = γ(X + Y) − γ(Y).

Taking the limit t → 0, shows that γ(X + Y) = γ(X) + γ(Y). Hence, γ defines a (smooth) 1-form and

A(X, Y) = \frac{1}{2}(A(X + Y, X + Y) − A(X, X) − A(Y, Y))
= \frac{1}{2}(γ(X)Y + γ(Y)X − γ(JX)(JY) − γ(JY)JX),

for any tangent vector X and Y as desired. □

Definition 2.1. Suppose that M is manifold of real dimension 2n ≥ 4.

(1) An almost c-projective structure on M consists of an almost complex structure J on M and a c-projective equivalence class [∇] of minimal complex connections.

(2) The torsion of an almost c-projective structure (M, J, [∇]) is the torsion T of one, hence any, of the connections in [∇], i.e. T = −\frac{1}{4}N^{J}.

(3) An almost c-projective structure (M, J, [∇]) is called a c-projective structure, if J is integrable. (This is the case if and only if some and hence all connections in the c-projective class are torsion-free.)

Remark 2.1. If M is a 2-dimensional manifold, any almost complex structure J is integrable and any two torsion free complex connections are c-projectively equivalent. Therefore, in this case one needs to modify the definition of a c-projective structure in order to have something nontrivial (cf. [23, 24]). We shall not pursue this here.

Remark 2.2. Recall that the geodesics of an affine connection can be also realised as the geodesics of a torsion-free connection; hence the definition of a projective structure as an equivalence class of torsion-free connections does not constrain the considered families of geodesics. The analogous statement for J-planar curves does not hold: the J-planar curves of a complex connection cannot in general be realised as the Jplanar curves of a minimal connection. We discuss the motivation for the restriction to minimal connections in the definition of almost c-projective manifolds in Remark 2.9.

Definition 2.2. Let (M, J_{M}, [∇^{M}]) and (N, J_{N}, [∇^{N}]) be almost c-projective manifolds of dimension 2n ≥ 4. A diffeomorphism Φ: M → N is called c-projective transformation or automorphism, if Φ is complex (i.e. T Φ ∘ J_{M} = J_{N} ∘ T Φ) and for a (hence any) connection ∇^{N} ∈ [∇^{N}] the connection Φ^{∗}∇^{N} is a connection in [∇^{M}].

From Proposition 2.1 one deduces straightforwardly that also the following characterisation of c-projective transformations holds:

Proposition 2.2. Let (M, J_{M}, [∇^{M}]) and (N, J_{N}, [∇^{N}]) be almost c-projective manifolds of dimension 2n ≥ 4. Then a complex diffeomorphism Φ: M → N is a cprojective transformation if and only if Φ maps J_{M}-planar curves to J_{N}-planar curves.

Suppose that (M, J, [∇]) is an almost c-projective manifold. Let ˆ∇ and ∇ be connections of the c-projective class [∇] that differ by Υ_{α} as in (11). Then ˆ∇ and ∇ give rise to linear connections on ℂT M = T^{1,0}M ⊕ T^{0,1}M that preserve the decomposition into types. Hence, they induce connections on the complex vector bundles T^{1,0}M and T^{0,1}M. To deduce the difference between the connections ˆ∇ and ∇ on T^{1,0}M (respectively T^{0,1}M), we just need to apply the splittings Π_{α}^{a} and Π_{a}^{α} (respectively their conjugates) from the previous section to (11). Using the identities (4), we obtain

Π_{a}^{α}Π_{b}^{β}υ_{αβ}^{γ}Π_{γ}^{c} = \frac{1}{2}Π_{a}^{α}Π_{b}^{β}(Υ_{α}δ_{β}^{γ} + δ_{α}^{γ}Υ_{β} − J_{α}^{δ}Υ_{δ}J_{β}^{γ} − J_{α}^{γ}Υ_{δ}J_{β}^{δ})Π_{γ}^{c}
= \frac{1}{2}Π_{a}^{α}Π_{b}^{β}((Υ_{α} − iJ_{α}^{δ}Υ_{δ})Π_{β}^{c} + (Υ_{β} − iJ_{β}^{δ}Υ_{δ})Π_{α}^{c})
= \frac{1}{2}Π_{a}^{α}(Υ_{α} − iJ_{α}^{δ}Υ_{δ})δ_{b}^{c} + \frac{1}{2}Π_{b}^{β}(Υ_{β} − iJ_{β}^{δ}Υ_{δ})δ_{a}^{c}
= Υ_{a}δ_{b}^{c} + Υ_{b}δ_{a}^{c}, where Υ_{a} ≡ Π_{a}^{α}Υ_{α} .

Similarly, we find that \overline{Π}α ; āΠ_{b}^{β}υ_{αβ}^{γ}Π_{γ}^{c} = 0. These identities are the key to the following:

Proposition 2.3. Suppose (M, J, [∇]) is an almost c-projective manifold of dimension 2n ≥ 4. Assume two connections ˆ∇ and ∇ in [∇] differ by Υ_{α} as in (11), and set Υ_{a} := Π_{a}^{α}Υ_{α} and Υā := Πα ; āΥ_{α}. Then we have the following transformation rules for the induced connections on T^{1,0}M and T^{0,1}M.

(1) ˆ∇_{a}X^{c} = ∇_{a}X^{c} + Υ_{a}X^{c} + δ_{a}^{c}Υ_{b}X^{b} and ˆ∇āX^{c} = ∇āX^{c},

(2) ˆ∇āXc̄ = ∇āXc̄ + ΥāXc̄ + δāc̄Υb̄Xb̄ and ˆ∇_{a}Xc̄ = ∇_{a}Xc̄.

Proof. We compute

ˆ∇_{a}X^{c} = Π_{a}^{α} ˆ∇_{α}(Π_{γ}^{c}Π_{b}^{γ}X^{b}) = Π_{a}^{α}Π_{γ}^{c} ˆ∇_{α}(Π_{b}^{γ}X^{b})
= Π_{a}^{α}Π_{γ}^{c}∇_{α}(Π_{b}^{γ}X^{b}) + Π_{a}^{α}Π_{γ}^{c}υ_{αβ}^{γ}Π_{b}^{β}X^{b}
= Π_{a}^{α}∇_{α}(Π_{γ}^{c}Π_{b}^{γ}X^{b}) + (Π_{a}^{α}Π_{b}^{β}υ_{αβ}^{γ}Π_{γ}^{c})X^{b}
= ∇_{a}X^{c} + (Υ_{a}δ_{b}^{c} + Υ_{b}δ_{a}^{c})X^{b} = ∇_{a}X^{c} + Υ_{a}X^{c} + δ_{a}^{c}Υ_{b}X^{b},

as required. The remaining calculations are similar. □

Remark 2.3. The differential operator ∇ā: T^{1,0}M → ∧^{0,1}M ⊗T^{1,0}M is c-projectively invariant, as is its conjugate ∇_{a}: T^{0,1}M → ∧^{1,0}M ⊗T^{0,1}M. (Here and throughout, the domain and codomain of a differential operator are declared as bundles, although the operator is a map between corresponding spaces of sections.) This is unsurprising: it is the usual ¯∂-operator on an almost complex manifold whose kernel (in the integrable case) comprises the holomorphic vector fields.

In contrast, the transformation rules for ∇_{a}: T^{1,0}M → ∧^{1,0}M ⊗ T^{1,0}M and its conjugate are analogues of projective equivalence (10) in the (1, 0) and (0, 1) directions respectively. When (M, J) is real-analytic and the c-projective class contains a real-analytic connection ∇, this can be made precise by extending J and [∇] to a complexification M^{ℂ} of M, so that T^{1,0}M and T^{0,1}M extend to distributions on M^{ℂ}. If J is integrable, these distributions integrate to two foliations of M^{ℂ}, and [∇] induces projective structures on the leaves of these foliations.

Taking the trace in equation (11) and in the formulae in Proposition 2.3, we deduce:

Corollary 2.4. On an almost c-projective manifold (M, J, [∇]), the transformation rules for the induced linear connections on ∧^{2n}T M, ∧^{n}T^{1,0}M, and ∧^{n}T^{0,1}M are:

(1) ˆ∇_{α}Σ = ∇_{α}Σ + (n + 1)Υ_{α}Σ, for Σ ∈ Γ(∧^{2n}T M)

(2) ˆ∇_{a}σ = ∇_{a}σ + (n + 1)Υ_{a}σ and ˆ∇āσ = ∇āσ, for σ ∈ Γ(∧^{n}T^{1,0}M)

(3) ˆ∇āσ̄ = ∇āσ̄ + (n + 1)Υāσ̄ and ˆ∇_{a}σ̄ = ∇_{a}σ̄, for σ̄ ∈ Γ(∧^{n}T^{0,1}M).

For the convenience of the reader, let us also record the transformation rules for the induced connections on T^{∗}M, respectively ∧^{1,0} and ∧^{0,1}. If two complex connections ˆ∇ and ∇ are related via Υ_{α} as in (11), then the induced connections on T^{∗}M are related by

ˆ∇_{α}ϕ_{γ} = ∇_{α}ϕ_{γ} − \frac{1}{2}(Υ_{α}ϕ_{γ} + Υ_{γ}ϕ_{α} − J_{α}^{β}Υ_{β}J_{γ}^{δ}ϕ_{δ} − J_{α}^{β}ϕ_{β}J_{γ}^{δ}Υ_{δ}).  (14)

Therefore, we obtain:

Proposition 2.5. Suppose (M, J, [∇]) is an almost c-projective manifold of dimension 2n ≥ 4. Assume two connections ˆ∇ and ∇ in [∇] differ by Υ_{α} as in (11) and set Υ_{a} := Π_{a}^{α}Υ_{α} and Υā := Πα ; āΥ_{α}. Then we have the following transformation rules for the induced connections on ∧^{1,0} and ∧^{0,1}:

(1) ˆ∇_{a}ϕ_{c} = ∇_{a}ϕ_{c} − Υ_{a}ϕ_{c} − ϕ_{a}Υ_{c} and ˆ∇āϕ_{c} = ∇āϕ_{c},

(2) ˆ∇āϕc̄ = ∇āϕc̄ − Υāϕc̄ − ϕāΥc̄ and ˆ∇_{a}ϕc̄ = ∇_{a}ϕc̄.

Note that the real line bundle ∧^{2n}T M is oriented and hence admits oriented roots. We denote (∧^{2n}T M)^{\frac{1}{n+1}} by E_{ℝ}(1, 1) and for any k ∈ ℤ we set E_{ℝ}(k, k) := E(1, 1)^{⊗k}, where E_{ℝ}(k, k)^{∗} = E_{ℝ}(−k, −k). It follows from Corollary 2.4 that for a section Σ of E_{ℝ}(k, k) we have

ˆ∇_{α}Σ = ∇_{α}Σ + kΥ_{α}Σ.  (15)

In particular, we immediately deduce the following result.

Proposition 2.6. Suppose (M, J, [∇]) is an almost c-projective manifold of dimension 2n ≥ 4. The map sending an affine connection to its induced connection on E_{ℝ}(1, 1) induces a bijection from connections in [∇] to linear connections on E_{ℝ}(1, 1).

Since ∧^{2n}T M and E_{ℝ}(1, 1) are oriented, they can be trivialised by choosing a positive section. Such a positive section τ of E_{ℝ}(1, 1) gives rise to a linear connection on E_{ℝ}(1, 1) by decreeing that τ is parallel and therefore, by Proposition 2.6, to a connection in the c-projective class. We call a connection ∇ ∈ [∇] that arises in this way a special connection. Suppose τ̂ and τ are two nowhere vanishing sections of E_{ℝ}(1, 1) and denote by ˆ∇ and ∇ the corresponding connections. Then τ̂ = e^{−f}τ for some smooth function f on M and any σ ∈ Γ(E_{ℝ}(1, 1)) can be written as σ = hτ = he^{f} τ̂ for a smooth function h on M. Since ∇σ = dh ⊗ τ, we have

ˆ∇σ = d(he^{f}) ⊗ τ̂ = dh ⊗ τ + df ⊗ σ = ∇σ + (∇f)σ.

Therefore, ˆ∇ and ∇ differ by an exact 1-form, namely Υ_{α} ≡ ∇_{α}f.

In some of the following sections, like for instance in Section 3.1, we shall assume also that the complex line bundle ∧^{n}T^{1,0}M admits a (n + 1)^{st} root and that we have chosen one, which we will denote by E(1, 0) (following a standard notation on ℂℙ^{n}). In that case we shall denote its conjugate bundle \overline{E(1, 0)} by E(0, 1) and the dual bundle E(1, 0)^{∗} by E(−1, 0). In general, we shall also write E(k, ℓ) := E(1, 0)^{⊗k} ⊗ E(0, 1)^{⊗ℓ} for (k, ℓ) ∈ ℤ × ℤ and refer to its sections as c-projective densities of weight (k, ℓ). By Corollary 2.4 we see that, for a c-projective density σ of weight (k, ℓ), we have

ˆ∇_{a}σ = ∇_{a}σ + kΥ_{a}σ ˆ∇āσ = ∇āσ + ℓΥāσ.  (16)

Our notion of c-projective density means, in particular, that we may identify ∧^{n,0} with E(−n − 1, 0) and it is useful to have a notation for this change of viewpoint. Precisely, we may regard our identification E(−n − 1, 0) ≃ ⟶ ∧^{n,0} as a tautological section ε_{ab···c} of ∧^{n,0}(n + 1, 0), such that a c-projective density ρ of weight (−n − 1, 0) corresponds to ρε_{ab···c}, a form of type (n, 0). Note that E(k, k) ≅ E_{ℝ}(k, k) ⊗ ℂ.

2.2. Parabolic geometries. For the convenience of the reader we recall here some basics of parabolic geometries; for a comprehensive introduction see [36].

A parabolic geometry on a manifold M is a Cartan geometry of type (G, P), where G is a semisimple Lie group and P ⊂ G a so-called parabolic subgroup. Hence, it is given by the following data:

• a principal P -bundle p: G → M

• a Cartan connection ω ∈ Ω^{1}(G, 𝔤)—that is, a P -equivariant 1-form on G with values in 𝔤 defining a trivialisation T G ≅ G × 𝔤 and reproducing the generators of the fundamental vector fields,

where 𝔤 denotes the Lie algebra of G. Note that the projection G → G/P, equipped with the (left) Maurer–Cartan form ω_{G} ∈ Ω^{1}(G, 𝔤) of G, defines a parabolic geometry on G/P, which is called the homogeneous or flat model for parabolic geometries of type (G, P).

The curvature of a parabolic geometry (G →^{p} M, ω) is a 2-form K on G with values in 𝔤, defined by

K(χ, ξ) = dω(χ, ξ) + [ω(χ), ω(ξ)] for vector fields χ and ξ on G,

where d denotes the exterior derivative and [ , ] the Lie bracket of 𝔤.

The curvature of the homogeneous model (G → G/P, ω_{G}) vanishes identically. Furthermore, the curvature K of a parabolic geometry of type (G, P) vanishes identically if and only if it is locally isomorphic to (G → G/P, ω_{G}). Thus, the curvature K measures the extent to which the geometry differs from its homogeneous model.

Given a parabolic geometry (G →^{p} M, ω) of type (G, P), any representation 𝔼 of P gives rise to an associated vector bundle E := G ×_{P} 𝔼 over M. These are the natural vector bundles on a parabolic geometry. Notice that the Cartan connection ω induces an isomorphism

G ×_{P} 𝔤/𝔭 ≅ T M
[u, X + 𝔭] ↦ T_{u}p(ω^{−1}(X)),

where 𝔭 denotes the Lie algebra of P and the action of P on 𝔤/𝔭 is induced by the adjoint action of G. Similarly, ω allows us to identify all tensor bundles on M with associated vector bundles. The vector bundles corresponding to P -modules obtained by restricting a representation of G to P are called tractor bundles. These bundles play an important role in the theory of parabolic geometries, since the Cartan connection induces linear connections, called tractor connections, on these bundles. An important example of a tractor bundle is the adjoint tractor bundle AM = G ×_{P} 𝔤, which has a canonical projection to T M corresponding to the P -equivariant projection 𝔤 → 𝔤/𝔭.

Remark 2.4. The abstract theory of tractor bundles and connections even provides an alternative description of parabolic geometries (see [30]).

By normalising the curvature of a parabolic geometry, the prolongation procedures of [35, 85, 95] leads to an equivalence of categories between so-called regular normal parabolic geometries and certain underlying structures, which may be described in more conventional geometric terms. Among the most prominent of these are conformal structures, projective structures, and CR-structures of hypersurface type. In the next section we shall see that almost c-projective manifolds form another class of examples.

From the defining properties of a Cartan connection it follows immediately that the curvature K of a parabolic geometry of type (G, P) is P -equivariant and horizontal. Hence, K can be identified with a section of the vector bundle ∧^{2}T^{∗}M ⊗ AM and therefore corresponds via ω to a section κ of the vector bundle

G ×_{P} ∧^{2}(𝔤/𝔭)^{∗} ⊗ 𝔤 ≅ G ×_{P} ∧^{2}𝔭_{+} ⊗ 𝔤,

where 𝔭_{+} is the nilpotent radical of 𝔭 and the latter isomorphism is induced by the Killing form of 𝔤. Now consider the complex for computing the Lie algebra homology H_{∗}(𝔭_{+}, 𝔤) of 𝔭_{+} with values in 𝔤:

0 ← 𝔤 ←^{∂^{∗}} 𝔭_{+} ⊗ 𝔤 ←^{∂^{∗}} ∧^{2}𝔭_{+} ⊗ 𝔤 ← …

Since the linear maps ∂^{∗} are P -equivariant, they induce vector bundle maps between the corresponding associated vector bundles. Moreover, the homology spaces H_{i}(𝔭_{+}, 𝔤) are naturally P -modules and therefore give rise to natural vector bundles. A parabolic geometry is called normal, if ∂^{∗}κ = 0. In this case, we can project κ to a section κ_{h} of G ×_{P} H_{2}(𝔭_{+}, 𝔤), called the harmonic curvature. The spaces H_{i}(𝔭_{+}, 𝔤) are completely reducible P -modules and hence arise as completely reducible representations of the reductive Levi factor G_{0} of P via the projection P → P/ exp(𝔭_{+}) = G_{0}. In particular, the harmonic curvature is a section of a completely reducible vector bundle, which makes it a much simpler object than the full curvature. Moreover, using the Bianchi identities of κ, it can be shown that the harmonic curvature is still a complete obstruction to local flatness:

Proposition 2.7 (see e.g. [36]). Suppose that (G → M, ω) is a regular normal parabolic geometry. Then κ ≡ 0 if and only if κ_{h} ≡ 0.

Remark 2.5. The machinery of BGG sequences shows that the curvature of a regular normal parabolic geometry can be reconstructed from the harmonic curvature by applying a BGG splitting operator (see [25]).

2.3. Almost c-projective manifolds as parabolic geometries. It is convenient for our purposes to realise the Lie algebra 𝔤 := 𝔰𝔩(n + 1, ℂ) of complex trace-free linear endomorphisms of ℂ^{n+1} as block matrices of the form

𝔤 = \{(\begin{matrix}−tr A & Z \\ X & A \end{matrix}) : A ∈ 𝔤𝔩(n, ℂ), X ∈ ℂ^{n}, Z ∈ (ℂ^{n})^{∗}\} ,  (17)

where tr: 𝔤𝔩(n, ℂ) → ℂ denotes the trace. The block form equips 𝔤 with the structure of a graded Lie algebra:

𝔤 = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1},

where 𝔤_{0} is the block diagonal subalgebra isomorphic to 𝔤𝔩(n, ℂ) and 𝔤_{−1} ≅ ℂ^{n}, respectively 𝔤_{1} ≅ (ℂ^{n})^{∗}, as 𝔤_{0}-modules. Note that the subspace 𝔭 := 𝔤_{0} ⊕ 𝔤_{1} is a subalgebra of 𝔤 (with 𝔭 ≅ 𝔤_{0} ⋉ 𝔤_{1} as Lie algebra). Furthermore, 𝔭 is a parabolic subalgebra with Abelian nilpotent radical 𝔭_{+} := 𝔤_{1} and Levi factor isomorphic to 𝔤_{0}. For later purposes let us remark here that we may conveniently decompose an element A ∈ 𝔤_{0} into its trace-free part and into its trace part as follows

(\begin{matrix}0 & 0 \\ 0 & A − \frac{tr A}{n} Id_{n} \end{matrix}) + \frac{n + 1}{n}tr A(\begin{matrix}−\frac{n}{n+1} & 0 \\ 0 & \frac{1}{n+1} Id_{n} \end{matrix}) .  (18)

Now set G := PSL(n + 1, ℂ) and let P be the stabiliser in G of the complex line generated by the first standard basis vector of ℂ^{n+1}. Let G_{0} be the subgroup of P that consists of all elements g ∈ P whose adjoint action Ad(g): 𝔤 → 𝔤 preserve the grading. Hence, it consists of equivalence classes of matrices of the form

(\begin{matrix}(det_{ℂ} C)^{−1} & 0 \\ 0 & C \end{matrix}) where C ∈ GL(n, ℂ),

and the adjoint action of G_{0} on 𝔤 induces an isomorphism

G_{0} ≅ GL(𝔤_{−1}, ℂ) ≅ GL(n, ℂ).

Obviously, the subgroups G_{0} and P of G have corresponding Lie algebras 𝔤_{0} and 𝔭, respectively.

From now on we shall view G_{0} ⊂ P ⊂ G as real Lie groups in accordance with the identification of GL(n + 1, ℂ) with the real subgroup of GL(2n + 2, ℝ) that is given by

GL(2n + 2, 𝕁_{2(n+1)}) = {A ∈ GL(2(n + 1), ℝ) : A𝕁_{2(n+1)} = 𝕁_{2(n+1)}A} ,

where 𝕁_{2(n+1)} is the following complex structure on ℝ^{2n+2}:

𝕁_{2(n+1)} = (\begin{matrix}𝕁_{2} \\ & . & . & . \\ & & & & 𝕁_{2} \end{matrix}) with 𝕁_{2} = (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) .

Suppose now that (M, J, [∇]) is an almost c-projective manifold of real dimension 2n ≥ 4. Then J reduces the frame bundle FM of M to a principal bundle p_{0} : G_{0} → M with structure group G_{0} corresponding to the group homomorphism

G_{0} ≅ GL(n, ℂ) ≅ GL(2n, 𝕁_{2n}) ↪ GL(2n, ℝ).

The general prolongation procedures of [35, 85, 95] further show that G_{0} → M can be canonically extended to a principal P -bundle p: G → M, equipped with a normal Cartan connection ω ∈ Ω^{1}(G, 𝔤) of type (G, P). Moreover, (G →^{p} M, ω) is uniquely defined up to isomorphism and these constructions imply:

Theorem 2.8 (see also [55, 103]). There is an equivalence of categories between almost c-projective manifolds of real dimension 2n ≥ 4 and normal parabolic geometries of type (G, P), where G and P are viewed as real Lie groups. The homogeneous model (G → G/P, ω_{G}) corresponds to the c-projective manifold

(ℂℙ^{n}, J_{can}, [∇^{g_{F S}}]),

where J_{can} denotes the canonical complex structure on ℂℙ^{n} and ∇^{g_{F S}} the Levi-Civita connection of the Fubini–Study metric g_{F S}.

Let us explain briefly how the Cartan bundle G and the normal Cartan connection ω of an almost c-projective manifold (M, J, [∇]) of dimension 2n ≥ 4 are constructed. The reduction G_{0} → FM is determined by the pullback of the soldering form on FM and hence can be encoded by a strictly horizontal G_{0}-equivariant 1-form θ ∈ Ω^{1}(G_{0}, 𝔤_{−1}). Recall also that any connection ∇ ∈ [∇] can be equivalently viewed as a principal connection γ^{∇} ∈ Ω^{1}(G_{0}, 𝔤_{0}) on G_{0}. Then G is defined to be the disjoint union ⊔_{u∈G_{0}}G_{u}, where

G_{u} := {θ(u) + γ^{∇}(u) : ∇ ∈ [∇]} for any u ∈ G_{0}.

The projection p := p_{0} ∘ q: G → M, where G →^{q} G_{0} →^{p_{0}} M, naturally acquires the structure of a P -principal bundle. Any element p ∈ P can be uniquely written as p = g_{0} exp(Z), where g_{0} ∈ G_{0} and Z ∈ 𝔤_{1}. The right action of an element g_{0} exp(Z) ∈ P on an element θ(u) + γ^{∇}(u) ∈ G_{u} is given by

(θ(u) + γ^{∇}(u)) · g_{0} exp(Z) := θ(u · g_{0})(·) + γ^{∇}(u · g_{0})(·) + [Z, θ(u · g_{0})(·)], (19)

where [ , ] denotes the Lie bracket 𝔤_{1} × 𝔤_{−1} → 𝔤_{0}.

Remark 2.6. The soldering form θ ∈ Ω^{1}(G_{0}, 𝔤_{−1}) gives rise to isomorphisms T M ≅ G_{0} ×_{G_{0}} 𝔤_{−1} and T^{∗}M ≅ G_{0} ×_{G_{0}} 𝔤_{1}. For elements X ∈ 𝔤_{−1} and Z ∈ 𝔤_{1}, the Lie bracket [Z, X] ∈ 𝔤_{0} ≅ 𝔤𝔩(𝔤_{−1}, 𝕁_{2n}) evaluated on an element Y ∈ 𝔤_{−1} is given by

[[Z, X], Y] = −(ZXY + ZY X − Z𝕁_{2n}X𝕁_{2n}Y − Z𝕁_{2n}Y 𝕁_{2n}X).  (20)

This shows that changing a connection form θ + γ^{∇} by a G_{0}-equivariant function Z: G_{0} → 𝔤_{1} according to (19) corresponds precisely to changing it c-projectively (cf. formula (11)).

The definition of G easily implies that the following holds.

Corollary 2.9. The projection q: G → G_{0} is a trivial principal bundle with structure group P_{+} := exp(𝔭_{+}) and its global G_{0}-equivariant sections, called Weyl structures, are in bijection with principal connections in the c-projective class. Moreover, any Weyl structure σ: G_{0} → G induces an vector bundle isomorphism

G_{0} ×_{G_{0}} 𝔼 ≅ G ×_{P} 𝔼
[u, X] ↦ [σ(u), X],

for any P -module 𝔼.

Note that there is a tautological 1-form ν ∈ Ω^{1}(G, 𝔤_{−1} ⊕ 𝔤_{0}) on G given by

ν(θ(u) + γ^{∇}(u))(ξ) := (θ(u) + γ^{∇}(u))((T q)ξ).  (21)

Extending this form to a normal Cartan connection ω ∈ Ω^{1}(G, 𝔤) establishes the equivalence of categories in Theorem 2.8.

Remark 2.7. In Section 2.1 we observed that there are always so-called special connections in the c-projective class. A Weyl structure corresponding to a special connection is precisely what in the literature on parabolic geometries is called an exact Weyl structure (see [36, 37]). The name is due to the fact that they form an affine space over the space of exact 1-forms on M.

Note also that the almost complex structure J on M induces an almost complex structure J^{G_{0}} on the complex frame bundle G_{0} of M. If J is integrable, so is J^{G_{0}} and G_{0} is a holomorphic vector bundle over M. Moreover, the complex structure on 𝔤 induces, by means of the isomorphism ω: T G ≅ G × 𝔤, an almost complex structure J^{G} on G, satisfying T p ∘ J^{G} = J ∘ T p and T q ∘ J^{G} = J^{G_{0}} ∘ T q. Note that the definition of the almost complex structure on J^{G_{0}} and J^{G} implies that θ and ω are of type (1, 0).

Let us also remark that an immediate consequence of Theorem 2.8 and the Liouville Theorem for Cartan geometries (see e.g. [36, Proposition 1.5.3]) is the following classical result.

Proposition 2.10. For n ≥ 2 the c-projective transformations of (ℂℙ^{n}, J_{can}, [∇^{g_{F S}}]) (which by Proposition 2.2 are the complex diffeomorphisms of ℂℙ^{n} that map complex lines to complex lines) are precisely given by the left multiplications of elements in PSL(n + 1, ℂ). Moreover, any local c-projective transformation of (ℂℙ^{n}, J_{can}, [∇^{g_{F S}}]) uniquely extends to a global one.

We finish this section by introducing some notation. The P -module 𝔤 admits an invariant filtration 𝔤 ⊃ 𝔭 ⊃ 𝔤_{1} and hence the adjoint tractor bundle AM := G ×_{P} 𝔤 is naturally filtered

AM = A^{−1}M ⊃ A^{0}M ⊃ A^{1}M,

with A^{1}M ≅ T^{∗}M and AM/A^{0}M ≅ T M. Hence, the associated graded vector bundle to AM is given by

gr(AM) = gr_{−1}(AM) ⊕ gr_{0}(AM) ⊕ gr_{1}(AM) = T M ⊕ 𝔤𝔩(T M, J) ⊕ T^{∗}M, (22)

which can be identified with G_{0} ×_{G_{0}} 𝔤.

2.4. The curvature of the canonical Cartan connection. Suppose σ: G_{0} → G is a Weyl structure and let γ^{∇} be the corresponding principal connection in the c-projective class. Since the normal Cartan connection ω is P -equivariant and σ is G_{0}-equivariant, the pullback σ^{∗}ω ∈ Ω^{1}(G_{0}, 𝔤) is G_{0}-equivariant and hence decomposes according to the grading on 𝔤 into three components. Since ω extends the tautological form ν on G, defined by (21), we deduce that

σ^{∗}ω = θ + γ^{∇} − p^{∇},  (23)

where p^{∇} ∈ Ω^{1}(G_{0}, 𝔤_{1}) is horizontal and G_{0}-equivariant and hence can be viewed as a section P^{∇} of T^{∗}M ⊗ T^{∗}M, called the Rho tensor of ∇. Via σ, the curvature κ ∈ Ω^{2}(M, AM) of ω can be identified with a section κ^{σ} of

∧^{2}T^{∗}M ⊗ gr(AM)

= (∧^{2}T^{∗}M ⊗ T M) ⊕ (∧^{2}T^{∗}M ⊗ 𝔤𝔩(T M, J)) ⊕ (∧^{2}T^{∗}M ⊗ T^{∗}M),

which decomposes according to this splitting into three components

κ^{σ} = T + W^{∇} − C^{∇}.

One computes straightforwardly that T ∈ Γ(∧^{2}T^{∗}M ⊗T M) is the torsion of the almost c-projective structure and that C^{∇} = d^{∇}P^{∇} ∈ Γ(∧^{2}T^{∗}M ⊗ T^{∗}M), where d^{∇} denotes the covariant exterior derivative on differential forms with values in T^{∗}M induced by ∇. The tensor C^{∇} is called the Cotton–York tensor of ∇. To describe the component W^{∇} ∈ Γ(∧^{2}T^{∗}M ⊗ 𝔤𝔩(T M, J)), called the (c-projective) Weyl curvature of ∇, let us denote by R^{∇} ∈ Ω^{2}(M, 𝔤𝔩(T M, J)) the curvature of ∇. Then one has

W^{∇} = R^{∇} − ∂P^{∇},

where

(∂P^{∇})_{αβ}^{γ}_{ϵ} := δ_{[α}^{γ}P_{β]ϵ}^{∇} − J_{[α}^{γ}P_{β]ζ}^{∇}J_{ϵ}^{ζ} − P_{[αβ]}^{∇}δ_{ϵ}^{γ} − J_{[α}^{ζ}P_{β]ζ}^{∇}J_{ϵ}^{γ}.  (24)

Remark 2.8. The map ∂: T^{∗}M ⊗ T^{∗}M → ∧^{2}T^{∗}M ⊗ 𝔤𝔩(T M, J) given by (24) is related to Lie algebra cohomology. It is easy to see that the Lie algebra differentials in the complex computing the Lie algebra cohomology of the Abelian real Lie algebra 𝔤_{−1} with values in the representation 𝔤 are G_{0}-equivariant and that ∂ is induced by the restriction to 𝔤_{−1}^{∗} ⊗ 𝔤_{1} ≅ 𝔤_{1} ⊗ 𝔤_{1} of half of the second differential in this complex.

The normal Cartan connection ω is characterised as the unique extension of ν to a Cartan connection such that ∂^{∗}κ^{σ} = 0 for all Weyl structures σ: G_{0} → G. Analysing ker ∂^{∗} shows that T_{αβ}^{γ} is in there, since forms of type (0, 2) are, and C^{∇} is, since ∧^{2}T^{∗}M ⊗ T^{∗}M ⊂ ker ∂^{∗}. Hence, P^{∇} is uniquely determined by requiring that W^{∇} be in the kernel of ∂^{∗}.

Remark 2.9. Recall that in Definition 2.1 we restricted our definition of almost cprojective structures to c-projective equivalence classes of minimal connections. Since the kernel of

∂^{∗}: ∧^{2}T^{∗}M ⊗ T M → T^{∗}M ⊗ 𝔤𝔩(T M, J)

consists precisely of all the 2-forms with values in T M of type (0, 2), the discussion of the construction of the Cartan connection above shows that the minimality condition is forced by the normalisation condition of the Cartan connection. The requirement for the almost c-projective structure to be minimal is however not necessary in order to construct a canonical Cartan connection. In fact, starting with any complex connection, one can show that there is a complex connection with the same J-planar curves whose torsion has only two components, namely the (0, 2)-component −\frac{1}{4}N_{J} and a component in the subspace of (1, 1)-tensors in ∧^{2}T^{∗}M ⊗ T M that are trace and J-trace free. Imposing this normalisation condition on an almost c-projective structure allows then analogously as above to associate a canonical Cartan connection (see [65]).

Proposition 2.11. Suppose (M, J, [∇]) is an almost c-projective manifold of dimension 2n ≥ 4. Let ∇ ∈ [∇] be a connection in the c-projective class. Then the Rho tensor corresponding to ∇ is given by

P_{αβ}^{∇} = \frac{1}{n+1}(Ric_{αβ}^{∇} + \frac{1}{n−1}(Ric_{(αβ)}^{∇} − J_{(α}^{γ}J_{β)}^{δ}Ric_{γδ}^{∇})),  (25)

where Ric_{αβ}^{∇} := R_{γα}^{∇}^{γ}_{β} is the Ricci tensor of ∇. Moreover, if ˆ∇ ∈ [∇] is another connection in the class, related to ∇ according to (11), then

P ˆ∇ ; αβ = P_{αβ}^{∇} − ∇_{α}Υ_{β} + \frac{1}{2}(Υ_{α}Υ_{β} − J_{α}^{γ}J_{β}^{δ}Υ_{γ}Υ_{δ}).  (26)

Proof. The map ∂^{∗}: ∧^{2}T^{∗}M ⊗ 𝔤𝔩(T M, J) → T^{∗}M ⊗ T^{∗}M is a multiple of a Ricci-type contraction. Hence, the normality of ω implies

R_{αβ}^{∇}^{α}_{ϵ} = (∂P^{∇})_{αβ}^{α}_{ϵ} = (n + \frac{1}{2})P_{βϵ}^{∇} − \frac{1}{2}P_{ϵβ}^{∇} + J_{(β}^{γ}J_{ϵ)}^{ζ}P_{γζ}^{∇}.  (27)

Therefore, Ric_{[βϵ]}^{∇} = (n + 1)P_{[βϵ]}^{∇} and Ric_{(βϵ)}^{∇} = nP_{(βϵ)}^{∇} + J_{(β}^{γ}J_{ϵ)}^{ζ}P_{γζ}, which implies that

Ric_{(βϵ)}^{∇} − J_{(β}^{γ}J_{ϵ)}^{ζ}Ric_{γζ}^{∇} = (n − 1)(P_{(βϵ)}^{∇} − J_{(β}^{γ}J_{ϵ)}^{ζ}P_{γζ}^{∇}).

Using these identities one verifies immediately that formula (25) holds. The formula (26) for the change of the Rho tensor can easily be verified directly or follows from the general theory of Weyl structures for parabolic geometries established in [37] taking into account that the Rho tensor in [37] is −\frac{1}{2} times the Rho tensor given by (25) and our conventions for the definition of Υ as in (11). □

As an immediate consequence (writing out (26) in terms of its components using the various projectors Π_{α}^{a}, … and the formulae (4)) we have:

Corollary 2.12.

• \overline{P_{ab}^{∇}} = P_{āb̄}^{∇} and \overline{P_{āb}^{∇}} = P_{ab̄}^{∇}

• P ˆ∇ ; ab = P_{ab}^{∇} − ∇_{a}Υ_{b} + Υ_{a}Υ_{b}

• P ˆ∇ ; āb = P_{āb}^{∇} − ∇āΥ_{b}

For any connection ∇ ∈ [∇], its Weyl curvature W^{∇} is, by construction, a section of ∧^{2}T^{∗}M ⊗ 𝔤𝔩(T M, J) that satisfies W_{αβ}^{∇}^{α}_{γ} ≡ 0. This implies that also J_{ζ}^{α}W_{αβ}^{∇}^{ζ}_{ϵ} = W_{αβ}^{∇}^{α}_{ζ}J_{ϵ}^{ζ} ≡ 0. In the sequel we will often simply write W instead of W^{∇}, and similarly for other tensors such as the Rho tensor, the dependence of ∇ being understood. Viewing W as a 2-form with values in the complex bundle vector bundle 𝔤𝔩(T M, J) ≅ 𝔤𝔩(T^{1,0}M, ℂ), it decomposes according to (p, q)-types into three components:

W_{ab}^{c}_{d} W_{ab̄}^{c}_{d} Wāb̄^{c}_{d}.

The vanishing of the trace and J-trace above, then imply that

W_{ab}^{a}_{d} = W_{ab̄}^{a}_{d} ≡ 0.

In these conclusions and in Corollary 2.12 we begin to see the utility of writing our expressions in using the barred and unbarred indices introduced in Section 1. In the following discussion we pursue this systematically, firstly by describing exactly how the curvature of a complex connection decomposes. We analyse these decompositions from the perspective of c-projective geometry: some pieces are invariant whilst others transform simply. For the convenience of the reader, we reiterate some of our previous conclusions in the following theorem (but prove them more easily using barred and unbarred indices, as just advocated).

Proposition 2.13. Suppose (M, J, [∇]) is an almost c-projective manifold of dimension 2n ≥ 4. Let Tāb̄^{c} denote its torsion (already observed to be a constant multiple of the Nijenhuis tensor of (M, J)). Then the curvature R of a connection ∇ in the c-projective class decomposes as follows:

\begin{aligned}R_{ab}^{c}_{d} & = W_{ab}^{c}_{d} + 2δ_{[a}^{c}P_{b]d} + β_{ab}δ_{d}^{c} \\ R_{ab̄}^{c}_{d} & = W_{ab̄}^{c}_{d} + δ_{a}^{c}P_{b̄d} + δ_{d}^{c}P_{b̄a} \\ W_{ab̄}^{c}_{d} & = H_{ab̄}^{c}_{d} − \frac{1}{2(n+1)}(δ_{a}^{c}T_{df} ēTēb̄^{f} + δ_{d}^{c}T_{af} ēTēb̄^{f}) − \frac{1}{2}T_{ad}ēTēb̄^{c} \\ Rāb̄^{c}_{d} & = Wāb̄^{c}_{d} = ∇_{d}Tāb̄^{c} \end{aligned}  (28)

where

W_{ab}^{c}_{d} = W_{[ab]}^{c}_{d} W_{[ab}^{c}_{d]} = 0 W_{ab}^{a}_{d} = 0 β_{ab} = −2P_{[ab]}
H_{ab̄}^{c}_{d} = H_{db̄}^{c}_{a} H_{ab̄}^{a}_{d} = 0.

Let ˆ∇ be another connection in the c-projective class, related to ∇ by (11), and denote its curvature components by Ŵ, Ĥ, and P̂. Then we have:

(1) Ŵ_{ab}^{c}_{d} = W_{ab}^{c}_{d} and Ŵ_{ab̄}^{c}_{d} = W_{ab̄}^{c}_{d} and Ĥ_{ab̄}^{c}_{d} = H_{ab̄}^{c}_{d},

(2) Ŵāb̄^{c}_{d} = Wāb̄^{c}_{d} + Tāb̄^{e}υ_{ed}^{c} and if J is integrable, then Wāb̄^{c}_{d} ≡ 0,

(3) W_{ab}^{c}_{c} ≡ 0,

(4) W_{ab̄}^{c}_{c} = T_{fa}ēTēb̄^{f},

whilst we recall that P̂_{ab} = P_{ab} − ∇_{a}Υ_{b} + Υ_{a}Υ_{b}, P̂_{b̄d} = P_{b̄d} − ∇b̄Υ_{d}.

The tensor β_{ab} = −2P_{[ab]} satisfies

∇_{[b}β_{ce]} = P_{f̄[b}T_{ce]}f̄ − \frac{1}{n+1}T_{[bc}f̄T_{e]a}d̄Td̄ ; f̄^{a}.  (29)

Finally, the Cotton–York tensors C_{abc} and C_{ab̄c} are defined as

C_{abc} := ∇_{a}P_{bc} − ∇_{b}P_{ac} + T_{ab}d̄P_{d̄c} and C_{ab̄c} := ∇_{a}P_{b̄c} − ∇b̄P_{ac}.  (30)

The first of these satisfies a Bianchi identity

∇_{a}W_{bc}^{a}_{e} − (n − 2)C_{bce}

= 2T_{a[b}f̄H_{c]f̄}^{a}_{e} + \frac{2}{n + 1}T_{bc}f̄T_{ea}d̄Td̄ ; f̄^{a} − \frac{n}{n + 1}T_{e[b}f̄T_{c]a}d̄Td̄ ; f̄^{a} (31)

and transforms as

Ĉ_{bce} = C_{bce} + Υ_{a}W_{bc}^{a}_{e}  (32)

under c-projective change (11). Another part of the Bianchi identity reads

C_{ab̄c} − C_{cb̄a} = \frac{1}{n+1}(Tb̄f̄^{d}∇_{d}T_{ac}f̄ + Rb̄f̄^{d}_{d}T_{ac}f̄ − 2Rb̄f̄^{d}_{[a}T_{c]d}f̄).  (33)

Proof. In this proof we also take the opportunity to develop various useful formulae for torsion and curvature and for how these quantities transform under c-projective change (11). As in the statement of Proposition 2.13, we express all these formulae in terms of the abstract indices on almost complex manifolds developed in Section 1. Firstly, recall that since we are working with minimal connections (cf. Definition 2.1), their torsions are restricted to being of type (0, 2) and this means precisely that

\begin{aligned}(∇_{a}∇_{b} − ∇_{b}∇_{a})f + T_{ab}c̄∇c̄f & = 0 & (∇ā∇_{b} − ∇_{b}∇ā)f & = 0 \\ (∇ā∇b̄ − ∇b̄∇ā)f + Tāb̄^{c}∇_{c}f & = 0 & (∇_{a}∇b̄ − ∇b̄∇_{a})f & = 0, \end{aligned}  (34)

where T_{ab}c̄ ≡ Π_{a}^{α}Π_{b}^{β}\overline{Π}_{γ}^{c}T_{αβ}^{γ}, equivalently its complex conjugate Tāb̄^{c} = \overline{T_{ab}c̄}, represents the Nijenhuis tensor as in (8). Notice that the second line of (34) is the complex conjugate of the first. In this proof, we take advantage of this general feature by listing only one of such conjugate pairs, its partner being implicitly valid. For example, here are characterisations of sufficiently many components of the general curvature tensor R_{αβ}^{γ}_{δ}.

(∇_{a}∇_{b} − ∇_{b}∇_{a})X^{c} + T_{ab}d̄∇d̄X^{c} = R_{ab}^{c}_{d}X^{d}
(∇_{a}∇b̄ − ∇b̄∇_{a})X^{c} = R_{ab̄}^{c}_{d}X^{d}
[or (∇ā∇_{b} − ∇_{b}∇ā)X^{c} = R_{āb}^{c}_{d}X^{d}, if preferred]
(∇_{a}∇_{b} − ∇_{b}∇_{a})Xc̄ + T_{ab}d̄∇d̄Xc̄ = R_{ab}c̄d̄Xd̄.

(35)

For convenience, the dual formulae are sometimes preferred: for example,

(∇_{a}∇_{b} − ∇_{b}∇_{a})ϕ_{d} + T_{ab}c̄∇c̄ϕ_{d} = −R_{ab}^{c}_{d}ϕ_{c} .  (36)

The tensor υ_{αβ}^{γ} employed in a c-projective change of connection (11) was already broken into irreducible pieces in deriving Proposition 2.3, e.g.

υ_{ab}^{c} = Π_{a}^{α}Π_{b}^{β}Π_{γ}^{c}υ_{αβ}^{γ} = Υ_{a}δ_{b}^{c} + Υ_{b}δ_{a}^{c} ⇒ ˆ∇_{a}X^{c} = ∇_{a}X^{c} + Υ_{a}X^{c} + Υ_{b}X^{b}δ_{a}^{c} (37)
and υ_{āb}^{c} = \overline{Π}α ; āΠ_{b}^{β}Π_{γ}^{c}υ_{αβ}^{γ} = 0  ⇒ ˆ∇āX^{c} = ∇āX^{c}.  (38)

It is an elementary matter, perhaps more conveniently executed in the dual formulation

ˆ∇_{a}ϕ_{b} = ∇_{a}ϕ_{b} − Υ_{a}ϕ_{b} − Υ_{b}ϕ_{a}  ˆ∇āϕ_{b} = ∇āϕ_{b} ,  (39)

to compute the effect of these changes on curvature, namely

R̂_{ab}^{c}_{d} = R_{ab}^{c}_{d} − 2δ_{[a}^{d}(∇_{b]}Υ_{c}) + 2δ_{[a}^{d}Υ_{b]}Υ_{c} + 2(∇_{[a}Υ_{b]})δ_{d}^{c}
R̂_{ab̄}^{c}_{d} = R_{ab̄}^{c}_{d} − δ_{a}^{c}∇b̄Υ_{d} − δ_{d}^{c}∇b̄Υ_{a}
R̂_{ab}c̄d̄ = R_{ab}c̄d̄ + T_{ab}c̄Υd̄ + ΥēT_{ab}ēδd̄c̄ = R_{ab}c̄d̄ + T_{ab}ēυēd̄c̄.

(40)

We shall also need the Bianchi symmetries derived from (35) or, more conveniently in the dual formulation, as follows. Evidently,

∇_{a}(∇_{b}∇_{c} − ∇_{c}∇_{b})f + ∇_{b}(∇_{c}∇_{a} − ∇_{a}∇_{c})f + ∇_{c}(∇_{a}∇_{b} − ∇_{b}∇_{a})f = (∇_{a}∇_{b} − ∇_{b}∇_{a})∇_{c}f + (∇_{b}∇_{c} − ∇_{c}∇_{b})∇_{a}f + (∇_{c}∇_{a} − ∇_{a}∇_{c})∇_{b}f,

which we may expand using (34) and (35) to obtain

(∇_{a}T_{bc}d̄ + ∇_{b}T_{ca}d̄ + ∇_{c}T_{ab}d̄)∇d̄f = (R_{ab}^{d}_{c} + R_{bc}^{d}_{a} + R_{bc}^{d}_{a})∇_{d}f

and hence that

∇_{[a}T_{bc]}d̄ = 0 R_{[ab}^{c}_{d]} = 0.  (41)

Similarly, by looking at different orderings for the indices of ∇_{a}∇b̄∇_{c}f, we find that

R_{ab̄}^{c}_{d} − R_{db̄}^{c}_{a} + T_{ad}ēTēb̄^{c} = 0 R_{ab}c̄d̄ = ∇d̄T_{ab}c̄.  (42)

Already, the final statement of (28) is evident and if T_{ab}c̄ = 0 then both R_{ab}c̄d̄ and its complex conjugate Rāb̄^{c}_{d} vanish. Notice that ∂P does not contribute to this piece of curvature. Specifically, from (24)

(∂P)āb̄^{c}_{d} = \overline{Π}α ; ā\overline{Π}β ; b̄Π_{γ}^{c}Π_{d}^{ϵ}(∂P)_{αβ}^{γ}_{ϵ} = −P_{[āb̄]}δ_{d}^{c} − P_{[b̄ā]}δ_{d}^{c} = 0.

It follows that Wāb̄^{c}_{d} = Rāb̄^{c}_{d} in general and that Wāb̄^{c}_{d} = Rāb̄^{c}_{d} = 0 in the integrable case. The rest of statement (2) also follows, either from the last line of (40) or, more easily, from the c-projective invariance of T_{ab}c̄ (depending only on the underlying almost complex structure), the second identity of (42), and the transformation rules (39).

Now let us consider the curvature R_{ab}^{c}_{d}. From (24), we compute that

(∂P)_{ab}^{c}_{d} = Π_{a}^{α}Π_{b}^{β}Π_{γ}^{c}Π_{d}^{ϵ}(∂P)_{αβ}^{γ}_{ϵ} = 2δ_{[a}^{c}P_{b]d} − 2P_{[ab]}δ_{d}^{c}

and from (25) that

P_{ab} = Π_{a}^{α}Π_{b}^{β}P_{αβ} = \frac{1}{n+1}(Ric_{ab} + \frac{2}{n−1}Ric_{(ab)}) = \frac{1}{n+1}(Ric_{ab} + \frac{2}{n−1}Ric_{(ab)}),

equivalently that Ric_{ab} = (n − 1)P_{ab} + 2P_{[ab]}. Bearing in mind the Bianchi symmetry (41) for R_{ab}^{c}_{d}, this means that we may write

R_{ab}^{c}_{d} = W_{ab}^{c}_{d} + 2δ_{[a}^{c}P_{b]d} + β_{ab}δ_{d}^{c},  (43)

where

W_{ab}^{c}_{d} = W_{[ab]}^{c}_{d} W_{[ab}^{c}_{d]} = 0 W_{ab}^{a}_{d} = 0 β_{ab} = −2P_{[ab]}.

Comparing this decomposition with the first line of (40) implies that W_{ab}^{c}_{d} is invariant and confirms that P_{ab} transforms according to Corollary 2.12. In summary,

Ŵ_{ab}^{c}_{d} = W_{ab}^{c}_{d}  P̂_{ab} = P_{ab} − ∇_{a}Υ_{c} + Υ_{b}Υ_{c}  β̂_{ab} = β_{ab} + 2∇_{[a}Υ_{b]}.

We have shown (3) and the first statement of (1).

The remaining statements concern the curvature R_{ab̄}^{c}_{d}. From (24), we compute that

(∂P)_{ab̄}^{c}_{d} = Π_{a}^{α}\overline{Π}β ; b̄Π_{γ}^{c}Π_{d}^{ϵ}(∂P)_{αβ}^{γ}_{ϵ} = δ_{a}^{c}P_{b̄d} + δ_{d}^{c}P_{b̄a}

and from (25) that

P_{b̄d} = \overline{Π}β ; b̄Π_{d}^{ϵ}P_{βϵ} = \frac{1}{n+1}Ric_{b̄d} = \frac{1}{n+1}R_{ab̄}^{a}_{d}.

From (42) it now follows that

R_{ab̄}^{c}_{d} + \frac{1}{2}T_{ad}ēTēb̄^{c} = H_{ab̄}^{c}_{d} − \frac{1}{2(n+1)}(δ_{a}^{c}T_{df} ēTēb̄^{f} + δ_{d}^{c}T_{af} ēTēb̄^{f}) + (∂P)_{ab̄}^{c}_{d},  (44)

where

H_{ab̄}^{c}_{d} = H_{db̄}^{c}_{a} H_{ab̄}^{a}_{d} = 0.

Recall that by definition

W_{ab̄}^{c}_{d} = R_{ab̄}^{c}_{d} − (∂P)_{ab̄}^{c}_{d}.

Therefore

W_{ab̄}^{c}_{d} = H_{ab̄}^{c}_{d} − \frac{1}{2(n+1)}(δ_{a}^{c}T_{df} ēTēb̄^{f} + δ_{d}^{c}T_{af} ēTēb̄^{f}) − \frac{1}{2}T_{ad}ēTēb̄^{c}.

Comparison with the formula for R̂_{ab̄}^{c}_{d} in (40) immediately shows that W_{ab̄}^{c}_{d} and H_{ab̄}^{c}_{d} are c-projectively invariant and also that

W_{ab̄}^{c}_{c} = −T_{af} ēTēb̄^{f},

as required to complete (1) and (4). Next we demonstrate the behaviour of the Cotton– York tensor. For this, we need a Bianchi identity with torsion, which may be established as follows. Evidently,

∇_{a}(∇_{b}∇_{c} − ∇_{c}∇_{b})ϕ_{e} + ∇_{b}(∇_{c}∇_{a} − ∇_{a}∇_{c})ϕ_{e} + ∇_{c}(∇_{a}∇_{b} − ∇_{b}∇_{a})ϕ_{e} = (∇_{a}∇_{b} − ∇_{b}∇_{a})∇_{c}ϕ_{e} + (∇_{b}∇_{c} − ∇_{c}∇_{b})∇_{a}ϕ_{e} + (∇_{c}∇_{a} − ∇_{a}∇_{c})∇_{b}ϕ_{e},

the left hand side of which may be expanded by (36) as

∇_{a}(−R_{bc}^{d}_{e}ϕ_{d} − T_{bc}d̄∇d̄ϕ_{e}) + ⋯ + ⋯,

where ⋯ represent similar terms where the indices abc are cycled around. On the other hand, the right hand side may be expanded as

⋯ − R_{bc}^{d}_{a}∇_{d}ϕ_{e} − R_{bc}^{d}_{e}∇_{a}ϕ_{d} − T_{bc}d̄∇d̄∇_{a}ϕ_{e} − · · · .

Comparison yields

(∇_{[a}R_{bc]}^{d}_{e})ϕ_{d} + (∇_{[a}T_{bc]}d̄)∇d̄ϕ_{e} − T_{[bc}d̄R_{a]d̄}^{f}_{e}ϕ_{f} = R_{[bc}^{d}_{a]}∇_{d}ϕ_{e}

and, from the Bianchi symmetries (41), we conclude that

∇_{[a}R_{bc]}^{d}_{e} = T_{[ab}f̄R_{c]f̄}^{d}_{e}.

Using (28) and tracing over a and d yields

∇_{a}W_{bc}^{a}_{e} − 2(n − 2)∇_{[b}P_{c]e} + 3∇_{[b}β_{ce]}

= 2T_{a[b}f̄H_{c]f̄}^{a}_{e}+\frac{1}{n + 1}T_{bc}f̄T_{ea}d̄Td̄ ; f̄^{a}−\frac{n + 2}{n + 1}T_{e[b}f̄T_{c]a}d̄Td̄ ; f̄^{a}+(n−2)T_{bc}f̄P_{f̄e}+3P_{f̄[b}T_{ce]}f̄.

Skewing this identity over bce gives (29) and substituting back gives

∇_{a}W_{bc}^{a}_{e} − 2(n − 2)∇_{[b}P_{c]e}

= 2T_{a[b}f̄H_{c]f̄}^{a}_{e} + \frac{2}{n + 1}T_{bc}f̄T_{ea}d̄Td̄ ; f̄^{a} − \frac{n}{n + 1}T_{e[b}f̄T_{c]a}d̄Td̄ ; f̄^{a} + (n − 2)T_{bc}f̄P_{f̄e}.

The contracted Bianchi identity (31) follows from the definition (30) of the Cotton– York tensor. Notice that the right hand side of (31) is c-projectively invariant. Also, by computing that

ˆ∇_{a}Ŵ_{bc}^{d}_{e} = ˆ∇_{a}W_{bc}^{d}_{e}

= ∇_{a}W_{bc}^{d}_{e} − 2Υ_{a}W_{bc}^{d}_{e} − Υ_{b}W_{ac}^{d}_{e} − Υ_{c}W_{ba}^{d}_{e} + δ_{a}^{d}Υ_{f}W_{bc}^{f}_{e} − Υ_{e}W_{bc}^{d}_{a}

and tracing over a and d, we see that

ˆ∇_{a}Ŵ_{bc}^{a}_{e} = ∇_{a}W_{bc}^{a}_{e} + (n − 2)Υ_{a}W_{bc}^{a}_{e}

and for n > 2 conclude that (32) is valid. The case n = 2 is somewhat degenerate. Although (32) is still valid, as we shall see below in Proposition 2.14, the Weyl curvature W_{bc}^{a}_{d} vanishes by symmetry considerations and (32) reads Ĉ_{bce} = C_{bce}, the straightforward verification of which is left to the reader. Similarly, by considering different orderings for the indices of ∇_{a}∇b̄∇_{c}ϕ_{e}, we are rapidly led to

∇_{a}R_{cb̄}^{d}_{e} − ∇_{c}R_{ab̄}^{d}_{e} + ∇b̄R_{ac}^{d}_{e} = T_{ac}f̄Rb̄f̄^{d}_{e}

as another piece of the Bianchi identity, which may then be further split into irreducible parts. In particular, tracing over d and e (equivalently, tracing over a and d and then skewing over c and e) gives (33). □

Proposition 2.14. Suppose that W_{ab}^{c}_{d} ∈ ∧^{1,0} ⊗ ∧^{1,0} ⊗ T^{1,0}M ⊗ ∧^{1,0} has the following symmetries:

W_{ab}^{c}_{d} = W_{[ab]}^{c}_{d} W_{[ab}^{c}_{d]} = 0 W_{ab}^{a}_{d} = 0.

If 2n = 4, then W_{ab}^{c}_{d} ≡ 0.

Proof. Fix a nonzero skew tensor V_{ab}. As W_{ab}^{c}_{d} is skew in a and b, it follows that W_{ab}^{c}_{d} = V_{ab}S^{c}_{d} for some unique tensor S^{c}_{d}. Now W_{ab}^{a}_{d} = V_{ab}S^{a}_{d} but V_{ab} is also nondegenerate so W_{ab}^{a}_{d} = 0 implies S^{c}_{d} = 0. □

Remark 2.10. When n = 2, the identity (31) is vacuous. Proposition 2.14 implies that the left hand side vanishes. For the right hand side, the vanishing of T_{a[b}f̄H_{c]f̄}^{a}_{e} follows by tracing the identity T_{[ab}f̄H_{c]f̄}^{d}_{e} = 0 over a and d, bearing in mind that H_{af̄}^{d}_{e} is trace-free in a and d. The remaining terms also evaporate because, when n = 2, the tensor T_{bc}f̄T_{ea}d̄ is symmetric in f̄ ; d̄ whereas Td̄ ; f̄^{a} is skew.

The torsion T_{ab}c̄ (equivalently, its complex conjugate Tāb̄^{c}) is c-projectively invariant. The same is true, not only of the Weyl curvature W_{ab̄}^{c}_{d}, but also of its trace-free symmetric part H_{ab̄}^{c}_{d} (which will be identified as part of the harmonic curvature in Section 2.7). The Weyl curvature W_{ab}^{c}_{d} is c-projectively invariant and forms the final piece of harmonic curvature except when 2n = 4, in which case W_{ab}^{c}_{d} necessarily vanishes, its role being taken by C_{abc}, the c-projectively invariant part of the Cotton– York tensor. In Section 2.7, we place this discussion in the context of general parabolic geometry but, before that, we collect in the following section some useful formulae for the various curvature operators on c-projective densities.

2.5. Curvature operators on c-projective densities. Suppose X^{cd···e} = X^{[cd···e]} is a section of E(n + 1, 0) = ∧^{n}T^{1,0}M and Y^{c̄d̄···ē} a section of E(0, n + 1) = ∧^{n}T^{0,1}M. Then it follows from (35) that

(∇ā∇_{b} − ∇_{b}∇ā)X^{cd···e} = R_{āb}^{c}_{f}X^{fd···e} + R_{āb}^{d}_{f}X^{cf···e} + ⋯ + R_{āb}^{e}_{f}X^{cd···f}
= R_{āb}^{f}_{f}X^{cd···e}.
(∇ā∇_{b} − ∇_{b}∇ā)Y^{c̄d̄···ē} = R_{āb}c̄f̄Y^{f̄d̄···ē} + R_{āb}d̄f̄Y^{c̄f̄···ē} + ⋯ + R_{āb}ēf̄Y^{c̄d̄···f̄}
= R_{āb}f̄f̄Y^{cd···e}.

However, from Proposition 2.13 part (4), we find that

R_{āb}^{f}_{f} = −R_{bā}^{f}_{f} = −W_{bā}^{f}_{f} − (∂P)_{bā}^{f}_{f} = −T_{fb}ēT_{ēā}^{f} − (n + 1)P_{āb}
R_{āb}f̄f̄ = W_{āb}f̄f̄ + (∂P)_{āb}f̄f̄ = T_{fb}ēT_{ēā}^{f} + (n + 1)P_{bā}.

We conclude immediately that for a section σ of E(k, ℓ) we have

(∇ā∇_{b} − ∇_{b}∇ā)σ = \frac{ℓ−k}{n+1} T_{fb}ēT_{ēā}^{f}σ + ℓP_{bā}σ − kP_{āb}σ.  (45)

Similarly, from (35) it also follows that

∇_{a}∇_{b} − ∇_{b}∇_{a})X^{cd···e} + T_{ab}f̄∇f̄X^{cd···e} = R_{ab}^{f}_{f}X^{cd···e}
(∇_{a}∇_{b} − ∇_{b}∇_{a})Y^{c̄d̄···ē} + T_{ab}f̄∇f̄Y^{c̄d̄···ē} = R_{ab}f̄f̄Y^{c̄d̄···ē}.

From Proposition 2.13 we conclude that

R_{ab}^{f}_{f} = 2P_{[ba]} + nβ_{ab} = (n + 1)β_{ab}
R_{ab}f̄f̄ = ∇f̄T_{ab}f̄.

Therefore, if σ is c-projective density of weight (k, ℓ), then

(∇_{a}∇_{b} − ∇_{b}∇_{a})σ + T_{ab}^{f̄}∇f̄σ = kβ_{ab}σ + \frac{ℓ}{n+1}(∇f̄T_{ab}^{f̄})σ  (46)

and, accordingly,

(∇ā∇b̄ − ∇b̄∇ā)σ + Tāb̄^{f}∇_{f}σ = ℓβāb̄ + \frac{k}{n+1}(∇_{f}Tāb̄^{f})σ.  (47)

Recall that for any connection ∇ ∈ [∇] its Rho tensor, by definition, satisfies P_{āb} = \frac{1}{n+1}Ric_{āb} and P_{[ab]} = \frac{1}{n+1}Ric_{[ab]}. Hence, the identities (45) and (46) imply that the Ricci tensor of a special connection ∇ ∈ [∇] satisfies

• Ric_{āb} = Ric_{bā}

• Ric_{[ab]} = \frac{1}{2}∇c̄T_{ab}c̄.

If ∇c̄T_{ab}c̄ vanishes, the special connection has symmetric Ricci tensor. In particular, if J is integrable all special connections have symmetric Ricci tensor.

2.6. The curvature of complex projective space. In Section 2.4, and especially in Proposition 2.13, the curvature of a complex connection on a general almost complex manifold was decomposed into various irreducible pieces (irreducibility to be further discussed in Section 3.3). Here, we pause to examine this decomposition on complex projective space ℂℙ^{n} with its standard Fubini–Study metric.

Lemma 2.15. The Riemannian curvature tensor for the Fubini–Study metric g_{αβ} on ℂℙ^{n} is given by

R_{αβγδ} = g_{αγ}g_{βδ} − g_{βγ}g_{αδ} + Ω_{αγ}Ω_{βδ} − Ω_{βγ}Ω_{αδ} + 2Ω_{αβ}Ω_{γδ}  (48)

where J_{α}^{β} is the complex structure and Ω_{αγ} ≡ J_{α}^{β}g_{βγ} (the Kähler form).

Proof. A direct calculation from the definition of the Fubini–Study metric (e.g. [27]) or by invariant theory noting that (up to scale) the right hand side of (48) is the only covariant expression in g_{αβ} and Ω_{αβ} such that

R_{αβγδ} = R_{[αβ][γδ]} R_{[αβγ]δ} = 0 R_{αβγ[δ}J_{ϵ]}^{γ} = 0

where the last condition is a consequence of the Kähler condition dΩ = 0 (or, more precisely, a consequence of ∇_{α}Ω_{βγ} = 0 as one can check, by direct computation in case the almost complex structure J_{α}^{β} is orthogonal (i.e. J_{α}^{β}g_{βγ} is skew), that

2∇_{α}Ω_{βγ} = 3∇_{[α}Ω_{βγ]} − 3J_{α}^{δ}J_{β}^{ϵ}∇_{[α}Ω_{δϵ]} − Ω_{αδ}N_{αβ}^{δ},

where recall that N_{αβ}^{γ} is the Nijenhuis tensor (1), which vanishes when the complex structure is integrable, as it is on ℂℙ^{n}). □

To apply the decompositions of Proposition 2.13 to (48) we should raise an index

R_{αβ}^{γ}_{ϵ} = δ_{α}^{γ}g_{βϵ} − δ_{β}^{γ}g_{αϵ} + J_{α}^{γ}Ω_{βϵ} − J_{β}^{γ}Ω_{αϵ} − 2Ω_{αβ}J_{ϵ}^{γ}

and then apply the various projectors such as Π_{a}^{α}Π_{b}^{β}Π_{γ}^{c}Π_{d}^{ϵ}. However, firstly note that applying Π_{a}^{α}Π_{c}^{γ} to J_{α}^{β}g_{βγ} + J_{γ}^{β}g_{βα} = 0 implies that g_{ac} = 0 (consequently Ω_{ac} = 0) whilst applying Π_{a}^{α}\overline{Π}γ ; c̄ to Ω_{αγ} = J_{α}^{β}g_{βγ} shows that Ω_{ac̄} = ig_{ac̄}. We conclude that R_{ab}^{c}_{d} = 0 and

R_{ab̄}^{c}_{d} ≡ Π_{a}^{α}\overline{Π}β ; b̄Π_{γ}^{c}Π_{d}^{ϵ}R_{αβ}^{γ}_{ϵ} = δ_{a}^{c}g_{db̄} − iδ_{a}^{c}Ω_{db̄} − 2iΩ_{ab̄}δ_{d}^{c} = 2δ_{a}^{c}g_{db̄} + 2δ_{d}^{c}g_{ab̄}.

Thus, with reference to Proposition 2.13, we see that all irreducible pieces of curvature vanish save for P_{b̄d} = 2g_{db̄}. In particular, all invariant pieces

Tāb̄^{c} H_{ab̄}^{c}_{d} W_{ab}^{c}_{d}

of harmonic curvature (as identified the following section) vanish. This is, of course, consistent with ℂℙ^{n}, equipped with its standard complex structure and Fubini–Study connection, being the flat model of c-projective geometry, as discussed in Section 2.3 and especially Theorem 2.8.

Finally, observe that if we regard ℂℙ^{n} as

SL(n + 1, ℂ)/ \{(\begin{matrix}λ & ∗ & ⋯ & ∗ \\ 0 & ∗ & ⋯ & ∗ \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & ∗ & ⋯ & ∗ \end{matrix})\} ,

rather than as a homogeneous PSL(n+1, ℂ)-space as in Section 2.3, then the character λ ↦ λ^{−k}\overline{λ}^{−ℓ} induces a homogeneous line bundle E(k, ℓ) on ℂℙ^{n} as we were supposing earlier and as we shall soon suppose in Section 3.1. This observation also explains our copacetic choice of notation: on ℂℙ^{n} it is standard to write O(k) for the holomorphic bundle that is E(k, 0) just as a complex bundle (and then \overline{E(k, 0)} = E(0, k)).

2.7. The harmonic curvature. A normal Cartan connection gives rise to a simpler local invariant than the Cartan curvature κ, called the harmonic curvature κ_{h}, which still provides a full obstruction to local flatness, as discussed in Section 2.2 (cf. especially Proposition 2.7). The harmonic curvature κ_{h} of an almost c-projective manifold is the projection of κ ∈ ker ∂^{∗} to its homology class in

G ×_{P} H_{2}(𝔤_{1}, 𝔤) ≅ G_{0} ×_{G_{0}} H_{2}(𝔤_{1}, 𝔤).

By Kostant’s version of the Bott–Borel–Weil Theorem [64] the G_{0}-module H_{2}(𝔤_{1}, 𝔤) can be naturally identified with a G_{0}-submodule in ∧^{2}𝔤_{1} ⊗_{ℝ} 𝔤 ≅ ∧^{2}𝔤_{−1}^{∗} ⊗_{ℝ} 𝔤 that decomposes into three irreducible components as follows:

• for n = 2

(∧^{0,2}𝔤_{−1}^{∗} ⊗_{ℂ} 𝔤_{−1}) ⊕ (∧^{1,1}𝔤_{−1}^{∗} ⊚_{ℂ} 𝔰𝔩(𝔤_{−1}, ℂ)) ⊕ (∧^{2,0}𝔤_{−1}^{∗} ⊗_{ℂ} 𝔤_{1})
• for n > 2
(∧^{0,2}𝔤_{−1}^{∗} ⊗_{ℂ} 𝔤_{−1}) ⊕ (∧^{1,1}𝔤_{−1}^{∗} ⊚_{ℂ} 𝔰𝔩(𝔤_{−1}, ℂ)) ⊕ (∧^{2,0}𝔤_{−1}^{∗} ⊚_{ℂ} 𝔰𝔩(𝔤_{−1}, ℂ)),

where these are complex vector spaces but regarded as real, and where ⊚ denotes the Cartan product. Correspondingly, we decompose the harmonic curvature as

κ_{h} = τ + ψ + χ

in case n = 2 and

κ_{h} = τ + ψ_{1} + ψ_{2}

in case n > 3.

Note that ∂^{∗} preserves homogeneities, i.e. ∂^{∗}(∧^{i}𝔤_{1} ⊗ 𝔤_{j}) ⊂ ∧^{i−1}𝔤_{1} ⊗ 𝔤_{j+1}. In particular, the induced vector bundle map ∂^{∗} maps ∧^{3}T^{∗}M ⊗ AM to ∧^{2}T^{∗}M ⊗ A^{0}M. Hence, we conclude that τ must equal the torsion T_{αβ}^{γ}. If n = 2, then ψ is the component H_{ab̄}^{c}_{d} in (∧^{1,1} ⊚_{ℂ} 𝔰𝔩(T^{1,0}M)) of the Weyl curvature of any connection in the c-projective class, and χ is the (2, 0)-part of the Cotton–York tensor. If n > 2, then ψ_{1}, respectively ψ_{2}, is the totally trace-free (1, 1)-part, respectively (2, 0)-part, of the Weyl curvature of any connection in the class.

We now give a geometric interpretation of the three harmonic curvature components.

Theorem 2.16. Suppose (M, J, [∇]) is an almost c-projective manifold of dimension 2n ≥ 4 and denote by κ_{h} the harmonic curvature of its normal Cartan connection. Then the following statements hold.

(1) κ_{h} ≡ 0 if and only if the almost c-projective manifold (M, J, [∇]) is locally isomorphic to (ℂP^{n}, J_{can}, [∇^{g_{F S}}]).

(2) τ is the torsion of (M, J, [∇]). In particular, τ ≡ 0 if and only if J is integrable, i.e. (M, J, [∇]) is a c-projective manifold. Moreover, in this case, J^{G} is integrable and the Cartan bundle p: G → M is a holomorphic principal P -bundle.

(3) Suppose τ ≡ 0. Then ψ_{1} ≡ 0 (resp. ψ ≡ 0) if and only if ω is a holomorphic Cartan connection on the holomorphic principal bundle p: G → M. This is the case if and only if [∇] locally admits a holomorphic connection, i.e. for any connection ∇ ∈ [∇] and any point x ∈ M there is an open neighbourhood U ∋ x such that ∇|_{U} is c-projectively equivalent to a holomorphic connection on U.

Proof. We have already observed (1) and the first two assertions of (2). To prove the last statement of (2) and (3), assume that τ ≡ 0, which says, in particular, that the Cartan geometry is torsion-free. Since P acts on the complex vector space ∧^{2}𝔤_{1}⊗_{ℝ}𝔤 by complex linear maps, P preserves the decomposition of this vector space into the three (p, q)-types. Therefore [28, Corollary 3.2] applies and hence κ_{h} has components of type (p, q) if and only if κ has components of type (p, q). Therefore, τ ≡ 0 implies that κ has no (0, 2)-part, which by the proof of [28, Theorem 3.4] (cf. [36, Proposition 3.1.17]) implies that J^{G} is integrable and p: G → M a holomorphic principal bundle. This finishes the proof of (2). We know that the component ψ_{1} (respectively ψ) vanishes identically if and only if κ is of type (2, 0), which by [28, Theorem 3.4] is the case if and only if ω ∈ Ω^{1,0}(G, 𝔤) is holomorphic, i.e. dω is of type (2, 0). Hence, it just remains to prove the last assertion of (3). Assume firstly that ψ_{1} (respectively ψ) vanishes identically and hence that (p: G → M, ω) is a holomorphic Cartan geometry. Then we can find around each point of M an open neighbourhood U ⊂ M such that G and G_{0} trivialise as holomorphic principal bundles over U. Having chosen such trivialisations, the holomorphic inclusion G_{0} ↪ P induces a holomorphic G_{0}- equivariant section σ: p_{0}^{−1}(U) → p^{−1}(U). Since dω is of type (2, 0) and σ is holomorphic

σ^{∗}dω = dσ^{∗}ω = dθ + dγ^{∇} − dp^{∇}

is also of type (2, 0). In particular, dγ^{∇} is of type (2, 0) and it follows that γ^{∇} ∈ Ω^{1,0}(p_{0}^{−1}(U), 𝔤_{0}) is a holomorphic principal connection in the c-projective class. Conversely, assume that U ⊂ M is an open set and that γ^{∇} ∈ Ω^{1,0}(p_{0}^{−1}(U), 𝔤_{0}) is a holomorphic principal connection that belongs to the c-projective class. Since the Lie bracket on 𝔤 is complex linear, the holomorphicity of γ^{∇} implies that its curvature dγ^{∇} + [γ^{∇}, γ^{∇}] is of type (2, 0). By definition of the Weyl curvature this implies that also its Weyl curvature is of type (2, 0) and hence so is κ_{h}|_{U}. By assumption there exists locally around any point a holomorphic connection and hence κ_{h} is of type (2, 0) on all of M. □

3. Tractor bundles and BGG sequences

The normal Cartan connection of an almost c-projective manifold induces a canonical linear connection on all associated vector bundles corresponding to representations of PSL(n + 1, ℂ) (cf. Section 2.2). These, in the theory of parabolic geometries, so-called tractor connections, provide an efficient calculus, especially well suited for explicit constructions of local invariants and invariant differential operators. We develop in this section the basics of the theory of tractor connections for almost c-projective manifolds, and explain their relation to geometrically significant overdetermined systems of PDE and sequences of invariant differential operators.

3.1. Standard complex tractors. Suppose that (M, J, [∇]) is an almost c-projective manifold of dimension 2n ≥ 4. Further, assume that the complex line bundle ∧^{n}T^{1,0}M admits an (n + 1)^{st} root and choose one, denoted E(1, 0), with conjugate E(0, 1). More generally, we write E(k, ℓ) = E(1, 0)^{⊗k} ⊗E(0, 1)^{ℓ} for any (k, ℓ) ∈ ℤ×ℤ (cf. Section 2.1). Note that such a choice of a root E(1, 0) is at least locally always possible and the assumption that such roots exists globally is a relatively minor constraint. The choice of E(1, 0) canonically extends the Cartan bundle of (M, J, [∇]) to a P̃-principal bundle p̃: G̃ → M, where P̃ is the stabiliser in SL(n+1, ℂ) of the complex line generated by the first basis vector in ℂ^{n+1}, and the normal Cartan connection of (M, J, [∇]) naturally extends to a normal Cartan connection on G̃ of type (SL(n + 1, ℂ), P̃), which we also denote by ω. The groups SL(n + 1, ℂ) and P̃ are here viewed as real Lie groups as in Section 2.3, and we obtains in this way, analogously to Theorem 2.8, an equivalence of categories between almost c-projective manifolds equipped with an (n + 1)st root E(1, 0) of ∧^{n}T^{1,0}M and normal Cartan geometries of type (SL(n + 1, ℂ), P̃). The homogeneous model of such structures is again ℂℙ^{n}, but now viewed as a homogeneous space SL(n + 1, ℂ)/P̃ with E(1, 0) being dual to the tautological line bundle O(−1), cf. Section 2.6.

The extended normal Cartan geometry of type (SL(n + 1, ℂ), P̃) allows us to form the standard complex tractor bundle

T := G̃ ×P̃ 𝕍

of (M, J, [∇], E(1, 0)), where 𝕍 = ℝ^{2n+2} is the defining representation of the real Lie group SL(2(n + 1), 𝕁_{2(n+1)}) ≅ SL(n + 1, ℂ). Note that the complex structure 𝕁_{2(n+1)} on 𝕍 induces a complex structure J^{T} on T . Analogously to the discussion of the tangent bundle of an almost complex manifold in Section 1, (T , J^{T}) can be identified with the (1, 0)-part of its complexification T_{ℂ}, on which J^{T} acts by multiplication by i. We will implement this identification in the sequel without further comment, and similarly for all the other tractor bundles with complex structures in the following Sections 3.1, 3.3 and 3.4. Since P̃ stabilises the complex line generated by the first basis vector in ℂ^{n+1}, this line defines a complex 1-dimensional submodule of ℂ^{n+1}. Correspondingly, the standard complex tractor bundle (identified with the (1, 0)-part of its complexification T_{ℂ}) is filtered as

T = T^{0} ⊃ T^{1}  (49)

where T^{1} ≅ E(−1, 0) and T^{0}/T^{1} ≅ T^{1,0}M(−1, 0). Since T is induced by a representation of SL(n+1, ℂ), the Cartan connection induces a linear connection ∇^{T} on T , called the tractor connection (see Section 2.2). Any choice of a linear connection ∇ ∈ [∇], splits the filtration of the tractor bundle T and the splitting changes by

\begin{matrix} \\ ( & \begin{matrix}X^{b} \\ ρ \end{matrix} & ) \end{matrix} = (\begin{matrix}X^{b} \\ ρ − Υ_{c}X^{c} \end{matrix}) , where \{ \begin{matrix}X^{b} ∈ T^{1,0}M(−1, 0) \\ ρ ∈ E(−1, 0), \end{matrix}  (50)

if one changes the connection according to (11). In terms of a connection ∇ ∈ [∇], the tractor connection is given by

∇_{α}^{T} (\begin{matrix}X^{b} \\ ρ \end{matrix}) = (\begin{matrix}∇_{α}X^{b} + ρΠ_{α}^{b} \\ ∇_{α}ρ − P_{αb}X^{b} \end{matrix}) .  (51)

Applying Π_{a}^{α} and \overline{Π}α ; ā to (51), we can write the tractor connection as

∇_{a}^{T} (\begin{matrix}X^{b} \\ ρ \end{matrix} ) = (\begin{matrix}∇_{a}X^{b} + ρδ_{a}^{b} \\ ∇_{a}ρ − P_{ab}X^{b} \end{matrix})  (52)

and

∇T ; ā (\begin{matrix}X^{b} \\ ρ \end{matrix}) = (\begin{matrix}∇āX^{b} \\ ∇āρ − P_{āb}X^{b} \end{matrix}) .  (53)

By (49) the dual (or “co-standard”) complex tractor bundle T^{∗} admits a natural subbundle isomorphic to ∧^{1,0}(1, 0) and the quotient T^{∗}/∧^{1,0}(1, 0) is isomorphic to E(1, 0). One immediately deduces from (52) and (53) that in terms of a connection ∇ ∈ [∇], the tractor connection on T^{∗} is given by

∇_{a}^{T^{∗}} (\begin{matrix}σ \\ μ_{b} \end{matrix}) = (\begin{matrix}∇_{a}σ − μ_{a} \\ ∇_{a}μ_{b} + P_{ab}σ \end{matrix})
∇T^{∗} ; ā (\begin{matrix}σ \\ μ_{b} \end{matrix}) = (\begin{matrix}∇āσ \\ ∇āμ_{b} + P_{āb}σ \end{matrix}) .

For a choice of connection ∇ ∈ [∇] consider now the following overdetermined system of PDE on sections σ of E(1, 0):

(i) ∇āσ = 0  (ii) ∇_{(a}∇_{b)}σ + P_{(ab)}σ = 0.  (54)

Suppose ˆ∇ ∈ [∇] is another connection in the c-projective class. Then the formulae (16) imply that ˆ∇āσ = ∇āσ. Moreover, we deduce from Proposition 2.5 and the formulae (16) that

ˆ∇_{a} ˆ∇_{b}σ = ∇_{a}∇_{b}σ + (∇_{a}Υ_{b})σ − Υ_{a}Υ_{b}σ,

which together with Corollary 2.12 implies that

ˆ∇_{a} ˆ∇_{b}σ + P̂_{ab}σ = ∇_{a}∇_{b}σ + P_{ab}σ.

This shows that the overdetermined system (54) is c-projectively invariant. Note also that by equation (46) we have

∇_{[a}∇_{b]}σ + P_{[ab]}σ = −\frac{1}{2}T_{ab}c̄∇c̄σ.  (55)

Therefore ∇_{a}∇_{b}σ + P_{ab}σ is symmetric provided that J is integrable or that σ satisfies equation (i) of (54). The following proposition shows that, if J is integrable, the tractor connection on T^{∗} can be viewed as the prolongation of (54):

Proposition 3.1. Suppose (M, J, [∇], E(1, 0)) is a c-projective manifold of dimension 2n ≥ 4. The projection π: T^{∗} → T^{∗}/∧^{1,0}(1, 0) ≅ E(1, 0) induces a bijection between sections of T^{∗} parallel for ∇^{T^{∗}} and sections σ of E(1, 0) that satisfy (54) for some (and hence any) connection ∇ ∈ [∇]. Moreover, suppose that σ ∈ E(1, 0) is a nowhere vanishing section, then for any connection ∇ ∈ [∇] the connection

ˆ∇_{a}σ′ = ∇_{a}σ′ − (∇_{a}σ)σ^{−1}σ′  (56)

is induced from a connection in the c-projective class, and σ with ∇āσ = 0 is a solution of (54) if and only if (56) is Ricci-flat.

Proof. Suppose s is a parallel section of T^{∗} and set σ := π(s) ∈ Γ(E(1, 0)). It follows from (50) that changing from connection to another in [∇] changes the splitting of T^{∗} by { \atop (σ, μ_{b})} = (σ, μ_{b} + Υ_{b}σ). Hence, for any connection ∇ ∈ [∇] we can identify s with a section of the form (σ, μ_{b}) for some μ_{b} ∈ Γ(∧^{1,0}(1, 0)) and by definition of the tractor connection σ ∈ Γ(E(1, 0)) satisfies (54) for any connection ∇ ∈ [∇]. So π induces a map from parallel sections of T^{∗} to solutions of (54).

Conversely, let us contemplate the differential operator L: E(1, 0) → T^{∗}, which, for a choice of connection in [∇], is given by Lσ = (σ, ∇_{b}σ). Suppose now σ is a solution of (54). Then obviously ∇_{a}^{T^{∗}}Lσ = 0, since (55) vanishes. By (45) we have

∇ā∇_{b}σ + P_{āb}σ = (∇ā∇_{b} − ∇_{b}∇ā)σ + P_{āb}σ  (57)
= \frac{1}{n+1}T_{bf} ēT_{ēā}^{f}σ − P_{āb}σ + P_{āb}σ
= \frac{1}{n+1}T_{bf} ēT_{ēā}^{f}σ ,

which vanishes, since J is integrable. Hence, we also have ∇T^{∗} ; āLσ = 0. Therefore L maps solutions σ ∈ Γ(E(1, 0)) of (54) to parallel sections of T^{∗} and defines an inverse to the restriction of π to parallel section. For the second statement, assume now that σ is a section of E(1, 0) that is nowhere vanishing and let ∇ ∈ [∇] be a connection in the c-projective class. If we change ∇ according to (11) by Υ_{a} = −(∇_{a}σ)σ^{−1} to a connection ˆ∇ ∈ [∇], then we deduce from Corollary 2.4 that the induced connection on E(1, 0) is given by (56). Moreover, ˆ∇_{a}σ = 0. Therefore, using that (57) vanishes, we deduce that σ with ˆ∇āσ = 0 satisfies (54) if and only if P̂_{ab}σ = 0 and P̂_{āb}σ = 0, and hence if and only if ˆ∇ is Ricci-flat. □

More generally, we immediately conclude from equation (57) that, in the case of an almost c-projective manifold, i.e. J is not necessarily integrable, the corresponding proposition reads as follows:

Proposition 3.2. Let (M, J, [∇], E(1, 0)) be an almost c-projective manifold of dimension 2n ≥ 4. Then sections σ of E(1, 0) that satisfy (54) are in bijection with sections of T^{∗} that are parallel for the connection given by

∇_{a}^{T^{∗}} and ∇T^{∗} ; ā (\begin{matrix}σ \\ μ_{b} \end{matrix}) − \frac{1}{n + 1} (\begin{matrix}0 \\ T_{bf} ēT_{ēā}^{f}σ \end{matrix}) .  (58)

Moreover, suppose σ ∈ E(1, 0) is a nowhere vanishing section with ∇āσ = 0. Then σ is a solution of (54) if and only if ˆ∇, defined as in (56), satisfies P̂_{ab} = 0 and P̂_{āb} = \frac{1}{n+1}T_{bf} ēT_{ēā}^{f}.

Remark 3.1. Recall that a parallel section of a linear connection of a vector bundle over a connected manifold, is already determined by its value at one point. The correspondences established in Propositions 3.1 and 3.2 between solutions of (54) and parallel sections of a linear connection on T^{∗} therefore implies, that on a connected almost c-projective manifold

U := {x ∈ M : σ(x) ≠ 0} ⊂ M

is a dense open subset for any nontrivial solution σ ∈ Γ(E(1, 0)) of (54). In particular, the second statement of Proposition 3.1, respectively of Proposition 3.2, holds always on the dense open subset U.

The equations (54) define an invariant differential operator of order two

D^{T^{∗}}: E(1, 0) → ∧^{0,1}(1, 0) ⊕ S^{2}∧^{1,0}(1, 0),

whose kernel are the solutions of (54). The differential operator D^{T^{∗}} is the first operator in the BGG sequence corresponding to the co-standard complex tractor bundle; see [25, 38]. The proof of Proposition 3.2 implies that in order for a (nonzero) parallel section of the tractor connection on T^{∗} to exist, it is necessary that Nijenhuis tensor of J satisfy N_{bf}^{J} ēN_{ēā}^{J}^{f} ≡ 0 and, in this case, parallel sections of the tractor connection are in bijection with sections in the kernel of D^{T^{∗}}.

Similarly, one may consider the first BGG operator in the sequence corresponding to the standard complex tractor bundle T , which is a first order invariant differential operator, defined, for a choice of connection ∇ ∈ [∇], by

D^{T} : T^{1,0}M(−1, 0) → (∧^{0,1} ⊗ T^{1,0}M(−1, 0)) ⊕ (∧^{1,0} ⊗_{∘} T^{1,0}M(−1, 0)),
X^{b} ↦ (∇āX^{b}, ∇_{a}X^{b} − \frac{1}{n}∇_{c}X^{c}δ_{a}^{b}),  (59)

where the subscript ∘ denotes the trace-free part.

Proposition 3.3. Suppose (M, J, [∇], E(1, 0)) is an almost c-projective manifold of dimension 2n ≥ 4. The projection π: T → T /E(−1, 0) ≅ T^{1,0}M(−1, 0) induces a bijection between sections of T that are parallel for the connection given by

∇_{a}^{T}  and ∇T ; ā (\begin{matrix}X^{b} \\ ρ \end{matrix}) − \frac{1}{n(n + 1)} (\begin{matrix}0 \\ T_{bf} ēT_{ēā}^{f}X^{b} \end{matrix})  (60)

and sections X^{b} ∈ Γ(T^{1,0}M(−1, 0)) that are in the kernel of D^{T}. In particular, elements X^{b} ∈ ker D^{T} with N_{bf}^{J} ēN_{ēā}^{J}^{f}X^{b} = 0 are in bijection with parallel sections of the tractor connection ∇^{T}.

Proof. Suppose firstly that s ∈ Γ(T ) is parallel for the connection (60) and set X^{b} := π(s). For a choice of connection ∇ ∈ [∇] we can identify s with an element of the form (X^{b}, ρ), where ρ ∈ Γ(E(−1, 0)). By assumption ∇āX^{b} = 0 and ∇_{a}X^{b} = −ρδ_{a}^{b}. Taking the trace of the latter equation shows that ρ = −\frac{1}{n}∇_{c}X^{c}. Hence, X^{b} is in the kernel of D^{T}.

Conversely, suppose X^{b} ∈ ker D^{T} and pick a connection ∇ ∈ [∇]. Then we deduce from Proposition 2.13 and equation (46) that

(∇_{a}∇_{b} − ∇_{b}∇_{a})X^{c} = (∇_{a}∇_{b} − ∇_{b}∇_{a})X^{c} + T_{ab}d̄∇d̄X^{c}  (61)
= R_{ab}^{c}_{d}X^{d} + 2P_{[ab]}X^{c} = W_{ab}^{c}_{d}X^{d} + 2δ_{[a}^{c}P_{b]d}X^{d}.

By assumption ∇_{a}X^{b} = \frac{1}{n}∇_{c}X^{c}δ_{a}^{b} and therefore (61) implies

\frac{1}{n}(∇_{a}∇_{d}X^{d}δ_{b}^{c} − ∇_{b}∇_{d}X^{d}δ_{a}^{c}) = W_{ab}^{c}_{d}X^{d} + 2δ_{[a}^{c}P_{b]d}X^{d}.  (62)

Taking the trace in (62) over a and c shows that −\frac{1}{n}∇_{b}∇_{d}X^{d} = P_{bd}X^{d}. Hence, (X^{b}, −\frac{1}{n}∇_{c}X^{c}) defines a section s of T that satisfies ∇_{a}^{T} s = 0. Similarly, since ∇āX^{b} = 0, Proposition 2.13 and equation (45) imply

∇ā∇_{b}X^{c} = (∇ā∇_{b} − ∇_{b}∇ā)X^{c}  (63)
= R_{āb}^{c}_{d}X^{d} − \frac{1}{n+1}T_{bf} ēT_{ēā}^{f}X^{c} + P_{āb}X^{c}
= W_{āb}^{c}_{d}X^{d} − 2P_{ā(b}δ_{d)}^{c}X^{d} + P_{āb}X^{c} − \frac{1}{n+1}T_{bf} ēT_{ēā}^{f}X^{c}.

Taking the trace in (63) over b and c implies that

−\frac{1}{n}∇ā∇_{c}X^{c} − P_{āc}X^{c} = \frac{1}{n(n+1)}T_{cf} ēT_{ēā}^{f}X^{c}.

Hence, s is parallel for the connection (60) and it follows immediately that the differential operator X^{b} ↦ (X^{b}, −\frac{1}{n}∇_{c}X^{c}) defines an inverse to the restriction of π to parallel sections of (60). □

3.2. Cone description of almost c-projective structures. For (real) projective structures there is an alternative description of the tractor connection as an affine connection on a cone manifold over the projective manifold [36, 46]. This point of view, which (at least in spirit) goes back to work of Tracy Thomas, is often convenient—it has for instance been used in [6] to classify holonomy reductions of projective structures. An analogue holds for almost c-projective manifolds, which we will now sketch, following the presentation in [36] of the projective case. This canonical cone connection was used in [6] to realise c-projective structures as holonomy reductions of projective structures. It also underlies metric cone constructions [78, 83] which we discuss later.

Let (M, J, [∇], E(1, 0)) be an almost c-projective manifold and, as in Section 3.1, let P̃ ⊂ G̃ = SL(n + 1, ℂ) be the stabiliser of the complex line 𝕍^{1} through the first basis vector e_{1} of 𝕍 = ℂ^{n+1}. Denote by Q̃ ⊂ P̃ the stabiliser of e_{1}, which is the derived group of P̃, hence a normal complex Lie subgroup. Now set

C := G̃/Q̃ = G̃ ×P̃ P̃/Q̃.

The natural projection p_{C}: G̃ → C defines a (real) principal bundle with structure group Q̃. Since the canonical Cartan connection ω of (M, J, [∇], E(1, 0)) can also be viewed as a Cartan connection of type (G̃, Q̃) for p_{C}, it induces an isomorphism

T C ≅ G̃ ×Q̃ 𝔤/𝔮̃.

Note that C inherits an almost complex structure J^{C} from the almost complex structure on G̃ characterised by T p_{C} ∘ J^{G} = J^{C} ∘ T p_{C}. Furthermore, the projection π_{C}: C → M defines a principal bundle with structure group P̃/Q̃ ≅ ℂ^{×}. Since P̃/Q̃ can be identified with the nonzero elements in the complex P̃-submodule 𝕍^{1} ⊂ 𝕍, we see that C can be identified with the space of nonzero elements in E(−1, 0) or with the complex frame bundle of E(−1, 0). Note that, by construction (recall that T p ∘ J^{G} = J ∘ T p), we have T π_{C} ∘ J^{C} = J ∘ T π_{C}. By the compatibility of the almost complex structures J^{G}, J^{C} and J with the various projections, it follows immediately that vanishing of the Nijenhuis tensor N^{J^{G}} of J^{G} implies vanishing of the Nijenhuis tensors N^{J^{C}}, which in turn implies vanishing of N^{J}. Conversely, Theorem 2.16 shows that N^{J} ≡ 0 implies N^{J^{G}} ≡ 0. This shows in particular that

N^{J} ≡ 0 ⟺ N^{J^{C}} ≡ 0,  (64)

in which case π_{C}: C → M is a holomorphic principal bundle with structure group ℂ^{×}.

Lemma 3.4. There are canonical isomorphisms T C ≅ G̃ ×Q̃ 𝕍 ≅ π_{C}^{∗}T .

Proof. From the block decomposition (17) of 𝔤 it follows that 𝔤/𝔮̃ = (𝕍^{1})^{∗} ⊗ 𝕍 and hence 𝔤/𝔮̃ ≅ 𝕍 as Q̃-modules, i.e.

T C ≅ G̃ ×Q̃ 𝔤/𝔮̃ ≅ G̃ ×Q̃ 𝕍 ≅ π_{C}^{∗}(G̃ ×P̃ 𝕍) = π_{C}^{∗}T .  □

Hence the standard tractor connection induces an affine connection ∇^{C} on C which preserves J^{C} and the complex volume form vol^{C} ∈ ∧^{n,0}T^{∗}C that is induced by the standard complex volume on 𝕍 = ℂ^{n+1}. Alternatively, note that ω can be extended to a principal connection on the principal G-bundle G̃ ×Q̃ G, and since 𝕍 = ℂ^{n+1} is a G-module, we obtain an induced connection on T C ≅ G̃ ×Q̃ G ×_{G} 𝕍.

If we identify a vector field Y ∈ 𝔛(C) with a Q̃-equivariant function f: G̃ → 𝕍 via Lemma 3.4, then by [36, Theorem 1.5.8], the equivariant function corresponding to ∇_{X}^{C}Y for a vector field X ∈ 𝔛(C) is given by

X̃ · f + ω(X̃)f  (65)

where X̃ ∈ 𝔛(G̃) is an arbitrary lift of X. Moreover [36, Corollary 1.5.7] shows that the curvature R^{C} ∈ ∧^{2}T^{∗}C ⊗ AC of ∇^{C} is given by

R^{C}(X, Y)(Z) = κ(X, Y) • Z,  (66)

where AC = G̃×Q̃𝔤 ≅ 𝔰𝔩(T C, J^{C}) and •: AC×T C → T C denotes the vector bundle map induced by the action of 𝔤 on 𝕍. Let us write T^{C} ∈ ∧^{2}T^{∗}C ⊗ T C for the torsion of ∇^{C}. It follows straightforwardly from (65) that T^{C} is the projection of κ ∈ Γ(∧^{2}T^{∗}C ⊗ AC) to ∧^{2}T^{∗}C ⊗ T C, i.e. it is the torsion of ω viewed as a Cartan connection of type (Q̃, G̃). In particular, like κ, the 2-forms T^{C} and R^{C} vanish upon insertion of sections of the vertical bundle of π_{C}, which is canonically trivialised by the fundamental vector fields E and J^{C}E generated by 1 and i respectively.

Proposition 3.5. Suppose (M, J, [∇], E(1, 0)) is an almost c-projective manifold. Then there is a unique affine connection ∇^{C} on the total space of the principal bundle π_{C}: C → M with the following properties:

(1) ∇^{C}J^{C} = 0 and ∇^{C}vol^{C} = 0;

(2) ∇_{X}^{C}E = X for all X ∈ 𝔛(C);

(3) L_{E}∇^{C} = 0 and L_{J^{C}E}∇^{C} = 0;

(4) i_{E}T^{C} = 0 and i_{J^{C}E}T^{C} = 0;

(5) T π ∘ T^{C} is of type (0, 2) and the (2, 0)-part of T^{C} vanishes;

(6) ∇^{C} is Ricci-flat;

(7) for any (local) section s of π_{C} the connection s^{∗}∇^{C} lies in [∇]. Moreover, if J is integrable, π_{C}: C → M is a holomorphic principal bundle and T^{C} ≡ 0.

Proof. We already observed that (1) and (4) hold and (2) is an immediate consequence of (65). Since we have in addition i_{E}R^{C} = 0 and i_{J^{C}E}R^{C} = 0 by (66), statement (3) follows from (78). The statements (5) and (6) are consequences of ∂^{∗}κ = 0. More explicitly, note that (5) can be simply read off Proposition 2.13, which also shows that if J is integrable, T^{C} ≡ 0. In this case N^{J^{C}}, which is up to a constant multiple the (0, 2)-part of T^{C}, vanishes and π_{C} is a holomorphic principal bundle as calimed. Statement (6) follows because T π ∘ T^{C}, viewed as a section of ∧^{0,2}T^{∗}M ⊗ T M, has vanishing trace and ∂^{∗}: ∧^{2}T^{∗}M ⊗ 𝔤𝔩(T M, J) → T^{∗}M ⊗ T^{∗}M is a multiple of a Riccitype contraction. The proof of statement (7) and the uniqueness of ∇^{C} we leave to the reader, but note that (1)–(6) imply that ∇^{C} descends to the normal Cartan connection on T C/ℂ^{×} ≅ T . □

3.3. BGG sequences. For a general parabolic geometry, it was shown in [25, 38] that there are natural sequences of invariant linear differential operators generalising the corresponding complexes on the flat model. These are the Bernstein–Gelfand–Gelfand (BGG) sequences, named after the constructors [10] of complexes of Verma modules, roughly dual to the current circumstances.

Here is not the place to say much about the general theory. Instead, we would like to like to present something of the theory as it applies in the c-projective case. The point is that the invariant operators that we have already encountered and are about to encounter, all can be seen as curved analogues of operators from the BGG complex on ℂℙ^{n} (as the flat model of c-projective geometry).

In fact, the main hurdle in presenting the BGG complex and sequences is in having a suitable notation for the vector bundles involved. Furthermore, this notation is already of independent utility since, as foretold in Remark 1.1, it neatly captures the natural irreducible bundles on an almost complex manifold.

Recall from Section 1 that the complexified tangent bundle on an almost complex manifold decomposes

ℂT M = T^{1,0}M ⊕ T^{0,1}M

as does its dual. An alternative viewpoint on these decompositions is that the tangent bundle T M on any 2n-dimensional manifold is tautologically induced from its framebundle by the defining representation of GL(2n, ℝ), that an almost complex structure is a reduction of structure group for the frame bundle to GL(n, ℂ) ⊂ GL(2n, ℝ), that the defining representation of GL(2n, ℝ) on ℝ^{2n} complexifies as GL(2n, ℝ) acting on ℂ^{2n} (as real matrices acting on complex vectors), and finally that this complex representation when restricted to GL(n, ℂ) decomposes into two irreducibles inducing the bundles T^{1,0}M and T^{0,1}M, respectively. Of course, the dual decomposition comes from the dual representation, namely GL(n, ℂ) acting on (ℂ^{2n})^{∗}. Our notation arises by systematically using the representation theory of GL(n, ℂ) as a real Lie group but adapted to its embedding

GL(n, ℂ) ≅ G_{0} ⊂ P ⊂ G = PSL(n + 1, ℂ)

as described in Section 2.3.

For relatively simple bundles, there is no need for any more advanced notation. In several complex variables, for example, it is essential to break up the complex-valued differential forms into types but that’s about it. Recall already with 2-forms

∧^{2} = ∧^{0,2} ⊕ ∧^{1,1} ⊕ ∧^{2,0}

that this complex decomposition is finer that the real decomposition

∧^{2}T^{∗}M = [∧^{0,2} ⊕ ∧^{2,0}]_{ℝ} ⊕ ∧_{ℝ}^{1,1}  (67)

already discussed in Section 1 following (6). Of course, as soon as one speaks of holomorphic functions on a complex manifold one is obliged to work with complexvalued differential forms. However, even if one is concerned only with real-valued forms and tensors, it is convenient firstly to decompose the complex versions and then impose reality as, for example, in (67). In fact, this is already a feature of representation theory in general.

For more complicated bundles, we shall use Dynkin diagrams from [36] decorated in the style of [8]. The formal definitions will not be given here but the upshot is that the general complex irreducible bundle on an almost complex manifold will be denoted as

× • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{p} ^{a} ^{b} ^{c} ^{d}_{q} _{e} _{f} _{g} _{h} (in the 10-dimensional case (2n nodes in general))  (68)

where a, b, c, d, e, f, g, h are nonnegative integers whilst, in the first instance, p, q are real numbers restricted by the requirement that

p + 2a + 3b + 4c + 5d = q + 2e + 3f + 4g + 5h mod 6  (69)

(again, in the 10-dimensional case). For example,

T^{1,0}M = × • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{1} ^{0} ^{0} ^{0} ^{1}^{0} ^{0} ^{0} ^{0} ^{0} ∧^{1,0} = × • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{−2} ^{1} ^{0} ^{0} ^{0}^{0} ^{0} ^{0} ^{0} ^{0} ∧^{0,2} = × • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{0} ^{0} ^{0} ^{0} ^{0}^{−3} ^{0} ^{1} ^{0} ^{0}

but the point is that this notation covers all bases and, in particular, the various awkward bundles that have already arisen and will now arise in this article. In general, the integrality condition (69) is needed, as typified by

det ∧^{1,0} = ∧^{5,0} = × • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{−6} ^{0} ^{0} ^{0} ^{0}^{0} ^{0} ^{0} ^{0} ^{0}  E(p, p) = × • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{p} ^{0} ^{0} ^{0} ^{0}^{p} ^{0} ^{0} ^{0} ^{0}

but, as already discussed at the end of Section 2.1, on an almost c-projective manifold we shall suppose that there is a bundle E(1, 0) and an identification E(n + 1, 0) := E(1, 0)^{n+1} = ∧^{n}T^{1,0}M, in which case we shall add

E(1, 0) = × • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{1} ^{0} ^{0} ^{0} ^{0}^{0} ^{0} ^{0} ^{0} ^{0}

to our notation and relax (69) to requiring merely that p − q be integral. In fact, all of p, q, a, b, c, d, e, f, g, h will be integral for the rest of this article.

Our Dynkin diagram notation is well suited to the barred and unbarred indices that we have already been using. Specifically, a section of

× • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{p} ^{a} ^{b} ^{c} ^{d}^{0} ^{0} ^{0} ^{0} ^{0}

may be realised as tensors with a + 2b + 3c + 4d unbarred covariant indices, having symmetries specified by the Young diagram

[figure: ⋯; ⋯; ⋯; ⋯; ⋯; ⋯; ⋯; ⋯; ⋯; ⋯; d; c; b; a]

and of c-projective weight (p + 2a + 3b + 4c + 5d, 0). Indeed, for those reluctant to trace through the conventions in [8], this suffices as a definition and then

× • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{0} ^{0} ^{0} ^{0} ^{0}^{q} ^{e} ^{f} ^{g} ^{h} is the complex conjugate of × • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{q} ^{e} ^{f} ^{g} ^{h}^{0} ^{0} ^{0} ^{0} ^{0}

corresponding to tensors with barred indices in the obvious fashion and

× • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{p} ^{a} ^{b} ^{c} ^{d}^{q} ^{e} ^{f} ^{g} ^{h} = × • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{p} ^{a} ^{b} ^{c} ^{d}^{0} ^{0} ^{0} ^{0} ^{0} ⊗ × • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{0} ^{0} ^{0} ^{0} ^{0}^{q} ^{e} ^{f} ^{g} ^{h}.

Already, these bundles provide locations for the tensors we encountered earlier. For example,

T_{ab}c̄ ∈ Γ( × • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{−3} ^{0} ^{1} ^{0} ^{0}^{1} ^{0} ^{0} ^{0} ^{1} ) and H_{ab̄}^{c}_{d} ∈ Γ( × • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{−3} ^{2} ^{0} ^{0} ^{1}^{−2} ^{1} ^{0} ^{0} ^{0} ).

Although the Dynkin diagram notation may at first seem arcane, it comes into its own when discussing invariant linear differential operators. The complex-valued de Rham complex

∧^{0,0} → \begin{matrix}∧^{1,0} \\ ⊕ \\ ∧^{0,1} \end{matrix} → \begin{matrix}∧^{2,0} \\ ⊕ \\ ∧^{1,1} \\ ⊕ \\ ∧^{0,2} \end{matrix} → ⋯  (70)

becomes

× • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{−3} ^{0} ^{1} ^{0} ^{0}^{0} ^{0} ^{0} ^{0} ^{0}
× • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{−2} ^{1} ^{0} ^{0} ^{0}^{0} ^{0} ^{0} ^{0} ^{0}
× • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{0} ^{0} ^{0} ^{0} ^{0}^{0} ^{0} ^{0} ^{0} ^{0} →  → × • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{−2} ^{1} ^{0} ^{0} ^{0}^{−2} ^{1} ^{0} ^{0} ^{0} → ⋯
× • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{0} ^{0} ^{0} ^{0} ^{0}^{−2} ^{1} ^{0} ^{0} ^{0}
× • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{0} ^{0} ^{0} ^{0} ^{0}^{−3} ^{0} ^{1} ^{0} ^{0}

and in either of them one sees the torsion T_{ab}c̄: ∧^{0,1} → ∧^{2,0} and its complex conjugate Tāb̄^{c}: ∧^{1,0} → ∧^{0,2} as the restriction of the exterior derivative d: ∧^{1} → ∧^{2} to the relevant bundles (note that

Hom(∧^{0,1}, ∧^{2,0}) = T^{0,1} ⊗ ∧^{2,0} = × • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{−3} ^{0} ^{1} ^{0} ^{0}^{1} ^{0} ^{0} ^{0} ^{1} ,

as expected). In the torsion-free case, the de Rham complex takes the form

\begin{aligned}& & & & ∧^{2,0} \\ & & ∧^{1,0} & \begin{matrix}↗ \\ ↘ \end{matrix} \\ ∧^{0,0} & \begin{matrix}↗ \\ ↘ \end{matrix} & & & ∧^{1,1} \\ & & ∧^{0,1} & \begin{matrix}↗ \\ ↘ \end{matrix} \\ & & & & ∧^{0,2} \end{aligned}  (71)

familiar from complex analysis and the remarkable fact about c-projectively invariant linear differential operators is firstly that this pattern is repeated on the flat model starting with any bundle (68) with p, q ∈ ℤ_{≥0}, for example

\begin{aligned}× & • & • & • & • & ↕ & ↕ & ↕ & ↕ & ↕ & × & • & • & • & • & −4 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ & & × • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{−3} ^{2} ^{0} ^{0} ^{1}^{2} ^{0} ^{0} ^{0} ^{0} & \begin{matrix}↗ \\ ↘ \end{matrix} \\ × • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{1} ^{0} ^{0} ^{0} ^{1}^{0} ^{0} ^{0} ^{0} ^{0} & \begin{matrix}↗ \\ ↘ \end{matrix} & & & × • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{−3} ^{2} ^{0} ^{0} ^{1}^{−2} ^{1} ^{0} ^{0} ^{0} \\ & & × • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{1} ^{0} ^{0} ^{0} ^{1}^{−2} ^{1} ^{0} ^{0} ^{0} & \begin{matrix}↗ \\ ↘ \end{matrix} \\ & & & & × • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{1} ^{0} ^{0} ^{0} ^{1}^{−3} ^{0} ^{1} ^{0} ^{0} & . \end{aligned}  (72)

The algorithm for determining the bundles in these patterns is detailed in [8] (it is the affine action λ ↦ w(λ + ρ) − ρ of the Weyl group for G along the Hasse diagram corresponding to the parabolic subgroup P). On G/P in general, these are complexes of differential operators referred to as Bernstein–Gelfand–Gelfand (BGG) complexes. In our case, i.e. on ℂℙ^{n}, they provide resolutions of the finite-dimensional representations

• • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} • • • • •^{p} ^{a} ^{b} ^{c} ^{d}_{q} _{e} _{f} _{g} _{h} (in case n = 5 (2n nodes in general))

of the group G = PSL(n + 1, ℂ) as a real Lie group. More precisely, any finitedimensional representation 𝔼 of G gives rise to a constant sheaf G/P × 𝔼, which may in turn be identified with the corresponding homogeneous bundle induced on G/P by means of

G/P ×_{P} 𝔼 ∋ [g, e] ↦ ([g], ge) ∈ G/P × 𝔼.  (73)

Since the first bundle in the BGG complex is a quotient of this bundle, we obtain a mapping of 𝔼 to the sections of this first bundle and to say that the complex is a resolution of 𝔼 is to say that these sections are locally precisely the kernel of the first BGG operator (just as the locally constant functions are precisely the kernel of the first exterior derivative d: ∧^{0} → ∧^{1}). In our example (72), this means that

× • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{−3} ^{2} ^{0} ^{0} ^{1}^{0} ^{0} ^{0} ^{0} ^{0}
0 → • • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} • • • • •^{1} ^{0} ^{0} ^{0} ^{1}^{0} ^{0} ^{0} ^{0} ^{0} ⟶ × • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{1} ^{0} ^{0} ^{0} ^{1}^{0} ^{0} ^{0} ^{0} ^{0}  \begin{matrix}↗ \\ ↘ \end{matrix}
× • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{1} ^{0} ^{0} ^{0} ^{1}^{−2} ^{1} ^{0} ^{0} ^{0}

is exact, the G-module 𝔼 in this case being the adjoint representation of PSL(n+1, ℂ) as a complex Lie algebra. More generally, the BGG resolutions on ℂℙ^{n} as a homogeneous space for PSL(n + 1, ℂ) begin

\begin{matrix}× & • & • & • & • & ↕ & ↕ & ↕ & ↕ & ↕ & × & • & • & • & • & −p−2 & p+a+1 & b & c & d & q & e & f & g & h \\ 0 → • • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} • • • • •^{p} ^{a} ^{b} ^{c} ^{d}^{q} ^{e} ^{f} ^{g} ^{h} ⟶ × • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{p} ^{a} ^{b} ^{c} ^{d}^{q} ^{e} ^{f} ^{g} ^{h} & \begin{matrix}↗ \\ ↘ \end{matrix} \\ × & • & • & • & • & ↕ & ↕ & ↕ & ↕ & ↕ & × & • & • & • & • & p & a & b & c & d & −q−2 & q+e+1 & f & g & h \end{matrix}  (74)

for nonnegative integers p, a, b, c, d, q, e, f, g, h constrained by (69). We may drop the constraint (69) by considering ℂℙ^{n} instead as a homogeneous space for SL(n+1, ℂ), as is perhaps more usual. Having done that, the standard representation of SL(n + 1, ℂ) on ℂ^{n+1} gives rise to the BGG resolution

\begin{matrix}× & • & • & • & • & ↕ & ↕ & ↕ & ↕ & ↕ & × & • & • & • & • & −2 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 → • • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} • • • • •^{0} ^{0} ^{0} ^{0} ^{1}^{0} ^{0} ^{0} ^{0} ^{0} ⟶ × • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{0} ^{0} ^{0} ^{0} ^{1}^{0} ^{0} ^{0} ^{0} ^{0} & \begin{matrix}∇_{a} \\ ↗ \\ ↘ \\ ∇ā \end{matrix} \\ × & • & • & • & • & ↕ & ↕ & ↕ & ↕ & ↕ & × & • & • & • & • & 0 & 0 & 0 & 0 & 1 & −2 & 1 & 0 & 0 & 0 \end{matrix}  (75)

where the operators ∇_{a} and ∇ā are, more explicitly and as noted in (59),

X^{b} ↦ (∇_{a}X^{b})_{∘} and X^{b} ↦ ∇āX^{b}  (76)

where X^{b} is a vector field of type (1, 0) and of c-projective weight (−1, 0) and the subscript ∘ means to take the trace-free part. Notice that these are exactly the operators implicitly encoded in the standard tractor connection (52) and (53). More precisely, the filtration (49) is equivalent to the short exact sequence of vector bundles

0 → × • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{−1} ^{0} ^{0} ^{0} ^{0}^{0} ^{0} ^{0} ^{0} ^{0} ⟶ • • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} • • • • •^{0} ^{0} ^{0} ^{0} ^{1}^{0} ^{0} ^{0} ^{0} ^{0} ⟶ × • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{0} ^{0} ^{0} ^{0} ^{1}^{0} ^{0} ^{0} ^{0} ^{0} → 0
∥  ∥  ∥
E(−1, 0)  T  T^{1,0}M(−1, 0)

and on the flat model, namely ℂℙ^{n} as a homogeneous space for SL(n+1, ℂ), the tractor connection is the exactly the flat connection induced by (73). In the c-projectively flat case, the remaining entries in (52) and (53), namely

∇_{a}ρ − P_{ab}X^{b} and ∇āρ − P_{āb}X^{b}

may be regarded as quantities whose vanishing are differential consequences of setting

∇_{a}X^{b} + ρδ_{a}^{b} = 0 and ∇āX^{b} = 0.

Hence, they add no further conditions to being in the kernel of the first BGG operator (76) and the exactness of (75) follows. The same reasoning pertains in the curved but torsion-free setting and leads to the standard tractor connection being obtained by prolongation of the first BGG operator. This is detailed in Proposition 3.3. For more complicated representations, the tractor connection may not be obtained by prolongation in the curved setting, even if torsion-free. This phenomenon will soon be seen in two key examples, specifically in the connection (85) and Proposition 3.9 concerned with infinitesimal automorphisms and in Proposition 4.5, Theorem 4.6, and Corollary 4.7 dealing with the metrisability of c-projective structures. With reference to the general first BGG operators (74), the following cases occur prominently in this article.

• • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} • • • • •^{0} ^{0} ^{0} ^{0} ^{1}^{0} ^{0} ^{0} ^{0} ^{0} ⟶^{π} × • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{0} ^{0} ^{0} ^{0} ^{1}^{0} ^{0} ^{0} ^{0} ^{0} This is the standard complex tractor bundle T

and its canonical projection to T^{1,0}M(−1, 0).

• • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} • • • • •^{1} ^{0} ^{0} ^{0} ^{1}^{0} ^{0} ^{0} ^{0} ^{0} ⟶^{π} × • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{1} ^{0} ^{0} ^{0} ^{1}^{0} ^{0} ^{0} ^{0} ^{0} This is the adjoint tractor bundle AM to be con-

sidered in Section 3.4 and its canonical projection to T^{1,0}M. A first BGG operator acting on T^{1,0}M is given in Remark 3.3.

• • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} • • • • •^{0} ^{0} ^{0} ^{0} ^{1}^{0} ^{0} ^{0} ^{0} ^{1} ⟶^{π} × • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{0} ^{0} ^{0} ^{0} ^{1}^{0} ^{0} ^{0} ^{0} ^{1} This is the tractor bundle arising in the metris-

ability of c-projective structures to be discussed in Section 4 and a first BGG operator is given in (121).

• • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} • • • • •^{1} ^{0} ^{0} ^{0} ^{0}^{1} ^{0} ^{0} ^{0} ^{0} ⟶^{π} × • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{1} ^{0} ^{0} ^{0} ^{0}^{1} ^{0} ^{0} ^{0} ^{0} This is the dual of the previous case and arises in

Section 4.6, which is concerned with the first BGG operator D^{W} defined in (167) and acting on c-projective densities of weight (1, 1). It is a second order and c-projectively invariant operator.

In fact, there is quite a bit of flexibility in what one might allow as BGG operators, already for the first ones (74). For example, the operator D^{A} in Remark 3.3 is rather different from the c-projectively invariant operators occurring as the left hand sides of (1) and (2) in Proposition 3.7. Even for the bundle T^{1,0}M(−1, 0) in (75) corresponding to standard complex tractors, there is the option of replacing the second operator in (76) by

X^{c} ↦ ∇āX^{c} + Tāb̄^{c}Xā

in line with equation (1) in Proposition 3.7. Only in the torsion-free case do these operators agree (with each other and the usual ¯∂-operator).

On the flat model, however, there is no choice. The operators occurring in the BGG complexes are unique up to scale. Moreover, there are no other c-projectively invariant linear differential operators: every such operator is determined by its symbol and the BGG operators comprise a classification. In the curved setting it is necessary to add curvature correction terms and there is almost always some choice. Regarding existence, it is shown in [25] and [38] that such curved analogues always exist. However, even for the BGG sequence associated to the trivial representation, the resulting operators are different if there is torsion. Specifically, the construction in [38] follows the Hasse diagram beginning as in (71). In particular, there is no place for the torsion as an operator ∧^{0,1} → ∧^{2,0} whereas, in [25], the first BGG sequence associated to the trivial representation for the case of |1|-graded geometry such as c-projective geometry, is just the de Rham complex (70).

In summary, the BGG operators on ℂℙ^{n} provide models for what one should expect in the curved setting. In the flat case, there is no choice. In the curved case, there is a certain degree of flexibility, more so when there is torsion. Finally, the general theory of parabolic geometry [36] provides a location for harmonic curvature, as already discussed in Sections 2.2 and 2.7 and Kostant’s Theorem [64] on Lie algebra cohomology provides the location for this curvature, namely the three bundles appearing in the second step of the BGG sequence (72) for the adjoint representation whilst the two bundles at the first step locate the infinitesimal deformations of an almost c-projective structure, in line with the general theory [29].

3.4. Adjoint tractors and infinitesimal automorphisms. For a vector field X on a manifold M we write L_{X} for the Lie derivative along X of tensor fields on M. Recall that there is also a notion of a Lie derivative of an affine connection ∇ along a vector field X ∈ Γ(T M). It is given by the tensor field

L_{X}∇: T M → T^{∗}M ⊗ T M

characterised by

(L_{X}∇)(Y) ≡ L_{X}(∇Y) − ∇L_{X}Y

for any vector field Y ∈ Γ(T M). In abstract index notation we adopt the convention that (L_{X}∇)_{αβ}^{γ}Y^{β} = L_{X}(∇_{α}Y^{γ}).

Definition 3.1. A c-projective vector field or infinitesimal automorphism of an almost c-projective manifold (M, J, [∇]) of dimension 2n ≥ 4 is a vector field X on M that satisfies

• L_{X}J ≡ 0 (i.e. [X, JY] = J[X, Y] for all vector fields Y ∈ Γ(T M))

• (L_{X}∇)_{αβ}^{γ} = υ_{αβ}^{γ}, where υ_{αβ}^{γ} ∈ Γ(S^{2}T^{∗}M ⊗ T M) is a tensor of the form (11).

Note that X ∈ Γ(T M) is an infinitesimal automorphism of an almost c-projective manifold precisely if its flow acts by local automorphisms thereof.

Let us rewrite the two conditions defining a c-projective vector field as a system of differential equations on a vector field X of M. Expressing the Lie bracket in terms of a connection ∇ ∈ [∇] and its torsion shows that L_{X}J = 0 is equivalent to

T_{αβ}^{γ}X^{α} = −\frac{1}{2}(∇_{β}X^{γ} + J_{ϵ}^{γ}J_{β}^{ζ}∇_{ζ}X^{ϵ}).  (77)

for one (and hence any) connection ∇ ∈ [∇]. Moreover, one deduces straightforwardly from the definition of the Lie derivative of a connection that for any connection ∇ ∈ [∇] we have

(L_{X}∇)_{βδ}^{γ} = R_{αβ}^{γ}_{δ}X^{α} + ∇_{β}∇_{δ}X^{γ} + ∇_{β}(T_{αδ}^{γ}X^{α}).  (78)

Via the isomorphism T M ≅ T^{1,0}M we may write the result as a differential equation on X^{a}: equation (77) then becomes

∇b̄X^{c} + Tāb̄^{c}Xā = 0.  (79)

Since a tensor υ_{βδ}^{γ} of the form (11) satisfies υ_{b̄d}^{c} = 0 = υb̄d̄^{c}, the equation (L_{X}∇)_{β}^{γ}_{δ} = υ_{βδ}^{γ} can be equivalently encoded by the three equations

(L_{X}∇)_{bd}^{c} = υ_{bd}^{c} (L_{X}∇)_{b̄d}^{c} ≡ 0 (L_{X}∇)b̄d̄^{c} ≡ 0,  (80)

or, alternatively, their complex conjugates.

Lemma 3.6. If X^{a} ∈ Γ(T^{1,0}M) satisfies the invariant differential equation (79), then

(L_{X}∇)_{b̄d}^{c} ≡ 0 (L_{X}∇)b̄d̄^{c} ≡ 0,

for any connection ∇ ∈ [∇].

Proof. Equation (78) and the formulae of Proposition 2.13 imply

(L_{X}∇)_{b̄d}^{c} = R_{ab̄}^{c}_{d}X^{a} + Rāb̄^{c}_{d}Xā + ∇b̄∇_{d}X^{c}
= −R_{b̄a}^{c}_{d}X^{a} + R_{b̄d}^{c}_{a}X^{a} + ∇_{d}∇b̄X^{c} + Rāb̄^{c}_{d}Xā
= 2R_{b̄[d}^{c}_{a]}X^{a} + ∇_{d}∇b̄X^{c} − (∇_{d}Tb̄ā^{c})Xā.

Hence, the Bianchi symmetry (42) shows that

(L_{X}∇)_{b̄d}^{c} = T_{da}ēTēb̄^{c}X^{a} + ∇_{d}∇b̄X^{c} − (∇_{d}Tb̄ā^{c})Xā,

which evidently vanishes if ∇b̄X^{c} = Tb̄ā^{c}Xā, and consequently also ∇_{d}Xē = T_{da}ēX^{a}. As (L_{X}∇)b̄d̄^{c} = ∇b̄∇d̄X^{c} + ∇b̄(Tād̄^{c}Xā), the second assertion is obvious. □

According to Lemma 3.6, it remains to rewrite (L_{X}∇)_{b}^{c}_{d} = υ_{bd}^{c} as a differential equation on X^{a}. Note that we have

(L_{X}∇)_{b}^{c}_{d} = ∇_{b}∇_{d}X^{c} + R_{ab}^{c}_{d}X^{a} + R_{āb}^{c}_{d}Xā.  (81)

The Bianchi symmetry (41) R_{[bd}^{c}_{a]} ≡ 0 implies R_{bd}^{c}_{a} = −2R_{a[b}^{c}_{d]}. Moreover,

∇_{b}∇_{d}X^{c} = ∇_{(b}∇_{d)}X^{c} + ∇_{[b}∇_{d]}X^{c} = ∇_{(b}∇_{d)}X^{c} + \frac{1}{2}(R_{bd}^{c}_{a}X^{a} − T_{bd}ē∇ēX^{c}).

Therefore, we may rewrite (81) as

(L_{X}∇)_{b}^{c}_{d} = ∇_{(b}∇_{d)}X^{c} + R_{a(b}^{c}_{d)}X^{a} + R_{ā(b}^{c}_{d)}Xā
− \frac{1}{2}T_{bd}ē∇ēX^{c} + \frac{1}{2}T_{bd}ēT_{ēā}^{c}Xā,  (82)

where we used the Bianchi symmetry (42) given by R_{ā[b}^{c}_{d]} = \frac{1}{2}T_{bd}ēT_{ēā}^{c}Xā. The torsion terms of (82) evidently cancel if X^{a} satisfies (79).

Suppose now that 2n ≥ 6. Then we deduce from Proposition 2.13 that

R_{a(b}^{c}_{d)}X^{a} = W_{a(b}^{c}_{d)}X^{a} + P_{(bd)}X^{c} + δ_{(b}^{c}P_{d)a}X^{a} − δ_{b}^{c}P_{ad}X^{a} − δ_{d}^{c}P_{ab}X^{a},  (83)

where the third term and the two last terms already define two tensors of the form (11). Moreover, we obtain by Proposition 2.13 that

R_{ā(b}^{c}_{d)}Xā = H_{āb}^{c}_{d}Xā + \frac{1}{n+1}δ_{(b}^{c}T_{d)f} ēT_{ēā}^{f}Xā − 2P_{ā(b}δ_{d)}^{c}Xā,  (84)

where the last two terms are again already of the form (11). Therefore, we conclude:

Proposition 3.7. Suppose (M, J, [∇]) is an almost c-projective manifold of dimension 2n ≥ 6. A vector field X^{a} ∈ Γ(T^{1,0}M) is c-projective if and only if it satisfies the following equations

(1) ∇b̄X^{c} + Tāb̄^{c}Xā = 0

(2) (∇_{(b}∇_{d)}X^{c} + P_{(bd)}X^{c} + W_{a(b}^{c}_{d)}X^{a} + H_{āb}^{c}_{d}Xā)_{∘} = 0, where the subscript ∘ denotes the trace-free part.

Due to Proposition 2.14 for 2n = 4 the equations take a simpler form:

Proposition 3.8. Suppose (M, J, [∇]) is an almost c-projective manifold of dimension 2n = 4. A vector field X^{a} ∈ Γ(T^{1,0}M) is c-projective if and only if it satisfies the following equations

(1) ∇b̄X^{c} + Tāb̄^{c}Xā = 0

(2) (∇_{(b}∇_{d)}X^{c} + P_{(bd)}X^{c} + H_{āb}^{c}_{d}Xā)_{∘} = 0, where the subscript ∘ denotes the trace-free part.

The equations in Propositions 3.7 and 3.8 define an invariant differential operator

D^{aut} : T^{1,0}M → (∧^{1,0} ⊗ T^{0,1}M) ⊕ (S^{2}∧^{1,0} ⊗ T^{1,0}M)_{∘},

whose kernel comprises the infinitesimal automorphisms of (M, J, [∇]).

Let us recall some facts about infinitesimal automorphisms of Cartan geometries.

Definition 3.2. Suppose (p: G → M, ω) is a Cartan geometry. A vector field X̃ ∈ Γ(T G) is called an infinitesimal automorphism of (p: G → M, ω), if X̃ is right-invariant for the principal right action on G and LX̃ω = 0.

A Cartan connection ω on p: G → M induces a bijection between right-invariant vector fields X̃ ∈ Γ(T G) and equivariant functions ω(X̃): G → 𝔤. Hence, right-invariant vector fields on G are in bijection with sections of the adjoint tractor bundle AM. A section s of AM corresponds to an infinitesimal automorphism of the Cartan geometry if and only if s is parallel for the linear connection

∇^{A}s + κ(Π(s), ·),  (85)

where ∇^{A} is the adjoint tractor connection, Π: AM → T M the natural projection, and κ ∈ Ω^{2}(M, AM) the curvature of the Cartan geometry; see [29, 36].

The equivalence of categories established in Theorem 2.8 implies that any infinitesimal automorphism X ∈ Γ(T M) of an almost c-projective manifold can be lifted uniquely to an infinitesimal automorphism of its normal Cartan geometry and conversely, any infinitesimal automorphism of the Cartan geometry projects to an infinitesimal automorphism of the underlying almost c-projective manifold. This implies, in particular, that Π induces a bijection between sections of the adjoint tractor bundle of the almost c-projective manifold that are parallel for the connection (85) and infinitesimal automorphisms of the almost c-projective manifold.

For the convenience of the reader let us explicitly compute the modified adjoint tractor connection (85). For these purposes let us identify the adjoint tractor bundle with the (1, 0)-part of its complexification. As such it is filtered as

AM = A^{−1}M ⊃ A^{0}M ⊃ A^{1}M,

where A^{−1}M/A^{0}M ≅ T^{1,0}M, A^{0}M/A^{1}M ≅ 𝔤𝔩(T^{1,0}M, ℂ) and A^{1}M ≅ ∧^{1,0}. Hence, for any choice of connection ∇ ∈ [∇], we can identify an element of AM with a triple

(\begin{matrix}X^{b} \\ ϕ_{b}^{c} \\ μ_{b} \end{matrix}) , where \{ \begin{matrix}X^{b} ∈ T^{1,0}M \\ ϕ_{b}^{c} ∈ 𝔤𝔩(T^{1,0}M, ℂ) \\ μ_{b} ∈ ∧^{1,0}. \end{matrix}

Note that ϕ_{b}^{c} may be decomposed further into its trace-free and trace parts according to the decomposition (18) 𝔤𝔩(T^{1,0}M, ℂ) ≅ 𝔰𝔩(T^{1,0}M, ℂ) ⊕ E(0, 0). However, we shall not make use of this decomposition. From the formulae (52) and (53) defining the tractor connection on the standard complex tractor bundle T one easily deduces that tractor connection on AM = 𝔰𝔩(T ) is given by

∇_{a}^{A} (\begin{matrix}X^{b} \\ ϕ_{b}^{c} \\ μ_{b} \end{matrix}) = (\begin{matrix}∇_{a}X^{b} − ϕ_{a}^{b} \\ ∇_{a}ϕ_{b}^{c} + δ_{a}^{c}μ_{b} + P_{ab}X^{c} + (μ_{a} + P_{ad}X^{d})δ_{b}^{c} \\ ∇_{a}μ_{b} − P_{ac}ϕ_{b}^{c} \end{matrix})  (86)

∇A ; ā (\begin{matrix}X^{b} \\ ϕ_{b}^{c} \\ μ_{b} \end{matrix}) = (\begin{matrix}∇āX^{b} \\ ∇āϕ_{b}^{c} + P_{āb}X^{c} + P_{ād}X^{d}δ_{b}^{c} \\ ∇āμ_{b} − P_{āc}ϕ_{b}^{c} \end{matrix}) .  (87)

From (85) we deduce that:

Proposition 3.9. Suppose (M, J, [∇]) is an almost c-projective manifold of dimension 2n ≥ 6. Then the projection Π: AM → AM/A^{0}M ≅ T^{1,0}M induces a bijection between sections of AM that are parallel for

∇_{a}^{A} (\begin{matrix}X^{b} \\ ϕ_{b}^{c} \\ μ_{b} \end{matrix}) + (\begin{matrix}0 \\ W_{da}^{c}_{b}X^{d} + W_{d̄a}^{c}_{b}Xd̄ + T_{af} ēTēd̄^{f}Xd̄δ_{b}^{c} \\ C_{cab}X^{c} + C_{c̄ab}Xc̄ \end{matrix})

∇A ; ā (\begin{matrix}X^{b} \\ ϕ_{b}^{c} \\ μ_{b} \end{matrix}) + (\begin{matrix}T_{c̄ā}^{b}Xc̄ \\ W_{dā}^{c}_{b}X^{d} + Wd̄ ; ā^{c}_{b}Xd̄ + (∇_{e}Td̄ ; ā^{e}Xd̄ − T_{df} ēT_{ēā}^{f}X^{d})δ_{b}^{c} \\ C_{cāb}X^{c} + C_{c̄āb}Xc̄ \end{matrix})

and infinitesimal automorphisms of the almost c-projective manifold.

Proposition 3.9 can, of course, also be obtained directly by prolonging cleverly the equations of Proposition 3.7. Note that the form of the equations in Proposition 3.7 immediately shows that Π maps parallel sections for the connection in Proposition 3.9 to c-projective vector fields. To see the converse, one may verify that that for a c-projective vector field X^{b} and for any choice of ∇ ∈ [∇] the section

(\begin{matrix}X^{b} \\ ϕ_{b}^{c} \\ μ_{b} \end{matrix}) = (\begin{matrix}X^{b} \\ ∇_{b}X^{c} \\ −\frac{1}{n+1}(∇_{a}∇_{b}X^{a} + 2P_{(ab)}X^{b}) \end{matrix})

is parallel for the connection given in Proposition 3.9 and observe that this differential operator indeed defines the inverse to the claimed bijection.

Remark 3.2. If the dimension of the almost c-projective manifold is 2n = 4, then Proposition 3.9 still holds taking into account that W_{ab}^{c}_{d} ≡ 0.

Remark 3.3. Note that the differential operator

D^{A}: T^{1,0}M → (∧^{1,0} ⊗ T^{0,1}M) ⊕ (S^{2}∧^{1,0} ⊗ T^{1,0}M)_{∘}
X^{c} ↦ (∇b̄X^{c}, (∇_{(b}∇_{d)}X^{c} + X^{c}P_{(bd)})_{∘})

is also invariant. It is the first operator in the BGG sequence of the adjoint tractor bundle. As for projective structures, this operator differs from D^{aut}, the operator that controls the infinitesimal automorphisms of the almost c-projective manifold. For a discussion of this phenomena in the context of general parabolic geometries see [29].

4. Metrisability of almost c-projective structures

On any (pseudo-)Kähler manifold (M, J, g) one may consider the c-projective structure that is induced by the Levi-Civita connection of g. The c-projective manifolds that arise in this way from a (pseudo-)Kähler metric are the most extensively studied cprojective manifolds; see [41, 56, 81, 90] and, more recently, [44, 77, 80]. A natural but difficult problem in this context is to characterise the c-projective structures that arise from (pseudo-)Kähler metrics or, more generally, the almost c-projective structures that arise from (2, 1)-symplectic (also called quasi-Kähler) metrics. In the following sections we shall show that, suitably interpreted, this problem is controlled by an invariant linear overdetermined system of PDE and we shall explicitly prolong this system. Under the assumptions that J is integrable and the c-projective manifold (M, J, [∇^{g}]) arose via the Levi-Civita connection ∇^{g} of a Kähler metric g, a prolongation of the system of PDE governing the Kähler metrics that are c-projectively equivalent to g was first given in [41, 81] and rediscovered in the setting of Hamiltonian 2-forms on Kähler manifolds in [2].

4.1. Almost Hermitian manifolds. We begin by recalling some basic facts.

Definition 4.1. Suppose (M, J) is an almost complex manifold of dimension 2n ≥ 4. A Hermitian metric on (M, J) is a (pseudo-)Riemannian metric g_{αβ} ∈ Γ(S^{2}T^{∗}M) that is J-invariant:

J_{α}^{γ}J_{β}^{δ}g_{γδ} = g_{αβ}.

We call such a triple (M, J, g) an almost Hermitian manifold, or, if J is integrable, a Hermitian manifold. Note that we drop the awkward (pseudo-) prefix.

To an almost Hermitian manifold (M, J, g) one can associate a nondegenerate Jinvariant 2-form Ω ∈ Γ(∧^{2}T^{∗}M) given by

Ω_{αβ} := J_{α}^{γ}g_{γβ}.  (88)

It is called the fundamental 2-form or Kähler form of (M, J, g). If Ω is closed (dΩ = 0), we say (M, J, g) is almost Kähler or almost pseudo-Kähler accordingly as g is Riemannian or pseudo-Riemannian; the “almost” prefix is dropped if J is integrable.

We write g^{αβ} for the inverse of the metric g_{αβ}:

g_{αγ}g^{γβ} = δ_{α}^{β}.

We raise and lower indices of tensors on an almost Hermitian manifold (M, J, g) with the metric and its inverse. The Poisson tensor on M is Ω^{αβ} = J_{γ}^{β}g^{αγ}, with

Ω_{αβ}Ω^{βγ} = −δ_{α}^{γ}.  (89)

Viewing ∧^{1,0} ⊗ ∧^{0,1} as a complex vector bundle equipped with the real structure given by swapping its factors, a Hermitian metric can, by definition, also be seen as a real nondegenerate section g_{ab̄} of ∧^{1,0} ⊗ ∧^{0,1}. We denote by g^{āb} ∈ Γ(T^{0,1}M ⊗ T^{1,0}M) its inverse, characterised by

g_{ab̄}g^{b̄c} = δ_{a}^{c} and g_{ab̄}g^{c̄a} = δb̄c̄.

Let us denote by ∇^{g} the Levi-Civita connection of a Hermitian metric g. Differentiating the identity J_{α}^{γ}J_{γ}^{β} = −δ_{α}^{β} shows that

(∇_{α}^{g}J_{β}^{ϵ})J_{ϵ}^{γ} + J_{β}^{ϵ}∇_{α}^{g}J_{ϵ}^{γ} = 0.  (90)

Since ∇_{α}^{g}Ω_{βγ} = g_{γϵ}∇_{α}^{g}J_{β}^{ϵ}, it follows immediately from (90) that

∇_{α}^{g}Ω_{βγ} + J_{β}^{ϵ}J_{γ}^{ζ}∇_{α}^{g}Ω_{ϵζ} = 0.  (91)

Viewing ∇_{α}^{g}Ω_{βγ} as 2-form with values in T^{∗}M, equation (91) says that the part of type (1, 1) vanishes identically. On the other hand, the vector bundle map

∧^{2}T^{∗}M ⊗ T^{∗}M → ∧^{3}T^{∗}M
Ψ_{αβγ} ↦ Ψ_{[αβγ]}

induces an isomorphism between 2-forms with values in T^{∗}M of type (0, 2) and 3-forms on M of type (2, 1) + (1, 2), i.e. real sections of ∧^{2,1} ⊕ ∧^{1,2}. Since

∇_{[α}^{g}Ω_{βγ]} = \frac{1}{3}(dΩ)_{αβγ},  (92)

the identity

2∇_{α}^{g}Ω_{βγ} = (dΩ)_{αβγ} − J_{β}^{ϵ}J_{γ}^{ζ}(dΩ)_{αϵζ} − N_{βγ}^{J}^{ϵ}Ω_{ϵα}  (93)

shows that type (0, 2) component of ∇_{α}^{g}Ω_{βγ} is identified with the (2, 1) + (1, 2) component of dΩ [48]. The type (3, 0) + (0, 3) component of dΩ is determined by the Nijenhuis tensor N^{J}, hence so is the type (2, 0) part of ∇_{α}^{g}Ω_{βγ} (which has type (0, 2) when viewed as a 2-form with values in T M using g).

If M has dimension 2n ≥ 6, ∇_{α}^{g}Ω_{βγ} can be decomposed into 4 components, which correspond to 4 real irreducible U(p, q)-submodules in ∧^{2}ℂ^{n}⊗ℂ^{n}, where U(p, q) denotes the (pseudo-)unitary group of signature (p, q) with p + q = n, the signature of g_{αβ}. If 2n = 4, then ∇_{α}^{g}Ω_{βγ} has only two components. The different possibilities of a subset of these invariants vanishing leads to the Gray–Hervella classification of almost Hermitian manifolds into 16, respectively 4, classes in dimension 2n ≥ 6, respectively 2n = 4, see [51]. In the following we shall be interested in the class of almost Hermitian manifolds which in the literature (at least in the case of metrics of definite signature) are referred to as quasi-Kähler or (2, 1)-symplectic, see [48, 51]. We extend this terminology to indefinite signature, as we have done for Hermitian metrics in general.

Definition 4.2. Suppose (M, J, g) is an almost Hermitian manifold of dimension 2n ≥ 4. Then (M, J, g) is called a quasi-Kähler or (2, 1)-symplectic manifold, if

∇_{α}^{g}Ω_{βγ} + J_{α}^{ϵ}J_{β}^{ζ}∇_{ϵ}^{g}Ω_{ζγ} = 0,  (94)

which is the case if and only if

∇_{α}^{g}J_{β}^{γ} = −J_{α}^{ϵ}J_{β}^{ζ}∇_{ϵ}^{g}J_{ζ}^{γ}.  (95)

Since ∇_{α}Ω_{βγ}, as a 2-form with values in T^{∗}M, has no component of type (1, 1), (94) means, equivalently, that ∇_{α}Ω_{βγ} has type (2, 0), i.e. has no (0, 2) part; equivalently dΩ has no component of type (2, 1) + (1, 2), i.e. it has type (3, 0) + (0, 3), which explains the “(2, 1)-symplectic” terminology. The class of (2, 1)-symplectic manifolds of dimension 2n ≥ 6 contains as a subclass the almost (pseudo-)Kähler manifolds, which are symplectic, and the subclass of nearly Kähler manifolds, i.e. those almost Hermitian manifolds that satisfy ∇_{α}^{g}Ω_{βγ} = −∇_{β}^{g}Ω_{αγ}, which is manifestly equivalent to 3∇_{α}^{g}Ω_{βγ} = (dΩ)_{αβγ}. Since in dimension 2n = 4 any 3-form has type (2, 1)+(1, 2), the condition for an almost Hermitian manifold of dimension 4 to be (2, 1)-symplectic is equivalent to the condition to be almost (pseudo-)Kähler. If J is integrable, i.e. (M, J, g) a Hermitian manifold, then (M, J, g) is (2, 1)-symplectic if and only if dΩ = 0, i.e. (M, J, g) is (pseudo-)Kähler.

Definition 4.3. Let (M, J, g) be an almost Hermitian manifold of dimension 2n ≥ 4. Then a Hermitian connection on M is an affine connection ∇ with ∇J = 0 and ∇g = 0.

Such Hermitian connections exist and are uniquely determined by their torsion. A discussion of Hermitian connections and of the freedom in prescribing their torsion can for instance be found in [48] (see also [69]). The following proposition shows that (2, 1)-symplectic manifolds can be characterised as those almost Hermitian manifolds which admit a minimal Hermitian connection; for a proof see [48].

Proposition 4.1. Suppose (M, J, g) is an almost Hermitian manifold of dimension 2n ≥ 4. Then (M, J, g) admits a (unique) Hermitian connection whose torsion T is of type (0, 2) as 2-form with values in T M, equivalently T = −\frac{1}{4}N^{J}, if and only if it is (2, 1)-symplectic.

For a a (2, 1)-symplectic manifold (M, J, g) we refer to the unique Hermitian connection ∇ of Proposition 4.1 as the canonical connection of (M, J, g). In terms of the Levi-Civita connection ∇^{g} of g it is given by

∇_{α}X^{β} = ∇_{α}^{g}X^{β} + \frac{1}{2}(∇_{α}^{g}J_{γ}^{β})J_{ϵ}^{γ}X^{ϵ}.  (96)

For the convenience of the reader let us check that this connection has the desired properties. For an arbitrary almost Hermitian manifold (M, J, g) the formula (96) is obviously a complex connection, since

∇_{α}(J_{γ}^{β}X^{γ}) = \frac{1}{2}(∇_{α}^{g}(J_{γ}^{β}X^{γ}) + J_{γ}^{β}∇_{α}^{g}X^{γ})
J_{γ}^{β}∇_{α}X^{γ} = \frac{1}{2}(∇_{α}^{g}(J_{γ}^{β}X^{γ}) + J_{γ}^{β}∇_{α}^{g}X^{γ}),

which implies (∇_{α}J_{β}^{γ})X^{β} = ∇_{α}(J_{γ}^{β}X^{γ})−J_{γ}^{β}∇_{α}X^{γ} = 0 for all vector fields X^{α}. Since ∇^{g} is a metric connection, the connection given by (96) is a metric connection if and only if (∇_{α}^{g}J_{ζ}^{ϵ})J_{β}^{ζ}g_{γϵ} = (∇_{α}^{g}Ω_{ζγ})J_{β}^{ζ} is skew in γ and β, which follows immediately from (93). Hence, on any almost Hermitian manifold formula (96) defines a Hermitian connection. Moreover, since ∇^{g} is torsion free, the torsion T of (96) satisfies

T_{αβ}^{γ} = \frac{1}{2}((∇_{α}^{g}J_{ϵ}^{γ})J_{β}^{ϵ} − (∇_{β}^{g}J_{ϵ}^{γ})J_{α}^{ϵ}).  (97)

Recall that the Nijenhuis tensor can be expressed in terms of ∇^{g} (actually in terms of any torsion free connection) as

N_{αβ}^{J}^{γ} = −(∇_{α}^{g}J_{ϵ}^{γ})J_{β}^{ϵ} + (∇_{β}^{g}J_{ϵ}^{γ})J_{α}^{ϵ} − J_{α}^{ϵ}∇_{ϵ}J_{β}^{γ} + J_{β}^{ϵ}∇_{ϵ}J_{α}^{γ},  (98)

which by (95) reduces in the case of a (2, 1)-symplectic manifold to the equation

N_{αβ}^{J}^{γ} = −2((∇_{α}^{g}J_{ϵ}^{γ})J_{β}^{ϵ} − (∇_{β}^{g}J_{ϵ}^{γ})J_{α}^{ϵ}).  (99)

Comparing (97) with (99) shows that on a (2, 1)-symplectic manifold the torsion T of (96) satisfies T = −\frac{1}{4}N^{J} as required. Note that, if the Levi-Civita connection ∇^{g} of (M, J, g) is a complex connection, then also ∇_{α}^{g}Ω_{βγ} = 0, which by (92) implies that Ω_{αβ} is closed. Moreover, the identity (93) shows that J is necessarily integrable in this case. Conversely, the same identity shows that, if J is integrable and the fundamental 2form closed, then the Levi-Civita connection is a complex connection, cf. Corollary 1.3. Hence, the connection in (96) coincides with the Levi-Civita connection of on an almost Hermitian manifold if and only if (M, J, g) is (pseudo-)Kähler.

Remark 4.1. We have already observed that formula (96) defines a Hermitian connection on any almost Hermitian manifold, which is usually referred to as the first canonical connection following [69]. In the case of a (2, 1)-symplectic manifold the first canonical connection coincides also with the second canonical connection of [69], which is also called Chern connection; see [48].

Let (M, J, g) be a (2, 1)-symplectic manifold and denote by R the curvature of its canonical connection ∇. Since ∇ is Hermitian, we have R ∈ Ω^{2}(M, 𝔲(T M)), where 𝔲(T M) ⊂ T^{∗}M ⊗ T M denotes the subbundle of unitary bundle endomorphisms of (T M, J, g). Setting R_{αβγδ} ≡ R_{ab}^{ϵ}_{δ}g_{ϵγ}, the property R ∈ Ω^{2}(M, 𝔲(T M)) of the curvature of a (2, 1)-symplectic manifold can be expressed as

R_{αβγδ} = R_{[αβ][γδ]} and R_{αβγ[δ}J_{ϵ]}^{γ} = 0.  (100)

Moreover, recall that for any linear connection the Bianchi symmetry holds. Hence, R satisfies

R_{[αβ}^{γ}_{δ]} = ∇_{[α}T_{βδ]}^{γ} + T_{ϵ[α}^{γ}T_{βδ]}^{ϵ},  (101)

where T_{αβ}^{γ} = −\frac{1}{4}N_{αβ}^{J}^{γ} is the torsion of ∇. Note that (101) for a minimal connection is of course precisely equivalent to the already established identities (41) and (42). Since ∇ is a complex connection, R decomposes as a 2-form with values in the complex endomorphism of T M into three components according to type as explained in Section 2.4. In barred and unbarred indices R can therefore be encoded by the three tensors

R_{ab}^{c}_{d} R_{ab̄}^{c}_{d} Rāb̄^{c}_{d},

or equivalently by their complex conjugates, where R_{ab}^{c}_{d} ≡ Π_{a}^{α}Π_{b}^{β}Π_{γ}^{c}Π_{d}^{δ}R_{αβ}^{γ}_{δ} and so on. Since ∇ preserves in addition a (pseudo-)Riemannian metric, the additional symmetry R_{αβγδ} = −R_{αβδγ} implies

R_{abc̄d} ≡ R_{ab}^{e}_{d}g_{ec̄} = −R_{ab}ēc̄g_{ēd} ≡ −R_{abdc̄} R_{āb̄c̄d} ≡ Rāb̄^{e}_{d}g_{ec̄} = −Rāb̄ēc̄g_{ēd} ≡ −R_{āb̄dc̄}

R_{ab̄c̄d} ≡ R_{ab̄}^{e}_{d}g_{ec̄} = −R_{ab̄}ēc̄g_{ēd} ≡ −R_{ab̄dc̄} R_{ābc̄d} ≡ R_{āb}^{e}_{d}g_{ec̄} = −R_{āb}ēc̄g_{ēd} ≡ −R_{ābdc̄}.

Note that the first two identities (which are conjugates of each other) show that for the canonical connection of a (2, 1)-symplectic manifold (in contrast to a general minimal complex connection) the curvature components R_{ab}^{c}_{d} and R_{ab}c̄d̄ = \overline{Rāb̄^{c}_{d}} are not independent of each other, since they are related by g. Hence, the curvature R of the canonical connection of a (2, 1)-symplectic manifold can be encoded by the two tensors

R_{abc̄d} = R_{[ab]c̄d} and R_{ab̄c̄d}

(or their complex conjugates). By (101), (100) and the fact that the torsion of ∇ has type (0, 2) one deduces straightforwardly that the curvature and the torsion of a (2, 1)-symplectic manifold satisfy the symmetries

R_{abc̄d} = −∇c̄T_{abd} R_{abc̄d} + R_{bdc̄a} + R_{dac̄b} = 0  (102)
∇_{[a}T_{bc]d} = 0  (103)
R_{ab̄c̄d} − R_{db̄c̄a} = −T_{ad}ēTēb̄c̄ R_{ab̄c̄d} − R_{ac̄b̄d} = −Tb̄c̄^{e}T_{ead}  (104)
R_{ab̄c̄d} − R_{c̄d̄ab̄} = Tāb̄c̄T_{da}ē + T_{fda}Tc̄b̄^{f},  (105)

where T_{abc} = T_{ab}d̄g_{cd̄} and Tāb̄c̄ = Tāb̄^{d}g_{dc̄} (cf. also (41) and (42)). Now let us consider the Ricci tensor Ric of the canonical connection of a (2, 1)-symplectic manifold. By definition we have

Ric_{ab} = R_{ca}^{c}_{b} Ricāb̄ = R_{c̄ā}c̄b̄ Ric_{ab̄} = R_{c̄a}c̄b̄ Ric_{āb} = R_{cā}^{c}_{b}.

From the identities (102) we we conclude that

Ric_{ab} = −∇c̄T c̄_{ab}  Ric_{[ab]} = \frac{1}{2}∇c̄T_{ab}c̄
Ricāb̄ = −∇_{c}T^{c}āb̄  Ric_{[āb̄]} = \frac{1}{2}∇_{c}Tāb̄^{c}. (106)

Moreover, taking a Ricci type contraction in (105) shows immediately that the Jinvariant part of the Ricci tensor of the canonical connection of a (2, 1)-symplectic manifold is symmetric:

Ric_{ab̄} = Ric_{b̄a}.  (107)

The canonical connection of a (2, 1)-symplectic manifold is special in the sense of Section 2.5, since it preserves the volume form of g; hence (106) and (107) confirm in particular what we deduced there for the Ricci curvature of special connections.

We already observed that (2, 1)-symplectic is equivalent to (pseudo-)Kähler when J is integrable. Hence, in this case, the canonical connection simply coincides with the Levi-Civita connection. Suppose (M, J, g) is now a (pseudo-)Kähler manifold. Then the identities (102)–(105) imply that R is determined by any of the following tensors

R_{ab̄cd̄} R_{ābcd̄} R_{ab̄c̄d} R_{ābc̄d}

which are now subject to the following symmetries

R_{ab̄cd̄} = −R_{b̄acd̄} R_{ab̄cd̄} = −R_{ab̄d̄c} R_{ab̄cd̄} = R_{cb̄ad̄} R_{ab̄cd̄} = R_{ad̄cb̄}  (108)

as well as

\overline{R}_{āb\overline{c}d} ≡ \overline{R_{ab̄cd̄}} = R_{bādc̄}.

Remark 4.2. Let us remark that for the curvature R of a (pseudo-)Kähler manifold, we have R_{[αβ}^{γ}_{δ]} = 0 which, together with the symmetries (100), implies

R_{αβγδ} = R_{γδαβ}  and J_{α}^{ϵ}J_{β}^{ζ}R_{ϵζ}^{γ}_{δ} = R_{αβ}^{γ}_{δ}.  (109)

The symmetries of (100) and (109) are precisely the ones in (108) expressed in real indices. Note also that (109) shows immediately that the Ricci tensor Ric_{αβ} = R_{ϵα}^{ϵ}_{β} is symmetric and J-invariant, which is consistent with (107).

Moreover, note that it is immediate that the rank of the bundle of (pseudo-)Kähler curvatures is (n(n + 1)/2)^{2} and that this bundle further decomposes under U(n) as

S^{2}∧^{1,0} ⊗ S^{2}∧^{0,1} = (S^{2}∧^{1,0} ⊗_{∘} S^{2}∧^{0,1}) ⊕ (∧^{1,0} ⊗_{∘} ∧^{0,1}) ⊕ ℝ,

where the subscript ∘ means trace-free part and ℝ stands for the trivial bundle. Under this decomposition, the (pseudo-)Kähler curvature splits as

R_{ab̄cd̄} = U_{ab̄cd̄} − 2(Ξ_{ab̄}g_{cd̄} + Ξ_{cd̄}g_{ab̄} + Ξ_{ad̄}g_{cb̄} + Ξ_{cb̄}g_{ad̄}) − 2Λ(g_{ab̄}g_{cd̄} + g_{ad̄}g_{cb̄}), (110)

where

U_{ab̄cd̄} = U_{cb̄ad̄} = U_{ad̄cb̄} g^{b̄c}U_{ab̄cd̄} = 0 g^{b̄a}Ξ_{ab̄} = 0.

This is a Kähler analogue of the usual decomposition of Riemannian curvature into the conformal Weyl tensor, the trace-free Ricci tensor, and the scalar curvature. The tensor U_{ab̄cd̄} is called the Bochner curvature (or tensor) and is the orthogonal projection of the conformal Weyl curvature onto the intersection of the space of Kähler curvatures with the space of conformal Weyl tensors [2]. The analogue of constant curvature in (pseudo-)Kähler geometry is to insist that R_{ab̄d̄c} = Λ(g_{ab̄}g_{cd̄} + g_{cb̄}g_{ad̄}), where the (a priori) smooth function Λ is constant by the Bianchi identity. This is called constant holomorphic sectional curvature and (for Λ > 0) locally characterises ℂℙ^{n} and its Fubini–Study metric as in Section 2.6 (where the normalisation is such that Λ = 1).

4.2. Other curvature decompositions. It will be useful, both in this article and elsewhere, to decompose the (pseudo-)Kähler curvature tensor from various different viewpoints, some of which ignore the complex structure. Without a complex structure, barred and unbarred indices are unavailable so firstly we should rewrite the irreducible decomposition (110) using only real indices. We recall that

R_{αβγδ} = R_{[αβ][γδ]} R_{[αβγ]δ} = 0 R_{αβγ[δ}J_{ϵ]}^{γ} = 0  (111)

and the real version of (110) will apply to any tensor satisfying these identities. Recalling that Ω_{αβ} = J_{αβ} = J_{α}^{γ}g_{βγ}, we obtain

R_{αβγδ} = U_{αβγδ}
+ g_{αγ}Ξ_{βδ} − g_{βγ}Ξ_{αδ} − g_{αδ}Ξ_{βγ} + g_{βδ}Ξ_{αγ}
+ Ω_{αγ}Σ_{βδ} − Ω_{βγ}Σ_{αδ} − Ω_{αδ}Σ_{βγ} + Ω_{βδ}Σ_{αγ} + 2Ω_{αβ}Σ_{γδ} + 2Ω_{γδ}Σ_{αβ}
+ Λ(g_{αγ}g_{βδ} − g_{βγ}g_{αδ} + Ω_{αγ}Ω_{βδ} − Ω_{βγ}Ω_{αδ} + 2Ω_{αβ}Ω_{γδ}),

(112)

where

• U_{αβγδ} is totally trace-free with respect to g^{αβ} and Ω^{αβ}
• Σ_{αβ} ≡ J_{α}^{γ}Ξ_{βγ} whilst Ξ_{αβ} is symmetric, trace-free, and of type (1, 1):
Ξ_{αβ} = Ξ_{(αβ)} Ξ_{α}^{α} = 0 Σ_{αβ} = Σ_{[αβ]}.

A simple way to see this is to check that all parts of this decomposition satisfy (111) as they should and then apply Π_{a}^{α}\overline{Π}β ; b̄Π_{c}^{γ}\overline{Π}δ ; d̄, using the various identities from Section 1.1 including (4), to recover (110). One can also read off from (112) the corresponding decomposition of the Ricci tensor in (pseudo-)Kähler geometry. Specifically,

Ric_{βδ} = 2(n + 2)Ξ_{βδ} + 2(n + 1)Λg_{βδ} Scal = 4n(n + 1)Λ

and, conversely,

Λ = \frac{1}{4n(n+1)}Scal Ξ_{αβ} = \frac{1}{2(n+2)} (Ric_{αβ} − \frac{1}{2n}Scal g_{αβ}) .

Other natural realms in which one may view (pseudo-)Kähler geometry are • projective

• conformal

• c-projective

• symplectic and in each case decompose the curvature accordingly. The projective Weyl curvature tensor [42] on a Riemannian manifold of dimension m is given by

R_{αβγδ} − \frac{1}{m−1}g_{αγ}Ric_{βδ} + \frac{1}{m−1}g_{βγ}Ric_{αδ}.

If this vanishes, then, in conjunction with the interchange symmetry R_{αβγδ} = R_{γδαβ}, we deduce that R_{αβγδ} = λ(g_{αγ}g_{βδ} − g_{βγ}g_{αδ}) where, if m ≥ 3, the (a priori) smooth function λ is constant by the Bianchi identity. This is Beltrami’s Theorem that the only projectively flat (pseudo-)Riemannian geometries are constant curvature (when m = 2 one instead uses that the projective Cotton–York tensor vanishes). In any case, comparison with (112) shows that for n ≥ 2 the only projectively flat (pseudo-)Kähler manifolds are flat. The conformal Weyl curvature is given by

R_{αβγδ} − g_{αγ}Q_{βδ} + g_{βγ}Q_{αδ} − g_{βδ}Q_{αγ} + g_{αδ}Q_{βγ},

where Q_{αβ} is the Riemannian Schouten tensor

Q_{αβ} = \frac{1}{m − 2} (Ric_{αβ} − \frac{1}{2(m − 1)}Scal g_{αβ}) .

Thus, if the conformal Weyl curvature vanishes on a (pseudo-)Kähler manifold, then

2R_{α}^{δ}_{γ[δ}J_{ϵ]}^{γ} = J_{ϵα}Q_{β}^{β} + 2(n − 2)J_{ϵ}^{γ}Q_{αγ}.

From (111), we see that for n ≥ 3 the only conformally flat (pseudo-)Kähler manifolds are flat. For n = 2 it follows only that the geometry is scalar flat and, in fact, Tanno [96] showed that 4-dimensional conformally flat Kähler manifolds are locally of the form ℂℙ^{1} ×Σ where ℂℙ^{1} has the Fubini–Study metric up to constant scale and the complex surface Σ has a constant negative scalar curvature of equal magnitude but opposite sign to that on ℂℙ^{1}.

From the c-projective viewpoint, if we compare the decomposition (112) with (24), then we conclude, firstly that W_{αβ}^{γ}_{δ} = H_{αβ}^{γ}_{δ} (see the proof of Proposition 4.4 for a barred/unbarred index proof of this), and then that

H_{αβγδ} = U_{αβγδ} − g_{αδ}Ξ_{βγ} + g_{βδ}Ξ_{αγ} + \frac{1}{n+1}(g_{βγ}Ξ_{αδ} − g_{αγ}Ξ_{βδ})

+ 2Ω_{αβ}Σ_{γδ} − Ω_{αδ}Σ_{βγ} + Ω_{βδ}Σ_{αγ} − \frac{1}{n+1}(2Ω_{γδ}Σ_{αβ} − Ω_{βγ}Σ_{αδ} + Ω_{αγ}Σ_{βδ}). (113)

Notice, in particular, that

H_{αβγ}^{β} = 2\frac{n(n+1)}{n+1}Ξ_{αγ}  (114)

from which we can deduce the following c-projective counterpart to Beltrami’s Theorem.

Theorem 4.2. Suppose a (pseudo-)Kähler metric is c-projectively flat. Then it has constant holomorphic sectional curvature.

Proof. To be c-projectively flat, the harmonic curvature tensor H_{αβ}^{γ}_{δ} must vanish. Then from (114) we find that Ξ_{αβ} = 0 and from (113) that also U_{αβγδ} = 0. According to (112) we find that R_{αβγδ} is of the required form. □

Finally, we may view (pseudo-)Kähler geometry from the purely symplectic viewpoint as follows. For any torsion-free connection preserving Ω_{αβ}, the tensor R_{αβ}^{ϵ}_{δ}Ω_{ϵγ} is symmetric in γδ and may be decomposed into irreducible pieces under Sp(2n, ℝ):

R_{αβ}^{ϵ}_{δ}Ω_{ϵγ} = V_{αβγδ} + Ω_{αγ}Φ_{βδ} − Ω_{βγ}Φ_{αδ} + Ω_{αδ}Φ_{βγ} − Ω_{βδ}Φ_{αγ} + 2Ω_{αβ}Φ_{γδ},  (115)

where

V_{αβγδ} = V_{[αβ](γδ)} V_{[αβγ]δ} = 0 Ω^{αβ}V_{αβγδ} = 0 Φ_{αβ} = Φ_{(αβ)}.

Proposition 4.3. On a (pseudo-)Kähler manifold, if the tensor V_{αβγδ} vanishes, then the metric has constant holomorphic sectional curvature.

Proof. From (115), we find that

Ω^{αβ}R_{αβ}^{ϵ}_{δ}Ω_{ϵγ} = Ω^{αβ} [Ω_{αγ}Φ_{βδ} − Ω_{βγ}Φ_{αδ} + Ω_{αδ}Φ_{βγ} − Ω_{βδ}Φ_{αγ} + 2Ω_{αβ}Φ_{γδ}] = 4(n + 1)Φ_{γδ}

whereas computing according to (112) leads to Ω^{αβ}R_{αβ}^{ϵ}_{δ}Ω_{ϵγ} = 4Ξ_{γδ}. We conclude that Ξ_{αβ} = (n + 1)Φ_{αβ} at which point we may compare (115) with (112) when V_{αβγδ} = 0 to conclude that U_{αβγδ} = 0 and Ξ_{αβ} = 0, as required. □

4.3. Metrisability of almost c-projective manifolds. Suppose (M, J, [∇]) is an almost c-projective manifold. It is natural to ask whether [∇] contains the canonical connection of a (2, 1)-symplectic metric on (M, J).

Definition 4.4. On an almost c-projective manifold (M, J, [∇]) a (2, 1)-symplectic metric g ∈ Γ(S^{2}T^{∗}M) on (M, J) is compatible with the c-projective class [∇] if and only if its canonical connection is contained in [∇]. The almost c-projective structure on M is said to be metrisable or (2, 1)-symplectic or quasi-Kähler (or Kähler or pseudo-Kähler when J is integrable) if it admits a compatible (2, 1)-symplectic metric g (respectively a Kähler or pseudo-Kähler metric g, if J is integrable).

The volume form vol(g) of g is a positive section of ∧^{2n}T^{∗}M, which we view as a c-projective density of weight (−(n + 1), −(n + 1)) under the identification of oriented real line bundles ∧^{2n}T^{∗}M = E_{ℝ}(−(n + 1), −(n + 1)) determined by

ε_{ab···c} ε̄d̄ē_{···}f̄ ∈ Γ(∧^{2n}T^{∗}(n + 1, n + 1)),

where ε_{ab···c} ∈ Γ(∧^{n,0}(n+1, 0)) is the tautological form from Section 2.1. We now write vol(g) = τ_{g}^{−(n+1)} uniquely to determine a positive section τ_{g} of E_{ℝ}(1, 1). The canonical connection ∇ of g is a special connection in the c-projective class, and for all ℓ ∈ ℤ, τ_{g}^{ℓ} = vol(g)^{−ℓ/(n+1)} ∈ Γ(E_{ℝ}(ℓ, ℓ)) is a ∇-parallel trivialisation of E_{ℝ}(ℓ, ℓ).

In the integrable case, the metrisability of a c-projective structure gives easily the following constraints on the harmonic curvature.

Proposition 4.4. Let (M, J, [∇]) be a c-projective manifold of dimension 2n ≥ 4. If [∇] is induced by the Levi-Civita of a (pseudo-)Kähler metric on (M, J), then the harmonic curvature only consists of the (1, 1)-part

W_{ab̄}^{c}_{d} = H_{ab̄}^{c}_{d}

of the (c-projective) Weyl curvature.

Proof. Suppose first that 2n ≥ 6. Then we have to show that W_{ab}^{c}_{d} vanishes. Recall that, by construction, W_{ab}^{c}_{d} is the connection-independent part of the (2, 0)-component of the curvature of any connection in the c-projective class. Hence, if [∇] is induced from the Levi-Civita connection of a (pseudo-)Kähler metric on (M, J), then W_{ab}^{c}_{d} vanishes identically, since the curvature of a (pseudo-)Kähler metric is J-invariant. If 2n = 4, then W_{ab}^{c}_{d} is always identically zero and the (2, 0)-part C_{abc} of the Cotton– York tensor is independent of the choice of connection in the c-projective class. Since the Ricci tensor Ric_{αβ} of a (pseudo-)Kähler metric g is J-invariant (109), we have P_{ab̄} = \frac{1}{n+1}Ric_{ab̄} and Ric_{ab} = P_{ab} = 0. Hence, if 2n = 4 and the c-projective structure is metrisable, then C_{abc} = ∇_{a}P_{bc}−∇_{b}P_{ac} vanishes identically, which proves the claim. □

We now link compatible metrics to solutions of the first BGG operator associated to a real analogue V of the standard complex tractor bundle T . Any almost c-projective manifold (M, J, [∇]) admits a complex vector bundle

V_{ℂ} = T ⊗ \overline{T} .

Although the construction of T and \overline{T} requires the existence and choice of an (n + 1)^{st} root E(1, 0) of ∧^{n}T^{1,0}M, the vector bundle T ⊗ \overline{T} is defined independently of such a choice. Moreover, swapping the two factors defines a real structure on T ⊗ \overline{T} and hence V_{ℂ} is the complexification of a real vector bundle V over M corresponding to that real structure. The filtration (49) of T induces filtrations on V and V_{ℂ} given by

V_{ℂ} = V_{ℂ}^{−1} ⊃ V_{ℂ}^{0} ⊃ V_{ℂ}^{1} ,

where

V_{ℂ}^{−1}/V_{ℂ}^{0} ≅ T^{0,1}M ⊗ T^{1,0}M(−1, −1)
V_{ℂ}^{0}/V_{ℂ}^{1} ≅ (T^{1,0}M ⊕ T^{0,1}M)(−1, −1)
V_{ℂ}^{1} ≅ E(−1, −1).

For any choice of connection ∇ ∈ [∇] we can therefore identify an element of V_{ℂ} with a quadruple

(\begin{matrix}η^{b̄c} \\ X^{b} | Y b̄ \\ ρ \end{matrix}) , where \{ \begin{matrix}η^{b̄c} ∈ T^{0,1}M ⊗ T^{1,0}M(−1, −1), \\ X^{b} ∈ T^{1,0}M(−1, −1), Y b̄ ∈ T^{0,1}M(−1, −1), \\ ρ ∈ E(−1, −1), \end{matrix}

and elements of V can be identified with the real elements of V_{ℂ}:

\overline{η^{c̄b}} = η^{b̄c},  \overline{X^{b}} = Y b̄ and ρ̄ = ρ.  (116)

The formulae (52) and (53) for the tractor connection on T immediately imply that the tractor connection on V_{ℂ} = T ⊗ \overline{T} is given by

∇_{a}^{V_{ℂ}} (\begin{matrix}η^{b̄c} \\ X^{b} | Y b̄ \\ ρ \end{matrix}) = (\begin{matrix}∇_{a}η^{b̄c} + δ_{a}^{c}Y b̄ \\ ∇_{a}X^{b} + ρδ_{a}^{b} − P_{ac̄}η^{c̄b} | ∇_{a}Y b̄ − P_{ac}η^{b̄c} \\ ∇_{a}ρ − P_{ab̄}Y b̄ − P_{ab}X^{b} \end{matrix})  (117)

∇V_{ℂ} ; ā (\begin{matrix}η^{b̄c} \\ X^{b} | Y b̄ \\ ρ \end{matrix}) = (\begin{matrix}∇āη^{b̄c} + δāb̄X^{c} \\ ∇āX^{b} − P_{āc̄}η^{c̄b} | ∇āY b̄ + ρδāb̄ − P_{āc}η^{b̄c} \\ ∇āρ − Pāb̄Y b̄ − P_{āb}X^{b} \end{matrix}) .  (118)

Note that the real structure on V_{ℂ} is parallel for this connection and that, consequently, the tractor connection on V is the restriction of (117) and (118) to real sections (116).

Now consider, for a section η^{b̄c} of T^{0,1}M ⊗ T^{1,0}M(−1, −1), the system of equations

∇_{a}η^{b̄c} + δ_{a}^{c}Y b̄ = 0,  ∇āη^{b̄c} + δāb̄X^{c} = 0  (119)

for some sections X^{c} of T^{1,0}M(−1, −1) and Y b̄ of T^{0,1}M(−1, −1). It follows immediately from the invariance of (59) that the system (119) is c-projectively invariant. In fact, if η^{b̄c} ∈ Γ(T^{0,1}M ⊗ T^{1,0}M(−1, −1)) satisfies (119) for some connection ∇ ∈ [∇], for some X^{c} ∈ Γ(T^{1,0}M(−1, −1)), and for some Y b̄ ∈ Γ(T^{1,0}M(−1, −1)), then η^{b̄c} satisfies (119) for ˆ∇ ∈ [∇] with

X̂^{c} = X^{c} − Υb̄η^{b̄c} and Ŷ b̄ = Y b̄ − Υ_{c}η^{b̄c}.  (120)

Moreover, if (119) is satisfied, one must have Y b̄ = −\frac{1}{n}∇_{a}η^{b̄a} and X^{c} = −\frac{1}{n}∇āη^{āc}. If η^{b̄c} is a real section, then the first equation in (119) is satisfied if and only if the second equation of (119) holds, in which case \overline{X^{b}} = Y b̄. We can reformulate these observations as follows. There is an invariant differential operator

D_{ℂ}^{V}: T^{0,1}M ⊗ T^{1,0}M(−1, −1) → \begin{matrix}(∧^{1,0} ⊗ T^{0,1}M ⊗ T^{1,0}M(−1, −1))_{∘} \\ ⊕ \\ (∧^{0,1} ⊗ T^{0,1}M ⊗ T^{1,0}M(−1, −1))_{∘} \end{matrix}  (121)

given by η^{b̄c} ↦ (∇_{a}η^{b̄c}− \frac{1}{n}δ_{a}^{c}∇_{d}η^{b̄d}, ∇āη^{b̄c}− \frac{1}{n}δāb̄∇d̄η^{d̄c}). Restricting D^{V_{ℂ}} to real sections η^{b̄c} = \overline{η^{c̄b}} gives an invariant differential operator D^{V}. It is the first operator in the BGG sequence corresponding to the tractor bundle V and D^{V_{ℂ}} is its complexification.

Proposition 4.5. Let (M, J, [∇]) be an almost c-projective manifold of dimension 2n ≥ 4. Then, when n is even, the map sending a Hermitian metric g_{bc̄} to the real section η^{āb} = g^{āb}τ_{g}^{−1} of T^{0,1}M ⊗ T^{1,0}M(−1, −1) restricts to a bijection between compatible (2, 1)-symplectic Hermitian metrics on (M, J, [∇]) and nondegenerate sections in the kernel of D^{V}. The inverse map sends η^{āb} to the Hermitian metric g_{bc̄} with g^{āb} = (det η)η^{āb}, where

det η := \frac{1}{n!} ε̄_{āc̄···ē} ε_{bd···f} η^{āb}η^{c̄d} ⋯ η^{ēf} ∈ Γ(E_{ℝ}(1, 1))  (122)

and ε_{ab···c} denotes the tautological section of ∧^{n,0}(n+1, 0). When n is odd, the mapping η^{āb} ↦ g^{āb} := (det η)η^{āb} is 2–1 and, conversely, the mapping g^{āb} ↦ η^{āb} := τ_{g}^{−1}g^{āb} picks a preferred sign for η^{āb} but, otherwise, the same conclusions hold.

Proof. Assume first that g_{bc̄} is a compatible (2, 1)-symplectic Hermitian metric, i.e. its canonical connection ∇ is contained in [∇]. Then η^{āb} = g^{āb}τ_{g}^{−1} is a real section of T^{0,1}M ⊗ T^{1,0}M(−1, −1), which satisfies (119) for ∇ with X^{c} = 0 and Y c̄ = \overline{X^{c}} = 0. Hence, η^{b̄c} is in the kernel of D^{V}, and det η = τ_{g}^{n+1}τ_{g}^{−n} = τ_{g}.

Conversely, suppose that η^{b̄c} ∈ Γ(T^{0,1}M ⊗ T^{1,0}M(−1, −1)) is a real nondegenerate section satisfying (119) for some connection ∇ ∈ [∇] with X^{b} ∈ Γ(T^{1,0}M(−1, −1)) and Y b̄ = Xb̄ ∈ Γ(T^{0,1}M(−1, −1)). Since η^{āb} is nondegenerate, there is a unique 1-form Υ_{b} such that η^{āb}Υ_{b} = Xā. Let us denote by ˆ∇ ∈ [∇] the connection obtained by cprojectively changing ∇ via Υ_{b}. Then we deduce form (120) that ˆ∇_{a}η^{b̄c} = ˆ∇āη^{b̄c} = 0. Since ε_{ab···c} is parallel for any connection in the c-projective class, det η is parallel for ˆ∇. Hence, g^{b̄c} = η^{b̄c} det η is a real nondegenerate section of T^{0,1}M ⊗ T^{1,0}M that is parallel for ˆ∇, i.e. its inverse g_{bc̄} is a (2, 1)-symplectic Hermitian metric whose canonical connection is ˆ∇ ∈ [∇]. □

The real vector bundle V can be realised naturally in two alternative ways as follows. First, let us view T as a real vector bundle T_{ℝ} equipped with a complex structure J^{T} (thus, equivalently, T_{ℝ} ⊗ ℂ ≅ T ⊕ \overline{T} ). Then we can identify V as the J^{T} -invariant elements in S^{2}T_{ℝ}. However, since J^{T} induces an isomorphism between J^{T} -invariant elements in S^{2}T_{ℝ} and such elements in ∧^{2}T_{ℝ}, cf. (88), we may, secondly, realise V as the latter. Realised as the bundle of J^{T} -invariant elements in S^{2}T_{ℝ} we can, for any choice of connection in the c-projective class, identify an element of V with a triple

(\begin{matrix}η^{βγ} \\ X^{β} \\ ρ \end{matrix}) , where \{ \begin{matrix}η^{βγ} ∈ S^{2}T M ⊗ E_{ℝ}(−1, −1) with J_{δ}^{β}J_{ϵ}^{γ}η^{δϵ} = η^{βγ} \\ X^{β} ∈ T M ⊗ E_{ℝ}(−1, −1) \\ ρ ∈ E_{ℝ}(−1, −1). \end{matrix}

In this picture the tractor connection becomes

∇_{α}^{V} (\begin{matrix}η^{βγ} \\ X^{β} \\ ρ \end{matrix}) = (\begin{matrix}∇_{α}η^{βγ} + δ_{α}^{(β}X^{γ)} + J_{α}^{(β}J_{ϵ}^{γ)}X^{ϵ} \\ ∇_{α}X^{β} + ρδ_{α}^{β} − P_{αγ}η^{βγ} \\ ∇_{α}ρ − P_{αβ}X^{β} \end{matrix}) .  (123)

The formulae (117) and (118) may be recovered from (123) by natural projection:

∇_{a}^{V_{ℂ}} = Π_{a}^{α}∇_{α}^{V}, ∇V_{ℂ} ; ā = \overline{Π}α ; ā∇_{α}^{V}, η^{b̄c} = \overline{Π}b̄ ; βΠ_{γ}^{c}η^{βγ}, X^{b} = Π_{β}^{b}X^{β}, Y b̄ = \overline{Π}b̄ ; βX^{β},

so that, for example,

Π_{a}^{α}Π_{b}^{β}(∇_{α}X^{β} + ρδ_{α}^{β} − P_{αγ}η^{βγ}) = ∇_{a}X^{b} + ρδ_{a}^{b} − P_{aγ}η^{γb}
= ∇_{a}X^{b} + ρδ_{a}^{b} − P_{ac̄}η^{c̄b},

as in (117). To pass explicitly to the second (skew) viewpoint on V described above, one can write Φ^{βγ} = J_{α}^{γ}η^{αβ} and Y^{β} = J_{α}^{β}X^{α}. Then, for any choice of connection in the c-projective class, an element of V may alternatively be identified with a triple

(\begin{matrix}Φ^{βγ} \\ Y^{β} \\ ρ \end{matrix}) , where \{ \begin{matrix}Φ^{βγ} ∈ ∧^{2}T M ⊗ E_{ℝ}(−1, −1) with J_{δ}^{β}J_{ϵ}^{γ}Φ^{δϵ} = Φ^{βγ} \\ Y^{β} ∈ T M ⊗ E_{ℝ}(−1, −1) \\ ρ ∈ E_{ℝ}(−1, −1). \end{matrix}

The tractor connection becomes

∇_{α}^{V} (\begin{matrix}Φ^{βγ} \\ Y^{β} \\ ρ \end{matrix}) = (\begin{matrix}∇_{α}Φ^{βγ} + δ_{α}^{[β}Y^{γ]} + J_{α}^{[β}J_{ϵ}^{γ]}Y^{ϵ} \\ ∇_{α}Y^{β} + ρJ_{α}^{β} + P_{αγ}Φ^{βγ} \\ ∇_{α}ρ + P_{αβ}J_{γ}^{β}Y^{γ} \end{matrix}) .  (124)

The formulae (117) and (118) are again projections of (124):

η^{b̄c} = i\overline{Π}b̄ ; βΠ_{γ}^{c}Φ^{βγ} X^{b} = −iΠ_{β}^{b}Y^{β} Y b̄ = i\overline{Π}b̄ ; βY^{β}.

4.4. The metrisability equation and mobility. Let (M, J, [∇]) be an almost cprojective manifold. By Proposition 4.5, solutions to the metrisability problem on M, i.e. compatible (2, 1)-symplectic metrics up to sign, correspond bijectively to nondegenerate solutions η of the equation D^{V}η = 0. We refer to this equation as the metrisability equation on (M, J, [∇]). It may be written explicitly in several ways.

First, viewing V as the real part of V_{ℂ}, η^{b̄c} satisfies, by (119), the conjugate equations:

∇_{a}η^{b̄c} + δ_{a}^{c}Xb̄ = 0 and ∇āη^{b̄c} + δāb̄X^{c} = 0  (125)

for some (and hence any) connection ∇ ∈ [∇] and some section X^{a} of T^{1,0}M ⊗ E_{ℝ}(−1, −1) with conjugate Xā. In the alternative realisation (123) of V, the metrisability equation for J-invariant sections η^{αβ} of S^{2}T M ⊗ E_{ℝ}(−1, −1) is

∇_{α}η^{βγ} + δ_{α}^{(β}X^{γ)} + J_{α}^{(β}J_{ϵ}^{γ)}X^{ϵ} = 0  (126)

for some section X^{α} of T M ⊗ E_{ℝ}(−1, −1). Similarly, using the realisation (124) of V, the metrisability equation for J-invariant sections Φ^{αβ} of ∧^{2}T M ⊗ E_{ℝ}(−1, −1) is

∇_{α}Φ^{βγ} + δ_{α}^{[β}Y^{γ]} + J_{α}^{[β}J_{ϵ}^{γ]}Y^{ϵ} = 0  (127)

for some section Y^{α} of T M ⊗ E_{ℝ}(−1, −1).

Definition 4.5. The (degree of ) mobility of an almost c-projective manifold, is the dimension of the space

𝔪_{c}[∇] := ker D^{V} = \{η^{αβ} | \begin{matrix}J_{γ}^{α}J_{ϵ}^{β}η^{γϵ} = η^{αβ} \\ ∇_{α}η^{βγ} + δ_{α}^{(β}X^{γ)} + J_{α}^{(β}J_{ϵ}^{γ)}X^{ϵ} = 0 for some X^{α} \end{matrix}\}

of solutions to the metrisability equation.

In the sequel, the notion of mobility will only be of interest to us when the metrisability equation has a nondegenerate solution. Then (M, J, [∇]) has mobility ≥ 1, and the mobility is the dimension of the space of compatible (2, 1)-symplectic metrics. For any (2, 1)-symplectic Hermitian metric g on a complex manifold (M, J), the mobility of the c-projective class [∇] of its canonical connection ∇ is ≥ 1, and will be called the mobility of g. If such a metric g has mobility one, i.e. the constant multiples of g are the only metrics compatible with c-projective class [∇], then most natural questions about the geometry of the c-projective manifold (M, J, [∇]) can be reformulated as questions about the Hermitian manifold (M, J, g). For example the c-projective vector fields of (M, J, [∇]) are Killing or homothetic vector fields for g. Hence, roughly speaking, there is essentially no difference between the geometry of the Hermitian manifold (M, J, g) and the geometry of the c-projective manifold (M, J, [∇]).

We will therefore typically assume in the sequel that (M, J, g), or rather, its cprojective class (M, J, [∇]), has mobility ≥ 2, and hence admits compatible metrics g̃ that are not proportional to g; we then say g and g̃ are c-projectively equivalent. Although all metrics in a given c-projective class are on the same footing, it will often be convenient to fix a background metric g, corresponding to a nondegenerate solution η of (125). Then any section η̃ of T^{0,1}M ⊗ T^{1,0}M(−1, −1) may be written

η̃^{āc} = η^{āb}A_{b}^{c}

for uniquely determined A_{b}^{c}—explicitly, we have:

A^{āb} = (det η)η̃^{āb} and A_{a}^{b} = g_{ac̄}A^{c̄b}.

Since η and η̃ are real, A_{b}^{c} is g-Hermitian (i.e. the isomorphism T^{0,1}M → Ω^{1,0} induced by g intertwines the transpose of A_{a}^{b} with its conjugate):

\overline{A_{a}^{b}} = Ab̄ā := g^{b̄d}A_{d}^{c}g_{cā}.

Using the canonical connection ∇ of g, the metrisability equation (125) for η̃ may be rewritten as an equation for A_{a}^{b}, which we call the mobility equation:

∇_{a}A_{b}^{c} = −δ_{a}^{c}Λ_{b}, or (equivalently) ∇c̄A_{a}^{b} = −g_{ac̄}Λ^{b},  (128)

where Λ^{b} = Π_{β}^{b}Λ^{β} with Λ^{β} real, and Λ_{b} = Π_{b}^{β}Λ_{β} = Π_{b}^{β}g_{βα}Λ^{α} = g_{bā}Λā with Λā = Πā ; αΛ^{α}. Taking a trace gives Λ_{c} = ∇_{c}λ and Λc̄ = ∇c̄λ, with λ = −A_{a}^{a} = −Aāā real. The metric g itself corresponds to the solution A_{a}^{b} = δ_{a}^{b} of (128), with Λ_{c} = 0.

Since the background metric g trivialises the bundles E(ℓ, ℓ) by ∇-parallel sections τ_{g}^{ℓ} = (det η)^{ℓ}, we shall often assume these bundles are trivial. We may also raise and lower indices using g to obtain further equivalent forms of the mobility equations:

∇_{a}A^{b̄c} = −δ_{a}^{c}Λb̄ or ∇āA^{b̄c} = −δāb̄Λ^{c},  (129)
∇_{a}A_{bc̄} = −g_{ac̄}Λ_{b} or ∇āA_{bc̄} = −g_{bā}Λc̄.  (130)

Like the metrisability equation, the mobility equation can be rewritten in explicitly real terms. If we let η̃^{αγ} = η^{αβ}A_{β}^{γ} and raise indices using g, then the metrisability equation (126) maybe rewritten as a mobility equation for the unweighted tensor A^{αβ} ∈ Γ(S_{J}^{2}(T M)):

∇_{α}A^{βγ} = −δ_{α}^{(β}Λ^{γ)} − J_{α}^{(β}J_{δ}^{γ)}Λ^{δ}.  (131)

We thus have that

Λ_{α} = ∇_{α}λ where λ = −\frac{1}{2}A_{β}^{β}.  (132)

Tracing back through the identifications, note that

A^{αβ} = (\frac{vol(g̃)}{vol(g)})^{1/(n+1)}g̃^{αβ},  (133)

where g̃^{αβ} = (det η̃)η̃^{αβ} is the inverse metric induced by η̃^{αβ}.

We may, of course, also lower indices to obtain:

∇_{α}A_{βγ} = −g_{α(β}Λ_{γ)} + Ω_{α(β}J_{γ)}^{δ}Λ_{δ}.  (134)

This is the form of the mobility equation used in [41, 94] and [44, Equation (3)] to study c-projectively equivalent Kähler metrics. This is a special case of Proposition 4.5, in which we suppose that there is a (pseudo-)Kähler metric in our c-projective class and we ask about other (pseudo-)Kähler metrics in the same c-projective class.

Finally, we may rewrite (127) as a mobility equation with respect to a background (2, 1)-symplectic metric g with fundamental 2-form Ω and canonical connection ∇. Trivialising E(1, 1) and lowering indices using g, we obtain

∇_{α}Φ_{βγ} + g_{α[β}Y_{γ]} − Ω_{α[β}J_{γ]}^{δ}Y_{δ} = 0

for a 2-form Φ_{αβ}. In the integrable case (i.e. when g is (pseudo-)Kähler) this is the equation for Hamiltonian 2-forms in the terminology of [2]. We extend this terminology to the (2, 1)-symplectic setting and refer to its c-projectively invariant version (127) as the equation for Hamiltonian 2-vectors Φ^{αβ} on an almost c-projective manifold.

Remark 4.3. If g is a Kähler metric, then applying the contracted differential Bianchi identity g^{eb̄}∇_{[e}R_{a]b̄cd̄} = 0 to the Bochner curvature decomposition (110), we deduce that if the Bochner curvature is coclosed, i.e. g^{eb̄}∇_{e}U_{ab̄cd̄} = 0, then A_{cd̄}:= (n + 2)Ξ_{cd̄}+ Λg_{cd̄} satisfies the mobility equation in the form (130). Equivalently, the corresponding Jinvariant 2-form, which is a modification of the Ricci form, is a Hamiltonian 2-form. This was one of the motivations for the introduction of Hamiltonian 2-forms in [2], and is explored further in [5].

Remark 4.4. Many concepts and results in c-projective geometry have analogues in real projective differential geometry. We recall that on a smooth manifold M of dimension m ≥ 2, a (real) projective structure is a class [∇] of projectively equivalent affine connections, cf. (10). It is shown in [43] that the operator

Γ(M, S^{2}T M(−2)) ∋ η^{βγ} ↦ (∇_{α}η^{βγ})_{∘},  (135)

where S^{2}T M(−2) denotes the bundle of contravariant symmetric tensors of projective weight −2 and ∘ denotes the trace-free part, is projectively invariant (it is a first BGG operator) and that, when n is even and otherwise up to sign, nondegenerate solutions are in bijection with compatible (pseudo-)Riemannian metrics, i.e. metrics whose Levi-Civita connection is in the projective class [∇]. We define the mobility of [∇], or of any compatible (pseudo-)Riemannian metric, to be the dimension of this space

𝔪[∇] := {η^{βγ} ∈ Γ(M, S^{2}T M) | ∇_{α}η^{βγ} = δ_{α}^{β}μ^{γ} + δ_{a}^{γ}μ^{β} for some μ^{α}}

of solutions to this projective metrisability or mobility equation, where we reserve the latter term for the case that the projective structure admits a compatible metric.

4.5. Prolongation of the metrisability equation. Suppose (M, J, [∇]) is an almost c-projective manifold and let us prolong the invariant system of differential equations on sections η^{b̄c} of T^{0,1}M ⊗ T^{1,0}M(−1, −1) given by (119). We have already observed that (119) implies that

X^{b} = −\frac{1}{n}∇āη^{āb} and Y b̄ = −\frac{1}{n}∇_{a}η^{b̄a}.  (136)

Moreover, we immediately deduce from (119) that

(∇_{a}∇_{b} − ∇_{b}∇_{a})η^{c̄d} + T_{ab}ē∇ēη^{c̄d} = 2δ_{[a}^{d}∇_{b]}Y c̄ − T_{ab}c̄X^{d}  (137)
(∇ā∇b̄ − ∇b̄∇ā)η^{c̄d} + Tāb̄^{e}∇_{e}η^{c̄d} = 2δ_{[ā}c̄∇_{b̄]}X^{d} − Tāb̄^{d}Y c̄.  (138)

The left hand sides of equations (137) and (138) equal

R_{ab}^{d}_{e}η^{c̄e} + R_{ab}c̄ēη^{ēd} + 2P_{[ab]}η^{c̄d} − \frac{1}{n+1}(∇ēT_{ab}ē)η^{c̄d}
= W_{ab}^{d}_{e}η^{c̄e} + 2δ_{[a}^{d}P_{b]e}η^{c̄e} + (∇ēT_{ab}c̄)η^{ēd} − \frac{1}{n+1}(∇ēT_{ab}ē)η^{c̄d}  (139)
Rāb̄c̄ēη^{ēd} + Rāb̄^{d}_{e}η^{c̄e} + 2P_{[āb̄]}η^{c̄d} − \frac{1}{n+1}(∇_{e}Tāb̄^{e})η^{c̄d}
= Wāb̄c̄ēη^{ēd} + 2δ_{[ā}c̄P_{b̄]ē}η^{ēd} + (∇_{e}Tāb̄^{d})η^{c̄e} − \frac{1}{n+1}(∇_{e}Tāb̄^{e})η^{c̄d},  (140)

where we have used Theorem 2.13 to rewrite the curvature tensors R_{ab}^{c}_{d}, Rāb̄c̄d̄, Rāb̄^{c}_{d}, and R_{ab}c̄d̄. We conclude from (137) and (139), taking a trace with respect to a and d, and from (138) and (140), taking a trace with respect to ā and c̄, that

∇_{b}Y c̄ = P_{be}η^{c̄e} + \frac{1}{n}U_{b}c̄  ∇b̄X^{d} = Pb̄ēη^{ēd} + \frac{1}{n}Vb̄^{d},  (141)

where

U_{b}c̄ := \frac{n}{n−1}T_{ab}c̄X^{a} + \frac{n}{n−1}(∇ēT_{ab}c̄)η^{ēa} − \frac{n}{(n+1)(n−1)}(∇ēT_{ab}ē)η^{c̄a}  (142)
Vb̄^{d} := \frac{n}{n−1}Tāb̄^{d}Y ā + \frac{n}{n−1}(∇_{e}Tāb̄^{d})η^{āe} − \frac{n}{(n+1)(n−1)}(∇_{e}Tāb̄^{e})η^{ād},  (143)

depend linearly on η^{b̄c} and on X^{a} respectively Y ā.

Remark 4.5. Suppose J is integrable. Then the equations (141) imply ∇_{b}Y c̄ = P_{be}η^{c̄e} and ∇b̄X^{d} = Pb̄ēη^{ēd}. Hence, in this case, the equalities between (137) and (139) and between (138) and (140) show that

W_{ab}^{d}_{e}η^{c̄e} ≡ 0 Wāb̄c̄ēη^{ēd} ≡ 0.  (144)

If η^{āb} is a nondegenerate solution of (119), then (144) implies that W_{ab}^{c}_{d} and its conjugate are identically zero, which confirms again Proposition 4.4 for 2n ≥ 6.

Now consider

(∇_{a}∇b̄ − ∇b̄∇_{a})η^{c̄d} = R_{ab̄}^{d}_{e}η^{c̄e} + R_{ab̄}c̄ēη^{ēd} + P_{ab̄}η^{c̄d} − P_{b̄a}η^{c̄d}.  (145)

By Equation (119) and Theorem 2.13 we may rewrite (145) as

−δb̄c̄∇_{a}X^{d} + δ_{a}^{d}∇b̄Y c̄ = W_{ab̄}^{d}_{e}η^{c̄e} + W_{ab̄}c̄ēη^{ēd} + δ_{a}^{d}P_{b̄e}η^{c̄e} − δb̄c̄P_{aē}η^{ēd}.  (146)

Taking the trace in (146) with respect to b̄ and c̄ shows that

∇_{a}X^{d} = P_{aē}η^{ēd} − \frac{1}{n}δ_{a}^{d}(P_{b̄e}η^{b̄e} − ∇b̄Y b̄) − \frac{1}{n}W_{ab̄}^{d}_{e}η^{b̄e}  (147)

and with respect a and d that

∇b̄Y c̄ = P_{b̄e}η^{c̄e} − \frac{1}{n}δb̄c̄(P_{aē}η^{ēa} − ∇_{a}X^{d}) + \frac{1}{n}W_{ab̄}c̄ēη^{ēa}.  (148)

As the contraction of (147) with respect to a and d and the contraction of (148) with respect b̄ and c̄ must lead to the same result, we see that

\frac{1}{n}(P_{b̄e}η^{b̄e} − ∇b̄Y b̄) = \frac{1}{n}(P_{aē}η^{ēa} − ∇_{a}X^{a}),  (149)

which we denote by ρ ∈ Γ(E(−1, −1)). Inserting (136) into (149) therefore implies that

ρ = \frac{1}{n^{2}} (∇ā∇_{b}η^{āb} + nP_{āb}η^{āb}) = \frac{1}{n^{2}} (∇_{b}∇āη^{āb} + nP_{ab̄}η^{b̄a}).  (150)

By Theorem 2.13 we have

(∇_{a}∇b̄ − ∇b̄∇_{a})X^{c} = R_{ab̄}^{c}_{d}X^{d} + P_{ab̄}X^{c} − P_{b̄a}X^{c}  (151)
= W_{ab̄}^{c}_{d}X^{d} + δ_{a}^{c}P_{b̄d}X^{d} + P_{ab̄}X^{c}.

Inserting the second equation of (141) and (147) into the left hand side of (151) one computes that

(∇_{a}∇b̄ − ∇b̄∇_{a})X^{a} = n∇b̄ρ + P_{ab̄}X^{a} − nPb̄ēY ē + C_{ab̄ē}η^{ēa} + Zb̄,  (152)

with

Zb̄ :=\frac{n}{(n+1)(n−1)}(∇_{a}Tēb̄^{a})Y ē + \frac{1}{n−1}Tēb̄^{a}P_{ad}η^{ēd} + \frac{1}{n(n−1)}Tēb̄^{a}U_{a}ē
− \frac{1}{(n+1)(n−1)}(∇_{a}∇_{d}Tēb̄^{d})η^{ēa} + \frac{1}{(n−1)}(∇_{a}∇_{d}Tēb̄^{a})η^{ēd},  (153)

where we have used (119), (142) and that W_{ab̄}^{a}_{d} is zero. Note again that Zb̄ depends linearly on η^{āb}, X^{a} and Y ā. From (151), the expression (152) must be equal to nP_{b̄d}X^{d}+ P_{ab̄}X^{a}, which implies that

∇b̄ρ = P_{b̄a}X^{a} + Pb̄ēY ē − \frac{1}{n}C_{ab̄ē}η^{ēa} − \frac{1}{n}Zb̄.  (154)

Rewriting (∇_{a}∇b̄ − ∇b̄∇_{a})Y c̄ analogously shows immediately that

∇_{a}ρ = P_{ad}X^{d} + P_{aē}Y ē + \frac{1}{n}C_{ab̄d}η^{b̄d} + \frac{1}{n}Q_{a},  (155)

where

Q_{a} :=\frac{n}{(n+1)(n−1)}(∇ēT_{da}ē)X^{d} + \frac{1}{n−1}T_{da}b̄Pb̄ēη^{ēd} + \frac{1}{n(n−1)}T_{da}b̄Vb̄^{d}
− \frac{1}{(n+1)(n−1)}(∇b̄∇ēT_{da}ē)η^{b̄d} + \frac{1}{(n−1)}(∇b̄∇ēT_{da}b̄)η^{ēd},  (156)

depends linearly on η^{āb}, X^{a} and Y ā. In summary, we have proved the following.

Theorem 4.6. Suppose (M, J, [∇]) is an almost c-projective manifold. The canonical projection π: V_{ℂ} := T ⊗ \overline{T} → T^{0,1}M ⊗ T^{1,0}M(−1, −1) induces a bijection between sections of V_{ℂ} that are parallel for the linear connection

∇_{a}^{V_{ℂ}} (\begin{matrix}η^{b̄c} \\ X^{b} | Y b̄ \\ ρ \end{matrix}) + \frac{1}{n} (\begin{matrix}0 \\ W_{ad̄}^{b}_{c}η^{d̄c} | − U_{a}b̄ \\ −C_{ab̄c}η^{b̄c} − Q_{a} \end{matrix})  (157)

and

∇V_{ℂ} ; ā (\begin{matrix}η^{b̄c} \\ X^{b} | Y b̄ \\ ρ \end{matrix}) + \frac{1}{n} (\begin{matrix}0 \\ −Vā^{b} | W_{āc}b̄d̄η^{d̄c} \\ −C_{ācb̄}η^{b̄c} − Z_{a} \end{matrix})  (158)

and elements in the kernel of D^{V_{ℂ}}, where U_{a}b̄, Vā^{b}, Q_{a} and Z_{a} are defined as in (142), (143), (153) and (156). The inverse of this bijection is induced by a differential operator L: T^{0,1}M ⊗ T^{1,0}M(−1, −1) → V_{ℂ}, which for a choice of connection ∇ ∈ [∇] can be written as

L: η^{b̄c} ↦ (\begin{matrix}η^{b̄c} \\ −\frac{1}{n}∇āη^{āb} | −\frac{1}{n}∇_{a}η^{b̄a} \\ \frac{1}{n^{2}} (∇ā∇_{b}η^{āb} + nP_{āb}η^{āb}) \end{matrix}) .

If J is integrable, W_{ab̄}^{c}_{d} = H_{ab̄}^{c}_{d} and W_{āb}c̄d̄ = H_{āb}c̄d̄ (by Theorem 2.13) and U_{a}b̄, Vā^{b}, Q_{a} and Zā are identically zero.

Let now V be the real form of the vector bundle V_{ℂ}, as defined in the previous section. Obviously, the connection in Theorem 4.6 preserves V and therefore Proposition 4.5 and Theorem 4.6 imply that:

Corollary 4.7. Suppose (M, J, [∇]) is an almost c-projective manifold of dimension 2n ≥ 4. Then, up to sign, there exists a bijection between compatible (2, 1)-symplectic Hermitian metrics and sections s of V that satisfy

• π(s) ≡ η^{b̄c} is nondegenerate

• s is parallel for the connection given by (157) and (158). Note, that since s is a real section, it is covariant constant for (157) if and only if it is covariant constant for (158).

Suppose s is a section of V that is parallel for the tractor connection. Then π(s) ≡ η^{b̄c} is still in the kernel of D^{V} and hence Theorem 4.6 implies that s is also parallel for the connection given by (157) and (158), i.e. π(s) ≡ η^{āb} must satisfy W_{ad̄}^{b}_{c}η^{d̄c} = 0, W_{āc}b̄d̄η^{d̄c} = 0, U_{a}b̄ = 0, Vā^{b} = 0, C_{ab̄c}η^{b̄c} + Q_{a} = 0 and C_{ābc̄}η^{c̄b} + Zā = 0. The following proposition gives a geometric interpretation of parallel sections of the tractor connection and hence of so-called normal solutions of the first BGG operator D^{V} in the terminology of [33].

Proposition 4.8. Suppose (M, J, [∇]) is an almost c-projective manifold of dimension 2n ≥ 4. Then, if n is even, there is a bijection between sections s of V such that

• π(s) ≡ η^{b̄c} is nondegenerate

• s is parallel for the tractor connection ∇^{V} on V and compatible (2, 1)-symplectic metrics g satisfying the generalised Einstein condition:

Ric_{ab} = 0 and Ric_{ab̄} = kg_{ab̄} for some constant k ∈ ℝ,  (159)

where Ric is the Ricci tensor of the canonical connection of g. If J is integrable, then (159) simply characterises (pseudo-)Kähler–Einstein metrics. If n is odd, the same conclusions are valid up to sign.

Proof. Suppose s ∈ Γ(V) is parallel for the tractor connection ∇^{V} and that π(s) ≡ η^{b̄c} ∈ ker D^{V} is nondegenerate. Then Proposition 4.5 implies that the inverse of g^{āb} ≡ η^{āb} det η is a compatible (2, 1)-symplectic Hermitian metric. Now let ∇ ∈ [∇] be the canonical connection of g_{ab̄}. With respect to the splitting of V determined by ∇ the section s corresponds to the section

(\begin{matrix}η^{b̄c} \\ X^{b} | Xb̄ \\ ρ \end{matrix}) = (\begin{matrix}η^{b̄c} \\ 0 | 0 \\ \frac{1}{n}P_{ab̄}η^{b̄a} \end{matrix}) .  (160)

From ∇^{V}s = 0 it therefore follows on the one hand that P_{ac}η^{b̄c} = 0, which implies P_{ac} = 0 by the nondegeneracy of η^{b̄c}. Since Ric_{ab} = (n − 1)P_{ab} + 2P_{[ab]}, we see that the first condition of (159) holds for g. On the other hand, we deduce from ∇^{V}s = 0 that P_{ac̄}η^{c̄b} = ρδ_{a}^{b} and ∇_{a}ρ = ∇āρ = 0. Since P_{ac̄} = \frac{1}{n+1}Ric_{ab̄}, we conclude that

Ric_{ac̄}g^{c̄b} = (n + 1)ρ(det η)δ_{a}^{b}.

Hence g_{ab̄} satisfies also the second condition of (159).

Conversely, suppose g_{ab̄} is a compatible (2, 1)-symplectic Hermitian metric satisfying (159). Let us write s ∈ Γ(V) for the corresponding parallel section of the prolongation connection given by (157) and (158). With respect to the splitting of V determined by the canonical connection ∇ of g_{ab̄}, the section s is again given by (160), where η^{āb} det(η) = g^{āb}. By assumption we have

Ric_{ab} = 0,  (161)

which is equivalent to P_{ab} = 0, and also that

P_{ab̄} = \frac{1}{n+1}Ric_{ab̄} = \frac{k}{n+1}g_{ab̄}  (162)

for some constant k. Moreover, (161) yields

0 = Ric_{ab} = −g^{c̄d}∇c̄T_{da}f̄g_{bf̄}  0 = Ric_{[ab]} = \frac{1}{2}∇c̄T_{ab}c̄  (163)

which shows immediately that (with respect to ∇) in (157) and (158) we have

U_{a}b̄ = 0 Vā^{b} = \overline{U_{a}b̄} = 0 Q_{a} = 0 Zā = \overline{Qā} = 0.

Hence, to prove that s is parallel for ∇^{V} it remains to show that W_{ad̄}^{b}_{c}g^{d̄c} and C_{ab̄c}g^{b̄c} (or equivalently their conjugates) are identically zero. From Theorem 2.13 and (162) we obtain

W_{ad̄}^{b}_{c}g^{d̄c} = R_{ad̄}^{b}_{c}g^{d̄c} − δ_{a}^{b}P_{d̄c}g^{d̄c} − P_{d̄a}g^{d̄b} = R_{ad̄}^{b}_{c}g^{d̄c} − kδ_{a}^{b}.  (164)

Therefore, if we lower the b index in (164) with the metric, we obtain

W_{ad̄b̄c}g^{d̄c} = R_{ad̄b̄c}g^{d̄c} − kg_{ab̄}.

Since ∇ preserves g, the tensors R_{ad̄b̄c} = −R_{d̄ab̄c} and −R_{ad̄cb̄} coincide. Hence, R_{ad̄b̄c}g^{d̄c} = R_{d̄acb̄}g^{d̄c} = Ric_{ab̄} = kg_{ab̄}, which shows that (164) vanishes identically. From (161) and (162) it follows immediately that C_{ab̄c} = ∇_{a}P_{b̄c} − ∇b̄P_{ac} vanishes identically, which completes the proof. □

Remark 4.6. As observed in Section 4.3, V_{ℂ} = T ⊗T̄, and sections of V may be viewed as Hermitian forms on T^{∗}. This has an interpretation in terms of the construction of the complex affine cone π_{C}: C → M described in Section 3.2: by Lemma 3.4, a Hermitian form on T^{∗} pulls back to a Hermitian form on T^{∗}C. If this form is nondegenerate, its inverse defines a Hermitian metric on C. Further, if the section of V is parallel with respect to a connection on V induced by a connection on T , then the latter connection induces a metric connection on C.

In particular, if we have a compatible metric satisfying the generalised Einstein condition of Proposition 4.8, then it generically induces a metric on C which is parallel for the connection ∇^{C} induced by the tractor connection on T .

4.6. The c-projective Hessian. Let us consider the dual W of the tractor bundle V of an almost c-projective manifold. Its complexification is given by W_{ℂ} = T^{∗} ⊗ \overline{T}^{∗}, which admits a filtration

W_{ℂ} = W_{ℂ}^{−1} ⊃ W_{ℂ}^{0} ⊃ W_{ℂ}^{1},

such that for any connection ∇ ∈ [∇] we can write an element of W_{ℂ} as

(\begin{matrix}σ \\ μ_{b} | λb̄ \\ ζ_{bc̄} \end{matrix}) , where \{ \begin{matrix}σ ∈ E(1, 1), \\ μ_{b} ∈ ∧^{1,0}M(1, 1), λb̄ ∈ ∧^{0,1}M(1, 1), \\ ζ_{bc̄} ∈ ∧^{1,1}M(1, 1), \end{matrix}

and the tractor connection as

∇_{a}^{W_{ℂ}} (\begin{matrix}σ \\ μ_{b} | λb̄ \\ ζ_{bc̄} \end{matrix}) = (\begin{matrix}∇_{a}σ − μ_{a} \\ ∇_{a}μ_{b} + P_{ab}σ | ∇_{a}λb̄ + P_{ab̄}σ − ζ_{ab̄} \\ ∇_{a}ζ_{bc̄} + P_{ac̄}μ_{b} + P_{ab}λc̄ \end{matrix})  (165)

and

∇W_{ℂ} ; ā (\begin{matrix}σ \\ μ_{b} | λb̄ \\ ζ_{bc̄} \end{matrix}) = (\begin{matrix}∇āσ − λā \\ ∇āμ_{b} + P_{āb}σ − ζ_{bā} | ∇āλb̄ + Pāb̄σ \\ ∇āζ_{bc̄} + P_{āc̄}μ_{b} + P_{āb}λc̄ \end{matrix}) .  (166)

The first BGG operator associated to W_{ℂ} or W is a c-projectively invariant operator of order two, which we call the c-projective Hessian. It can be written as

D^{W}: E(1, 1) →S^{2}∧^{1,0}M(1, 1) ⊕ S^{2}∧^{0,1}M(1, 1)  (167)
D^{W}σ = (∇_{(a}∇_{b)}σ + P_{(ab)}σ , ∇_{(ā}∇_{b̄)}σ + P_{(āb̄)}σ),

or alternatively as

D^{W}σ = ∇_{(α}∇_{β)}σ + P_{(αβ)}σ − J_{(α}^{γ}J_{β)}^{δ}(∇_{γ}∇_{δ}σ + P_{γδ}σ),  (168)

for any connection ∇ ∈ [∇]. The reader might easily verify the c-projective invariance of D^{W} directly using Proposition 2.5, the identities (16), and the formulae for the change of Rho tensor in Corollary 2.12. The following Proposition gives a geometric interpretation of nonvanishing real solutions σ = σ̄ ∈ Γ(E(1, 1)) of the invariant overdetermined system D^{W}σ = 0.

Proposition 4.9. Let (M, J, [∇]) be an almost c-projective manifold and σ ∈ Γ(E(1, 1)) a real nowhere vanishing section. Then D^{W}σ = 0 if and only if the Ricci tensor of the special connection ∇^{σ} ∈ [∇] associated to σ satisfies Ric_{(ab)} = 0. In particular, if J is integrable, then D^{W}σ = 0 if and only if the Ricci tensor of ∇^{σ} satisfies Ric_{ab} = 0, i.e. the Ricci tensor is symmetric and J-invariant.

Proof. Let σ = σ̄ ∈ Γ(E(1, 1)) be nowhere vanishing. Recall that the Ricci tensor of the special connection ∇^{σ} associated to σ satisfies

Ric_{āb} = Ric_{bā} Ric_{[ab]} = \frac{1}{2}∇σ ; c̄T_{ab}c̄.

With respect to ∇^{σ} the equation D^{W_{ℂ}}σ = 0 reduces to

P_{(ab)}σ = 0 P_{(āb̄)}σ = 0,

i.e. to Ric_{(ab)} = \frac{1}{n−1}P_{(ab)} = 0 and Ric_{(āb̄)} = \frac{1}{n−1}P_{(āb̄)} = 0, since σ is nonvanishing. □

It follows immediately that if a c-projective manifold (M, J, [∇]) admits a compatible (pseudo-)Kähler metric g, then τ_{g} = vol(g)^{−1/(n+1)} ∈ Γ(E(1, 1)) satisfies D^{W}τ_{g} = 0. By Proposition 4.5, τ_{g} = det η, where η is the nondegenerate solution of the metrisability equation corresponding to g. This observation continues to hold without the nondegeneracy assumption.

Proposition 4.10. Let (M, J, [∇]) be a c-projective manifold and suppose that η^{b̄c} ∈ Γ(T^{0,1}M ⊗ T^{1,0}M(−1, −1)) is a real section satisfying (125). Then σ ≡ det η ∈ Γ(E(1, 1)) is a real section in the kernel of the c-projective Hessian (which might be identically zero).

Proof. Let U ⊂ M be the open subset (possibly empty), where σ is nowhere vanishing or equivalently where η^{b̄c} is invertible. By Proposition 4.5 the section η^{b̄c}(det η) ∈ Γ(T^{0,1}M ⊗ T^{1,0}M) defines the inverse of a compatible (pseudo-)Kähler metric on U and its Levi-Civita connection on U is ∇^{σ}. Since the Ricci tensor of a (pseudo-)Kähler metric is J-invariant (109), i.e. Ric_{ab} = Ricāb̄ = 0, we deduce from Proposition 4.9 that σ satisfies D^{W}σ = 0 on U whence, by continuity, on \overline{U}. Since σ vanishes identically on the open set M \ \overline{U}, we obtain that D^{W}σ is identically zero on all of M. □

Remark 4.7. For an almost c-projective manifold admitting a compatible (2, 1)- symplectic metric g, the section τ_{g} ∈ Γ(E(1, 1)) is in the kernel of the c-projective Hessian, if the Ricci tensor of the canonical connection ∇ of g satisfies

Ric_{(ab)} = −∇c̄T c̄_{(ab)} = \frac{1}{4}∇c̄N c̄_{(ab)} = 0,

where we use g to raise and lower indices. It is well known that nearly Kähler manifolds can be characterised as (2, 1)-symplectic manifolds such that T_{abc} is totally skew (see e.g. [63]). It then follows straightforwardly from the identities (102)–(103) that the canonical connection of a nearly Kähler manifold preserves its torsion, i.e. ∇T = −\frac{1}{4}∇N = 0 (see [58, 87]), and R_{abc̄d} vanishes identically. Hence, Proposition 4.10 extends to the nearly Kähler setting.

4.7. Prolongation of the c-projective Hessian. The c-projective Hessian will play a crucial role in the sequel. We therefore prolong the associated equation. Suppose σ ∈ Γ(E(1, 1)) is in the kernel of the c-projective Hessian:

∇_{(a}∇_{b)}σ + P_{(ab)}σ = 0 ∇_{(ā}∇_{b̄)}σ + P_{(āb̄)}σ = 0,  (169)

Then we deduce from (46) that (169) is equivalent to

∇_{a}∇_{b}σ + P_{ab}σ = ∇_{[a}∇_{b]}σ + P_{[ab]}σ = \frac{1}{2(n+1)}(∇c̄T_{ab}c̄)σ − \frac{1}{2}T_{ab}c̄∇c̄σ  (170)
∇ā∇b̄σ + Pāb̄σ = ∇_{[ā}∇_{b̄]}σ + P_{[āb̄]}σ = \frac{1}{2(n+1)}(∇_{c}Tāb̄^{c})σ − \frac{1}{2}Tāb̄^{c}∇_{c}σ,  (171)

where we abbreviate the left-hand sides by Φ_{ab} respectively Ψāb̄, which depend linearly on σ and on λā := ∇āσ respectively μ_{a} := ∇_{a}σ. From (45) we moreover deduce that

∇_{a}λb̄ + P_{ab̄}σ = ∇_{a}∇b̄σ + P_{ab̄}σ = ∇b̄∇_{a}σ + P_{b̄a}σ = ∇b̄μ_{a} + P_{b̄a}σ,

which we shall denote by ζ_{ab̄} ∈ ∧^{1,1}M(1, 1). Consequently, we have

∇_{a}∇c̄μ_{b} − ∇c̄∇_{a}μ_{b} =  (172)
∇_{a}ζ_{bc̄} − (∇_{a}P_{c̄b})σ − P_{c̄b}μ_{a} + (∇c̄P_{ab})σ + P_{ab}λc̄ + Σ_{abc̄},

where

Σ_{abc̄} := − \frac{1}{2(n+1)}((∇c̄∇d̄T_{ab}d̄)σ + (∇d̄T_{ab}d̄)λc̄)  (173)
+ \frac{1}{2}((∇c̄T_{ab}d̄)λd̄ − T_{ab}d̄Pc̄d̄ + T_{ab}d̄Ψc̄d̄)

depends linearly on σ, μ_{a} and λā. From Proposition 2.13 and the identity (45) we obtain that the expression (172) must be also equal to

∇_{a}∇c̄μ_{b} − ∇c̄∇_{a}μ_{b} = −W_{ac̄}^{d}_{b}μ_{d} − P_{c̄b}μ_{a} − P_{ac̄}μ_{b},  (174)

which shows that

∇_{a}ζ_{bc̄} = −P_{ab}λc̄ − P_{ac̄}μ_{b} − W_{ac̄}^{d}_{b}μ_{d} + C_{ac̄b}σ − Σ_{abc̄}.  (175)

Similarly, one shows that

∇āζ_{bc̄} = −P_{āb}λc̄ − P_{āc̄}μ_{b} − W_{āb}d̄c̄λd̄ + C_{ābc̄}σ − Ξ_{ābc̄},  (176)

where

Ξ_{ābc̄} := − \frac{1}{2(n+1)}((∇_{b}∇_{d}T_{āc̄}^{d})σ + (∇_{d}T_{āc̄}^{d})μ_{b})  (177)
+ \frac{1}{2}((∇_{b}T_{āc̄}^{d})μ_{d} − T_{āc̄}^{d}P_{bd}σ + T_{āc̄}^{d}Φ_{bd})

depends linearly on σ, μ_{a} and λā. In summary, we have shown the following theorem:

Theorem 4.11. Suppose (M, J, [∇]) is a c-projective manifold. Then the canonical projection π: W_{ℂ} → E(1, 1) induces a bijection between sections of W_{ℂ} that are parallel for the linear connection

∇_{a}^{W_{ℂ}} (\begin{matrix}σ \\ μ_{b} | λb̄ \\ ζ_{bc̄} \end{matrix}) + (\begin{matrix}0 \\ −Φ_{ab} | 0 \\ W_{ac̄}^{d}_{b}μ_{d} − C_{ac̄b}σ + Σ_{abc̄} \end{matrix})  (178)

∇W_{ℂ} ; ā (\begin{matrix}σ \\ μ_{b} | λb̄ \\ ζ_{bc̄} \end{matrix}) + (\begin{matrix}0 \\ 0 | − Ψāb̄ \\ W_{āb}d̄c̄λd̄ − C_{ābc̄}σ + Ξ_{ābc̄} \end{matrix}) .  (179)

and sections σ ∈ Γ(E(1, 1)) in the kernel of the c-projective Hessian, where Φ_{ab}, Ψāb̄, Σ_{abc̄} and Ξ_{ābc̄} are defined as in (170), (171), (173) and (177). The inverse of this bijection is induced by a linear differential operator L, which, for a choice of connection ∇ ∈ [∇], can be written as

L: E(1, 1) → W_{ℂ}
L(σ) = (\begin{matrix}σ \\ ∇_{a}σ | ∇āσ \\ ∇_{a}∇b̄σ + P_{ab̄}σ \end{matrix}) .

The following Proposition characterises normal solutions of D^{W}(σ) = 0, i.e. real sections σ = σ̄ ∈ Γ(E(1, 1)) in the kernel of the c-projective Hessian that in addition satisfy:

Φ_{ab} = 0  Ψāb̄ = 0  (180)
W_{ac̄}^{d}_{b}∇_{d}σ − C_{ac̄b}σ + Σ_{abc̄} = 0  W_{āb}d̄c̄∇d̄σ − C_{ābc̄}σ + Ξ_{ābc̄} = 0,  (181)

where Φ, Ψ, Σ and Ξ depend linearly on σ and ∇σ.

Proposition 4.12. Let (M, J, [∇]) be an almost c-projective manifold and suppose that σ ∈ Γ(E(1, 1)) is a real nowhere vanishing section in the kernel of the c-projective Hessian. Then σ satisfies (180) if and only if the Ricci tensor Ric_{αβ} of the special connection ∇^{σ} ∈ [∇] corresponding to σ satisfies

Ric_{ab} = 0 and ∇_{a}^{σ}Ric_{bc̄} = 0 = ∇σ ; āRic_{bc̄}.

If the Ricci tensor Ric_{bc̄} = Ric_{c̄b} is, in addition, nondegenerate, then it defines a (2, 1)- symplectic Hermitian metric satisfying the generalised Einstein condition (159) with canonical connection ∇^{σ}.

Proof. Let σ = σ̄ ∈ Γ(E(1, 1)) be a real nowhere vanishing section in the kernel of (167). With respect to the special connection ∇^{σ} ∈ [∇] corresponding to σ, the equations (180) reduce to

0 = \frac{1}{2(n+1)}∇σ ; c̄T_{ab}c̄ = \frac{1}{n+1}Ric_{[ab]} = P_{[ab]}
0 = \frac{1}{2(n+1)}∇_{c}^{σ}Tāb̄^{c} = \frac{1}{n+1}Ric_{[āb̄]} = P_{[āb̄]},

which, since σ is in the kernel of the c-projective Hessian, is equivalent to Ric_{ab} = 0 = Ricāb̄. If these equations are satisfied, is follows immediately that also Σ_{abc̄} and Ξ_{ābc̄} are identically zero (with respect to ∇^{σ}) and that the equations (181) reduces to

C_{ac̄b}σ = (∇_{a}^{σ}P_{c̄b})σ = (∇_{a}^{σ}P_{bc̄})σ = \frac{1}{n+1}(∇_{a}^{σ}Ric_{bc̄})σ = 0
C_{ābc̄}σ = (∇σ ; āP_{bc̄})σ = \frac{1}{n+1}(∇σ ; āRic_{bc̄})σ = 0

which proves the claim, since σ is nowhere vanishing. □

5. Metrisability, conserved quantities and integrability

In this section we investigate the implications of mobility ≥ 2 for the geodesic flow of a (pseudo-)Kähler manifold (M, J, g): we show that any metric g̃ c-projectively equivalent, but not homothetic, to g gives rise to families of commuting linear and quadratic integrals for the geodesic flow of g, and characterise when this implies integrability of the flow.

5.1. Conserved quantities for the geodesic flow. For any smooth manifold M, the total space of its cotangent bundle p: T^{∗}M → M has a canonical exact symplectic structure dΘ, where Θ: T T^{∗}M → ℝ is the tautological 1-form defined by Θ_{α}(X) = α(T p(X)). The Poisson bracket of smooth functions on T^{∗}M preserves the subalgebra

C_{pol}^{∞}(T^{∗}M, ℝ) ≅ ⨁_{k≥0} C^{∞}(M, S^{k}T M)

of functions which are polynomial on the fibres of p, where a symmetric tensor Q of valence (k, 0), i.e. a section of S^{k}T M, is identified with the function α ↦ Q(α, …, α) on T^{∗}M (which is homogeneous of degree k on each fibre of p). The induced bracket

{·, ·}: C^{∞}(M, S^{j}T M) × C^{∞}(M, S^{k}T M) → C^{∞}(S^{j+k−1}T M)

on symmetric multivectors is sometimes called the (symmetric) Schouten–Nijenhuis bracket. It may be computed using any torsion-free connection ∇ on T M as

{Q, R}^{α···ϵ} = j Q^{ζ(α···β}∇_{ζ}R^{γδ···ϵ)} − k R^{ζ(δ···ϵ}∇_{ζ}Q^{α···βγ)}.  (182)

When j = 1 and Q is a vector field, {Q, R} is just the Lie derivative L_{Q}R.

Now suppose g is a (pseudo-)Riemannian metric on M. Then the inverse metric g^{αβ} induces a function on T^{∗}M which is quadratic on each fibre. The flow of the corresponding Hamiltonian vector field on T^{∗}M is the image of the geodesic flow on T M under the vector bundle isomorphism T M → T^{∗}M defined by g.

Definition 5.1. A smooth function I: T M → ℝ on a (pseudo-)Riemannian manifold (M, g) is called an integral of the geodesic flow (or an integral) of g, if for any affinely parametrised geodesic γ, the function s ↦ I(γ̇(s)) is constant.

The interpretation of the geodesic flow as a Hamiltonian flow on T^{∗}M allows us to describe integrals as functions on T^{∗}M.

Proposition 5.1. Q: T^{∗}M → ℝ defines an integral I of the geodesic flow of g if and only if it is a conserved quantity for g^{αβ} i.e. has vanishing Poisson bracket with g^{αβ}.

We shall only consider integrals defined by Q ∈ C_{pol}^{∞}(T^{∗}M, ℝ). Without loss of generality, we may assume such an integral is homogeneous, hence given by a symmetric tensor Q^{α···γ} ∈ C^{∞}(M, S^{k}T M). Using the Levi-Civita connection of g to compute the Schouten–Nijenhuis bracket, we obtain

{g, Q}^{βγ···ϵ} = 2 g^{α(β}∇_{α}Q^{γ···ϵ)},

which is obtained from ∇_{(α}Q_{γ···ϵ)} by raising all indices (using g) and multiplying by 2. When k = 1, {g, Q} = 0 if and only if Q^{α} is a Killing vector field. Thus we recover Clairaut’s Theorem, that Killing vector fields define integrals of the geodesic flow. More generally, a symmetric Killing tensor of valence (0, ℓ) on a (pseudo-)Riemannian manifold (M, g) is a tensor H_{αβ···δ} ∈ S^{ℓ}T^{∗}M that satisfies

∇_{(α}H_{βγ···ϵ)} = 0,  (183)

where ℓ ≥ 1 can be any integer and ∇ is the Levi-Civita connection of g.

Corollary 5.2. Q^{α···γ} ∈ C^{∞}(M, S^{k}T M) defines an integral of the geodesic flow of g if and only if Q_{α···γ} is a symmetric Killing tensor of g.

5.2. Holomorphic Killing fields. Let (M, J, g) be a (pseudo-)Kähler manifold with Levi-Civita connection ∇ and Kähler form Ω_{αβ} = J_{α}^{γ}g_{γβ}.

Definition 5.2. A vector field X on (M, J, g) is called a holomorphic Killing field if it preserves the complex structure J and the metric g, i.e. L_{X}J = 0 and L_{X}g = 0.

In terms of the Levi-Civita connection ∇ the defining properties of a holomorphic Killing field can be rewritten as:

∇_{α}X^{β} = −J_{α}^{γ}J_{δ}^{β}∇_{γ}X^{δ} and ∇_{α}X_{β} + ∇_{β}X_{α} = 0.  (184)

It follows immediately from the definition of a holomorphic Killing field X that X also preserves the Kähler form, which means that L_{X}Ω = d(i_{X}Ω) = 0 or equivalently

∇_{α}(Ω_{γβ}X^{γ}) − ∇_{β}(Ω_{γα}X^{γ}) = 0.  (185)

In particular, this equation is satisfied if there exists a smooth function f: M → ℝ such that −i_{X}Ω = df, i.e. Ω_{αγ}X^{γ} = ∇_{α}f, or, using the Poisson structure Ω^{αβ},

X^{β} = Ω^{αβ}∇_{α}f = J_{α}^{β}∇^{α}f,  (186)

in which case X is said to be the symplectic gradient of f.

Proposition 5.3. If X and Y are symplectic gradients of functions f and h, then L_{X}h = 0 if and only if L_{Y} f = 0 if and only if Ω^{αβ}(∇_{α}f)(∇_{β}h) = 0 if and only if Ω_{αβ}X^{α}Y^{β} = 0. These equivalent conditions imply that X and Y commute: [X, Y] = 0.

Proof. i_{X}dh = −i_{X}(i_{Y} Ω) = i_{Y} (i_{X}Ω) = −i_{Y} df and so the equivalences are trivial. Now L_{X}h = 0 implies 0 = L_{X}dh = −L_{X}(i_{Y} Ω) = −i_{[X,Y]}Ω, since L_{X}Ω = 0. Hence [X, Y] = 0, since Ω is nondegenerate. □

In this situation, X and Y have isotropic span with respect to Ω, and they are said to Poisson commute, since f and h have vanishing Poisson bracket.

We now return to holomorphic Killing fields.

Proposition 5.4. Let f: M → ℝ be a smooth function. Then the symplectic gradient X^{β} = Ω^{αβ}∇_{α}f is a holomorphic Killing field if and only if the Hessian ∇^{2}f is Jinvariant, i.e.

∇_{a}∇_{b}f = 0 = ∇ā∇b̄f.  (187)

Proof. Since any two equations of (184) and (185) imply the third, we deduce that a vector field of the form X^{β} = Ω^{αβ}∇_{α}f is a holomorphic Killing field if and only if

∇_{α}J_{β}^{γ}∇_{γ}f + ∇_{β}J_{α}^{γ}∇_{γ}f = 0  (188)

or equivalently

∇_{α}∇_{β}f = J_{α}^{γ}J_{β}^{δ}∇_{γ}∇_{δ}f,  (189)

which is equivalent to (187). □

We call f in this case a Killing potential or a Hamiltonian for the holomorphic Killing field X. Note that a holomorphic Killing field always admits such a potential locally (and on any open subset U with H^{1}(U, ℝ) = 0).

Suppose now that g is a compatible (pseudo-)Kähler metric on a c-projective manifold (M, J, [∇]). Then we may write any real section σ ∈ Γ(E(1, 1)) as σ = hτ_{g} for some function h: M → ℝ, where τ_{g} is the trivialisation of E(1, 1) determined by g.

Proposition 5.5. Let (M, J, [∇]) be a c-projective manifold and h ∈ C^{∞}(M, ℝ).

(1) If τ_{g} is the (real) trivialisation of E(1, 1) corresponding to a compatible metric g, then σ = hτ_{g} is in the kernel of the c-projective Hessian D^{W}σ = 0 if and only if h is a Killing potential with respect to (g, J).

(2) If g and g̃ are compatible metrics whose corresponding trivialisations of E(1, 1) are related by τg̃ = e^{−f}τ_{g}, then h is a Killing potential with respect to (g, J) if and only if e^{f}h is a Killing potential with respect to (g̃, J).

Proof. For the first part, compute D^{W}σ using the Levi-Civita connection ∇^{g}. Since τ_{g} is parallel, and the Ricci tensor of g is J-invariant, D^{W}σ = 0 if and only if the J-invariant part of the Hessian of h is zero, and Proposition 5.4 applies. The second part follows from the first. □

These observations may be generalised to (possible degenerate) solutions η of the metrisability equation. Given any J-invariant section η^{αβ} of S^{2}T M ⊗ E_{ℝ}(−1, −1) and any section σ of E_{ℝ}(1, 1), we define vector fields Λ(η, σ) and K(η, σ) by

Λ^{γ}(η, σ) = η^{αγ}∇_{α}σ − \frac{1}{n}σ∇_{α}η^{αγ}  (190)
K^{β}(η, σ) = J_{γ}^{β}Λ^{γ}(η, σ) = Φ^{αβ}∇_{α}σ − \frac{1}{n}σ∇_{α}Φ^{αβ},  (191)

where Φ^{αβ} = J_{γ}^{β}η^{αγ}.

Proposition 5.6. Λ(η, σ) and K(η, σ) are c-projectively invariant, and if η is a nondegenerate solution of the metrisability equation corresponding to a metric g and σ = h det η is in the kernel of the c-projective Hessian, then Λ(η, σ) is holomorphic, and K(η, σ) is the holomorphic Killing field of g with Killing potential h.

Proof. For a c-projectively equivalent connection ˆ∇ ∈ [∇], we have

η^{αγ} ˆ∇_{α}σ − \frac{1}{n}σ ˆ∇_{α}η^{αγ} = η^{αγ}∇_{α}σ + η^{αγ}Υ_{α}σ − \frac{1}{n}σ∇_{α}η^{αγ} − η^{αγ}Υ_{α}σ

and the Υ terms cancel, showing that Λ(η, σ)—and hence also K(η, σ)—is independent of the choice of ∇ ∈ [∇].

Now if η is nondegenerate, corresponding to a compatible metric g with τ_{g} = det η, we use ∇^{g} to compute

K^{β}(η, σ) = Φ^{αβ}∇_{α}^{g}(hτ_{g}) = Ω^{αβ}∇_{α}h,

which is the holomorphic Killing field associated to h. □

Remark 5.1. Suppose that (M, J, [∇]) is an almost c-projective manifold and consider the tensor product

V_{ℂ} ⊗ W_{ℂ} = T^{∗} ⊗ \overline{T}^{∗} ⊗ T ⊗ \overline{T} .

Since T^{∗} ⊗ T = AM ⊕ E(0, 0), there is a natural projection

Π: V_{ℂ} ⊗ W_{ℂ} →\begin{matrix}AM ⊕ E(0, 0) \\ ⨁ \\ \overline{AM} ⊕ E(0, 0) \end{matrix}  (192)

or equivalently a natural projection

Π: V ⊗ W → AM ⊕ E(0, 0).  (193)

Hence, the results in [25] imply that there are two invariant bilinear differential operators

Λ: T^{0,1}M ⊗ T^{1,0}M(−1, −1) × E(1, 1) → T^{1,0}M ⊕ T^{0,1}M  (194)
c: T^{0,1}M ⊗ T^{1,0}M(−1, −1) × E(1, 1) → E(0, 0) ⊕ E(0, 0),  (195)

which are constructed as follows. Consider the two differential operators L: T^{0,1}M ⊗ T^{1,0}M(−1, −1)) → V_{ℂ} and L: E(1, 1)) → W_{ℂ} from Theorem 4.6 respectively 4.11. Recall that in terms of a connection ∇ ∈ [∇] they can be written as

L(η^{b̄c}) = (\begin{matrix}η^{b̄c} \\ −\frac{1}{n}∇āη^{āb} | −\frac{1}{n}∇_{a}η^{b̄a} \\ \frac{1}{n}(∇ā∇_{b}η^{āb} + P_{āb}η^{āb}) \end{matrix}) L(σ) = (\begin{matrix}σ \\ ∇_{a}σ | ∇āσ \\ ∇_{a}∇b̄σ + P_{ab̄}σ \end{matrix}) .

Then Λ and c are defined as the projections to T_{ℂ}M = A_{ℂ}M/A_{ℂ}^{0}M respectively to E(0, 0) ⊕ E(0, 0) of Π(L(η^{b̄c}) ⊗ L(σ)).

In particular, for a choice of connection ∇ ∈ [∇], the invariant differential operator Λ is given by

Λ(η, σ) = (η^{b̄c}∇b̄σ − \frac{1}{n}σ∇b̄η^{b̄c} | η^{b̄c}∇_{c}σ − \frac{1}{n}σ∇_{c}η^{b̄c}).  (196)

Note that if η^{b̄c} and σ are real sections, the two components of (196) are conjugate to each other. In this case we may identify η^{b̄c} with a J-invariant section η^{βγ} of S^{2}T M.

5.3. Hermitian symmetric Killing tensors. Suppose (M, J) is an almost complex manifold and k ≥ 1. Then we call a symmetric tensor H_{αβ···ϵ} ∈ Γ(S^{2k}T^{∗}M) Hermitian, if it satisfies

J_{(α}^{β}H_{βγ···ϵ)} = 0.  (197)

Since, by definition, H_{βγ···ϵ} = H_{(βγ···ϵ)}, equation (197) is equivalent to

J_{α}^{β}H_{βγ···ϵ} + J_{γ}^{β}H_{αβ···ϵ} + ⋯ + J_{ϵ}^{β}H_{αγ···β} = 0.  (198)

Viewing a symmetric tensor H of valence (0, 2k) as an element in S^{2k}T^{∗}M ⊗ℂ = S^{2k}∧^{1} via complexification, we can use the projectors from Section 1 to decompose H into components according to the decomposition of S^{2k}∧^{1} into irreducible vector bundles:

S^{2k}∧^{1} = ⨁_{j=0}^{2k} S^{2k−j}∧^{1,0} ⊗ S^{j}∧^{0,1}.  (199)

Since this decomposition is in particular invariant under the action of J, all the components of a tensor H_{αβ···γδ} ∈ S^{2k}∧^{1} that satisfies (198) must independently satisfy (198). If H_{ab···d ēf̄···h̄} is a section of S^{2k−j}∧^{1,0} ⊗ S^{j}∧^{0,1} that satisfies (198), then this equation says that 2(k − j)iH = 0, which implies that H ≡ 0 unless j = k. We conclude that Hermitian symmetric tensors of valence (0, 2k) can be viewed as real sections of the vector bundle

S^{k}∧^{1,0} ⊗ S^{k}∧^{0,1},

which is the complexification of the vector bundle that consists of those elements in S^{2k}T^{∗}M that satisfy (197).

Remark 5.2. Note that, if H is a symmetric tensor of valence (0, 2k + 1) satisfying (197), then the above reasoning immediately implies that H ≡ 0. The same arguments apply, mutatis mutandis, to symmetric tensors Q^{αβ···ϵ} of valence (2k, 0), and to weighted tensors of valence (0, 2k) and (2k, 0).

Suppose now (M, J, g) is a (pseudo-)Kähler manifold and ℓ = 2k is even, then we can restrict equation (183) for symmetric Killing tensors of valence (0, 2k) to Hermitian tensors. If we complexify (183), we obtain the following system of differential equations on tensors H ∈ Γ(S^{k}∧^{1,0} ⊗ S^{k}∧^{0,1}):

∇_{(a}H_{bc···d)ēf̄···h̄} = 0 and ∇_{(ā}H_{|bc···d|ēf̄···ḡ)} = 0,  (200)

where | ⋯ | means that one does not symmetrise over these indices. Real solutions of (200) thereby correspond to Hermitian symmetric Killing tensors of valence (0, 2k) and obviously for real solutions the two equations of (200) are conjugates of each other. The following proposition shows that (suitably interpreted) the Killing equation for Hermitian symmetric tensors of valence (0, 2k) is c-projectively invariant.

Proposition 5.7. Suppose (M, J, [∇]) is an almost c-projective manifold of dimension 2n ≥ 4. If H_{ab···d ēf̄···h̄} ∈ Γ(S^{k}∧^{1,0} ⊗ S^{k}∧^{0,1}(2k, 2k)) satisfies

∇_{(a}H_{bc···d)ēf̄···h̄} = 0 ∇_{(ā}H_{|bc···d|ēf̄···h̄)} = 0,  (201)

for some connection in ∇ ∈ [∇], then it does so for any other connection in the cprojective class.

Proof. Suppose H ∈ Γ(S^{k}∧^{1,0} ⊗ S^{k}∧^{0,1}(2k, 2k)) satisfies (201) for some connection ∇ ∈ [∇] and let ˆ∇ ∈ [∇] be another connection in the c-projective class. Then it follows from Proposition 2.5 and Corollary 2.4 that

ˆ∇_{a}H_{b···dē···h̄} = ∇_{a}H_{b···dē···h̄} − kΥ_{a}H_{b···dē···h̄} − Υ_{b}H_{a···dē···h̄} − ⋯ − Υ_{d}H_{b···aē···h̄}
+ 2kΥ_{a}H_{b···dē···h̄}
= ∇_{a}H_{b···dē···h̄} + kΥ_{a}H_{b···dē···h̄} − Υ_{b}H_{a···dē···h̄} − ⋯ − Υ_{d}H_{b···aē···h̄}.

Since ∇_{(a}H_{b···d)ē···h̄} = 0 by assumption and

Υ_{(a}H_{b···d)ē···h̄} = \frac{1}{k+1}(Υ_{a}H_{b···dē···h̄} + Υ_{b}H_{a···dē···h̄} + ⋯ + Υ_{d}H_{b···aē···h̄}),

we conclude that the symmetrisation over the unbarred indices on the right hand side is zero, which proves that the first equation of (201) is independent of the connection. Analogous reasoning shows that this is also true for the second equation of (201). □

We refer to solutions of the c-projectively invariant equation (201) as c-projective Hermitian symmetric Killing tensors of valence (0, 2k).

Corollary 5.8. Suppose (M, J, [∇]) is a metrisable c-projective manifold with compatible (pseudo-)Kähler metric g. Then a real section H ∈ Γ(S^{k}∧^{1,0} ⊗ S^{k}∧^{0,1}) is a Hermitian symmetric Killing tensor of g (i.e. a solution of (200) with respect to ∇^{g}) if and only if τ_{g}^{2k}H is a c-projective Hermitian symmetric Killing tensor. In particular, in this case, if g̃ is another compatible (pseudo-)Kähler metric, then e^{2kf}H is a Hermitian symmetric Killing tensor of g̃, where f is given by τg̃ = e^{−f}τ_{g}.

The differential equation (201) gives rise to a c-projectively invariant operator, which is the first BGG operator

\begin{matrix}× & • & • & • & • & ↕ & ↕ & ↕ & ↕ & ↕ & × & • & • & • & • & −2 & k+1 & 0 & 0 & 0 & 0 & k & 0 & 0 & 0 \\ × • • • •^{↕} ^{↕} ^{↕} ^{↕} ^{↕} × • • • •^{0} ^{k} ^{0} ^{0} ^{0}^{0} ^{k} ^{0} ^{0} ^{0} & \begin{matrix}↗ \\ ↘ \end{matrix} \\ × & • & • & • & • & ↕ & ↕ & ↕ & ↕ & ↕ & × & • & • & • & • & 0 & k & 0 & 0 & 0 & −2 & k+1 & 0 & 0 & 0 \end{matrix}  (202)

corresponding to the tractor bundle W, where W is the Cartan product of k copies of ∧^{2}T^{∗} and k copies of ∧^{2}\overline{T}^{∗}. As for the BGG operators discussed in previous sections, this implies (see [20, 52, 88]), that there is a linear connection on W whose parallel sections are in bijection to solution of (201). Hence, the dimension of the solution space is bounded by the rank of W.

Proposition 5.9. Suppose (M, J, g) is a (pseudo-)Kähler manifold of dimension 2n ≥ 4 and let k ≥ 1 be an integer. Then the space of Hermitian symmetric Killing tensors of valence (0, 2k) of (M, J, g) has dimension at most

(\frac{(k + 1)(k + 2)^{2} ⋯ (k + (n − 1))^{2}(k + n)}{(n − 1)!n!})^{2} .  (203)

In the sequel, we shall be interested in the case k = 1, where any compatible metric g defines a c-projective Hermitian symmetric Killing tensor H_{bā} = τ_{g}^{2}g_{bā} by Corollary 5.8. This has the following c-projectively invariant formulation.

Proposition 5.10. Let (M, J, [∇]) be an almost c-projective manifold and let η^{āb} be a real section of T^{0,1}M ⊗ T^{1,0}M(−1, −1). Then

H_{bā} := \frac{1}{(n−1)!} ε̄_{āc̄···ē} ε_{bd···f} η^{c̄d} ⋯ η^{ēf}  (204)

is a real section of ∧^{1,0} ⊗ ∧^{0,1}(2, 2) with H_{bā}η^{āc} = σδ_{b}^{c}, where σ = det η. If η^{āb} satisfies (119) for some X^{d}, Y c̄ (depending on ∇) then H_{bā} is a c-projective Hermitian symmetric Killing tensor and

η^{c̄d}∇_{d}H_{bā} = Y ēH_{bē}δāc̄ − Y c̄H_{bā},  η^{c̄d}∇c̄H_{bā} = X^{e}H_{eā}δ_{b}^{d} − X^{d}H_{bā}.  (205)

Proof. The first statement is straightforward. For the rest, suppose first that η^{āb} is nondegenerate, hence parallel with respect to some connection ˆ∇ in [∇], related to ∇ by Υ with Υ_{b} = H_{bā}Y ā and Υā = H_{bā}X^{b}. Then H_{bā} is parallel with respect to ˆ∇, hence a Hermitian symmetric Killing tensor, and equation (205) follows by rewriting this condition in terms of ∇. At each point, these are statements about the 1-jet of H, which depends polynomially on the 1-jet of η. They hold when the 0-jet of η is invertible (at a given point, hence in a neighbourhood of that point), hence in general by continuity. □

5.4. Metrisability pencils, Killing fields and Killing tensors. Suppose we have two (real) linearly independent solutions η^{āb} and η̃^{āb} of the metrisability equation (125). Since the metrisability equation is linear, the one parameter family

η̃^{āb}(t) := η̃^{āb} − tη^{āb}  (206)

also satisfies (125), and we refer to such a family as a pencil of solutions of the metrisability equation, or metrisability pencil for short.

By Proposition 4.10, the determinant

σ̃(t) := det η̃(t)  (207)

of the pencil (206) lies in the kernel of the c-projective Hessian for all t ∈ ℝ (as does σ := det η). If η̃(t) is degenerate for all t, then σ̃(t) is identically zero. Otherwise, we may assume, at least locally:

Condition 5.1. η is nondegenerate, i.e. σ = det η is nonvanishing, and hence g^{αβ} = (det η)η^{αβ} is inverse to a compatible metric g.

Assuming Condition 5.1, we may write η̃^{āc} = η^{āb}A_{b}^{c} as in Section 4.4, where the (g, J)-Hermitian metric A satisfies (128). Setting A_{a}^{b}(t) := A_{a}^{b} − tδ_{a}^{b}, we have

η̃^{āc}(t) = η^{āb}A_{b}^{c}(t) and σ̃(t) = (det η)(det A(t)).

Thus σ̃(t) is essentially the characteristic polynomial det A(t) of A_{a}^{b}, regarded as a complex linear endomorphism of the complex bundle T^{1,0}M.

Remark 5.3. A pencil is another name for a projective line: if we make a projective change s = (at + b)/(ct + d) of parameter (with ad − bc ≠ 0) then the pencil may be rewritten, up to overall scale, as aη̃ + bη − s(cη̃ + dη) = (η̃ − tη)(ad − bc)/(ct + d). Assuming that cη̃ + dη is nondegenerate, the rescaled and reparameterized pencil is thus (cη̃ + dη)(Ã − s Id), where Ã = (cA + d Id)^{−1}(aA + b Id).

We next set H_{bā} := \frac{1}{(n−1)!} ε̄_{āc̄···ē} ε_{bd···f} η^{c̄d} ⋯ η^{ēf} as in (204) and introduce

H̃_{bā}(t) := \frac{1}{(n−1)!} ε̄_{āc̄···ē} ε_{bd···f} η̃^{c̄d}(t) ⋯ η̃^{ēf}(t) = (adj A(t))_{b}^{c}H_{cā},  (208)

where adj B denotes the endomorphism adjugate to B, with B adj B = (det B)I.

Proposition 5.10 implies that for all t ∈ ℝ, H̃_{bā}(t) is a c-projective Hermitian symmetric Killing tensor of (M, J, [∇]). Hence for any s ∈ ℝ with η̃(s) nondegenerate, σ̃(s)^{−2}H̃_{bā}(t) defines a family of Hermitian symmetric Killing tensors for the corresponding metric.

Similarly, Proposition 5.6 implies that if η̃(s) is nondegenerate (for s ∈ ℝ), then for all t ∈ ℝ, K(η̃(s), σ̃(t)) is a holomorphic Killing field with respect to the corresponding metric (hence an inessential c-projective vector field). Now observe that

K(η̃(s), σ̃(t)) = K(η̃(t) + (t − s)η, σ̃(t)) = (t − s)K(η, σ̃(t)),

since K is bilinear and K(η̃(t), σ̃(t)) = 0. By continuity, the vector fields

K̃(t) := K(η, σ̃(t)),  i.e. K̃^{β}(t) = Ω^{αβ}∇_{α} det A(t),  (209)

which are holomorphic Killing fields with respect to g, preserve η̃(s) for all s, t ∈ ℝ, i.e. L_{K̃(t)}η̃(s) = 0, and hence also L_{K̃(t)}H̃(s) = 0 = L_{K̃(t)}σ̃(s). Thus K̃(t) preserves the Killing potential det A(s) of K̃(s) with respect to g, so Proposition 5.3 implies that K̃(s) and K̃(t) Poisson-commute. We summarise what we have proven as follows.

Theorem 5.11. Let (M, J, [∇]) be a c-projective manifold with metrisability solutions η and η̃ corresponding to compatible (pseudo-)Kähler metric metrics g and g̃ that are not homothetic. Let η̃(t) be the corresponding metrisability pencil (206).

(1) The vector fields K̃(t) : t ∈ ℝ defined by (207)–(209) are Poisson-commuting holomorphic Killing fields with respect to g and g̃.

(2) The tensors H̃(t) : t ∈ ℝ defined by (207)–(208) are c-projective Hermitian symmetric Killing tensors, invariant with respect to K̃(s) for any s ∈ ℝ. In particular, by Corollary 5.8, they induce Hermitian symmetric Killing tensors of g respectively g̃ (by tensoring with τ_{g}^{−2} respectively τ −2 ; g̃).

We call the vector fields K̃(t) and tensor densities H̃(t) the canonical Killing fields and canonical Killing tensors (respectively) for the pair (g, g̃); the former are Killing vector fields with respect to any nondegenerate metric in the family (206), and the latter give rise to symmetric Killing tensor fields for any such metric by tensoring with the corresponding trivialisation of E(−2, −2).

Since the canonical Killing fields K̃(t) are holomorphic with [K̃(t), K̃(s)] = 0 for all s, t ∈ ℝ, we also have [K̃(t), JK̃(s)] = 0. Since J is integrable, JK̃(t) are also holomorphic vector fields, and [JK̃(t), JK̃(s)] = 0 for all s, t ∈ ℝ.

The fact that for all t ∈ ℝ, K̃(t) is a holomorphic Killing field means equivalently (by linearity) that the coefficients of K̃(t) are holomorphic Killing fields, whose Killing potentials with respect g are the coefficients of the characteristic polynomial det A(t). Up to scale, the nontrivial coefficients of det A(t) can be written

σ̃_{1} := A_{b}^{b}, σ̃_{2} := A_{b}^{[b}A_{c}^{c]}, … σ̃_{n} := A_{b}^{[b}A_{c}^{c} ⋯ A_{d}^{d]},  (210)

which are real-valued because A is Hermitian with respect to g. Raising an index in (128), we have

∇c̄A_{a}^{b} = −g^{c̄b}Λ_{a}, or equivalently, ∇^{c}A_{a}^{b} = −δ_{a}^{c}Λ^{b}.  (211)

Hence, applying the (1, 0)-gradient operator ∇^{a} = Π_{α}^{a}g^{αβ}∇_{β} to the canonical potentials in (210), we obtain (up to sign) holomorphic vector fields

Λ_{(1)}^{a} := Λ^{a}, Λ_{(2)}^{a} := 2Λ^{[a}A_{b}^{b]}, … Λ_{(n)}^{a} := nΛ^{[a}A_{b}^{b}A_{c}^{c} ⋯ A_{d}^{d]}  (212)

whose imaginary parts are (up to scale) the coefficients of K̃(t). In particular, Π_{a}^{α}Λ^{a} = \frac{1}{2}(Λ^{α} −iK^{α}), where the holomorphic Killing field K^{α} = J_{β}^{α}Λ^{β} is the leading coefficient of K̃(t). In general, the coefficients satisfy the recursive relation

Λ_{(k+1)}^{a} = A_{b}^{a}Λ_{(k)}^{b} + σ̃_{1}Λ^{a}.  (213)

Proposition 5.12. Let g and g̃ be compatible metrics on (M, J, [∇]) related by a (real) solution A of (128). Then there is an integer ℓ, with 0 ≤ ℓ ≤ n, such that Λ_{(1)}^{a}, …, Λ_{(ℓ)}^{a} are linearly independent on a dense open subset of M, and dim span K̃(t) ≤ ℓ on M.

Proof. Suppose for some p ∈ M and 1 ≤ k ≤ n, Λ_{(k)}^{a} is a linear combination of Λ_{(1)}^{a}, …, Λ_{(k−1)}^{a} at p. Then A_{b}^{a}Λ_{(k)}^{b} is a linear combination of A_{b}^{a}Λ_{(1)}^{b}, …, A_{b}^{a}Λ_{(k−1)}^{a}, hence of Λ_{(1)}^{a}, …, Λ_{(k)}^{a} by (213). Applying (213) once more, we see that Λ_{(k+1)}^{a} is a linear combination of Λ_{(1)}^{a}, …, Λ_{(k)}^{a}. Hence at each p ∈ M, dim span{Λ_{(1)}^{a}, …, Λ_{(n)}^{a}} = dim span K̃(t) is the largest integer ℓ such that Λ_{(1)}^{a}, …, Λ_{(ℓ)}^{a} are linearly independent at p. However, for any integer k, Λ_{(1)}^{a}, …, Λ_{(k)}^{a} are linearly dependent if and only if the holomorphic k-vector Λ_{(1)}^{[a}Λ_{(2)}^{b} ⋯ Λ_{(k)}^{e]} is zero. Hence the set where Λ_{(1)}^{a}, …, Λ_{(k)}^{a} are linearly independent is empty (for k > ℓ) or dense (for k ≤ ℓ). The result follows. □

Following [2], the integer ℓ of this Proposition will be called the order of the pencil.

Proposition 5.13. Let g and g̃ be compatible metrics on (M, J, [∇]) related by a (real) solution A of (128). Then the endomorphisms ∇Λ and A commute, i.e.

A_{a}^{c}∇_{c}Λ^{b} − A_{c}^{b}∇_{a}Λ^{c} = 0.  (214)

Proof. We first give a proof using Theorem 5.11, which implies that K^{α} is a holomorphic Killing field with respect to both g and g̃. It follows that L_{K}A = 0. However, ∇_{K}A = 0 (since Λc̄∇c̄A_{a}^{b} = −Λ_{a}Λ^{b} = Λ^{c}∇_{c}A_{a}^{b}) and so [∇K, A] = 0. Equation (214) is obtained by taking (1, 0)-parts.

We now give a more direct proof of (214), starting from the observation that

∇_{a}Λb̄ = −∇_{a}∇b̄A_{c}^{c} = −∇b̄∇_{a}Āc̄c̄ = ∇b̄Λ̄_{a},

(i.e. ∇_{a}Λb̄ is real). Now expand (∇_{a}∇b̄−∇b̄∇_{a})A^{c̄d} by curvature and also by using (129) to obtain

R_{ab̄}c̄ēA^{ēd} + R_{ab̄}^{d}_{e}A^{c̄e} = δ_{a}^{d}∇b̄Λ̄c̄ − δb̄c̄∇_{a}Λ^{d}.  (215)

Transvect with Ab̄c̄ to conclude that

A^{b̄c}R_{ab̄cē}Aēd̄ + Ab̄c̄A^{c̄e}R_{ab̄d̄e} = g_{ad̄}Ab̄c̄∇b̄Λ̄c̄ − Ab̄b̄∇_{a}Λd̄

and hence that

A^{b̄c}R_{ab̄cē}Aēd̄ − A^{c̄b}R_{bd̄ec̄}A_{a}^{e} = 0  (216)

(i.e. A^{b̄c}R_{ab̄cē}Aēd̄ is real). Now transvect (215) with A_{f}^{a}δc̄b̄ to conclude that

−A_{f}^{a}Ric_{aē}Aēd̄ + A_{f}^{a}R_{ac̄d̄e}A^{c̄e} = A_{fd̄}∇c̄Λ̄c̄ − nA_{f}^{a}∇_{a}Λd̄

and hence that

A_{a}^{b}R_{bc̄d̄e}A^{c̄e} − Ab̄d̄R_{b̄caē}A^{ēc} = n(Ab̄d̄∇b̄Λ̄_{a} − A_{a}^{b}∇_{b}Λd̄).

But from (216) the left hand side vanishes whence

A_{a}^{b}∇_{b}Λ^{d} = A^{b̄d}∇b̄Λ̄_{a} = A^{b̄d}∇_{a}Λb̄ = A_{b}^{d}∇_{a}Λ^{b},

as required. □

Note that (214) can be used to provide an alternative proof that the holomorphic vector fields (212) commute. If V^{a} and W^{b} are two such fields, we must show that

0 = [V, W]^{b} = V^{a}∇_{a}W^{b} − W^{a}∇_{a}V^{b}.  (217)

Let us take V^{a} = Λ^{a} and W^{b} = 2Λ^{[b}A_{c}^{c]}. Then (128) yields

∇_{a}W^{b} = (∇_{a}Λ^{b})A_{c}^{c} + Λ^{b}∇_{a}A_{c}^{c} − (∇_{a}Λ^{c})A_{c}^{b} − Λ^{c}∇_{a}A_{c}^{b}
= (∇_{a}Λ^{b})A_{c}^{c} − Λ^{b}Λ̄_{a} − (∇_{a}Λ^{c})A_{c}^{b} + δ_{a}^{b}Λ^{c}Λ̄_{c}

whence V^{a}∇_{a}W^{b} = Λ^{a}(∇_{a}Λ^{b})A_{c}^{c} − Λ^{a}(∇_{a}Λ^{c})A_{c}^{b} and

V^{a}∇_{a}W^{b} − W^{a}∇_{a}V^{b} = Λ^{a}(A_{a}^{c}∇_{c}Λ^{b} − A_{c}^{b}∇_{a}Λ^{c}).

Similar computations show that all the fields in (212) commute.

Remark 5.4. It is interesting to compare Proposition 5.13 with what happens in the real projective setting. The mobility equations (128) are replaced by

∇^{α}A_{β}^{γ} = −δ_{β}^{α}Λ^{γ} − g^{αγ}Λ_{β}

and the development runs in parallel. These equations control the existence of another metric in the projective class other than the assumed background metric and, from the coefficients of the characteristic polynomial of A_{α}^{β}, a solution gives rise to n canonically defined potentials for n canonically defined vector fields. These are counterparts to the fields (212) and, as such, need not be Killing. Nevertheless, they commute and to see this it is necessary to employ the alternative reasoning that we encountered near the end of the proof just given. The key observation, like (214), is that the endomorphisms A_{α}^{β} and ∇_{α}Λ^{β} commute and its proof follows exactly the course just given.

5.5. Conserved quantities on c-projective manifolds. On a c-projective manifold (M, J, [∇]), the construction of an integral from a compatible metric and Killing tensor has a c-projectively invariant formulation: in particular, given a nondegenerate solution η^{αβ} of the metrisability equation, and a c-projective Hermitian symmetric Killing tensor H_{γδ} of valence (0, 2), the Hermitian (2, 0)-tensor η^{αγ}η^{βδ}H_{γδ} defines an integral of the geodesic flow of the metric corresponding to η; if H_{γδ} is associated to η^{αβ} by Proposition 5.10, then this integral is the Hamiltonian associated to the inverse metric g^{αβ} = (det η) η^{αβ}.

Definition 5.3. Let (M, J, [∇]) be a c-projective manifold, and let η̃^{αβ}(t) := η̃^{αβ} −tη^{αβ} be a metrisability pencil satisfying Condition 5.1, so that g^{αβ} = (det η)η^{αβ} is inverse to a (nondegenerate) compatible metric g, and we may write η̃^{āc}(t) = η^{āb}A_{b}^{c}(t). Then the linear and quadratic integrals L_{t}, I_{t}: T M → ℝ of (the geodesic flow of) g are defined by L_{t}(X) := g(K̃(t), X) = g_{αβ}K̃^{α}(t)X^{β} and I_{t}(X) := τ_{g}^{−2}H̃(t)(X, X) = τ_{g}^{−2}H̃_{αβ}(t)X^{α}X^{β}, where K̃(t) and H̃(t) are the holomorphic Killing fields and c-projective Hermitian symmetric Killing tensors associated to g by Theorem 5.11.

Proposition 5.14. The integrals I_{t}, L_{t} of g (for all t ∈ ℝ) mutually commute under the Poisson bracket on T M induced by g.

Proof. Since, by Theorem 5.11, the canonical Killing fields commute and Lie preserve the canonical Killing tensors, it remains only to show that for any s, t ∈ ℝ the quadratic integrals I_{s} and I_{t} commute. The Hermitian symmetric tensors of valence (2, 0) corresponding to I_{t} are Q_{(t)}^{āb} := η^{ād}η^{c̄b}H_{dc̄}^{(t)}, where we write H_{dc̄}^{(t)} instead of H̃_{dc̄}(t). It thus suffices to show that for all s ≠ t ∈ ℝ, Q_{(s)}^{āb} and Q_{(t)}^{āb} have vanishing Schouten–Nijenhuis bracket, which (in barred and unbarred indices) means

(∇_{b}Q_{(t)}^{d(ā}) Q_{(s)}^{c̄)b} − (∇_{b}Q_{(s)}^{d(ā}) Q_{(t)}^{c̄)b} = 0, (∇āQ_{(t)}^{c̄(b}) Q_{(s)}^{d)ā} − (∇āQ_{(s)}^{c̄(b}) Q_{(t)}^{d)ā} = 0.

We prove the first equation (the second is analogous); taking for ∇ the Levi-Civita connection of g (so ∇η = 0), this reduces to:

η^{c̄f} (∇_{f}H_{ā(b}^{(t)}) H_{d)c̄}^{(s)} − η^{c̄f} (∇_{f}H_{ā(b}^{(s)}) H_{d)c̄}^{(t)} = 0.

The key trick is to multiply the left hand side by s − t and observe that (s − t)η^{c̄f} = η̃^{c̄f}(t) − η̃^{c̄f}(s). Now using equation (205) for η̃^{āb}(t) and H_{dc̄}^{(t)}, we obtain

η̃^{c̄f}(t) (∇_{f}H_{ā(b}^{(t)}) H_{d)c̄}^{(s)} = Y ēδāc̄H_{ē(b}^{(t)}H_{d)c̄}^{(s)} − Y c̄H_{ā(b}^{(t)}H_{d)c̄}^{(s)} = Y c̄H_{c̄(b}^{(t)}H_{d)ā}^{(s)} − Y c̄H_{ā(b}^{(t)}H_{d)c̄}^{(s)}

for some Y ā. The same reasoning applies to η̃^{āb}(s) and H_{dc̄}^{(s)} with the same vector field Y ā to obtain the same expression with s and t interchanged. These two expressions sum to zero and hence

(η̃^{c̄f}(t) − η̃^{c̄f}(s)) ((∇_{f}H_{ā(b}^{(t)}) H_{d)c̄}^{(s)} − (∇_{f}H_{ā(b}^{(s)}) H_{d)c̄}^{(t)}) = −σ̃(s)∇_{(d}H_{b)ā}^{(s)} − σ̃(t)∇_{(d}H_{b)ā}^{(t)}

which vanishes because H̃(s) and H̃(t) are c-projective Hermitian symmetric Killing tensors. □

We now discuss the question how many of the functions L_{t} and I_{t} (t ∈ ℝ) are functionally independent on T M, i.e. have linearly independent differentials. Since T M has dimension 4n, and the functions L_{t} and I_{t} mutually commute (i.e. they span an Abelian subalgebra under the Poisson bracket induced by g), at most 2n of these functions can be functionally dependent at each point of T M. If equality holds on the fibres of T M over a dense open subset of M, the geodesic flow of g is said to be integrable.

Since A_{a}^{b}(t) = A_{a}^{b} − tδ_{a}^{b}, integrability turns out to be related to the spectral theory of the field A_{a}^{b} of endomorphisms of T^{1,0}M. In particular, using the trivialisation of E(1, 1) determined by g, the determinant σ̃(t) := det η̃(t) becomes the characteristic polynomial χ_{A}(t) := det A(t) of A_{a}^{b}. Since A is Hermitian, the coefficients of χ_{A}(t) are smooth real-valued functions on M. Any complex-valued function μ on an open subset U ⊆ M has an associated algebraic multiplicity m_{μ}: U → ℕ, where m_{μ}(p) is the multiplicity of μ(p) as a root of χ_{A}(t) at p ∈ U, or equivalently the rank of the generalised μ(p)-eigenspace of A_{a}^{b} in T_{p}^{1,0}M; additionally, its geometric multiplicity d_{μ}(p) is the dimension of the μ(p)-eigenspace of A_{a}^{b} in T_{p}^{1,0}M, and its index h_{μ}(p) is the multiplicity of μ(p) in the minimal polynomial of A_{a}^{b} at p.

Remark 5.5. If μ: U → ℂ is smooth with m_{μ} constant on U, then the restriction of A_{a}^{b} − μδ_{a}^{b} to the generalised μ-eigendistribution, defines, using an arbitrary local frame of this distribution, a family of nilpotent m_{μ} × m_{μ} matrices N. There are only finitely many conjugacy classes of such matrices, parametrised by partitions of m_{μ}: we can either use the Segre characteristics, which are the sizes of the Jordan blocks of N, or the dual partition by the Weyr characteristics dim ker N^{k} − dim ker N^{k−1}, k ∈ ℤ^{+}. The index h_{μ} is the first Segre characteristic (i.e. the size of the largest Jordan block), while the geometric multiplicity is the first Weyr characteristic dim ker N (so max{d_{μ}, h_{μ}} ≤ m_{μ} with equality if and only if d_{μ} = 1 or h_{μ} = 1). The unique (hence dense) open orbit consists of nilpotent matrices of degree m_{μ}, whose Jordan normal forms have a single Jordan block, and in general, the orbit closures stratify the nilpotent matrices (with the partial ordering of strata corresponding to the dominance ordering of partitions). Thus N maps a sufficiently small neighbourhood of a point p ∈ U into a unique minimal stratum, and for generic p ∈ U, this stratum is the orbit closure of N(p). In other words, the type of N may be assumed constant in neighbourhood of any point in a dense open subset of U.

The general theory of families of matrices is considerably simplified here by Proposition 5.12 and the following two lemmas.

Lemma 5.15. Suppose U is an open subset of M and T^{1,0}U = E ⊕ F where E and F are smooth A-invariant subbundles over U such that the restriction of A to E has a single Jordan block with smooth eigenvalue μ: U → ℂ. Then the gradient of μ is a section over U of E ⊕ F^{⊥} ⊆ T^{1,0}U ⊕ T^{0,1}U = T U ⊗ ℂ, where F^{⊥} denotes the subspace of T^{0,1}U orthogonal to F with respect to g.

Proof. In a neighbourhood of any point in U, we may choose a frame Z^{a}(1), …, Z^{a}(m) of E such that A is in Jordan normal form on E. We identify E^{∗} with the annihilator of F in Ω^{1,0} and let Z_{a}(1), …, Z_{a}(m) be the dual frame (with Z_{a}(i)Z^{a}(j) = δ_{ij}). Then the transpose of A is in Jordan normal form with respect to this dual frame in reverse order: for k = 1, … m we thus have

(A_{a}^{b} − μ δ_{a}^{b})Z^{a}(k) = Z^{b}(k − 1),  (218)
(A_{a}^{b} − μ δ_{a}^{b})Z_{b}(k) = Z_{b}(k + 1),  (219)

where Z^{b}(−1) = 0 = Z_{b}(m + 1). By (128), the (0, 1)-derivative of (218) yields

(g_{ac̄}Λ^{b} + δ_{a}^{b}∇c̄μ)Z^{a}(k) = (A_{a}^{b} − μ δ_{a}^{b})∇c̄Z^{a}(k) − ∇c̄Z^{b}(k − 1),  (220)

which we may contract with Z_{b}(k), using (219), to obtain

−Λ^{b}Z_{b}(k)g_{ac̄}Z^{a}(k) + ∇c̄μ = Z_{b}(k + 1)∇c̄Z^{b}(k) − Z_{b}(k)∇c̄Z^{b}(k − 1).

Summing from k = 1 to m, the right hand side sums to zero, and hence

m∇^{a}μ = ∑_{k=1}^{m} Λ^{b}Z_{b}(k)Z^{a}(k)

so the (1, 0)-gradient of μ is a linear combination of Z^{a}(1), …, Z^{a}(m), hence a section of E. Since A is Hermitian, its restriction to \overline{F}^{⊥} also has a single Jordan block, with eigenvalue μ̄, and so ∇āμ = \overline{∇^{a}μ̄} belongs to F^{⊥} by the same argument. □

It follows that if there is more than one Jordan block with eigenvalue μ, then μ is constant—equivalently, all nonconstant eigenvalues of A have geometric multiplicity one. In fact, a stronger result holds.

Lemma 5.16. Let μ be a smooth function on M and let U ⊂ M be a nonempty open subset on which μ has constant algebraic multiplicity m. If μ is constant and M is connected, then μ has algebraic multiplicity m_{μ} ≥ m on M. Conversely, if m ≥ 2 then μ is locally constant on U.

Proof. Since A is Hermitian, χ_{A}(t) has real coefficients, μ̄ is an eigenvalue of A with the same algebraic multiplicity as μ, and ∇āμ = \overline{∇^{a}μ̄}. By assumption χ_{A}(t) = (t−μ)^{m}q(t) on U, where q is smooth with q(μ) nonvanishing. Since χ_{A}(t) is a Killing potential for g, its gradient ∇^{a}χ_{A}(t) is a holomorphic vector field on M for all t.

Suppose first that μ is constant: we shall show by induction on k that if m ≥ k then m_{μ} ≥ k on M, which is trivially true for k = 0. So suppose that m ≥ k+1 and m_{μ} ≥ k, so that p(t) = χ_{A}(t)/(t − μ)^{k} is a polynomial in t. Since ∇^{a}p(μ) is holomorphic on M and vanishing on U, it vanishes on M, since M is connected. Similarly for \overline{p(μ)}, so that p(μ) is locally constant on M, hence zero, since M is connected and p(μ) vanishes on U. Thus m_{μ} ≥ k + 1 as required.

For the second part, the (m − 2)nd derivative in t of χ^{a}(t) is also a Killing potential, which may be written

∇^{a}χ_{A}^{(m−2)}(t) = m!(t − μ)q(t)∇^{a}μ + (t − μ)^{2}X^{a}(t)

for some polynomial of vector fields X^{a}(t). Applying ∇^{b} = g^{bc̄}∇c̄ and evaluating at t = μ yields ∇^{a}μ∇^{b}μ = 0, i.e. ∇^{a}μ = 0. Replacing μ by μ̄, we deduce that μ is locally constant on U. □

In contrast, in the analogous real projective theory of geodesically equivalent pseudo-Riemannian metrics, Jordan blocks with nonconstant eigenvalues can occur: see [15].

In order to apply the above lemmas at a point p ∈ M, we need p to be stable for A in the following sense. First, we need to suppose that the number of distinct eigenvalues of A_{a}^{b} is constant on some neighbourhood of p. This condition on p is clearly open, and it is also dense: if the number of distinct eigenvalues is not constant near p, then there are points arbitrarily close to p where the number of distinct eigenvalues is larger; repeating this argument, there are points arbitrarily close to p where the number of distinct eigenvalues is locally maximal, hence locally constant. Now, on the dense open set where this condition holds, the eigenvalues of A_{a}^{b} are smoothly defined, and their algebraic multiplicities are locally constant (since they are all upper semicontinuous). Now a point p in this dense open set is stable if in addition the Jordan type (Segre or Weyr characteristics) of each generalised eigenspace of A_{a}^{b} is constant on a neighbourhood of p. The stable points are open and dense by Remark 5.5.

Definition 5.4. We say p ∈ M is a regular point for the pencil η̃(t) = η^{āb}(A_{a}^{b} − tδ_{a}^{b}) if it is stable for A_{a}^{b}, and for each smooth eigenvalue μ on an open neighbourhood of p, either dμ_{p} ≠ 0 or μ is constant on an open neighbourhood of p.

Equivalently, the regular points are the open subset of the stable points where the rank of the span of the canonical Killing fields associated to the pencil is maximal, i.e. equal to the order ℓ. Consequently, by Proposition 5.12, the regular points form a dense open subset of M.

Corollary 5.17. Let μ be a smooth eigenvalue of A over the set of stable points. Then

A_{a}^{b}∇_{b}μ = μ∇_{a}μ and Aāb̄∇b̄μ = μ∇āμ.  (221)

If μ is constant, its algebraic multiplicity is constant on the set of regular points.

Indeed, where μ has algebraic multiplicity m_{μ} = 1, Lemma 5.15 implies that ∇_{a}μ generates the eigenspace of μ, whereas where m_{μ} ≥ 2, Lemma 5.16 implies that μ is locally constant, and hence equations (221) are trivially satisfied. Furthermore, it implies that the algebraic multiplicities of the constant eigenvalues are upper semicontinuous on M, hence constant on the connected set of regular points.

Theorem 5.18. Let (M, J, [∇]) be a c-projective manifold that admits (pseudo-)Kähler metrics g_{ab̄} and g̃_{ab̄} associated to linearly independent solutions η^{āc} and η̃^{āc} = η^{āb}A_{b}^{c} of the metrisability equation (125).

(1) The number of functionally independent linear integrals L_{s} is equal to the number of nonconstant eigenvalues of A at any regular point of M.

(2) The number of functionally independent quadratic integrals I_{t} is equal to the degree of the minimal polynomial of A at any stable point of M.

(3) The integrals I_{t} are functionally independent from the integrals L_{s}.

Proof. Integrals of the form L_{s} or I_{t} are functionally independent near X ∈ T_{p}M if their derivatives are linearly independent at X. Since L_{s} is linear along the fibres of T M → M, the restriction of dL_{s} to T_{X}(T_{p}M) ≅ T_{p}M is g(K̃(s), ·) (at p). Similarly, I_{t} is quadratic along fibres, and the restriction of dI_{t} to T_{X}(T_{p}M) ≅ T_{p}M is τ_{g}^{−2}H̃(t)(X, ·) with τ_{g} = det η. Hence, for generic X ∈ T_{p}M, the quadratic integrals I_{t} are functionally independent from the linear integrals L_{s}, and the number of functionally independent linear, respectively quadratic, integrals is at least the dimension of the span of K̃(s) at p, respectively the dimension of the span of H̃(t) at p.

The geodesic flow preserves the integrals and therefore the property of the integrals to be functionally independent. Since any two points of M can be connected by a piecewise geodesic curve, it suffices to compute the dimensions of these spaces at a regular point of p, where the dimensions of the spans of K̃(s) and H̃(t) are maximal.

At such a point, the number of linearly independent Killing vector fields K̃(s) is the number of nonconstant eigenvalues of A, so it remains to compute the number of linearly independent Killing tensors H̃(t). For this, recall that H̃_{bā}(t) = (adj A(t))_{b}^{c}H_{cā}, with adj A(t) = A(t)^{−1} det A(t). Now write A(t) in Jordan canonical form: on an h × h Jordan block with eigenvalue μ, (t − μ)^{h}A(t)^{−1} is a polynomial of degree h − 1 in t with h linearly independent coefficients. Hence on the generalised μ-eigenspace, (t − μ)^{m_{μ}}A(t)^{−1} is a polynomial in t with h_{μ} linearly independent coefficients, where m_{μ} is the geometric multiplicity of μ, and h_{μ} the index (the multiplicity of μ in the minimal polynomial, i.e. the size of the largest Jordan block). It follows readily that the dimension of the span of adj A(t) is the degree of the minimal polynomial of A. □

5.6. The local complex torus action. For a c-projective manifold M^{2n} admitting a metrisability pencil with no constant eigenvalues, Theorem 5.18 shows that any metric in the pencil is integrable, i.e. its geodesic flow admits 2n functionally independent integrals. Furthermore n of the independent integrals are linear, inducing Hamiltonian Killing vector fields. Hence if M is compact, it is toric (i.e. has an isometric Hamiltonian n-torus action).

When the pencil has constant eigenvalues, there are only ℓ independent linear integrals, where ℓ is the order of the pencil (the number of nonconstant eigenvalues), and at most n independent quadratic integrals. In this case the flows of the Hamiltonian Killing vector fields K̃(t) generate a foliation of M whose generic leaves are ℓ-dimensional. If M is compact, one can prove (see [2]) that these leaves are the orbits of an isometric Hamiltonian action of an ℓ-torus U(1)^{ℓ}, and it is convenient to assume this locally. The complexified action, generated by the commuting holomorphic vector fields K̃(t) and JK̃(t), is then a local holomorphic action of (ℂ^{×})^{ℓ}, and the leaves of the foliation, which are locally J-invariant submanifolds with generic dimension 2ℓ, will be called complex orbits.

Lemma 5.19. The complex orbits through regular points are totally geodesic and their tangent spaces are A-invariant. The c-projectively equivalent metrics g and g̃ restrict to nondegenerate c-projectively equivalent metrics (with respect to the induced complex structure) on any regular complex orbit O^{c}. The metrisability pencil η̃(t) restricts to a metrisability pencil of order ℓ on O^{c}, using g and its restriction to trivialise E_{ℝ}(1, 1), and then g̃ is a constant multiple of the metric induced by η̃.

Proof. Since the complex orbit of any regular point contains only regular points and the tangent space to the orbit is spanned by holomorphic vector fields, it suffices to prove that the (1, 0)-tangent spaces to regular points are closed under the Levi-Civita connection ∇ of g. These tangent spaces are spanned by eigenvectors Z^{a} of A_{a}^{b} with nonconstant eigenvalues μ, and by differentiating the eigenvector equation using (128), as in the proof of Lemma 5.15, we obtain

(A_{a}^{b} − μ δ_{a}^{b})∇_{c}Z^{a} = (−Λ_{a}δ_{c}^{b}+, δ_{a}^{b}∇_{c}μ)Z^{a} = −(Λ_{a}Z^{a})δ_{c}^{b} + (∇_{c}μ)Z^{b}.

Clearly if we contract the right hand side with a (1, 0)-vector X^{c} tangent to a complex orbit, we obtain another such vector. Hence ∇_{X}Z is a (1, 0)-vector tangent to the complex orbit as required: the complex orbits are thus totally geodesic.

The tangent spaces to a regular complex orbit O^{c} are clearly J-invariant and Ainvariant, so that g induces a Kähler metric O^{c}, with a metrisability pencil spanned by the restrictions of η and η̃, where we use g and its restriction to trivialise E_{ℝ}(1, 1). Since η̃ = η ∘A, and the generalised eigenspaces of A which are not tangent to O^{c} have constant eigenvalues, the metric induced by the restriction of η̃ is a constant multiple of the restriction of g̃. □

Also of interest is the local (ℝ^{+})^{ℓ} action whose local orbits are the leaves of the foliation generated by the vector fields JK̃(t), which will be called real orbits.

Lemma 5.20. The real orbits through regular points are totally geodesic, and their tangent spaces are A-invariant and generated by the gradients of the nonconstant eigenvalues of A. The c-projectively equivalent metrics g and g̃ restrict to nondegenerate projectively equivalent metrics on any regular real orbit O, and the restriction of A is a constant multiple of the (1, 1)-tensor (\frac{vol(g̃|_{O})}{vol(g̃|_{O})})^{1/(ℓ+1)}(g̃|_{O})^{−1}g|_{O}.

Proof. At a regular point p, X ∈ T_{p}M is tangent to the real orbit through p if and only if it is tangent to the complex orbit through p and orthogonal to the Killing vector fields K̃(t) at p. Since both properties are preserved along geodesics, the real orbits are totally geodesic with respect to g (hence also g̃).

Let O^{c} be the complex orbit through the regular real orbit O, so that g and g̃ restrict to c-projectively equivalent Kähler metrics on O^{c}. Furthermore (vol(g̃|_{O^{c}}))^{1/2(ℓ+1)}g̃^{−1}|_{O^{c}} is a constant multiple of (vol(g|_{O^{c}}))^{1/2(ℓ+1)}g^{−1} ∘A|_{O^{c}}. The tangent spaces to O are generated by the vector fields ∇^{a}μ, for nonconstant eigenvalues μ, which are mutually orthogonal and non-null. Hence T_{p}O^{c} is the orthogonal direct sum of T_{p}O and JT_{p}O (with respect to both g and g̃). Hence g and g̃ restrict to nondegenerate metrics on O and A restricts to a constant multiple of (\frac{vol(g̃|_{O})}{vol(g̃|_{O})})^{1/(ℓ+1)}(g̃|_{O})^{−1}g|_{O}. The Levi-Civita connections of g and g̃ on O^{c} are related by (11) for some 1-form Υ_{α}. If we now restrict to O (which is totally geodesic in O^{c}), it follows that the induced Levi-Civita connections ∇ and ˜∇ are related by

˜∇_{α}X^{γ} − ∇_{α}X^{γ} = \frac{1}{2}(Υ_{α}δ_{β}^{γ} + δ_{α}^{γ}Υ_{β}),

i.e. the metrics on O are projectively equivalent. □

5.7. Local classification. Let (M, J, [∇]) be a c-projective 2n-manifold admitting two compatible non-homothetic (pseudo-)Kähler metrics, and hence a pencil of solutions of the metrisability equation of order 0 ≤ ℓ ≤ n. Lemma 5.20 shows that the real orbits yield a foliation of the set M^{0} of regular points which is transverse and orthogonal to the common level sets of the nonconstant eigenvalues ξ_{1}, … ξ_{ℓ} of A; these are also the levels of the elementary symmetric functions σ_{1}, … σ_{ℓ} of ξ_{1}, … ξ_{ℓ}, which are Hamiltonians for Killing vector fields generating the local isometric Hamiltonian ℓ-torus action on M^{0}. Indeed, on M^{0}, χ_{A}(t) = χ_{c}(t)χ_{nc}(t), where χ_{nc}(t) = ∏_{i=1}^{ℓ}(t − ξ_{i}) = ∑_{r=0}^{ℓ}(−1)^{r}σ_{r}t^{ℓ−r}, χ_{c}(t) has constant coefficients, and σ_{0} = 1. The leaf space S of the foliation of M^{0} by the complex orbits may then be identified with the Kähler quotient of M^{0} by this local ℓ-torus action.

It is convenient to write χ_{c}(t) = ∏_{u} ρ_{u}(t)^{m_{u}}, where ρ_{u}(t) are the distinct irreducible real factors (with deg ρ_{u} = 1 or 2) and m_{u} their multiplicities. Then if S is a manifold, its universal cover is a product of complex manifolds S_{u} of (real) dimension 2m_{u} deg ρ_{u}.

These observations lead to a local classification of (pseudo-)Kähler metrics which belong to a metrisability pencil (i.e. admit a c-projectively equivalent metric, or equivalently, a Hamiltonian 2-form), which was obtained in [2] in the Kähler case, and in [16] for general (pseudo-)Kähler metrics. We state the result as follows.

Theorem 5.21. Let (M, J, [∇]) be a c-projective 2n-manifold, and suppose that g is a (pseudo-)Kähler metric in a metrisability pencil of order ℓ, which we may write as η̃^{āb}(t) = η^{āb}(A_{a}^{b} − tδ_{a}^{b}), where η^{āb} corresponds to g. Then on any open subset of M^{0} for which the leaf space of the complex orbits is a manifold S, we may write:

g = ∑_{u} g_{u}(χ_{nc}(A_{S})·, ·) + ∑_{i=1}^{ℓ} \frac{Δ_{j}}{Θ_{j}(ξ_{j})}dξ_{j}^{2} + ∑_{j=1}^{ℓ} \frac{Θ_{j}(ξ_{j})}{Δ_{j}}(∑_{r=1}^{ℓ} σ_{r−1}(ξ̂_{j})θ_{r})^{2},  (222)

ω = ∑_{u} ω_{u}(χ_{nc}(A_{S})·, ·) + ∑_{r=1}^{ℓ} dσ_{r} ∧ θ_{r}, with dθ_{r} = ∑_{u}(−1)^{r}ω_{u}(A_{S}^{ℓ−r}·, ·), (223)

Jdξ_{j} = \frac{Θ_{j}(ξ_{j})}{Δ_{j}} ∑_{r=1}^{ℓ} σ_{r−1}(ξ̂_{j}) θ_{r},  Jθ_{r} = (−1)^{r} ∑_{j=1}^{ℓ} \frac{ξ_{j}^{ℓ−r}}{Θ_{j}(ξ_{j})}dξ_{j}.  (224)

The ingredients appearing here are as follows, where we lift objects on S to M by identifying the horizontal distribution ker(dξ_{1}, … dξ_{ℓ}, θ_{1}, … θ_{ℓ}) with the pullback of T S.

• ξ_{1}, … ξ_{ℓ} are the nonconstant roots of A, which are smooth complex-valued functions on M^{0}, functionally independent over ℝ, such that for any j ∈ {1, … ℓ}, \overline{ξ}_{j} = ξ_{k} for some (necessarily unique) k.

• χ_{nc}(t) = ∏_{i=1}^{ℓ}(t − ξ_{i}) = ∑_{r=0}^{ℓ}(−1)^{r}σ_{r}t^{ℓ−r}, σ_{r−1}(ξ̂_{j}) is the (r − 1)st elementary symmetric function of {ξ_{k} : k ≠ j}, and Δ_{j} = ∏_{k≠j}(ξ_{j} − ξ_{k}).

• For j ∈ {1, … ℓ}, Θ_{j} is a smooth nonvanishing complex function on the image of ξ_{j} such that if \overline{ξ}_{j} = ξ_{k} then \overline{Θ}_{j} = Θ_{k}.

• For each distinct irreducible real factor ρ_{u} of χ_{c}, the metric g_{u} is induced by a (pseudo-)Kähler metric on the factor S_{u} of the universal cover of S.

• A_{S} is a parallel Hermitian endomorphism with respect to the local product metric ∑_{u} g_{u} on S, preserving the distributions induced by T S_{u}, on which it has characteristic polynomial ρ_{u}(t)^{m_{u}}.

Any such (pseudo-)Kähler metric admits a metrisability pencil of order ℓ, with

A = A_{S} + ∑_{i=1}^{ℓ} ξ_{i}(dξ_{i} ⊗ \frac{∂}{∂ξ_{i}} + Jdξ_{i} ⊗ J \frac{∂}{∂ξ_{i}} ).

In other words (M, g, J, ω) is locally a bundle over a product S of (pseudo-)Kähler whose fibres (the complex orbits) are totally geodesic toric (pseudo-)Kähler manifolds of a special kind, called “orthotoric”. The proof in [2] proceeds by establishing the orthotoric property of the fibres and the special structure of the base S. In contrast, the proof in [16] relies upon the observation (generalising Lemma 5.20) that the local quotient of (M, g) by the real isometric ℓ-torus action admits a projectively equivalent metric: the first two sums in (222) are the general form of such a metric when the nonconstant eigenvalues of the projective pencil have algebraic multiplicity one.

In the Riemannian case, the expression (222) provides a complete local description of the metric: locally, we may assume S = ∏_{u} S_{u} is product of open subsets S_{u} ⊆ ℝ^{2m_{u}}, and then A_{S} is a constant multiple of the identity on each factor. In the pseudo-Riemannian, it remains only to describe explicitly the parallel Hermitian endomorphism A_{S} on S = ∏_{u} S_{u}, for which we refer to [18].

Remark 5.6. In order to understand the compatible metrics corresponding to the general element η̃− tη of the metrisability pencil, it is convenient to make a projective change s = (at + b)/(ct + d) of parameter, as in Remark 5.3. The metric corresponding to cη̃ + dη (assuming this is nondegenerate) must have the same form (222) as g, with respect to the coordinates ξ̃_{j} = (aξ_{j} + b)/(cξ_{j} + d), and with A replaced by Ã = (cA + d)^{−1}(aA + b). We find in particular that the new functions Θ̃_{j} are related to the old functions by Θ̃_{j}(s)(ct + d)^{ℓ+1} = (ad − bc)^{ℓ+1}Θ_{j}(t)—in other words they transform like polynomials of degree ℓ + 1 (sections of O(ℓ + 1) over the projective parameter line).

Remark 5.7. It is straightforward to show that the restriction of the metric (222) to any complex orbit (a totally geodesic integral submanifold of ∂_{ξ_{j}}, J∂_{ξ_{j}} : j ∈ {1, … ℓ}) has constant holomorphic sectional curvature if and only if each Θ_{j}(t) is a polynomial independent of j, of degree at most ℓ + 1: the curvature computations in [2] extend readily to the (pseudo-)Kähler case. If we write Θ_{j}(t) = Θ(t) := ∑_{r=−1}^{ℓ} a_{r}t^{ℓ−r}, then the complex orbits have constant holomorphic sectional curvature B = \frac{1}{4}a_{−1}.

Following [2], we may introduce holomorphic coordinates u_{r} + it_{r} on the complex orbits by writing θ_{r} = dt_{r} +α_{r} and Jdu_{r} = dt_{r} for r ∈ {1, … ℓ}, where α_{r} are pullbacks of 1-forms on S. Thus

Jdu_{r} = −α_{r} − (−1)^{r} ∑_{j=1}^{ℓ} \frac{ξ_{j}^{ℓ−r}}{Θ_{j}(ξ_{j})}Jdξ_{j}

where dα_{r} = ∑_{u}(−1)^{r}ω_{u}(A_{S}^{ℓ−r}·, ·), and these formulae extend to any r ≤ ℓ. For r ≥ 1, dJdu_{r} = 0, whereas dJdu_{0} = −ω and dJdu_{−1} = ϕ + σ_{1}ω, where ϕ = g(JA·, ·).

In particular, if Θ_{j}(t) = Θ(t), then

∑_{r=−1}^{ℓ}(−1)^{r}a_{r}(Jdu_{r} + α_{r}) = −Jdσ_{1}

and hence

dJdσ_{1} = a_{−1}(ϕ + σ_{1}ω) + a_{0}ω − ∑_{u} ω_{u}(Θ(A)·, ·).

However, σ_{1} differs from trace A = A_{a}^{a} by an additive constant, so dσ_{1} = −Λ and hence dJdσ_{1} = −2∇JΛ, i.e.

2∇Λ = (a_{−1} + a_{0}σ_{1})g + a_{−1}g(A·, ·) − ∑_{u} g_{u}(Θ(A)·, ·).  (225)

6. Metric c-projective structures and nullity

Henceforth, we assume that (M, J) is a complex manifold (i.e. with J integrable) of real dimension 2n ≥ 4, equipped with a metric c-projective structure, i.e. a cprojective structure [∇] containing the Levi-Civita connection of a (pseudo-)Kähler metric g, which we denote by ∇^{g}, or ∇ if g is understood. We may also consider a metric c-projective structure as an equivalence class [g] of (pseudo-)Kähler metrics on (M, J) having the same J-planar curves.

By Proposition 4.5, the map sending a metric g ∈ [g] to η = τ_{g}^{−1}g^{−1} embeds [g] into 𝔪_{c} = 𝔪_{c}[∇] as an open subset of the nondegenerate solutions to the metrisability equation (126). We refer to dim 𝔪_{c} as the mobility of g for any g ∈ [g], cf. Section 4.4, and we are interested in the case that dim 𝔪_{c} ≥ 2. In Section 5, we obtained some consequences of this assumption for the geodesic flow of g on M. We now turn to the relationship between mobility and curvature.

As explained in Section 4.5, 𝔪_{c} may be identified with the space of parallel sections of the real tractor bundle V with respect to the prolongation connection (157)–(158). However, in [44, Theorem 5], it was shown that if dim 𝔪_{c} ≥ 3, then 𝔪_{c} may also be identified with the space of parallel sections of V with respect to the connection

∇_{α} (\begin{matrix}A^{βγ} \\ Λ^{β} \\ ρ \end{matrix}) = (\begin{matrix}∇_{α}A^{βγ} + δ_{α}^{(β}Λ^{γ)} + J_{α}^{(β}J_{ϵ}^{γ)}Λ^{ϵ} \\ ∇_{α}Λ^{β} + ρδ_{α}^{β} − 2Bg_{αγ}A^{βγ} \\ ∇_{α}ρ − 2Bg_{αβ}Λ^{β} \end{matrix})

for some uniquely determined constant B. In this section we explore this phenomenon, and its implications for the curvature of M. First, as a warm-up, we consider the analogous situation in real projective geometry.

6.1. Metric projective geometry and projective nullity. A metric projective structure on a smooth manifold M of dimension n ≥ 2 is a projective structure [∇] containing the Levi-Civita connection of a (pseudo-)Riemannian metric, or (which amounts to the same thing) an equivalence class [g] of (pseudo-)Riemannian metrics with the same geodesic curves. As in the c-projective case (see Section 4.3 and Remark 4.4), up to sign, [g] embeds into the space 𝔪 = 𝔪[∇] of solutions to the projective metrisability equation (135) as the open subset of nondegenerate solutions.

A metric projective structure has mobility dim 𝔪 ≥ 1, and we are interested in the case that dim 𝔪 ≥ 2. However, it is shown in [57] that, on a connected projective manifold (M, [∇]) with mobility dim 𝔪 ≥ 3, there is a constant B such that solutions A^{βγ} of the mobility equations may be identified with parallel sections for the connection

∇_{α} (\begin{matrix}A^{βγ} \\ μ^{β} \\ ρ \end{matrix}) = (\begin{matrix}∇_{α}A^{βγ} + 2δ_{α}^{(β}μ^{γ)} \\ ∇_{α}μ^{β} + ρδ_{α}^{β} − Bg_{αγ}A^{βγ} \\ ∇_{α}ρ − 2Bg_{αβ}μ^{β} \end{matrix})  (226)

on the tractor bundle associated to the metrisability equation. This connection is the main tool used in [45] to determine all possible values of the mobility of an ndimensional simply-connected Lorentzian manifold.

This result is an example of a general phenomenon: in metric projective geometry, solutions to first BGG equations are often in bijection with parallel sections of tractor bundles for a much simpler (albeit somewhat mysterious) connection than the prolongation connection. We illustrate this with a toy example. The operator

Γ(T M(−1)) ∋ η^{β} ↦ (∇_{α}η^{β})_{∘} = ∇_{α}η^{β} − \frac{1}{n}δ_{α}^{β}∇_{γ}η^{γ}

is projectively invariant, where T M(−1) denotes the bundle of vector fields of projective weight −1; its kernel consists of solutions to the concircularity equation

(∇_{α}η^{β})_{∘} = 0,  (227)

called concircular vector fields. This equation is especially congenial in that its prolongation connection coincides with the Cartan connection. Indeed, following [7], for any solution η^{α} of (227) there is a unique function ρ (of projective weight −1) such that ∇_{α}η^{β} = −δ_{α}^{β}ρ, namely ρ = −\frac{1}{n}∇_{γ}η^{γ}. We then have

R_{αβ}^{γ}_{δ}η^{δ} = (∇_{α}∇_{β} − ∇_{β}∇_{α})η^{γ} = δ_{α}^{γ}∇_{β}ρ − δ_{β}^{γ}∇_{α}ρ,  (228)

and tracing over αγ yields Ric_{βδ}η^{δ} = (n − 1)∇_{β}ρ. We conclude that η^{α} lifts uniquely to parallel section of the standard tractor bundle for the connection

∇_{α} (\begin{matrix}η^{β} \\ ρ \end{matrix}) = (\begin{matrix}∇_{α}η^{β} + δ_{α}^{β}ρ \\ ∇_{α}ρ − P_{αβ}η^{β} \end{matrix})  (229)

induced by the (normal) Cartan connection, where P_{αβ} ≡ \frac{1}{n−1}Ric_{αβ}.

The simpler connection arising in the metric projective case is described as follows.

Theorem 6.1. Let (M, [∇]) be a metric projective manifold, and for any p ∈ M, let N_{p} be the dimension of the span at p of the local solutions of (227). Then for any metric g with Levi-Civita connection ∇^{g} ∈ [∇], there is a function B on M, which is uniquely determined and smooth where N_{p} ≥ 1, such that every concircular vector field lifts uniquely to a parallel section of the standard tractor bundle for the connection

∇_{α} (\begin{matrix}η^{β} \\ ρ \end{matrix}) = (\begin{matrix}∇_{α}^{g}η^{β} + δ_{α}^{β}ρ \\ ∇_{α}^{g}ρ − Bg_{αβ}η^{β} \end{matrix}) .  (230)

Moreover B is locally constant on the open set where N_{p} ≥ 2, which is empty or dense in each connected component of M. If M is connected and B is locally constant on a dense open set, it may be assumed constant on M.

Proof. We take ∇ = ∇^{g} and use g to raise and lower indices. Suppose that

∇_{α}η^{β} + δ_{α}^{β}ρ = 0 and ∇_{α}η̃^{β} + δ_{α}^{β}ρ̃ = 0

for solutions η^{α}, η̃^{α} of (227). Then (228) implies that

R_{αβγδ}^{g}η^{δ} = g_{αγ}∇_{β}ρ − g_{βγ}∇_{α}ρ and R_{αβγδ}^{g}η̃^{δ} = g_{αγ}∇_{β}ρ̃− g_{βγ}∇_{α}ρ̃,
and so 2η̃_{[α}∇_{β]}ρ = R_{αβγδ}^{g}η̃^{γ}η^{δ} = −R_{αβγδ}^{g}η^{γ}η̃^{δ} = 2η_{[β}∇_{α]}ρ̃. (231)

In particular, η_{[α}∇_{β]}ρ = 0 and so there is a unique smooth function B on the open set where η^{α} ≠ 0 such that

∇_{α}ρ − Bg_{αβ}η^{β} = 0  (232)

on M for any extension of B over the zero-set of η^{α} (since ∇_{α}ρ also vanishes there). Equation (231) now implies that any two concircular vector fields have the same function B where both functions are determined. Thus B is uniquely determined and smooth where N_{p} ≥ 1. Differentiating (232) on the open set where B is smooth gives

∇_{β}∇_{α}ρ − η_{α}∇_{β}B + Bg_{αβ}ρ = 0,

and so η_{[α}∇_{β]}B = 0. Hence ∇_{α}B = 0 on the open set where N_{p} ≥ 2. This subset is empty or dense in each component of M, since two solutions of (227) that are pointwise linearly dependent on an open set are linearly dependent on that open set.

It remains to show that if M is connected and B is locally constant on a dense open subset U (which could be disconnected), then it may be assumed constant. To see this, we use only that

P_{βγ}η^{γ} = Bη_{β} and ∇_{α}B = 0

on U, for then we may differentiate once more to conclude that

(∇_{α}P_{βγ})η^{γ} + P_{βγ}∇_{α}η^{γ} = B∇_{α}η_{β}

and hence that

(∇_{α}P_{βγ})η^{γ} − P_{αβ}ρ = −Bg_{αβ}ρ

on U. Tracing over αβ yields

(∇_{α}P^{α}_{γ})η^{γ} − P_{α}^{α}ρ = −nBρ

and hence that

(\begin{matrix}P^{α}_{β} & 0 \\ −\frac{1}{n}∇_{α}P^{α}_{β} & \frac{1}{n}P_{α}^{α} \end{matrix}) (\begin{matrix}η^{β} \\ ρ \end{matrix}) = B (\begin{matrix}η^{α} \\ ρ \end{matrix})  (233)

on U. Although this equation was derived on U, it is a valid stipulation everywhere on M. Moreover, the tractor

(\begin{matrix}η^{β} \\ ρ \end{matrix})

is nowhere vanishing on M (else in (229), the vector field η^{β} would vanish identically). From this point of view, we see that B extends as a smooth function on M. Finally, since B is locally constant on U, it is locally constant and hence constant on M. □

The connection (230) of Theorem 6.1 differs from the tractor connection (229) by the endomorphism-valued 1-form

(\begin{matrix}0 & 0 \\ P_{αβ} − Bg_{αβ} & 0 \end{matrix}) : X^{α} ⊗ (\begin{matrix}η^{β} \\ ρ \end{matrix}) ↦ (\begin{matrix}0 \\ (P_{αβ} − Bg_{αβ})X^{α}η^{β} \end{matrix})

The connections agree on the flat model. Specifically, on the unit sphere we have

R_{αβγδ} = g_{αγ}g_{βδ} − g_{βγ}g_{αδ} whence P_{αβ} = g_{αβ},

so that the connections coincide with B = 1.

The proof of Theorem 6.1 may be broken down into two steps. First, one shows that the connection (230) has the required lifting property for some function B, which may only be uniquely determined and smooth on an open set. Secondly, one establishes sufficient regularity to determine the connection globally on M (in this case, with B constant). In the literature, the second step has often been carried out by probing M with geodesics. In the above proof we advocate an alternative line of argument that we believe to be simpler and more generally applicable.

Remark 6.1. For example, we may apply the same technique to the mobility equations (226), where the replacement for (233) has the form

(\begin{matrix}R & 0 & 0 \\ ∇R & R & 0 \\ ∇∇R + R ▷◁ R & ∇R & R \end{matrix}) (\begin{matrix}A^{βγ} \\ μ^{β} \\ ρ \end{matrix}) = B [(\begin{matrix}A^{βγ} \\ μ^{β} \\ ρ \end{matrix}) − \frac{A_{δ}^{δ}}{n} (\begin{matrix}g^{βγ} \\ 0 \\ \frac{1}{n}P_{γ}^{γ} \end{matrix})] .

As above, this is sufficient to show that B is constant if it is locally constant on a dense open set. One striking difference between this case and Theorem 6.1, however, is that the connection (230) actually has the same covariant constant sections as does the standard Cartan or prolongation connection (229). For the mobility equations, however, not only is the resulting connection (226) different from the prolongation connection [43] but also their covariant constant sections are generally different. Nevertheless, all solutions of the mobility equations lift uniquely as covariant constant sections with respect to either of these connections (and this is their crucial property).

We next seek to elucidate the first step in the proof of Theorem 6.1. Here we observe that the key equations (231) used to establish the uniqueness of B may be viewed as a characterisation of B in terms of the curvature R^{g} of g, namely that

R_{αβγδ}^{g}η^{δ} = B(g_{αγ}η_{β} − g_{βγ}η_{α}).

This motivates the introduction of some terminology, following Gray [50].

Definition 6.1. Let (M, g) be a (pseudo-)Riemannian manifold and suppose that the tensor R_{αβγδ} has the symmetries of the Riemannian curvature of g. Then a nullity vector of R at p ∈ M is a tangent vector v^{α} ∈ T_{p}M with R_{αβγδ}v^{δ} = 0, and the nullity space of R at p is the set of such nullity vectors. We say R has nullity at p if the nullity space is nonzero, i.e. the nullity index is positive.

In particular, if R^{g} is the Riemannian curvature of g, then at each p ∈ M, there is at most one scalar B ∈ ℝ such that R_{αβγδ}^{B} := R_{αβγδ}^{g} − B(g_{αγ}g_{βδ} − g_{βγ}g_{αδ}) has nullity at p. Indeed if v^{α} and ṽ^{α} are nullity vectors for R^{B} and RB̃ respectively then

0 = (B − B̃)(g_{αγ}g_{βδ} − g_{βγ}g_{αδ})ṽ^{β}v^{δ} = (B − B̃)(v_{β}ṽ^{β}g_{αγ} − v_{α}ṽ_{γ}),

which implies that B = B̃ unless v^{α} or ṽ^{α} are zero.

Definition 6.2. Let (M, g) be a (pseudo-)Riemannian manifold. Then the (projective) nullity distribution of g is the union of the nullity spaces of R_{αβγδ}^{B} over B ∈ ℝ and p ∈ M. We say that g has (projective) nullity at p ∈ M if there is a nonzero v^{α} ∈ T_{p}M in the nullity distribution of g, i.e.

(R_{αβγδ}^{g} − B(g_{αγ}g_{βδ} − g_{βγ}g_{αδ}))v^{δ} = 0,  (234)

for some B ∈ ℝ, uniquely determined by p.

The definition of B is reminiscent of an eigenvalue; indeed, the αγ trace of (234) is

P_{α}^{β}v^{α} = Bv^{β},

so B is an eigenvalue of the endomorphism P_{α}^{β}. On the other hand the trace-free part of (234) provides a projectively invariant characterisation, using the projective Weyl tensor P_{αβ}^{γ}_{δ} := R_{αβ}^{γ}_{δ} − δ_{α}^{γ}P_{βδ} + δ_{β}^{γ}P_{αδ}, as follows (cf. [49]).

Proposition 6.2. Let (M, g) be a (pseudo-)Riemannian manifold of dimension n ≥ 2, and let v^{δ} ∈ T_{p}M be nonzero. Then the following statements are equivalent:

(1) v^{δ} is a projective nullity vector at p

(2) there exists B ∈ ℝ such that P_{αβ}^{γ}_{δ}v^{β} = (P_{αδ} − Bg_{αδ})v^{γ}

(3) P_{αβ}^{γ}_{δ}v^{δ} = 0.

Proof. (1)⇒(2). Since P_{αβ}v^{β} = Bg_{αβ}v^{β}, R_{αβγδ} = R_{γδαβ} and P_{αβ} = P_{βα}, we have

P_{αβγδ}v^{β} = R_{γδαβ}v^{β} − g_{αγ}P_{βδ}v^{β} + g_{βγ}P_{αδ}v^{β}

= B(g_{αγ}g_{βδ} − g_{βγ}g_{αδ})v^{β} − Bg_{αγ}g_{βδ}v^{β} + g_{βγ}P_{αδ}v^{β} = (P_{αδ} − Bg_{αδ})g_{βγ}v^{β},

and (2) follows by raising the index γ.

(2)⇒(3). Since P_{[αβ}^{γ}_{δ]} = 0, which follows easily from R_{[αβ}^{γ}_{δ]} = 0,

P_{αβ}^{γ}_{δ}v^{δ} = (P_{αδ}^{γ}_{β} − P_{βδ}^{γ}_{α})v^{δ},

which vanishes by (2), since P_{αβ} − Bg_{αβ} is symmetric in αβ.

(3)⇒(1). Observe that 0 = R_{αβγδ}v^{γ}v^{δ} = (g_{αγ}P_{βδ} − g_{βγ}P_{αδ})v^{γ}v^{δ} = v_{[α}P_{β]δ}v^{δ}. Hence there exists B ∈ ℝ such that P_{βδ}v^{δ} = Bg_{βδ}v^{δ}, and hence

R_{αβγδ}v^{δ} = (g_{αγ}P_{βδ} − g_{βγ}P_{αδ})v^{δ} = B(g_{αγ}g_{βδ} − g_{βγ}g_{αδ})v^{δ},

i.e. v^{δ} is in the projective nullity at p. □

In particular, this shows that the projective nullity distribution is a metric projective invariant, as is the expression P_{αβ} − Bg_{αβ} wherever there is projective nullity, hence so is the special tractor connection (230). The above argument for this fact is given in [49], where the special connection on the standard tractor bundle is also discussed.

Remark 6.2. In the 2-dimensional case, all metrics have nullity at all points and B is the Gaußian curvature. On the unit n-sphere the nullity distribution is the tangent bundle and B ≡ 1. Condition (234) may be written as

C_{αβγδ}v^{δ} = (\frac{1}{(n−1)(n−2)}Scal − \frac{1}{n−2}B)(g_{αγ}v_{β} − g_{βγ}v_{α}) − \frac{1}{n−2}(Ric_{αγ}v_{β} − Ric_{βγ}v_{α})

where C_{αβγδ} is conformal Weyl curvature tensor. For a Riemannian metric, we may orthogonally diagonalise the Ricci tensor to see that if C_{αβγδ} = 0 (as it is in three dimensions or in the conformally flat case in higher dimensions) then R_{αβγδ} has nullity if and only if all but possibly one of the eigenvalues of P_{α}^{β} coalesce with B being the possible exception. So in the three-dimensional Riemannian case R_{αβγδ} has nullity if and only if the discriminant of the characteristic polynomial of P_{α}^{β} vanishes:

(P_{α}^{α})^{6} − 9(P_{α}^{α})^{4}(P_{β}^{γ}P_{γ}^{β}) + 21(P_{α}^{α})^{2}(P_{β}^{γ}P_{γ}^{β})^{2} − 3(P_{β}^{γ}P_{γ}^{β})^{3}

+ 8(P_{α}^{α})^{3}(P_{δ}^{ϵ}P_{ϵ}^{γ}P_{γ}^{δ}) − 36(P_{α}^{α})(P_{β}^{γ}P_{γ}^{β})(P_{δ}^{ϵ}P_{ϵ}^{ζ}P_{ζ}^{δ}) + 18(P_{δ}^{ϵ}P_{ϵ}^{γ}P_{γ}^{δ})^{2} = 0.

Indeed, in three dimensions (where R_{αβγδ} is determined by P_{αβ}) it is also the case in Lorentzian signature that R_{αβγδ} has nullity if and only if P_{α}^{β} is diagonalisable with eigenvalues distributed in this manner. In any case, in three dimensions it follows that B is a continuous function and is smooth except perhaps at points where P_{αβ} is a multiple of g_{αβ}. In the four-dimensional Riemannian case, one can check that if R_{αβγδ} has nullity and the eigenvalues of P_{α}^{β} are B, λ_{2}, λ_{3}, λ_{4}, then

I ≡ C_{αβγδ}C^{αβγδ} = 6((λ_{2} − λ_{3})^{2} + (λ_{3} − λ_{4})^{2} + (λ_{4} − λ_{2})^{2})

and if this expression is nonzero, then

B = \frac{1}{4}P_{α}^{α} + \frac{1}{4I} (C_{αβ}^{γδ}C_{γδ}^{ϵζ}C_{ϵζ}^{αβ} − 18C_{αβ}^{γδ}P_{γ}^{α}P_{δ}^{β}).

It follows that B is smooth on {I ≠ 0} whilst on {I = 0} three of the four eigenvalues of P_{α}^{β} merge as above and B is the odd one out unless P_{αβ} ∝ g_{αβ}. Therefore, as in three dimensions, it follows that B extends as a continuous function that is smooth except where P_{αβ} is a multiple of g_{αβ}. We anticipate similar behaviour in general but, for the moment, the regularity of B remains unknown.

6.2. C-projective nullity. We return now to metric c-projective geometry, where we seek to develop analogous interconnections between curvature and special tractor connections to those in the metric projective case. In order to do this, we first develop a notion of c-projective nullity for (pseudo-)Kähler metrics, modelled on the curvature of complex projective space (48) in the same way that projective nullity for (pseudo-)Riemannian metrics is modelled on the curvature of the unit sphere.

We suppose therefore that (M, J, g) is a (pseudo-)Kähler manifold with ∇ the Levi-Civita connection of g_{αβ} and Ω_{αβ} = J_{α}^{γ}g_{γβ} the Kähler form. Further let us write

S_{αβγδ} ≡ g_{αγ}g_{βδ} − g_{βγ}g_{αδ} + Ω_{αγ}Ω_{βδ} − Ω_{βγ}Ω_{αδ} + 2Ω_{αβ}Ω_{γδ}  (235)

for the Kähler curvature tensor of constant sectional holomorphic curvature 4. As in the (pseudo-)Riemannian case, at each p ∈ M, there is at most one scalar B ∈ ℝ such that G_{αβγδ}^{B} := R_{αβγδ} − BS_{αβγδ} has nullity at p. Indeed, if v^{α} and ṽ^{α} are nullity vectors for G_{αβγδ}^{B} and GB̃ ; αβγδ respectively then

0 = (B − B̃)S_{αβγδ}ṽ^{β}v^{δ}
= (B − B̃)(v_{β}ṽ^{β}g_{αγ} − v_{α}ṽ_{γ} + J_{β}^{δ}v_{δ}ṽ^{β}Ω_{αγ} + J_{α}^{δ}v_{δ}J_{γ}^{β}ṽ_{β} + 2J_{γ}^{δ}v_{δ}J_{α}^{β}ṽ_{β})

which implies that B = B̃ unless v^{α} or ṽ^{α} are zero. By analogy with Definition 6.2, and again following Gray [50] (who used the term “holomorphic constancy”), we therefore define c-projective nullity as follows.

Definition 6.3. The (c-projective) nullity distribution N of a (pseudo-)Kähler manifold (M, J, g) is the union of the nullity spaces of G_{αβγδ}^{B} over B ∈ ℝ and p ∈ M, and for each p ∈ M, we write N_{p} for the (c-projective) nullity space N ∩ T_{p}M. We say that (J, g) has (c-projective) nullity at p ∈ M if N_{p} is nonzero, i.e.

(R_{αβγδ} − BS_{αβγδ})v^{δ} = 0,  (236)

for some B ∈ ℝ, uniquely determined by p, and some nonzero v^{α} ∈ T_{p}M.

Thus N_{p} is the kernel of the linear map

v^{δ} ↦ G_{αβγδ}^{B}v^{δ},

for some B ∈ ℝ depending on p. Let us remark that, since G = G^{B} has the symmetries of the curvature tensor of a Kähler metric, N_{p} is a J-invariant subspace of T_{p}M (i.e. v^{δ} ∈ N_{p} implies J_{α}^{δ}v^{α} ∈ N_{p}), hence is even dimensional.

Bearing in mind the discussion of Section 4.1, we may write (236) in barred and unbarred indices. We find that

\begin{matrix}(R_{ab̄cd̄} + 2B(g_{ab̄}g_{cd̄} + g_{cb̄}g_{ad̄}))vd̄ = 0 \\ (R_{ab̄c̄d} − 2B(g_{ac̄}g_{db̄} + g_{ab̄}g_{dc̄}))v^{d} = 0. \end{matrix}  (237)

As in the projective case, tracing (236) over αγ yields an eigenvalue equation

Ric^{β}_{δ}v^{δ} = 2(n + 1)Bv^{β}, equivalently P^{β}_{δ}v^{δ} = 2Bv^{β},  (238)

since P_{αβ} = \frac{1}{n+1}Ric_{αβ} by (25) and (109). This can equivalently be expressed in barred and unbarred indices as

P^{b}_{d}v^{d} = 2Bv^{b},  or as Pb̄d̄vd̄ = 2Bvb̄.  (239)

Of course, we may derive (239) also directly by tracing the second equation of (237), respectively its conjugate, with respect to ac̄, respectively āc. Further, note that the symmetries of the Ricci tensor of a (pseudo-)Kähler metric show that (239) can be also equivalently written as P_{d}^{b}v^{d} = 2Bv^{b}, respectively Pd̄b̄vd̄ = 2Bvb̄.

Now assume that (237) is satisfied and decompose R_{ab̄}^{c}_{d} according to (28) as

R_{ab̄}^{c}_{d} = H_{ab̄}^{c}_{d} + δ_{a}^{c}P_{b̄d} + δ_{d}^{c}P_{b̄a}.

Then equation (239) implies

H_{ab̄}^{c}_{d}vb̄ = (R_{ab̄}^{c}_{d} − δ_{a}^{c}P_{b̄d} − δ_{d}^{c}P_{b̄a})vb̄ = 2B(δ_{a}^{c}v_{d} + δ_{d}^{c}v_{a}) − (δ_{a}^{c}P_{b̄d} + δ_{d}^{c}P_{b̄a})vb̄ = 0.

Furthermore,

H_{ab̄}^{c}_{d}v^{d} = (R_{ab̄}^{c}_{d} − δ_{a}^{c}P_{b̄d} − δ_{d}^{c}P_{b̄a})v^{d}
= (2B(g_{ab̄}v^{c} + δ_{a}^{c}vb̄) − 2Bδ_{a}^{c}vb̄ − P_{b̄a}v^{c})
= (2Bg_{ab̄} − P_{b̄a})v^{c},

which implies H_{ab̄}^{c}_{d}v^{a}v^{d} = 0. It fact these two conditions are also sufficient for nullity.

Proposition 6.3. Let (M, J, g) be a (pseudo-)Kähler manifold of dimension 2n ≥ 4, and let v^{d} ∈ T_{p}^{1,0}M ≅ T_{p}M be a nonzero tangent vector. Then the following statements are equivalent:

(1) v^{d} ∈ N_{p}

(2) there exists B ∈ ℝ such that H_{ab̄}^{c}_{d}v^{d} = (2Bg_{ab̄} − P_{ab̄})v^{c}

(3) H_{ab̄}^{c}_{d}v^{a}v^{d} = 0 and H_{ab̄}^{c}_{d}vb̄ = 0,

where, as in (28), H_{ab̄}^{c}_{d} is the trace-free part of R_{ab̄}^{c}_{d} ≡ −g^{ēc}R_{ab̄dē}.

Proof. We have just observed that (1) implies (2) and (3). Note, moreover that taking the trace with respect to a and c in (2) gives (239), which shows immediately that (2) implies (1). Hence, it remains to show that (3) implies (1). If (3) holds, then

R_{ab̄}^{c}_{d}v^{a}v^{d} = (δ_{a}^{c}P_{b̄d} + δ_{d}^{c}P_{b̄a})v^{a}v^{d} ⇒ R_{ab̄c̄d}v^{a}v^{d} = 2vc̄P_{b̄d}v^{d}

so 0 = v_{[c̄}P_{b̄]d}v^{d} and we conclude that P_{c̄d}v^{d} = 2B̄vc̄, for some constant B̄. Substituting the conjugate conclusion P_{d̄c}vd̄ = 2Bv_{c} into R_{ab̄}^{c}_{d}vb̄ gives

R_{ab̄}^{c}_{d}vb̄ = (H_{ab̄}^{c}_{d} + δ_{a}^{c}P_{b̄d} + δ_{d}^{c}P_{b̄a})vb̄ = 2Bδ_{a}^{c}v_{d} + 2Bδ_{d}^{c}v_{a}

which, after lowering the index c is equivalent to (237), as required. (Note that B is necessarily real, since R_{ab̄}^{c}_{d} and S_{ab̄}^{c}_{d} are real tensors.) □

Corollary 6.4. At any p ∈ M, the nullity distribution of N_{p} is a metric c-projective invariant, i.e. the same for c-projectively equivalent (pseudo-)Kähler metrics g_{ab̄} and g̃_{ab̄}. Furthermore, if N_{p} is nonzero, and B, B̃ ∈ ℝ are the corresponding scalars in the definition of N_{p} with respect to g, g̃ respectively, then P̃_{ab̄} − 2B̃g̃_{ab̄} = P_{ab̄} − 2Bg_{ab̄}.

Proof. By Proposition 2.13, criterion (3) of Proposition 6.3 is c-projectively invariant. In fact, by Proposition 4.4, H_{ab̄}^{c}_{d} is precisely the harmonic curvature of the underlying c-projective structure. The last part follows immediately from criterion (2). □

Remark 6.3. For later use, we apply the projectors of Section 1 to reformulate the equivalent conditions of Proposition 6.3 directly in terms of v^{δ} ∈ T_{p}M as follows:

(1) v^{δ} ∈ N_{p}

(2) there exists a constant B ∈ ℝ such that H_{αβ}^{γ}_{δ}v^{δ} = (J_{α}^{ϵ}P_{ϵβ} − 2BΩ_{αβ})J_{δ}^{γ}v^{δ} (3) H_{αβ}^{γ}_{δ}v^{α}v^{δ} = 0 and (H_{αβ}^{γ}_{δ} + J_{β}^{ϵ}J_{ζ}^{γ}H_{αϵ}^{ζ}_{δ})v^{β} = 0,

where H_{αβ}^{γ}_{δ} ≡ R_{αβ}^{γ}_{δ} − δ_{[α}^{γ}P_{β]δ} + J_{[α}^{γ}P_{β]ζ}J_{δ}^{ζ} + J_{α}^{ζ}P_{βζ}J_{δ}^{γ} and P_{βδ} ≡ \frac{1}{n+1}R_{αβ}^{α}_{δ}.

Proposition 6.5. Let (M, J, g) be a (pseudo-)Kähler manifold of dimension 2n ≥ 4, and B a smooth function on an open subset U. Then for any (real) vector field v in the nullity of G = G^{B} on U, if v is non-null at p ∈ U, then dB = 0 there.

Proof. The differential Bianchi identity ∇_{[a}R_{b]c̄d}^{e} = 0 on U implies that

∇_{[a}G_{b]c̄d}^{e} = 2(∇_{[a}B)g_{b]c̄}δ_{d}^{e} + 2(∇_{[a}B)δ_{b]}^{e}g_{dc̄}.

Since v^{a} and vā belong to the nullity of G, we may contract with vc̄v^{d} to obtain

0 = 2(∇_{[a}B)g_{b]c̄}vc̄v^{e} + 2(∇_{[a}B)δ_{b]}^{e}g_{dc̄}v^{d}vc̄.

A further contraction with v_{e} = g_{ef̄}vf̄ yields (∇_{[a}B)v_{b]}g_{dc̄}v^{d}vc̄ = 0, so if v is non-null at p, (∇_{[a}B)v_{b]} = 0 there; hence (∇_{[a}B)δ_{b]}^{e} = 0, which implies ∇_{a}B = 0, i.e. dB = 0. □

6.3. Mobility, nullity, and the special tractor connection. Our aim in this section is to show that, under certain conditions, the solutions of the mobility equation (129) on a (pseudo-)Kähler manifold (M, J, g) lift uniquely to parallel sections of V ⊆ V_{ℂ} for the special tractor connection:

∇_{a}^{V_{ℂ}} (\begin{matrix}A^{b̄c} \\ Λ^{b} | Λb̄ \\ μ \end{matrix}) = (\begin{matrix}∇_{a}A^{b̄c} + δ_{a}^{c}Λb̄ \\ ∇_{a}Λ^{b} + μδ_{a}^{b} − 2BA_{a}^{b} | ∇_{a}Λb̄ \\ ∇_{a}μ − 2BΛ_{a} \end{matrix})

∇V_{ℂ} ; ā (\begin{matrix}A^{b̄c} \\ Λ^{b} | Λb̄ \\ μ \end{matrix}) = (\begin{matrix}∇āA^{b̄c} + δāb̄Λ^{c} \\ ∇āΛ^{b} | ∇āΛb̄ + μδāb̄ − 2BAb̄ā \\ ∇āμ − 2BΛā \end{matrix}) ,

(240)

where ∇ is the Levi-Civita connection for g_{ab̄} and B is a smooth function on M. Here V_{ℂ} is identified via g_{ab̄} with a direct sum of unweighted tensor bundles, and we write the connection in barred and unbarred indices, so that for sections of V ⊆ V_{ℂ}, the two lines of (240) are conjugate.

Remark 6.4. By Theorem 4.6, we know already that any solution A^{āb} of the mobility equation (129) lifts uniquely to a parallel section of V for the more complicated prolongation connection (157)–(158). If it also lifts to a parallel section for (240), then (cf. Remark 6.1) the two lifts may differ, albeit only in the last component. More precisely the last component μ of the parallel lift for the special tractor connection is given by μ = μ′ − \frac{1}{n}(P_{ab̄} − 2Bg_{ab̄})A^{āb}, where μ′ is the last component of the parallel lift for the prolongation connection. Note that if the metric g^{āb} itself lifts to a parallel section for (240), then B must be locally constant.

In [44, Theorem 5], it is shown that if the mobility of (M, g, J) is at least three, then there is a constant B such that all solutions of the mobility equation lift uniquely to parallel sections of V for (240). Before developing this, and related results, it will be useful to establish some basic properties of special tractor connections (240) and their parallel sections. Throughout this section we set, for a given function B,

G_{ab̄cd̄}:= R_{ab̄cd̄} + 2B(g_{ab̄}g_{cd̄} + g_{cb̄}g_{ad̄}).  (241)

The equations satisfied by parallel sections of (240) are

∇_{a}A_{b}^{c} = −δ_{a}^{c}Λ_{b}  (242a)
∇_{a}Λc̄ = −μg_{ac̄} + 2BA_{ac̄} and ∇_{a}Λ_{b} = 0,  (242b)
∇_{a}μ = 2BΛ_{a}  (242c)

and their complex conjugates. Of course, the first line is simply the mobility equation. In particular, from (145)–(146) (and the symmetry of P_{bc̄}) we have

\begin{aligned}g_{db̄}∇_{a}Λc̄ − g_{ac̄}∇b̄Λ_{d} & = R_{ab̄ec̄}A_{d}^{e} − R_{ab̄dē}Aēc̄ \\ & = W_{ab̄ec̄}A_{d}^{e} − W_{ab̄dē}Aēc̄ − g_{ac̄}P_{eb̄}A_{d}^{e} + g_{db̄}P_{aē}Aēc̄, \end{aligned}  (243)

where W_{ab̄ec̄} = H_{ab̄ec̄}, since J is integrable.

Lemma 6.6. If A_{b}^{c} and Λ_{b} satisfy (242a)–(242b) for smooth functions B, μ, then

G_{ab̄c}^{e}A_{e}^{d} = G_{ab̄e}^{d}A_{c}^{e}  (244)
G_{ab̄c}^{d}Λ_{d} = −g_{cb̄}(∇_{a}μ − 2BΛ_{a}) + 2(∇_{a}B)A_{cb̄}.  (245)

Proof. We substitute (242b) into (243) to obtain

R_{ab̄ec̄}A_{d}^{e} − R_{ab̄dē}Aēc̄ = g_{ac̄}(μg_{db̄} − 2BA_{db̄}) − g_{db̄}(μg_{ac̄} − 2BA_{ac̄})
= 2B(g_{db̄}g_{aē}Aēc̄ − g_{ac̄}g_{eb̄}A_{d}^{e}),

and (244) follows from (241). To obtain (245), we apply (241) instead to the identity

R_{ab̄c}^{d}Λ_{d} = (∇_{a}∇b̄ − ∇b̄∇_{a})Λ_{c} = ∇_{a}(−μg_{cb̄} + 2BA_{cb̄})
= 2(∇_{a}B)A_{cb̄} − 2Bg_{ab̄}Λ_{c} − g_{cb̄}(∇_{a}μ).  □

In Theorem 6.11, we will show that if (M, g, J) of mobility ≥ 2 has c-projective nullity, then all solutions of the mobility equation lift uniquely to parallel sections of V for (240), where B is characterised by nullity. First we establish the following converse and regularity result.

Theorem 6.7. Let (M, J, g) be a connected (pseudo-)Kähler manifold with a nonparallel solution A^{āb} of the mobility equation, which lifts, over a dense open subset U of M, to a real parallel section (A^{āb}, Λ^{a}, μ) for (240) with B locally constant. Then:

(1) B is constant and (A^{āb}, Λ^{a}, μ) extends to a parallel section over M;

(2) G_{ab̄c}^{d}Λ_{d} = 0, and hence (J, g) has c-projective nullity on the dense open subset where Λ^{a} is nonzero.

Proof. As noted in Remark 6.4, Theorem 4.6 provides a real section (A^{āb}, Λ^{a}, μ′) of V (defined on all of M) which is parallel for the connection given by (157) and (158). On U we compute, using (242b) and (243), that

R_{ab̄}^{d}_{f}A^{c̄f} + R_{ab̄}c̄ēA^{ēd} = 2B(δ_{a}^{d}Ac̄b̄ − δb̄c̄A_{a}^{d}),  (246)

which implies

\frac{1}{n}(Ric_{b̄f}A^{c̄f} + R_{ab̄}c̄ēA^{ēa}) = 2B(Ac̄b̄ − \frac{1}{n}δb̄c̄A_{a}^{a}).  (247)

Applying ∇d̄ to (247) and taking the trace with respect to d̄ and c̄ shows that

\frac{1}{(1−n)(n+1)}((∇d̄Ric_{b̄f})A^{d̄f} + (∇d̄R_{ab̄}d̄ē)A^{ēa} + (1 − n)Ric_{b̄f}Λ^{f}) = 2BΛb̄.  (248)

Recall that

−μδ_{a}^{b} + 2BA_{a}^{b} = ∇_{a}Λ^{b} = −μ′δ_{a}^{b} + P_{ac̄}A^{c̄b} − \frac{1}{n}H_{ac̄}^{b}_{d}A^{c̄d}

and that P_{ab̄} = \frac{1}{n+1}Ric_{ab̄}. Hence, applying ∇_{h} to (248) and taking trace shows, together with the identities (247) and (248), that we have an identity of the form

(\begin{matrix}R & 0 & 0 \\ ∇R & R & 0 \\ ∇∇R + R ▷◁ R & ∇R & R \end{matrix}) (\begin{matrix}A^{āb} \\ Λ^{a} \\ μ′ \end{matrix}) = 2B [(\begin{matrix}A^{āb} \\ Λ^{a} \\ μ′ \end{matrix}) − \frac{A_{c}^{c}}{n} (\begin{matrix}g^{āb} \\ 0 \\ \frac{1}{n}P_{d}^{d} \end{matrix})] .  (249)

Since the left-hand side of (249) is defined on all of M and (A^{āb}, Λ^{a}, μ′) is a nowhere vanishing section on M, the identity (249) can be used to extend B smoothly as a function to all of M. Since B is locally constant on U and M is connected, B is actually a constant and (A^{āb}, Λ^{a}, μ) extends smoothly to a parallel section of the connection (240) on all of M.

The second part is immediate from (245) with ∇_{a}B = 0 and ∇_{a}μ = 2BΛ_{a}. □

Remark 6.5. When (J, g) has c-projective nullity, P_{ab̄}−2Bg_{ab̄} (with B given by (236)) is a metric c-projective invariant by Corollary 6.4, and hence the connection (240) is metric c-projectively invariant. In particular, by Theorem 6.7, the connection is metric c-projectively invariant if B is constant and it admits a parallel section with Λ_{a} nonzero.

On the other hand, if the connection (240) admits a parallel section with Λ_{a} = 0, then (242b) shows that B = 0 unless the corresponding solution A_{ab̄} of the mobility equation is a (necessarily locally constant) multiple of g_{ab̄}. Thus a parallel solution of the mobility equation which is not a multiple of g lifts to a parallel section for (240) if and only if B = 0.

Theorem 6.8. Let (M, J, [∇]) be a connected metric c-projective manifold of dimension 2n ≥ 4 arising from a (pseudo-)Kähler metric g with mobility ≥ 3. Then either (J, g) has c-projective nullity on a dense open subset U ⊆ M, with B constant in (236), or 2n ≥ 6 and all metrics c-projectively equivalent to g are affinely equivalent to g (i.e. have the same Levi-Civita connection).

To prove this theorem, we use a couple of lemmas, the first of which is a purely algebraic (pointwise) result.

Lemma 6.9. Suppose that R_{ab̄cd̄} is a tensor which has Kähler symmetries (108) with respect to g_{ab̄}. Let A_{ab̄}, Ã_{ab̄}, Λ_{ab̄} and Λ̃_{ab̄} be (real) tensors that satisfy

R_{ab̄c̄e}A_{d}^{e} + R_{ab̄df̄}Af̄c̄ = g_{ac̄}Λ_{db̄} − g_{db̄}Λ_{ac̄}  (250)
R_{ab̄c̄e}Ã_{d}^{e} + R_{ab̄df̄}Ãf̄c̄ = g_{ac̄}Λ̃_{db̄} − g_{db̄}Λ̃_{ac̄}.  (251)

If A_{ab̄}, Ã_{ab̄} and g_{ab̄} are linearly independent, then Λ_{ab̄}, respectively Λ̃_{ab̄}, are linear combinations of g_{ab̄} and A_{ab̄}, respectively g_{ab̄} and Ã_{ab̄}, with the same second coefficient.

Proof. Note first that these equations remain unchanged if we add scalar multiplies of g_{ab̄} to the tensors A_{ab̄}, Ã_{ab̄}, Λ_{ab̄} and Λ̃_{ab̄}. Hence, we can assume without loss of generality that the trace of these tensors vanishes. We then have to show that Λ_{ab̄} and Λ̃_{ab̄} are a common scalar multiple of A_{ab̄} and Ã_{ab̄} respectively.

From equation (250) it follows immediately that

Ã_{a}^{h}(R_{hb̄c̄e}A_{d}^{e} + R_{hb̄df̄}Af̄c̄) − Ãīb̄(R_{aīc̄e}A_{d}^{e} + R_{aīdf̄}Af̄c̄)

= Ã_{ac̄}Λ_{db̄} + Ã_{db̄}Λ_{ac̄} − g_{ac̄}Ãīb̄Λ_{dī} − g_{db̄}Ã_{a}^{h}Λ_{hc̄}. (252)

By the symmetries (108) of R_{ab̄cd̄}, the left-hand side of identity (252) equals

(Ã_{a}^{h}R_{hb̄c̄e} − Ãīb̄R_{aīc̄e})A_{d}^{e} + (Ã_{a}^{h}R_{hb̄df̄} − Ãīb̄R_{aīdf̄})Af̄c̄
= (Ã_{a}^{h}R_{hb̄c̄e} + Ãīb̄R_{ec̄aī})A_{d}^{e} − (Ã_{a}^{h}R_{df̄b̄h} + Ãīb̄R_{df̄aī})Af̄c̄
= A_{ac̄}Λ̃_{db̄} + A_{db̄}Λ̃_{ac̄} − g_{ac̄}Aīb̄Λ̃_{dī} − g_{db̄}A_{a}^{h}Λ̃_{hc̄}  (253)

where the last equality follows from (252). From (252) and (253) we therefore obtain

g_{ac̄}τ_{db̄} + g_{db̄}τ_{ac̄} = A_{db̄}Λ̃_{ac̄} + A_{ac̄}Λ̃_{db̄} − Ã_{db̄}Λ_{ac̄} − Ã_{ac̄}Λ_{db̄},  (254)

where τ_{ab̄} = A_{a}^{e}Λ̃_{eb̄} − Ãf̄b̄Λ_{af̄}. Taking the trace with respect to a and c yields

nτ_{db̄} + g_{db̄}τ_{a}^{a} = 0,  (255)

which shows that τ_{ab̄} = 0. Therefore, we conclude from (254) that

A_{db̄}Λ̃_{ac̄} + A_{ac̄}Λ̃_{db̄} = Ã_{db̄}Λ_{ac̄} + Ã_{ac̄}Λ_{db̄}.

Since any nonzero tensor of this form determines its factors up to scale, and A_{ab̄} and Ã_{ab̄} are linearly independent, we conclude that Λ_{ab̄} and Λ̃_{ab̄} are the same multiple of A_{ab̄} and Ã_{ab̄} respectively. □

We next relate linear dependence to pointwise linear dependence.

Lemma 6.10. Let (M, J, g) be a connected (pseudo-)Kähler 2n-manifold (n ≥ 2) and let A_{a}^{b} be a solution of (128) such that Ã_{a}^{b} := pδ_{a}^{b} + qA_{a}^{b} is also a solution for real functions p and q. Then p and q are constant or A_{a}^{b} = ξδ_{a}^{b} for constant ξ.

Proof. By assumption, we have ∇_{a}A_{b}^{c} = −δ_{a}^{c}Λ_{b} and ∇_{a}Ã_{b}^{c} = −δ_{a}^{c}Λ̃_{b}, hence

∇_{a}p δ_{b}^{c} + ∇_{a}q A_{b}^{c} = −(Λ̃_{b} − qΛ_{b})δ_{a}^{c}

If ∇_{a}q = 0, it follows easily that p and q are locally constant hence constant. Otherwise, contracting this expression with a nonzero tangent vector X^{a} in the kernel of ∇_{a}q, we deduce that Λ̃_{b} = qΛ_{b} and ∇_{a}p = −ξ∇_{a}q for some function ξ. Thus ∇_{a}q (A_{b}^{c} − ξδ_{b}^{c}) = 0. If A_{b}^{c} = ξδ_{b}^{c}, it follows from what we have already proven that ξ is constant. Otherwise, we deduce that p and q are constant. □

Proof of Theorem 6.8. Suppose that A_{ab̄} and Ã_{ab̄} are nondegenerate solutions of the mobility equation such that g_{ab̄}, A_{ab̄} and Ã_{ab̄} are linearly independent. At each point of M, (243) implies that A_{ab̄} and Ã_{ab̄} satisfy (250)–(251), with Λ_{ac̄} = ∇_{a}Λc̄ and Λ̃_{ac̄} = ∇_{a}Λ̃c̄. By Lemma 6.10, g_{ab̄}, A_{ab̄} and Ã_{ab̄} are pointwise linearly independent on a dense open set U′, and hence, on U′, Lemma 6.9 implies that A_{ab̄} and Ã_{ab̄} lift to smooth solutions (A_{ab̄}, Λ_{a}, μ) and (Ã_{ab̄}, Λ̃_{a}, μ̃) of (242a)–(242b) for the same smooth function B. Thus we may apply Lemma 6.6.

The trace-free parts of A_{a}^{b} and Ã_{a}^{b} are pointwise linearly independent on U′, hence if n = 2, their common centraliser at each p ∈ U′ consists only of multiples of the identity. By (244), G_{ab̄c}^{d} is a multiple α_{ab̄} of δ_{a}^{d}, hence zero, since G_{ab̄c}^{d} = G_{cb̄a}^{d}. Thus g has constant holomorphic sectional curvature, which proves the theorem for 2n = 4.

To prove the theorem for 2n ≥ 6, we substitute (245) into G_{ab̄c}^{d} = G_{cb̄a}^{d} to obtain

g_{cb̄}(∇_{a}μ − 2BΛ_{a}) − 2(∇_{a}B)A_{cb̄} = g_{ab̄}(∇_{c}μ − 2BΛ_{c}) − 2(∇_{c}B)A_{ab̄}.

If we contract this equation with a vector Y^{c} in the kernel of ∇_{c}B, then since n ≥ 3, we obtain a degenerate Hermitian form on the left hand side, equal to a multiple of g_{ab̄}. Hence both sides vanish, i.e. Y^{c} is in the kernel of ∇_{c}μ − 2BΛ_{c} and we have

A_{cb̄}Y^{c} = ξg_{cb̄}Y^{c} and ∇_{a}μ − 2BΛ_{a} = 2ξ(∇_{a}B)

for some function ξ on U′. Hence (245) now reads

G_{ab̄c}^{d}Λ_{d} = 2(∇_{a}B)(A_{cb̄} − ξg_{cb̄}) = 2(∇_{c}B)(A_{ab̄} − ξg_{ab̄}).

If ∇_{c}B is nonzero on an open subset of U′, it follows that A_{a}^{b} − ξδ_{a}^{b} has (complex) rank at most one there, with image spanned by ∇^{a}B and kernel containing the kernel of ∇_{a}B. Since the same holds for Ã_{a}^{b} − ξ̃δ_{a}^{b} for some function ξ̃, we have that δ_{a}^{b}, A_{a}^{b} and Ã_{a}^{b} are linearly dependent, a contradiction. Hence ∇_{c}B is identically zero on U′, i.e. B is locally constant. The result now follows from Theorem 6.7. □

Remark 6.6. The above proof shows (for mobility ≥ 3) that any solution of the mobility equation (129) lifts to a parallel section for (240), where B is given by (236), unless all solutions are parallel (i.e. affine equivalent to g). However, in the latter case, any solution of (129) lifts to a parallel section for (240) with B = 0 (cf. Remark 6.5). This establishes [44, Theorem 5]; the next result may be seen as a strengthening of this theorem in which c-projective nullity is brought to the fore, cf. also [26, Theorem 2].

Theorem 6.11. Let (M, J, g) be a connected (pseudo-)Kähler manifold admitting a solution of the mobility equation that is not a constant multiple of g. Assume that there is a dense open subset U ⊆ M on which (J, g) has c-projective nullity and denote by B the function in (236). Then the following hold:

• B is constant

• any solution A^{āb} of the mobility equation lifts uniquely to a section of V which is parallel for the special tractor connection (240).

We divide the proof of Theorem 6.11 into several propositions.

Proposition 6.12. Under the assumptions of Theorem 6.11, there is a dense open subset U′ ⊆ U on which B is smooth, and for any solution A_{bc̄} of the mobility equation:

(1) there is a smooth real-valued function μ on U′ such that (242b) holds, and if B is locally constant then (242c) also holds on U′;

(2) for any vector v^{a} in the nullity distribution of (J, g),

2(∇_{a}B)A_{cb̄}v^{c} + vb̄(2BΛ_{a} − ∇_{a}μ) = 0.  (256)

In particular, if v^{a} is not in any eigenspace of A_{a}^{b} then B is locally constant.

Proof. To see that (1) holds for A_{bc̄}, first recall that Λ^{a}, given by (242a), is holomorphic. Next, by assumption, at any p ∈ U there is a nonzero tangent vector vb̄ such that vb̄G_{ab̄cd̄} = 0. Hence, by equation (243), on U we have

\begin{aligned}−v_{c}∇_{a}Λd̄ + g_{ad̄}vb̄∇b̄Λ_{c} & = vb̄R_{ab̄cē}Aēd̄ − vb̄R_{ab̄ed̄}A_{c}^{e} \\ & = −2B(g_{aē}v_{c} + g_{cē}v_{a})Aēd̄ + 2B(g_{ad̄}v_{e} + g_{ed̄}v_{a})A_{c}^{e} \\ & = 2B(g_{ad̄}vb̄A_{cb̄} − v_{c}A_{ad̄}) \end{aligned}  (257)

and so

v_{c}V_{ad̄} − g_{ad̄}vb̄\overline{V}_{b̄c} = 0,

where V_{ab̄} ≡ ∇_{a}Λb̄ − 2BA_{ab̄}. As vb̄ ≠ 0 on U, it follows that V_{ab̄} is pure trace, i.e. the second equation of (242b) holds pointwise. By assumption, there is a dense open subset U′ ⊆ U on which g_{ab̄} and A_{ab̄} are pointwise linearly independent for some solution A_{ab̄}, from which it follows that B is a smooth real-valued function on U′. Hence (242a)– (242b) hold on U′ for any solution, with μ smooth on U′.

By Lemma 6.6, any solution satisfies (245), which implies (256). Now if ∇_{a}B = 0 it follows immediately from the existence of nullity that (242c) holds on U′. □

Proposition 6.12 and Theorem 6.7 have the following immediate consequence.

Corollary 6.13. Theorem 6.11 holds unless the nullity distribution is contained in an eigendistribution of every solution of the mobility equations.

It remains to show that Theorem 6.11 also holds when the nullity distribution is contained in an eigendistribution of every solution of the mobility equations, and for this it suffices to show that ∇_{a}B = 0 on a dense open set. Suppose then that v^{b} is a nonzero nullity vector such that A_{a}^{b}v^{a} = ξv^{b} for some smooth function ξ, so that ∇_{a}μ − 2BΛ_{a} = 2ξ∇_{a}B by (256) and hence (245) reads

G_{ab̄c}^{d}Λ_{d} = 2(∇_{a}B)(A_{cb̄} − ξg_{cb̄}) = 2(∇_{c}B)(A_{ab̄} − ξg_{ab̄})  (258)

as in the proof of Theorem 6.8. Since v^{a} is an eigenvector of A_{a}^{b} with eigenvalue ξ, vb̄ is an eigenvector of Aāb̄ with eigenvalue ξ̄. However vb̄ is in the nullity of G, so the contraction of (258) with vb̄ yields (∇_{c}B)(A_{ab̄} − ξg_{ab̄})vb̄ = 0. If we now combine these observations with Proposition 6.5, we obtain that either ∇_{a}B = 0 on a dense open set (and we are done) or there is an open set on which Ã_{a}^{b} := A_{a}^{b} − ξδ_{a}^{b} has (complex) rank at most one, v^{a} is a null vector in its kernel, and ξ is real. Hence the generalised ξ-eigenspace of A_{a}^{b} is nondegenerate, and so has (complex) dimension at least two, which implies that ξ is locally constant by Lemma 5.16. Now Ã_{a}^{b} is a rank one solution of the mobility equation with a nonzero (but null) nullity vector in its kernel, and so Theorem 6.11 is a consequence of the following proposition.

Proposition 6.14. Suppose (M, J, g) is a connected (pseudo-)Kähler manifold of dimension 2n ≥ 4 admitting a non-parallel solution of the mobility equation A_{ab̄} such that A_{a}^{b} is a complex endomorphism of rank 1. Assume that g has nullity on some dense open set U ⊆ M and that there is a nonzero vector in the nullity distribution that is in the kernel of A_{a}^{b}. Then the function B defined as in (236) is a constant and the conclusions of Theorem 6.11 hold.

Before we give a proof of Proposition 6.14 we collect some crucial information about solutions of the mobility equation of rank 1:

Lemma 6.15. Suppose (M, J, g) is a connected (pseudo-)Kähler manifold of dimension 2n ≥ 4 admitting a non-parallel solution of the mobility equation A_{ab̄} such that A_{a}^{b} is a complex endomorphism of rank 1. Assume that g has nullity on some dense open set U ⊆ M and that there is a nonzero vector in the nullity distribution that is in the kernel of A_{a}^{b}. Denote by B the function defined as in (236) and let Λ_{a} = ∇_{a}λ with λ = −A_{a}^{a}. Then the following holds on a dense open subset U′ ⊆ U:

(1) the triple (A_{a}^{b}, Λ^{a}, μ) satisfies system (242) (and its conjugate) for some smooth nonvanishing real-valued function μ;

(2) A_{ab̄} = μ^{−1}Λ_{a}Λb̄ and −μλ = Λ_{a}Λ^{a};

(3) ∇_{a}B is proportional to Λ_{a}, and at any x ∈ U′ either ∇_{a}B = 0 or the nullity space of g at x lies in the kernel of A_{a}^{b}.

Proof. Statement (1) follows immediately from (132), Proposition 6.12 and the existence of a nullity vector in the kernel of A_{a}^{b}. Since A_{a}^{b} has rank 1, its nonzero eigenvalue is −λ, and Λ^{a} is a nonzero section of the corresponding eigenspace by Corollary 5.17. Thus on the dense open subset U′ ⊆ U where Λ^{a} ≠ 0, A_{a}^{b} = ξΛ_{a}Λ^{b}, with ξ = −λ/(Λ_{a}Λ^{a}), and differentiating this identity using (242) yields

(∇_{a}ξ + 2Bξ^{2}Λ_{a})Λc̄ = (ξμ − 1)g_{ac̄}.

Since the left hand side is simple and g_{ab̄} nondegenerate both sides must vanish, which shows that ξ = μ^{−1}, and hence (2) holds. The identity (245) may now be written

G_{ab̄c}^{d}Λ_{d} = 2(∇_{a}B)A_{cb̄} = 2μ^{−1}(∇_{a}B)Λ_{c}Λb̄

This immediately implies the second statement of (3), while the first statement follows from the symmetry of G_{ab̄c}^{d} in a and c. □

Proof of Proposition 6.14. We have already noted that to prove Proposition 6.14 it suffices to show that B is locally constant. By Lemma 6.15, A_{α}^{β} is of the form

A_{α}^{β} = \frac{1}{2μ}(Λ_{α}Λ^{β} + Ω_{γα}Λ^{γ}J_{δ}^{β}Λ^{δ}).  (259)

Let us write D ⊂ T M for the distribution defined by the kernel of A_{α}^{β} and

P_{α}^{β} = δ_{α}^{β} − \frac{1}{Λ_{γ}Λ^{γ}} (Λ_{α}Λ^{β} + Ω_{δα}Λ^{δ}J_{ζ}^{β}Λ^{ζ}) = δ_{α}^{β} + \frac{1}{2μλ}(Λ_{α}Λ^{β} + Ω_{δα}Λ^{δ}J_{ζ}^{β}Λ^{ζ})

for the orthogonal projection P: T M → D, where we use Lemma 6.15(2) to rewrite Λ_{γ}Λ^{γ} = 2Λ_{c}Λ^{c} = −2λμ. Note that (J, g) induces by restriction a complex structure J^{D} and a J^{D}-invariant metric g_{D} on D. The projection P also determines a linear connection on D by

∇_{α}^{D}X^{β} = P_{γ}^{β}∇_{α}X^{γ}, for X ∈ Γ(D)

which preserves this Hermitian structure on D. Since Λ and JΛ commute and preserve D, L_{Λ}P = 0 and Γ(D) is generated by sections commuting with Λ and JΛ, which are called basic. For any basic X ∈ Γ(D), Lemma 6.15(1) implies

∇_{Λ}X = ∇_{X}Λ = −μX  (260)

and hence for any other basic element Y ∈ Γ(D) we compute

(L_{Λ}g_{D})(X, Y) = L_{Λ}(g_{D}(X, Y)) = ∇_{Λ}(g_{D}(X, Y)) = −2μg_{D}(X, Y).

Since L_{Λ}λ = ∇_{Λ}λ = g(Λ, Λ) = −2λμ, it follows that L_{Λ}(λ^{−1}g_{D}) = 0. Let us now regard ∇^{D} as a partial connection on D, i.e. an operator ∇^{D}: Γ(D) → Γ(D^{∗} ⊗ D). Since ∇_{X}^{D}λ = ∇_{X}λ = 0 for any X ∈ Γ(D), the partial connection ∇^{D} preserves λ^{−1}g_{D}. Furthermore, its partial torsion, given by

∇_{X}^{D}Y − ∇_{Y}^{D}X − P ([X, Y]) for X, Y ∈ Γ(D),

vanishes. It follows that L_{Λ}∇^{D} is a section of D^{∗} ⊗ D^{∗} ⊗ D ≅ D^{∗} ⊗ D^{∗} ⊗ D^{∗}, which is symmetric in the first two entries and skew in the last two entries, which implies it vanishes identically. We conclude that L_{Λ}R^{D} = 0, where the horizontal curvature R^{D} of ∇^{D} is defined, for X, Y, Z ∈ Γ(D), by

R^{D}(X, Y)(Z) = ∇_{X}^{D}∇_{Y}^{D}Z − ∇_{Y}^{D}∇_{X}^{D}Z − ∇_{P ([X,Y])}^{D}Z.

For basic X, Y, Z ∈ Γ(D) we compute via (260) that

∇_{X}^{D}∇_{Y}^{D}Z = P (∇_{X}∇_{Y} Z) − \frac{1}{2λ}g(∇_{Y} Z, Λ)X − \frac{1}{2λ}g(∇_{Y} Z, JΛ)JX
∇_{∇_{X}^{D}Y}^{D} Z = P (∇_{∇_{X}^{Y}}Z) − \frac{1}{2λ}g(∇_{X}Y, Λ)Z − \frac{1}{2λ}g(∇_{X}Y, JΛ)JZ.

Using g(Y, Λ) = 0 = g(Y, JΛ) for Y ∈ Γ(D), we also obtain, for X ∈ Γ(D), that

g(∇_{X}Y, Λ) = μg(X, Y) and g(∇_{X}Y, JΛ) = μg(Y, JX),

from which we deduce, for (basic) X, Y, Z ∈ Γ(D), that

R^{D}(X, Y)Z = P (R(X, Y)Z) − \frac{μ}{2λ}S(X, Y)Z,  (261)

where S is the constant holomorphic sectional curvature tensor defined as in (235).

Let us write Ric^{D}(Y, Z) = trace(X ↦ R^{D}(X, Y)Z) and Ric^{P} (Y, Z) = trace(X ↦ P (R(X, Y)Z)) for the Ricci-type contractions of R^{D} and P (R(X, Y)Z). Via the inverse g_{D}^{−1} of g_{D}, we view Ric^{D} and Ric^{P} as endomorphism of D, from which viewpoint equation (261) implies that they are related as follows:

Ric^{D} = Ric^{P} − \frac{nμ}{λ} Id_{D} .  (262)

By assumption, at each point of a dense open subset, there is a vector V in D that lies in the nullity distribution of g. Inserting V into equation (261) yields

R^{D}(X, V)Z = (B − \frac{μ}{2λ})S(X, V)Z,  (263)

which implies that

λRic^{D}(V) = 2n(Bλ − \frac{μ}{2})V.  (264)

Set C := Bλ− \frac{μ}{2}. By (1) and (3) of Lemma 6.15 we see that ∇_{X}C = 0 for all X ∈ Γ(D) and that ∇_{JΛ}C = 0. Equation (264) shows that V is an eigenvector of λRic^{D} with eigenvalue 2C. Since L_{Λ}R^{D} = 0 and L_{Λ}(λg_{D}^{−1}) = 0, it follows that L_{Λ}(λRic^{D}) = 0, and hence ∇_{Λ}C = L_{Λ}C = 0 as well. Thus C is locally constant, which implies that B is locally constant by Lemma 6.15(1), and this completes the proof. □

Remark 6.7. Proposition 6.14 may alternatively be proved as follows. Starting with the usual equations

\begin{matrix}∇_{α}Λ_{β} = −μg_{αβ} + 2BA_{αβ} \\ ∇_{α}μ = 2BΛ_{α} \\ ∇_{α}λ = Λ_{α} \end{matrix} where \begin{matrix}A_{αβ} = \frac{Λ_{α}Λ_{β} + K_{α}K_{β}}{2μ} \\ K^{α} = J_{β}^{α}Λ^{β} \\ Λ_{α}Λ^{α} + 2λμ = 0 \end{matrix}  (265)

we may consider the new metric

g̃^{αβ} ≡ λg^{αβ} + (1 + 2μ)A^{αβ}  (266)

and verify from (265) that • L_{Λ}g̃^{αβ} = 0,

• for any v^{β} such that A_{αβ}v^{β} = 0, we have

R̃ic_{αβ}v^{β} = Ric_{αβ}v^{β} − (2B + \frac{nμ}{λ} + \frac{1}{2λ})v_{α},  (267)

where R̃ic_{αβ} is the Ricci tensor of g̃^{αβ}. Hence, if v^{β} is a nullity vector for g_{αβ} so that in addition Ric_{αβ}v^{β} = 2(n + 1)Bv_{α}, then

g̃^{αγ}R̃ic_{γβ}v^{β} = (n(2Bλ − μ) − \frac{1}{2})v^{α}.  (268)

Now, since L_{Λ}(g̃^{αγ}R̃ic_{γβ}) = 0, it follows that any eigenvalue of this endomorphism is preserved by the flow of Λ^{α}. Therefore,

0 = L_{Λ}(n(2Bλ − μ) − \frac{1}{2}) = n(2λL_{Λ}B + Λ^{α}(2B∇_{α}λ − ∇_{α}μ)) = 2nλL_{Λ}B.

But from (265) we see that 0 = ∇_{[α}∇_{β]}μ = 2(∇_{[α}B)Λ_{β]} whence ∇_{α}B = 0, as required.

The only drawback with this proof is that verifying (267), though straightforward, is computationally severe, whereas the corresponding identity (264) in the previous proof is more easily established. The previous proof may be seen as a limiting case of the reasoning just given. Specifically, for any constant c ≠ 0, consider the metric

g̃_{αβ} ≡ \frac{1}{λ}g_{αβ} + \frac{1}{λ^{2}} (1 + \frac{c}{μ})A_{αβ} with inverse g̃^{αβ} ≡ λg^{αβ} + (1 + \frac{μ}{c})A^{αβ}  (269)

to arrive at

R̃ic_{αβ}v^{β} = Ric_{αβ}v^{β} − (2B + \frac{nμ}{λ} + \frac{c}{λ})v_{α}  (270)

instead of (267), an equation in which one can take a sensible limit as c → 0 essentially to arrive at (264) instead of (268). The metrics (269) and their invariance L_{Λ}g̃_{αβ} = 0 can also be recognised in the previous proof. More precisely, the first equation from (265) can be expressed as L_{Λ}g_{αβ} = −2μg_{αβ} + 4BA_{αβ} or, more compactly, as

L_{Λ}(λ^{−1}g_{αβ}) = 4λ^{−1}BA_{αβ},

which implies, using our earlier terminology, that the metric λ^{−1}g_{αβ} restricted to D is invariant under the flow of Λ^{α}. We also observed in the previous proof that orthogonal projection P_{α}^{β} = δ_{α}^{β} + λ^{−1}A_{α}^{β} onto D is invariant under this flow. We are therefore led to invariance of the covariant quadratic form

P_{α}^{γ}P_{β}^{ϵ}λ^{−1}g_{γϵ} = λ^{−1}(g_{αβ} + λ^{−1}A_{αβ}),

which is the limit of (269) as c → 0 whilst the nondegenerate metric g̃_{αβ} is obtained by decreeing that the remaining vectors Λ^{α} and K^{α} at each point be orthogonal to D and each other and satisfy g̃_{αβ}Λ^{α}Λ^{β} = g̃_{αβ}K^{α}K^{β} = 2c. The metric (266) is the case that Λ^{α} and K^{α} are taken to be orthonormal. In any case, it follows that L_{Λ}g̃_{αβ} = 0. □

6.4. The standard tractor bundle for metric c-projective structures. The metric theory of the standard tractor bundle T turns out to be rather degenerate. For a metric c-projective structure (M, J, [∇]) induced by the Levi-Civita connection ∇ of a Kähler metric g, we have P_{ab} = 0 and so the standard tractor connection (52)–(53) is given by

∇_{a}^{T} (\begin{matrix}X^{b} \\ ρ \end{matrix}) = (\begin{matrix}∇_{a}X^{b} + ρδ_{a}^{b} \\ ∇_{a}ρ \end{matrix})  ∇T ; ā (\begin{matrix}X^{b} \\ ρ \end{matrix}) = (\begin{matrix}∇āX^{b} \\ ∇āρ − P_{āb}X^{b} \end{matrix}) .  (271)

The kernel ker D^{T} of the first BGG operator (59) consists of vector fields X^{b} with c-projective weight (−1, 0) which satisfy

∇_{a}X^{b} + ρδ_{a}^{b} = 0 ∇āX^{b} = 0  (272)

for some section ρ of E(−1, 0); then ρ = −\frac{1}{n}∇_{a}X^{a}, and, setting the torsion to zero in Proposition 3.3, (X^{a}, ρ) defines a parallel section for the tractor connection (271). This is similar to the projective case, with the following distinction: although the tensor η^{α} in Theorem 6.1 is projectively weighted, the bundle E(1) is canonically trivialised by a choice of metric; here, in contrast, it is the real line bundle E(1, 1) that enjoys such a trivialisation, and not the complex line bundle E(1, 0).

However, taking care to use (45) (see also Proposition 2.13), it follows that any solution of (272) satisfies

\begin{aligned}−δ_{b}^{c}∇āρ = (∇ā∇_{b} − ∇_{b}∇ā)X^{c} & = R_{āb}^{c}_{d}X^{d} + P_{āb}X^{c} \\ & = H_{āb}^{c}_{d}X^{d} − δ_{b}^{c}P_{ād}X^{d}, \end{aligned}  (273)

where H_{āb}^{c}_{d} = H_{ād}^{c}_{b} and H_{āb}^{b}_{d} = 0. We may rearrange this as

δ_{b}^{c}∇āρ = δ_{b}^{c}P_{ād}X^{d} − H_{āb}^{c}_{d}X^{d};  (274)

then the trace over b and c shows that ∇āρ = P_{āb}X^{b} (as in Proposition 3.3) and hence that H_{āb}^{c}_{d}X^{d} = 0. Following the projective case (Theorem 6.1), we lower an index in (273) and (274) to obtain

R_{ābc̄d}X^{d} = −g_{bc̄}∇āρ − P_{āb}Xc̄.  (275)

It follows that for any solutions (X^{a}, ρ) and (X̃^{a}, ρ̃) of (272),

R_{ābc̄d}X̃^{b}X^{d} = −X̃c̄∇āρ − P_{āb}X̃^{b}Xc̄ = −X̃c̄∇āρ − Xc̄∇āρ̃

and hence, by symmetry,

X_{[ā}∇_{c̄]}ρ̃ = X̃_{[c̄}∇_{ā]}ρ.

As in Theorem 6.1, by first taking X̃ = X, we conclude that there is a real function B, uniquely determined and smooth on the union of the open sets where some solution X^{a} of (272) is nonzero, such that for any solution (X^{a}, ρ) of (272),

∇āρ = P_{āb}X^{b} = 2BXā.  (276)

Theorem 6.16. Let (M, J, [∇]) be a connected c-projective manifold, where ∇ preserves a (pseudo-)Kähler metric g_{ab̄}. Suppose that dim ker D^{T} ≥ 2. Then there is a unique constant B such that any element of the kernel of D^{T} lifts to a parallel section of T for the connection

∇_{a} (\begin{matrix}X^{b} \\ ρ \end{matrix}) = (\begin{matrix}∇_{a}X^{b} + ρδ_{a}^{b} \\ ∇_{a}ρ \end{matrix})  ∇ā (\begin{matrix}X^{b} \\ ρ \end{matrix}) = (\begin{matrix}∇āX^{b} \\ ∇āρ − 2Bg_{āb}X^{b} \end{matrix}) .  (277)

Proof. By Proposition 3.3 and (276), it remains only to show that the smooth function B is actually a constant. Differentiating the equation 2Bg_{c̄b}X^{b} = ∇c̄ρ and using (272) gives

2(∇_{a}B)Xc̄ − 2Bg_{c̄a}ρ = ∇_{a}∇c̄ρ and 2(∇āB)Xc̄ = ∇ā∇c̄ρ.  (278)

With (46), the second equation of (278) implies that X_{[c̄}∇_{ā]}B = 0. Where there are two nonzero solutions X^{a} and X̃^{a}, it follows from (272) that the sections X^{a}, X̃^{b} ∈ Γ(M, T^{1,0}(−1, 0)) and therefore X^{[a}X̃^{b]} ∈ Γ(M, T^{2,0}(−2, 0)) are holomorphic. Consequently, U = {X^{[a}X̃^{b]} ≠ 0} is the complement of an analytic subvariety and is thus connected. On U we also have X̃_{[c̄}∇_{ā]}B = 0 whence ∇āB = 0, i.e. B is locally constant on U, hence constant. □

By analogy with the projective case, one might now expect c-projective nullity to appear. However, when we combine (275) and (276), we obtain

(R_{ābc̄d} + P_{āb}g_{c̄d} + 2Bg_{c̄b}g_{ād})X^{d} = 0,

which is a halfway house on the way to (237). Underlying this degeneracy is the fact that T is associated to a holomorphic representation of SL(n + 1, ℂ).

Nevertheless the constant B in Theorem 6.16 is generically characterised by cprojective nullity in the following degenerate sense.

Theorem 6.17. Let (M, J, g) be a connected (pseudo-)Kähler manifold admitting a non-parallel solution X^{a} of (272). For any function B, the following are equivalent:

(1) B is characterised by c-projective nullity (237) on a dense open subset;

(2) B is constant and X^{b} lifts to a section of T parallel for (277);

(3) P_{āb} = 2Bg_{āb}. In particular, g is an Einstein metric, and the connections (271) and (277) coincide.

Proof. (1)⇒(2). This follows from Theorem 6.11 because Xā ⊗ X^{b} is a solution of the mobility equation which is not a constant multiple of g^{āb}, and by contracting (273) by a nullity vector v^{b}. (2)⇒(3). The identity (45) implies

P_{āb}ρ = (∇ā∇_{b} − ∇_{b}∇ā)ρ = −2Bg_{āc}∇_{b}X^{c} = 2Bg_{āb}ρ,

which establishes (3) on the dense open subset {ρ ≠ 0}, hence everywhere. (3)⇒(1). Since ∇āρ = P_{ād}X^{d} = 2Bg_{ād}X^{d}, equation (273) implies

−2Bg_{ād}X^{d}δ_{b}^{c} = R_{āb}^{c}_{d}X^{d} + P_{āb}X^{c} = R_{āb}^{c}_{d}X^{d} + 2Bg_{āb}X^{c},  (279)

and we deduce that G_{āb}^{c}_{d}X^{d} = 0. Hence, X^{d}/ρ is a nullity vector for g on the dense open subset {ρ ≠ 0}. □

6.5. Special tractor connections and the complex cone. Let (M, J, [∇]) be a metric c-projective structure. Then for any compatible metric g and any function B, there is a special tractor connection on T defined by (277). We first observe that the induced connection on V_{ℂ} = \overline{T} ⊗ T is the special tractor connection (240) (for the given g and B). This can be seen easily by taking A^{b̄c} = \overline{X^{b}}X^{c}, Λ^{a} = ρ̄X^{a} and μ = ρ̄ρ in (240). Consequently, parallel sections for the special tractor connection on V define parallel Hermitian forms on T^{∗}. This was used in [26] to characterise, for Kähler manifolds (M, J, g), the presence of nontrivial parallel sections for (240) in terms of the local classification of [2] (see Section 5.7). Using the extension of this classification (pseudo-)Kähler manifolds [16], together with Remark 5.7, and Theorems 6.7 and 6.11, we have the following more general characterisation.

Theorem 6.18. Let (M, J, g) be a connected (pseudo-)Kähler manifold admitting a solution A of the mobility equation which is not parallel (i.e. Λ ≠ 0), and let B be a function on M. Then the following are equivalent.

(1) For the given function B, (J, g) has c-projective nullity on a dense open set.

(2) Any solution to the mobility equation lifts to a global parallel section for (240) with B constant.

(3) On a dense open subset of M, A lifts to a parallel section for (240) with B locally constant.

(4) Λ lies in the c-projective nullity of (J, g) with constant B.

(5) B is constant and its c-projective nullity distribution contains the complex span (tangent to the complex orbits) of the Killing vector fields of the pencil A+t Id.

(6) Near any regular point, (J, g) is given by (222), where for all j ∈ {1, … ℓ}, Θ_{j}(t) = Θ(t), a polynomial of degree ℓ + 1, with constant coefficients, leading coefficient 4B, and divisible by any constant coefficient factor of the minimal polynomial of A.

Proof. (1) ⇒ (2) by Theorem 6.11, (2) ⇒ (3) trivially, and (3) ⇒ (4) by Theorem 6.7, and (4) ⇒ (1) because Λ is nonzero on a dense open subset of M. Clearly (5) ⇒ (4), and conversely, (1)–(4) imply (244), i.e. G_{ab̄c}^{d} = R_{ab̄c}^{d} + 2B(g_{ab̄}δ_{c}^{d} + g_{cb̄}δ_{a}^{d}) commutes with A_{a}^{b}: therefore, since G_{ab̄c}^{d}Λ_{d} = 0 and Λ^{α} is the sum of the gradients of the nonconstant eigenvalues ξ_{1}, … ξ_{ℓ} of A_{a}^{b}, which are sections of the corresponding eigenspaces, it follows that grad_{g}ξ_{j} and Jgrad_{g}ξ_{j} are in the nullity for j ∈ {1, … ℓ}; this is the tangent distribution to the complex orbits.

Now (5) implies that the restriction of the metric to the complex orbits has constant holomorphic sectional curvature, and hence the functions Θ_{j}(t) are equal to a common polynomial Θ(t) of degree ℓ + 1. Now comparing (242b) with (225), we conclude that Θ(t) has leading coefficient a_{−1} = 4B, and that g_{u}(Θ(A)·, ·) = 0 for all irreducible constant coefficient factors ρ_{u} of χ_{A}; thus all constant coefficient factors of the minimal polynomial of A are also factors of Θ.

Conversely, given (6), (225) implies (242b) with B constant, and hence, by equation (245) of Lemma 6.6, and the Bianchi symmetry G_{ab̄c}^{d} = G_{cb̄a}^{d}, we have

Yb̄(∇_{a}μ − 2BΛ_{a}) = −G_{ab̄c}^{d}Λ_{d}Y^{c} = g_{ab̄}(∇_{c}μ − 2BΛ_{c})Y^{c}

for any (1, 0)-vector Y^{c}. Since the left hand side is degenerate, whereas g_{ab̄} is nondegenerate, we conclude that ∇_{a}μ = 2BΛ_{a}, thus establishing (3). □

Secondly, we observe that by Lemma 3.4, the special tractor connection on T induces a complex affine connection on the complex affine cone π_{C}: C → M described in Section 3.2. Combining these observations, as in Remark 4.6, any solution A^{b̄c} of the mobility equation which lifts to a parallel section of V for the special tractor connection on V, and is nondegenerate as a Hermitian form on T^{∗}, induces a Hermitian metric on C which is parallel for the induced complex affine connection on C.

In particular, if B is constant, then the metric g itself induces a parallel section of V with A^{b̄c} = g^{b̄c}, Λ^{a} = 0 and μ = 2B, which is clearly nondegenerate on T if and only if B ≠ 0.

Proposition 6.19. Let (M, J, [∇]) be a metric c-projective structure. Then any compatible metric g and any real constant B ≠ 0 induce a Hermitian metric on the complex affine cone C → M defined by the c-projective structure.

For B > 0, this Hermitian metric is, up to scale, a metric cone dr^{2} + r^{2}ĝ over a (pseudo-)Sasakian metric ĝ = g + (dψ + α)^{2} on a (local) circle bundle over (M, J, g) whose curvature dα is a multiple of the Kähler form ω of g (see e.g. [19]). In the present context, as observed by [78, 81], any solution of the mobility equation which lifts to parallel section for the special tractor connection (240) on V (with B ≠ 0 constant) induces a Hermitian (0, 2) tensor on M which is parallel for the cone metric of g, B or −g, −B. Combining this with Theorem 6.18, we have the following extension of one of the key results of [78] to the (pseudo-)Kähler case.

Theorem 6.20. Let (M, J, g) be a connected (pseudo-)Kähler manifold admitting a non-parallel solution of the mobility equation, and assume that there is a dense open subset U ⊆ M on which (J, g) has c-projective nullity. Then, perhaps after replacing g by a c-projectively equivalent metric, there is a nonzero constant B such that solutions of the mobility equation on M are in bijection with parallel Hermitian (0, 2) tensors on the cone C.

Proof. Under these assumptions, the equivalent conditions of Theorem 6.18 apply. Hence the constant 4B is the leading coefficient of the common polynomial Θ_{j}(t) = Θ(t) of degree ℓ + 1 appearing in (222), which therefore vanishes if and only if Θ has a root at infinity. However, by Remark 5.6, Θ transforms as a polynomial of degree ℓ + 1 over projective changes of pencil parameter, and since Θ is not identically zero, we may change the pencil parameter so that ∞ is not a root, while keeping the metrisability solution nondegenerate. We thus obtain a c-projectively equivalent metric with nullity constant B ≠ 0, and the rest follows from Theorem 6.18(2), Proposition 6.19 and the subsequent observations (above) applied to this metric. □

By Theorem 6.8, the hypotheses of this theorem are satisfied when (M, J, g) has mobility at least 3 and the compatible metrics are not all affinely equivalent. In the case that g is positive definite, this allowed V. Matveev and S. Rosemann [78] to obtain the following classification of the possible mobilities of Kähler metrics, using the de Rham (or de Rham–Wu) decomposition of the cone.

Theorem 6.21. Let (M, J, g) be a simply connected Kähler manifold of dimension 2n ≥ 4 admitting a non-parallel solution of the mobility equation. Then the mobility of (g, J) has the form k^{2} + ℓ for k, ℓ ∈ ℕ with 0 ≤ k ≤ n − 1, 1 ≤ ℓ ≤ (n + 1 − k)/2 and (k, ℓ) ≠ (0, 1), unless (g, J) has constant holomorphic sectional curvature. Furthermore, any such value arises in this way.

6.6. The c-projective Hessian, nullity, and the Tanno equation. Let (M, J, g) be a (pseudo-)Kähler manifold of dimension 2n ≥ 4 and denote by ∇ the Levi-Civita connection of g. Recall that for any solution A_{ab̄} of the mobility equation of g the gradient, respectively the skew gradient, of the function A_{a}^{a} = −λ is a holomorphic vector field, respectively a holomorphic Killing field, which is, by Corollary 5.5, equivalent to the real section σ = λvol(g)^{− \frac{1}{n+1}} ∈ E(1, 1) being in the kernel of the c-projective Hessian. With respect to the Levi-Civita connection and the trivialisation vol(g)^{− \frac{1}{n+1}} of E(1, 1), the c-projective Hessian equation reads as

∇_{a}∇_{b}λ = 0, respectively ∇ā∇b̄λ = 0.  (280)

In Section 4.7 we prolonged the c-projective Hessian equation and have seen that any (real) solution of this equation lifts to a unique section of the connection (178)–(179), which shows that (280) implies that the function λ = λ̄ satisfies also the following system of equations

\begin{matrix}∇_{a}∇_{b}∇_{c}λ = 0 ∇_{a}∇b̄∇_{c}λ = −P_{cb̄}∇_{a}λ − P_{ab̄}∇_{c}λ − H_{ab̄}^{d}_{c}∇_{d}λ \\ ∇ā∇_{b}∇_{c}λ = 0 ∇ā∇b̄∇_{c}λ = −P_{cb̄}∇āλ − P_{cā}∇b̄λ − H_{āc}d̄b̄∇d̄λ. \end{matrix}  (281)

Suppose now that g has mobility ≥ 2 and that g has nullity on a dense open subset of M. Then Theorem 6.11 shows that the function B defined as in (236), is actually constant and any solution A_{ab̄} lifts to a (real) section of V_{ℂ} for the connection (240). The connection (240) induces a connection on the dual vector bundle W_{ℂ} = V_{ℂ}^{∗}, which is given by

∇_{a}^{W_{ℂ}} (\begin{matrix}λ \\ μ_{b} | νb̄ \\ ζ_{bc̄} \end{matrix}) = (\begin{matrix}∇_{a}λ − μ_{a} \\ ∇_{a}μ_{b} | ∇_{a}νb̄ + 2Bg_{ab̄}λ − ζ_{ab̄} \\ ∇_{a}ζ_{bc̄} + 2Bg_{ac̄}μ_{b} \end{matrix})  (282)

∇W_{ℂ} ; ā (\begin{matrix}λ \\ μ_{b} | νb̄ \\ ζ_{bc̄} \end{matrix}) = (\begin{matrix}∇āλ − νā \\ ∇āμ_{b} + 2Bg_{āb}λ − ζ_{bā} | ∇āνb̄ \\ ∇āζ_{bc̄} + 2Bg_{āb}νc̄ \end{matrix}) ,  (283)

where W_{ℂ} is identified via g with a direct sum of unweighted tensor bundles. By Proposition 2.13 the two equations on the right-hand side of (281) can be also written as

∇_{a}∇b̄∇_{c}λ = −R_{ab̄}^{d}_{c}∇_{d}λ and ∇ā∇b̄∇_{c}λ = −R_{āc}d̄b̄∇d̄λ.

Comparison of these equations with (282) and (283), shows immediately that any function λ satisfying (280) lifts to a parallel section for the special connection (282)– (283) if and only if the gradient of λ lies in the nullity distribution of g. Note that, by Theorem 6.7, for any solution of the mobility equation A_{ab̄} the function A_{a}^{a} = −λ has this property. Moreover, in fact, the following Proposition holds:

Proposition 6.22. Let 0 ≠ B ∈ ℝ be some constant. Suppose λ be a smooth realvalued function and write Λ_{a} = ∇_{a}λ for its derivative. Then the following statements are equivalent:

(1)

G_{ab̄c}^{d}Λ_{d} = 0 and ∇_{a}Λ_{b} = 0

(2) λ lifts uniquely to a parallel section of the connection (282) and (283) (and B is characterised by (236))

(3)

∇_{a}∇_{b}Λ_{c} = 0 and ∇_{a}∇b̄Λ_{c} = −2B(Λ_{a}g_{cb̄} + Λ_{c}g_{ab̄})

∇ā∇_{b}Λ_{c} = 0 and ∇ā∇b̄Λ_{c} = −2B(Λāg_{cb̄} + Λb̄g_{cā}), (284)

where G_{ab̄cd̄} ≡ R_{ab̄cd̄} + 2B(g_{ab̄}g_{cd̄} + g_{cb̄}g_{ad̄}).

Proof. In the discussion above we have already observed that (1) is equivalent to (2). Let us now show that (1) is equivalent to (3), as shown also by Tanno [97, Prop. 10.3]. Suppose first that (1) holds. Then, obviously also the first two equations on the lefthand side of (284) hold. Moreover, we immediately deduce from (1) that

−2B(Λ_{a}g_{cb̄} + Λ_{c}g_{ab̄}) = R_{ab̄c}^{d}Λ_{d} = ∇_{a}∇b̄Λ_{c},  (285)

which shows that the first equation of the right-hand side of (284) holds. The conjugate of (285) implies the second equation of the right-hand side of (284), since ∇ā∇b̄∇_{c}λ = ∇ā∇_{c}∇b̄λ. Conversely, suppose now that (3) holds. Then, obviously the identity (285) is satisfied, which shows that G_{ab̄c}^{d}Λ_{d} = 0. Hence, it remains to show that Λ^{a} is a holomorphic vector field. From (284) we deduce that

∇_{a}∇_{b}∇c̄Λ_{d} = −2B((∇_{a}Λ_{b})g_{cd̄} + (∇_{a}Λ_{d})g_{bc̄}).

Since R_{ab}^{c}_{d} = 0 and R_{ab}c̄d̄ = 0, skewing in a and b yields

0 = −B((∇_{a}Λ_{d})g_{bc̄} − (∇_{b}Λ_{d})g_{ac̄})),

which implies 0 = −B(n − 1)∇_{a}Λ_{d}. □

Remark 6.8. If a function λ satisfies (1) for B = 0, then this is still equivalent to (2), and the equivalent statements (1) and (2) imply (3), but the implication from (3) to (1) is not necessarily true.

The system of equations (284) can also be written as

∇_{α}∇_{β}∇_{γ}λ = −B(2Λ_{α}g_{βγ} + g_{αβ}Λ_{γ} + g_{αγ}Λ_{β} − Ω_{αβ}J_{γ}^{δ}Λ_{δ} − Ω_{αγ}J_{β}^{δ}Λ_{δ}),  (286)

where Λ_{α} = ∇_{α}λ. Since on Kähler manifolds equation (286) (respectively (284)) was intensively studied by Tanno in [97], we refer to this equation as the Tanno equation.

7. Global results

We now turn to the global theory of (pseudo-)Kähler manifolds of mobility at least two. In Section 5.7 we presented a local classification [2, 16] which shows that such a (pseudo-)Kähler manifold (M, J, g) is locally a bundle of toric (in fact, “orthotoric”) (pseudo-)Kähler manifolds over a local product S of (pseudo-)Kähler manifolds. When M is compact, and g is positive definite, this is not far from being true globally.

Indeed, several simplifications occur. First, any compact smooth orthotoric Kähler 2ℓ-manifold is biholomorphic (though not necessarily isometric) to ℂℙ^{ℓ} [3]. Secondly, when g is positive definite, the Hermitian endomorphism A_{a}^{b} is diagonalisable, and the local classification (222) simplifies to give

g = ∑_{u} χ_{nc}(η_{u})g_{u} + ∑_{i=1}^{ℓ} \frac{Δ_{j}}{Θ_{j}(ξ_{j})}dξ_{j}^{2} + ∑_{j=1}^{ℓ} \frac{Θ_{j}(ξ_{j})}{Δ_{j}}(∑_{r=1}^{ℓ} σ_{r−1}(ξ̂_{j})θ_{r})^{2},

ω = ∑_{u} χ_{nc}(η_{u})ω_{u} + ∑_{r=1}^{ℓ} dσ_{r} ∧ θ_{r}, with dθ_{r} = ∑_{u}(−1)^{r}η_{u}^{ℓ−r}ω_{u}

where η_{u} are the (real) constant eigenvalues of A, while the nonconstant eigenvalues ξ_{j}, and functions Θ_{j}, are all real valued. Thirdly, the eigenvalues are globally ordered and do not cross. However, if η_{u} is a root of Θ_{j} (for some u, j) and g_{u} is a Fubini–Study metric on ℂℙ^{m_{u}}, it is possible to have ξ_{j} = η_{u} along a critical submanifold of ξ_{j}, in which case χ_{nc}(η_{u}) = 0 along that submanifold, and the corresponding factor of the base manifold S collapses. We thus have the following global description [3].

Theorem 7.1. Let (M, J, g) be a compact connected Kähler manifold admitting a cprojectively equivalent Kähler metric that generates a metrisability pencil η∘(A−t Id) of order ℓ. Then the blow-up of M, along the subvarieties where nonconstant and constant eigenvalues of A coincide, is a toric ℂℙ^{ℓ}-bundle over a complex manifold S covered by a product, over the distinct constant eigenvalues, of complete Kähler manifolds with integral Kähler classes.

Conversely, such complex manifolds do admit Kähler metrics of mobility at least two, but we refer to [3] for a more precise description. In particular, examples are plentiful, and have been used to construct explicit extremal Kähler metrics [4], including in particular, weakly Bochner-flat Kähler metrics [5].

As soon as we impose nullity—for instance, by requiring metrics of mobility at least three—this plenitude disappears, even in the (pseudo-)Kähler case: a compact connected (pseudo-)Kähler 2n-manifold satisfying the equivalent conditions of Theorem 6.18 is isometric to ℂℙ^{n}, equipped with a constant multiple of the Fubini–Study metric [44], and this rigidity result extends to compact orbifolds (see [26]).

In the remainder of this section, we focus on positive definite complete Kähler metrics, and begin by showing, in Section 7.1, that the rigidity result for Kähler metrics with nullity also obtains in this case. In Section 7.2, we then discuss the group of cprojective transformations and the Yano–Obata Conjecture. This was established for compact (pseudo-)Kähler metrics in [77], and here we show it also holds for complete Kähler metrics.

7.1. Complete Kähler metrics with nullity. In this section we prove the following.

Theorem 7.2. Let g be a complete Kähler metric on a connected complex manifold (M, J) of real dimension 2n ≥ 4 that has nullity on a dense open subset of M. Then any complete Kähler metric on (M, J) that is c-projectively equivalent to g, is affinely equivalent to g, unless there is a positive constant c ∈ ℝ such that the Kähler manifold (M, J, cg) is isometric to (ℂℙ^{n}, J_{can}, g_{F S}).

Since, by Theorem 6.8, a connected Kähler manifold (M, J, g) of degree of mobility at least 3 has nullity on a dense open set unless all c-projectively equivalent metrics are affinely equivalent to g, we obtain the following immediate corollary, which in the case of closed Kähler manifolds was proved in [44, Theorem 2].

Corollary 7.3. Let g be a complete Kähler metric on a connected complex manifold (M, J) of real dimension 2n ≥ 4 with mobility at least 3. Then any complete Kähler metric on (M, J) that is c-projectively equivalent to g, is affinely equivalent to g, unless there is a positive constant c ∈ ℝ such that the Kähler manifold (M, J, cg) is isometric to (ℂℙ^{n}, J_{can}, g_{F S}).

Remark 7.1. It is easy to construct a complete Kähler manifold (M, J, g) of nonconstant sectional holomorphic curvature with mobility ≥ 3 such that all complete Kähler metrics on (M, J) are affinely equivalent to g. Indeed, take the direct product

(M_{1}, g_{1}, J_{1}) × (M_{2}, g_{2}, J_{2}) × (M_{3}, g_{3}, J_{3})

of three Kähler manifolds. It is again a Kähler manifold (M, J, g) with complex structure J := J_{1} + J_{2} + J_{3} and Kähler metric g := g_{1} + g_{2} + g_{3}. Obviously, (M, J, g) has mobility ≥ 3, since c_{1}g_{1} + c_{2}g_{2} + c_{3}g_{3} is again a Kähler metric on (M, J) for any constants c_{1}, c_{2}, c_{3} > 0. Note also that all these Kähler metrics are affinely equivalent to g and that, if (M_{i}, g_{i}, J_{i}) is complete for i = 1, 2, 3 the metric c_{1}g_{1} + c_{2}g_{2} + c_{3}g_{3} is also complete for any constants c_{1}, c_{2}, c_{3} > 0.

Remark 7.2. Suppose that g̃ is (pseudo-)Kähler metric that is compatible with [∇^{g}], then we may write

vol(g̃) = e^{(n+1)f}vol(g) and τg̃ = e^{−f}τ_{g},  (287)

where f = \frac{1}{n+1} log|\frac{vol(g̃)}{vol(g)} |. We have seen in Section 2.1 that the Levi-Civita connections

˜∇ and ∇ of g̃ and g are related by (11) with Υ_{α} = ∇_{α}f. For later use, note that that ˜∇_{α}g̃_{βγ} = 0 implies that the derivative ∇_{α}f of f satisfies the equation

∇_{α}g̃_{βγ} = (∇_{α}f)g̃_{βγ} + g̃_{α(β}∇_{γ)}f − J_{α}^{δ}g̃_{δ(β}J_{γ)}^{ϵ}∇_{ϵ}f.  (288)

Thus ∇ = ˜∇, i.e. g and g̃ are affinely equivalent, if and only if ∇_{α}f = 0.

In order to prove Theorem 7.2 suppose that g is a complete Kähler metric on a complex connected manifold (M, J) of dimension 2n ≥ 4 with nullity on a dense open set. Further we may assume that g has mobility at least 2, since otherwise Theorem 7.2 is trivially satisfied. Then Theorem 6.11 implies that the function B defined as in (237) is actually a constant. We shall see that for B > 0, the theory of the Tanno equation implies that (M, J, g) has positive constant holomorphic sectional curvature. For B ≤ 0, we will show that any complete Kähler metric that is cprojectively equivalent to g, is necessarily affinely equivalent to g. The proof will make essential use of the following two Lemmas. Recall for this purpose that a geodesic which is orthogonal to a Killing vector field at one point is orthogonal to it at all points.

Lemma 7.4. Suppose (M, J, g) is a Kähler manifold of dimension 2n ≥ 4 with mobility ≥ 2 and nullity on a dense open subset of M. Let g̃ be a Kähler metric that is cprojectively equivalent to g and f the function defined as in (287), whose derivative Υ_{α} ≡ ∇_{α}f relates the Levi-Civita connections of g̃ and g as in (11). Consider a geodesic c = c(t) that is orthogonal to the canonical Killing fields of the pair (g, g̃), defined as in Theorem 5.11. Then, for the constant B ∈ ℝ defined as in Theorem 6.11, the function f(t) = f(c(t)) satisfies the following ordinary differential equation:

{... \atop f}(t) = −4Bg(ċ, ċ)ḟ(t) + 3ḟ(t)f̈(t) − (ḟ(t))^{3}.  (289)

Proof. By Corollary 6.4 and identity (26) we have

∇_{α}Υ_{β} = −2B̃g̃_{αβ} + 2Bg_{αβ} + \frac{1}{2}(Υ_{α}Υ_{β} − J_{α}^{γ}J_{β}^{δ}Υ_{γ}Υ_{δ}).  (290)

Differentiating (290) and inserting (288) yields

∇_{α}∇_{β}Υ_{γ} =

− 2B̃(Υ_{α}g̃_{βγ} + g̃_{α(β}Υ_{γ)} − J_{α}^{δ}g̃_{δ(β}J_{γ)}^{ϵ}Υ_{ϵ}) + \frac{1}{2}(∇_{α}(Υ_{β}Υ_{γ}) − ∇_{α}(J_{β}^{δ}J_{γ}^{ϵ}Υ_{δ}Υ_{ϵ})).

Substituting for −2B̃g̃ the expression (290) we therefore obtain

∇_{α}∇_{β}Υ_{γ} = (∇_{β}Υ_{γ})Υ_{α} − 2Bg_{βγ}Υ_{α} − \frac{1}{2}(Υ_{β}Υ_{γ} − J_{β}^{δ}J_{γ}^{ϵ}Υ_{δ}Υ_{ϵ})Υ_{α}
+ (∇_{α}Υ_{(β})Υ_{γ)} − 2Bg_{α(β}Υ_{γ)} − \frac{1}{2}(Υ_{α}Υ_{β}Υ_{γ} − J_{α}^{δ}Υ_{δ}Υ_{ϵ}J_{(β}^{ϵ}Υ_{γ)})

− J_{α}^{δ}(∇_{δ}Υ_{(β})J_{γ)}^{ζ}Υ_{ζ} − 2Bg_{δ(β}J_{γ)}^{ζ}Υ_{ζ} − \frac{1}{2}(Υ_{δ}Υ_{(β}J_{γ)}^{ζ}Υ_{ζ} − J_{δ}^{η}Υ_{η}Υ_{ϵ}J_{(β}^{ϵ}J_{γ)}^{ζ}Υ_{ζ}))

+ \frac{1}{2}(∇_{α}(Υ_{β}Υ_{γ}) − ∇_{α}(J_{β}^{δ}J_{γ}^{ϵ}Υ_{δ}Υ_{ϵ})).

Note that the determinant of the complex endomorphism A_{a}^{b} relating g and g̃ as in (133) is given by e^{−f}. Hence the canonical Killing field K̃^{β}(0) = J^{αβ}∇_{α} det A = J^{αβ}∇_{α}e^{−f}, defined as in (209), is proportional to J^{αβ}Υ_{α} = J^{αβ}(∇_{α}f) and we have Υ_{β}J_{α}^{β}ċ^{α} = 0. Thus, contracting the above equation with ċ^{α}ċ^{β}ċ^{γ} all terms involving the complex structure J disappear and we derive that f(t) satisfies the desired ODE. □

Lemma 7.5. Let g and g̃ be c-projectively equivalent metrics on a complex manifold (M, J) of dimension 2n ≥ 4. Let c be a geodesic of g that is orthogonal to all canonical Killing fields of (g, g̃) (as defined in Theorem 5.11). Then there is a reparametrisation ϕ such that c̃(t) = c(ϕ(t)) is a geodesic of g̃. The inverse τ of the reparametrisation ϕ satisfies the formula

\frac{d}{dt}f(t) = \frac{d}{dt} log|\frac{dτ}{dt} |,  (291)

where f(t) = f(c(t)) with f defined as in (287).

Proof. Let c be a geodesic of g that is orthogonal to the canonical Killing fields associated to (g, g̃). Then formula (11) for the difference of the Levi-Civita connections ˜∇ and ∇ implies

˜∇ċċ = Υ(ċ)ċ − Υ(Jċ)Jċ = Υ(ċ)ċ,

where the last identity follows from the fact that Υ_{β}J^{αβ} is proportional to a canonical Killing field, as explained in proof of Lemma 7.4. Hence, ˜∇ċċ is a multiple of ċ and therefore there exists a reparametrisation ϕ of c such that c̃(t) = c(ϕ(t)) is geodesic of g̃.

Differentiating c(t) = c̃(τ(t)), where τ denotes the inverse of ϕ, gives ċ(t) = τ̇(t)˙c̃(τ(t)) and hence

0 = ∇_{ċ(t)}ċ(t) = ∇_{ċ(t)}(τ̇(t)˙c̃(τ(t))) = τ̇(t)^{2}∇_{˙c̃(τ(t))} ˙c̃(τ(t)) + τ̈(t)˙c̃(τ(t))
= (τ̈(t) − τ̇(t)^{2}Υ(˙c̃(τ(t)))) ˙ c̃(τ(t)),

since c̃ is a geodesic of g̃. This implies that τ̈(t) = τ̇(t)^{2}Υ(˙c̃(τ(t))) = τ̇(t)Υ(ċ(t)), which is equivalent to (291), since Υ_{α} = ∇_{α}f. □

Proof of Theorem 7.2. Let (M, J, g) be a connected complete Kähler manifold with nullity on a dense open set. We may assume that (M, J, g) has mobility at least 2. Hence, by Theorem 6.11 any solution of the mobility equation lifts uniquely to a parallel section of the connection (240) with B constant. This in turn, by Theorem 6.7, shows that for any solution A^{b̄c} of the mobility equation of g the function λ = −A_{a}^{a} satisfies (1) of Proposition 6.22. Hence, by Proposition 6.22 and Remark 6.8 λ satisfies the Tanno equation (284). Tanno showed in [97] that on a complete connected Kähler manifold and for a constant B > 0, the existence of a nonconstant solution λ of the Tanno equation (284) implies that (M, J, g) has positive constant holomorphic sectional curvature, which in turn implies that (M, J, g) is actually closed and isometric to (ℂℙ^{n}, J_{can}, cg_{F S}) for some positive constant c. Since any metric that is c-projectively but not affinely equivalent to g gives rise to a non-parallel solution of the mobility equation of g and hence to a nonconstant solution of the Tanno equation, Theorem 7.2 holds provided B is positive.

It remains to consider the case that the constant B defined as in Theorem 6.11 is nonpositive. Let g̃ be another complete Kähler metric on (M, J), which is c-projectively equivalent to g. Denote by f again the function defined as in (287), which has the property that Υ_{α} = ∇_{α}f relates the Levi-Civita connections of g and g̃ as in (11). We will show that B ≤ 0 implies Υ_{α} ≡ 0, that is, g and g̃ are necessarily affinely equivalent.

Note first that, since the canonical Killing fields associated to (g, g̃) are Killing for both metrics by Theorem 5.11 and f is constructed in a natural way only from the pair (g, g̃), the local flows of the canonical Killing fields preserve f. Hence, the canonical Killing fields lie in the kernel of Υ. To show that Υ is identically zero, it therefore remains to show that Υ vanishes when inserting vector fields orthogonal to the canonical Killing fields.

Consider a parametrised geodesic c of g, which at one (and hence at all points) is orthogonal to the canonical Killing fields. Since g and g̃ are complete, c is defined for all times and τ from Lemma 7.5 is a diffeomorphism of ℝ. Without loss of generality we assume that τ̇ is positive, otherwise replace t by −t. By Lemma 7.5, the function τ: ℝ → ℝ satisfies (291), which we rewrite as

f(t) = log(τ̇(t)) + const_{0}.  (292)

Now let us consider equation (289) and set τ̇(t) = (p(t))^{−1}. Substituting (292) into (289) yields

\begin{matrix}... \\ p \end{matrix} = −4Bg(ċ, ċ)ṗ.  (293)

If B = 0, the equation simplifies to \begin{matrix}... \\ p \end{matrix} = 0 and its general solution is of the form

p(t) = C_{2}t^{2} + C_{1}t + C_{0},

where C_{i} is a real constant for i = 0, 1, 2. Hence, we get

τ(t) = ∫_{t_{0}}^{t} \frac{dξ}{C_{2}ξ^{2} + C_{1}ξ + C_{0}} + const.  (294)

If the polynomial p(t) = C_{2}t^{2} + C_{1}t + C_{0} has real roots (which is always the case if C_{2} = 0, C_{1} ≠ 0), then the integral starts to be infinite in finite time. If the polynomial has no real roots, but C_{2} ≠ 0, the function τ is bounded. Thus, the only possibility for τ to be a diffeomorphism is C_{2} = C_{1} = 0 implying τ̇ = \frac{1}{C_{0}} , which shows that f is constant along the geodesic c.

If B < 0, the general solution of equation (293) is

C + C_{+}e^{2\sqrt{−Bg(ċ,ċ)}·t} + C_{−}e^{−2\sqrt{−Bg(ċ,ċ)}·t},  (295)

for real constants C, C_{+} and C_{−}. Hence, τ is of the form

τ(t) = ∫_{t_{0}}^{t} \frac{dξ}{C + C_{+}e^{2\sqrt{−Bg(ċ,ċ)}ξ} + C_{−}e^{−2\sqrt{−Bg(ċ,ċ)}ξ}} + const.  (296)

If one of the constants C_{+}, C_{−} is not zero, the integral (296) is bounded from one side, or starts to be infinite in finite time. In both cases, τ is not a diffeomorphism of ℝ. The only possibility for τ: ℝ → ℝ to be a diffeomorphism is when C_{+} = C_{−} = 0, in which case τ̇ is constant implying f is constant along the geodesic c. Hence, in both cases (B = 0 and B < 0) the one form Υ_{α} = ∇_{α}f vanishes when inserting vector fields orthogonal to the canonical Killing fields. □

7.2. The Yano–Obata Conjecture for complete Kähler manifolds. For a Kähler manifold (M, J, g) let us write Isom(J, g), Aff(J, g) and CProj(J, g) for the group of complex isometries, the group of complex affine transformations (i.e. of complex diffeomorphisms preserving the Levi-Civita connection) and the group of c-projective transformations of (M, J, g) respectively. By definition of these groups we obtain the following inclusions

Isom(J, g) ⊆ Aff(J, g) ⊆ CProj(J, g)

and consequently we also have

Isom_{0}(J, g) ⊆ Aff_{0}(J, g) ⊆ CProj_{0}(J, g),

where subscript 0 denotes the connected component of the identity.

Recall that Lie groups of affine transformations of complete Riemannian manifolds are well understood; see for example [68]. As explained there, if a connected Lie group G acts on a simply-connected complete Riemannian manifold (M^{n}, g) by affine transformations, then there exists a Riemannian decomposition

(M^{n}, g) = (M_{1}^{n_{1}}, g_{1}) × (ℝ^{n_{2}}, g_{euc})

of (M^{n}, g) into a direct product of a Riemannian manifold (M_{1}^{n_{1}}, g_{1}) and a Euclidean space (ℝ^{n_{2}}, g_{euc}) such that G acts componentwise. Specifically, it acts on (M_{1}^{n_{1}}, g_{1}) by isometries and on (ℝ^{n_{2}}, g_{euc}) by affine transformations (i.e. compositions of linear isomorphisms and parallel translations). Note that this implies that for closed simplyconnected Riemannian manifolds one always has Isom_{0}(J, g) = Aff_{0}(J, g), which holds in fact for any closed (not necessarily simply-connected) Riemannian manifold; see [102, Theorem 4]. If in addition (M^{n}, g) is Kähler for a complex structure J, and G is a connected Lie group of complex affine transformations, then (M_{1}^{n_{1}}, J_{1}, g_{1}) and (ℝ^{n_{2}}, J_{can}, g_{euc}) are also Kähler; furthermore, G acts on (M_{1}^{n_{1}}, J_{1}, , g_{1}) by complex isometries and on (ℝ^{n_{2}}, J_{can}, g_{euc}) by complex affine transformations.

For the complex projective space ℂℙ^{n} equipped with its natural complex structure J_{can} and the Fubini–Study metric g_{F S} we have Aff(J_{can}, g_{F S}) ≠ CProj(J_{can}, g_{F S}). To see this recall from the introduction that the J-planar curves of (ℂℙ^{n}, J_{can}, g_{F S}) are precisely those smooth regular curves that lie within complex lines. Moreover, recall that any complex linear isomorphism of ℂ^{n+1} induces a complex transformation of ℂℙ^{n} that sends complex lines to complex lines and hence induces a c-projective transformation of (ℂℙ^{n}, J_{can}, g_{F S}) by Proposition 2.2. In fact, Proposition 2.10 shows that all c-projective transformations of (ℂℙ^{n}, J_{can}, g_{F S}) arise in this way, i.e. CProj(J_{can}, g_{F S}) can be identified with the connected Lie group PGL(n + 1, ℂ) ≅ PSL(n + 1, ℂ). Note that an element in GL(n + 1, ℂ) induces a complex isometry on (ℂℙ^{n}, J_{can}, g_{F S}) if and only if it is proportional to a unitary automorphism of ℂ^{n+1}, which shows that Isom(J_{can}, g_{F S}) can be identified with the connected Lie group PU(n+1) = U(n+1)/U(1) of projective unitary transformations. For (ℂℙ^{n}, J_{can}, g_{F S}) we also clearly have Isom(J_{can}, g_{F S}) = Aff(J_{can}, g_{F S}).

In [77] in conjunction with [44] it was shown that any closed connected Kähler manifold (M, J, g) of dimension 2n ≥ 4 with the property that CProj_{0}(J, g) contains Isom_{0}(J, g) = Aff_{0}(J, g) as a proper subgroup is actually isometric to (ℂℙ^{n}, J, cg_{F S}) for some positive constant c. This rigidity result answers affirmatively in the case of closed Kähler manifolds the so-called Yano–Obata Conjecture, which is a c-projective analogue of the Projective and Conformal Lichnerowicz–Obata Conjectures; see the introductions of the papers [74, 77] for a historical overview. Most recently the conjecture has been proved for closed (pseudo-)Kähler manifolds of all signatures in [16]. We now show that the Yano–Obata Conjecture also holds for complete connected Kähler manifolds.

Theorem 7.6 (Yano–Obata Conjecture). Let (M, g, J) be a complete connected Kähler manifold of real dimension 2n ≥ 4. Then Aff_{0}(g, J) = CProj_{0}(g, J), unless (M, g, J) is actually compact and isometric to (ℂℙ^{n}, J_{can}, cg_{F S}) for some positive constant c ∈ ℝ.

Remark 7.3. In the projective case, a stronger version of the analogous Lichnerowicz– Obata result has recently been established [76]: on a complete Riemannian manifold (M, g) of dimension n ≥ 2, the quotient of the projective group Proj(g) by the affine group Aff(g) has at most two elements unless (M, g) has constant positive sectional curvature. It would be natural to establish such a result in the c-projective case.

7.3. The proof of the Yano–Obata Conjecture. Note that if the mobility of (M, J, g) is 1, then any metric g̃ that is c-projectively equivalent to g is homothetic to g. In particular, any c-projective transformation has to preserve the Levi-Civita connection of g and hence Aff_{0}(g, J) = CProj_{0}(g, J) in this case. On the other hand, since the pullback of a complete Kähler metric by a c-projective transformation is again a complete Kähler metric, Theorem 7.6 follows from Corollary 7.3 or Theorem 7.2 in the case that g has mobility ≥ 3 or has mobility ≥ 2 and nullity on a dense open set.

For the rest of this section we will therefore assume that (M, J, g) is a connected complete Kähler manifold of dimension 2n ≥ 4 with mobility 2. To show that Theorem 7.6 holds in this case (which for closed Kähler manifolds was proved in [77]), let us write Sol(g) for the 2-dimensional solution space of the mobility equation of g, which we view as a linear subspace of the space of J-invariant sections in S^{2}T M(−1, −1).

Suppose now that Aff_{0}(g, J) does not coincide with CProj_{0}(g, J). Then there exists a complete c-projective vector field V that is not affine. Since the flow Φ_{t} of V acts on (M, J, g) by c-projective transformations, for any t ∈ ℝ and for any η ∈ Sol(g) the pullback Φ_{t}^{∗}η is an element of the vector space Sol(g). Hence, the Lie derivative L_{V} η = \frac{d}{dt}|_{t=0}Φ_{t}^{∗}η can also be identified with an element of Sol(g), which implies that L_{V} induces a linear endomorphism of Sol(g). By the Jordan normal form, in a certain basis η, η̃ ∈ Sol(g), the linear endomorphism L_{V} : Sol(g) → Sol(g) corresponds to a matrix of one of the following three forms:

( \begin{matrix}a & 0 \\ 0 & b \end{matrix} ) ( \begin{matrix}a & b \\ −b & a \end{matrix} ) ( \begin{matrix}a & 1 \\ 0 & a \end{matrix} ) ,  (297)

where a, b ∈ ℝ. We will deal with these three cases separately. The last two cases are easy and will be considered in Section 7.3.1 The challenging case is the first one, which will be treated in Section 7.3.2.

7.3.1. L_{V} has complex-conjugate eigenvalues or a nontrivial Jordan block. Suppose first that the endomorphism L_{V} : Sol(g) → Sol(g) has two complex-conjugated eigenvalues. Hence, in some basis η, η̃ ∈ Sol(g) the endomorphism L_{V} corresponds to a matrix of the second type in (297). If b = 0, then V acts by homotheties on any element in Sol(g) and hence preserves in particular the Levi-Civita connection of g, which contradicts our assumption that V is not affine. Therefore, we can assume that b ≠ 0. Then the evolution of the solutions η, η̃ along the flow Φ_{t} of V is given by

Φ_{t}^{∗}η = e^{at} cos(bt)η + e^{at} sin(bt)η̃ Φ_{t}^{∗}η̃ = −e^{at} sin(bt)η + e^{at} cos(bt)η̃ .

Write g^{−1}vol(g)^{\frac{1}{n+1}} ∈ Sol(g) as cη + dη̃ for some real constants c and d. Then one has

Φ_{t}^{∗}(cη + dη̃) = c(e^{at} cos(bt)η + e^{at} sin(bt)η̃)
+d(−e^{at} sin(bt)η + e^{at} cos(bt)η̃)
= e^{at}\sqrt{c^{2} + d^{2}}(cos(bt + α)η + sin(bt + α)η̃),

where α = arccos(\frac{c}{\sqrt{c^{2}+d^{2}}} ). Since g is a Riemannian metric, for any point x ∈ M there exists a basis of T_{x}M in which η and η̃ are diagonal matrices. Hence, in this basis the i-th entry of Φ_{t}^{∗}(cη + dη̃) is given by e^{at}\sqrt{c^{2} + d^{2}}(cos(bt + α)e_{i} + sin(bt + α)ẽ_{i}), where e_{i} and ẽ_{i} are the i-th diagonal entries of η and η̃. Therefore, we see that Φ_{t}^{∗}(cη + dη̃) is degenerate for some t, which contradicts the fact that cη + dη̃ = g^{−1}vol(g)^{\frac{1}{n+1}} is nondegenerate. The obtained contradiction shows that L_{V} can not have two complexconjugate eigenvalues.

Suppose now that L_{V} : Sol(g) → Sol(g) is with respect to some basis η, η̃ ∈ Sol(g) a matrix of the third type in (297). Then the evolution of η and η̃ along Φ_{t} is given by

Φ_{t}^{∗}η = e^{at}η + te^{at}η̃
Φ_{t}^{∗}η̃ = e^{at}η̃.

We assume again that g^{−1}vol(g)^{\frac{1}{n+1}} = cη + dη̃. Then we obtain

\begin{aligned}Φ_{t}^{∗}(cη + dη̃) & = c(e^{at}η + e^{at}tη̃) + d(e^{at}η̃) \\ & = e^{at}(cη + (d + ct)η̃). \end{aligned}

Hence, we see again that for c ≠ 0 there exists t such that Φ_{t}^{∗}(cη + dη̃) is degenerate which contradicts the fact that g is nondegenerate. Now, if c = 0, then Φ_{t} acts by homotheties on g, which contradicts our assumption that V is not affine. Thus L_{V} can also not be of the third type in (297).

7.3.2. L_{V} has two real eigenvalues. Now we consider the remaining case, namely the one where L_{V} has two different real eigenvalues a and b (the case where the two eigenvalues are equal was already excluded in the previous section). Without loss of generality, we can assume that at least one of the eigenvalues is positive, since we can otherwise just replace V by −V. Hence, we can assume without loss of generality that a > b and that a > 0. Since ϕ_{t}^{∗}η = e^{at}η and ϕ_{t}^{∗}η̃ = e^{bt}η̃, we see that neither η nor η̃ can equal g^{−1}vol(g)^{\frac{1}{n+1}} , since otherwise ϕ_{t} acts by homotheties on g, which contradicts our assumption that V is not affine. Hence, g^{−1}vol(g)^{\frac{1}{n+1}} = cη + dη̃ for constant c, d ≠ 0. By rescaling η and η̃, we can therefore assume without loss of generality that

g^{αβ}vol(g)^{\frac{1}{n+1}} = η^{αβ} + η̃^{αβ}.

Let us write D^{αβ} = η^{αβ}vol(g)^{− \frac{1}{n+1}} and D̃^{αβ} = η̃^{αβ}vol(g)^{− \frac{1}{n+1}} such that

g^{αβ} = D^{αβ} + D̃^{αβ}.

Note that for a Kähler manifold (M, J, g) of mobility 2, the dense open subset of regular points (Definition 5.4) does not depend on the choice of non-proportional cprojectively equivalent metrics from the c-projective class of g. In particular, the set of regular points is invariant under c-projective transformations and hence under the action of the flow of a c-projective vector field. Thus if we fix a regular point x_{0} ∈ M and consider the integral curve ϕ_{t}(x_{0}) of our complete c-projective vector field V through x_{0}, then there is an open neighbourhood U of the curve ϕ_{t}(x_{0}) in the set of regular points, and, since g is positive definite, a frame of T U, in which g corresponds to the identity matrix and D and D̃ to diagonal matrices:

D = ( \begin{matrix}d_{1} \\ & d_{1} \\ & & . & . & . \\ & & & & & d_{n} \\ & & & & & & d_{n} \end{matrix} ) , D̃ = ( d̃_{1} ; d̃_{1} ; ... ; d̃_{n} ; d̃_{n} ) ,  (298)

where d_{i} and d̃_{i} are smooth real-valued functions on U such that d_{i} + d̃_{i} = 1 for i = 1, …, n. Then in the local frame the tensor A_{t}^{αβ} = ϕ_{t}^{∗}(η^{αβ} + η̃^{αβ})vol(g)^{− \frac{1}{n+1}} corresponds to the following diagonal matrix:

A_{t} = ( \begin{matrix}e^{at}d_{1} + e^{bt}d̃_{1} \\ & e^{at}d_{1} + e^{bt}d̃_{1} \\ & & . & . & . \\ & & & & & e^{at}d_{n} + e^{bt}d̃_{n} \\ & & & & & & e^{at}d_{n} + e^{bt}d̃_{n} \end{matrix} ) . (299)

Since g and ϕ_{t}^{∗}g are positive definite, all diagonal entries of (299) are positive for all t ∈ ℝ. Hence, d_{i} + e^{(b−a)t}d̃_{i} > 0 respectively e^{(a−b)t}d_{i} + d̃_{i} > 0 for all t and taking the limit t → ∞ respectively t → −∞ shows that d_{i}, d̃_{i} ≥ 0 for all i = 1, …, n. Since d_{i} + d̃_{i} = 1, we conclude that

0 ≤ d_{i} ≤ 1 and 0 ≤ d̃_{i} ≤ 1 for all i = 1, …, n.

Now consider the (1, 1)-tensor field D_{α}^{β} = g_{αγ}η^{γβ}vol(g)^{− \frac{1}{n+1}} and its pullback ϕ_{t}^{∗}(D_{α}^{β}) = ϕ_{t}^{∗}(g_{αγ}η^{γβ}vol(g)^{− \frac{1}{n+1}} ). Since g_{αβ}vol(g)^{− \frac{1}{n+1}} is inverse to η^{αβ} + η̃^{αβ}, we conclude that ϕ_{t}^{∗}(D_{α}^{β}) is given by a block diagonal matrix whose i-th block is given by the following 2 × 2 matrix

\frac{e^{at}d_{i}}{e^{at}d_{i} + e^{bt}(1 − d_{i})} Id_{2} .  (300)

By definition ϕ_{t} acts on the endomorphism D_{α}^{β} as ϕ_{t}^{∗}(D_{α}^{β}) = (T ϕ_{t})^{−1} ∘ D_{α}^{β} ∘ T ϕ_{t}, which implies that the eigenvalues of ϕ_{t}^{∗}(D_{α}^{β}) at a point x ∈ M are the same as the eigenvalues of D_{α}^{β} at ϕ_{t}(x). Therefore, it follows from (300) that the only possible constant eigenvalues of D_{α}^{β} on U are 0 and 1 (i.e. the only possible constant diagonal entries in (298) are 0 or 1). Note that d_{i} = 0 (respectively d_{i} = 1) on some open set implies that d̃_{i} = 1 (respectively d̃_{i} = 0). Hence, the only possible constant eigenvalues of A_{t} defined as in (299) are e^{at} and e^{bt}. Since U consists of regular points, the distinct eigenvalues of A_{t} (for any fixed t) are smooth real-valued functions with constant algebraic multiplicities on U. Let us write 2m, respectively 2m̃, for the multiplicity of the eigenvalues e^{a} and e^{b} of A_{1} on U. By Lemma 5.16, the number of distinct nonconstant eigenvalues of A_{1} is given by n − m − m̃, and m, m̃ are constant on the set of regular points by Corollary 5.17. We allow, of course, that m, respectively m̃, are zero.

Lemma 7.7. If at least one of the following two inequalities,

(n − m̃)a + (m̃ + 1)b ≤ 0 and (m + 1)a + (n − m)b ≥ 0.  (301)

is not satisfied, then the vector field Λ^{α} given by the gradient of λ = −\frac{1}{2}D_{β}^{β} lies in the nullity space of M at x_{0}.

Proof. Set G_{t} := det_{ℝ}(A_{t})^{− \frac{1}{2}} A_{t}^{−1} and note that ϕ_{t}^{∗}g = g(G_{t}·, ·). We may assume that the first 2ℓ := 2n − 2m − 2m̃ elements of D are not constant (which is equivalent to assuming that d_{i}(x_{0}) ≠ 0, 1 for i = 1, …, ℓ), the next 2m elements are equal to 1, and the remaining 2m̃ elements are zero on U. Then, we deduce from (299) that G_{t} on U is a block diagonal matrix of block sizes 2ℓ × 2ℓ, 2m × 2m and 2m̃ × 2m̃ respectively, where the three blocks are given by

Ψ(t) ( \frac{1}{d_{1}e^{at}+(1−d_{1})e^{bt}} ; \frac{1}{d_{1}e^{at}+(1−d_{1})e^{bt}} ; ... ; \frac{1}{d_{ℓ}e^{at}+(1−d_{ℓ})e^{bt}} ; \frac{1}{d_{ℓ}e^{at}+(1−d_{ℓ})e^{bt}} ) (302)

Ψ(t)e^{−at} Id_{2m}, respectively Ψ(t)e^{−bt} Id_{2m̃},

where

Ψ(t) := e^{−amt}e^{−bm̃t} ∏_{i=1}^{ℓ} \frac{1}{d_{i}e^{at} + (1 − d_{i})e^{bt}} .

Let us write ν_{1}, …, ν_{ℓ}, ν and ν̃ for the eigenvalues of these respective diagonal matrices. Note that their asymptotic behaviour for t → +∞ respectively for t → −∞ is as follows

\begin{matrix}t → +∞ & ν_{i}(t) ∼ \frac{e^{−((n−m̃+1)a+m̃b)t}}{d_{i} ∏ d_{j}} & ν(t) ∼ \frac{e^{−((n−m̃+1)a+m̃b)t}}{∏ d_{j}} & ν̃(t) ∼ \frac{e^{−((n−m̃)a+(m̃+1)b)t}}{∏ d_{j}} \\ t → −∞ & ν_{i}(t) ∼ \frac{e^{(ma+(n−m+1)b)t}}{(1−d_{i}) ∏(1−d_{j})} & ν(t) ∼ \frac{e^{((m+1)a+(n−m)b)t}}{∏(1−d_{j})} & ν̃(t) ∼ \frac{e^{(ma+(n−m+1)b)t}}{∏(1−d_{j})} . \end{matrix}

(303)

Let us now assume that (301) is not satisfied. We can assume without loss of generality that the first inequality of (301) is not satisfied, that is to say we can assume that (n − m̃)a + (m̃ + 1)b > 0, since otherwise we can change the sign of V, which interchanges the inequalities. Then it follows from (303) that all eigenvalues of G_{t} decay exponentially as t → ∞. Consider now the sequence (ϕ_{k}(x_{0}))_{k∈ℤ_{≥0}}. We claim that it is a Cauchy sequence. Indeed, note that the distance d(ϕ_{k}(x_{0}), ϕ_{k+1}(x_{0})) between ϕ_{k}(x_{0}) and ϕ_{k+1}(x_{0}) satisfies

d(ϕ_{k}(x_{0}), ϕ_{k+1}(x_{0})) ≤ ∫_{0}^{1} \sqrt{g(V (ϕ_{k+t}(x_{0})), V (ϕ_{k+t}(x_{0})))}dt  (304)
= ∫_{0}^{1} \sqrt{(ϕ_{k}^{∗}g)(V (ϕ_{t}(x_{0})), V (ϕ_{t}(x_{0})))}dt.

Since ϕ_{k}^{∗}g = g(G_{k}·, ·) and all eigenvalues of G_{t} decay exponentially as t → ∞, the inequality (304) shows that d(ϕ_{k}(x_{0}), ϕ_{k+1}(x_{0})) decays geometrically as k → ∞ (i.e. for sufficiently large k we have d(ϕ_{k}(x_{0}), ϕ_{k+1}(x_{0})) ≤ const q^{k} for some q < 1). Hence, (ϕ_{k}(x_{0}))_{k∈ℤ_{≥0}} is a Cauchy sequence and completeness of M implies that (ϕ_{k}(x_{0}))_{k∈ℤ_{≥0}} converges. We denote the limit of (ϕ_{k}(x_{0}))_{k∈ℤ_{≥0}} by p ∈ M. Now consider the smooth real-valued function F on M given by

F = H_{αβ}^{γ}_{δ}H_{ϵζ}^{θ}_{η}g_{γθ}g^{αϵ}g^{βζ}g^{δη},

where H_{αβ}^{γ}_{δ} denotes the harmonic curvature of (M, J, [∇^{g}]) defined as in Proposition 4.4, respectively Proposition 2.13. Since H_{αβ}^{γ}_{δ} is c-projectively invariant, we deduce that (ϕ_{k}^{∗}F)(x_{0}) = F (ϕ_{k}(x_{0})) equals

F (ϕ_{k}(x_{0})) = (H_{αβ}^{γ}_{δ}H_{ϵζ}^{θ}_{η} ϕ_{k}^{∗}g_{γθ} ϕ_{k}^{∗}g^{αϵ} ϕ_{k}^{∗}g^{βζ} ϕ_{k}^{∗}g^{δη})(x_{0}).  (305)

Moreover, since F is continuous, we have lim_{k→∞} F (ϕ_{k}(x_{0})) = F (p).

Since in the frame we are working the matrices corresponding to g and G_{t} are diagonal, we see that the function F (ϕ_{t}(x_{0})) is a sum of the form

∑_{1≤i,j,k,ℓ≤2n} C(ijkℓ; t) (H_{α_{i}α_{j}}^{α_{k}}_{α_{ℓ}}(x_{0}))^{2} ,  (306)

where the coefficient C(ijkℓ; t) is the product of the k-th diagonal entry and the reciprocals of the i-th, j-th and ℓ-th diagonal entry of the diagonal matrix that corresponds to G_{t} (in our chosen frame). The coefficients C(ijkℓ; t) depend on t and their asymptotic behaviour for t → ±∞ can be read off from (303). Note moreover that all coefficients C(ijkℓ; t) are positive.

We claim that, if at least one of the indices i, j or ℓ is less or equal than 2n−2m−2m̃, then H_{α_{i}α_{j}}^{α_{k}}_{α_{ℓ}}(x_{0}) vanishes. Indeed, note that, by (303), ϕ_{t}^{∗}g decays exponentially at least as e^{−((n−m̃+1)a+m̃b)t}, which is up to a constant the smallest eigenvalue of G_{t}, and ϕ_{t}^{∗}g^{−1} goes exponentially to infinity at least as e^{((n−m̃)a+(m̃+1)b)t} as t → ∞. Suppose now that at least one of the indices i, j or ℓ is less or equal than 2n − 2m − 2m̃. Then we deduce that up to multiplication by a positive constant C(ijkℓ; t) behaves asymptotically as t → ∞ at least as

e^{((n−m̃)a+(m̃+1)b)t}e^{((n−m̃)a+(m̃+1)b)t}e^{((n−m̃+1)a+m̃b)t}e^{−((n−m̃+1)a+m̃b)t} = e^{2((n−m̃)a+(m̃+1)b)t}.

Since (n−m̃)a+(m̃+1)b > 0 by assumption, we therefore conclude that the coefficient

C(ijkℓ; t) → ∞ as t → ∞.

Since all terms in the sum (306) are nonnegative and the sequence F (ϕ_{ℓ}(x_{0})) converges, we therefore deduce that H_{α_{i}α_{j}}^{α_{k}}_{α_{ℓ}}(x_{0}) = 0 provided that at least one of the indices i, j or ℓ is less or equal than 2n − 2m − 2m̃.

Observe now that Λ^{α} equals the negative of the sum of the gradients of the distinct nonconstant eigenvalues d_{1}, …, d_{n−m−m̃} of D. We therefore conclude that at x_{0}

H_{αβ}^{γ}_{δ}Λ^{α} = 0, H_{αβ}^{γ}_{δ}Λ^{β} = 0 and H_{αβ}^{γ}_{δ}Λ^{δ} = 0.

It follows that (3) of Remark 6.3 is satisfied, which implies that the vector field Λ^{α} lies in the nullity space of M at x_{0}. □

Since x_{0} is an arbitrary regular point and the conditions (301) do not depend on the choice of regular point, Lemma 7.7 shows that, if the inequalities (301) are not satisfied, then (M, J, g) has nullity on the dense open subset of regular points, since Λ^{α} does not vanish at regular points (otherwise V is necessarily affine, which contradicts our assumption). Hence, by Theorem 7.2, we have established Theorem 7.6, except when the inequalities (301) are satisfied.

Therefore, assume now that the inequalities (301) are satisfied. Subtracting the second inequality from the first shows that

(n − m − m̃ − 1)(a − b) ≤ 0.

Since a − b > 0 by assumption, we must have n − m − m̃ = 0 or n − m − m̃ = 1. In the first case ϕ_{t}^{∗}g is parallel for the Levi-Civita connection of g for all t and hence V is affine, which contradicts our assumption. Therefore, we must have n − m − m̃ = 1. Now substituting this identity back into (301) shows that (m + 1)a + (m̃ + 1)b ≤ 0 and (m + 1)a + (m̃ + 1)b ≥ 0, which implies (m + 1)a = −(m̃ + 1)b. Hence, we conclude that we must have

n − m − m̃ = 1 and (m + 1)a = −(m̃ + 1)b.  (307)

Therefore, locally around any regular point D has precisely one nonconstant eigenvalue, which we denote by ρ, and the constant eigenvalues 1 and 0 with multiplicity 2m, respectively 2m̃. Hence, for any regular point x_{0} ∈ M we can find an open neighbourhood U of the curve ϕ_{t}(x_{0}) in the set of regular points and a frame of T U such that D corresponds to a matrix of the form

D = ( \begin{matrix}ρ \\ & ρ \\ Id & 2m \\ & & & 0_{2m̃} \end{matrix} ) ,  (308)

where ρ is a smooth function on U with ρ(x) ≠ 0, 1 for x ∈ U. Note that we have λ = −\frac{1}{2}D_{α}^{α} = −(ρ + m) and consequently

Λ_{α} = −∇_{α}ρ,

which is nowhere vanishing on U. Recall also that the pair (D, Λ) satisfies the mobility equation (131), and that by Corollary 5.17 Λ^{α} is an eigenvector of D_{α}^{β} with eigenvalue ρ. Hence, Λ^{α} and J_{β}^{α}Λ^{β} form a basis for the eigenspace of D_{α}^{β} corresponding to the eigenvalue ρ.

For later use, let us also remark that by (300) the action of the flow ϕ_{t} on D preserves its block structure (308). This implies, in particular, that L_{V} Λ = [V, Λ] is an eigenvector of D with eigenvalue ρ. Since furthermore ∇_{α}ρ = −Λ_{α} vanishes in direction of all vector fields orthogonal to Λ^{α} and [Λ, JΛ] = 0, we therefore conclude that [V, JΛ] · ρ = 0 implying that the vector field [V, JΛ] = J[V, Λ] is actually a proportional to JΛ. Hence, the c-projective vector field V preserves the orthogonal projection from T M to the 1-dimensional subspace spanned by Λ^{α}, which is defined on the dense open subset on which Λ^{α} is not vanishing (hence in particular on U).

Lemma 7.8. Assume n − m − m̃ = 1 and (m + 1)a = −(m̃ + 1)b. Then in a neighbourhood of any regular point there exists a real positive constant B and a smooth real-valued function μ such that

∇_{α}Λ_{β} = −μg_{αβ} + 2BD_{αβ}  (309)
∇_{α}μ = 2BΛ_{α}.  (310)

Proof. Fix a regular point x_{0} ∈ M, an open neighbourhood U of ϕ_{t}(x_{0}) in the set of regular points, and a frame of T U with respect to which g corresponds to the identity and D is of the form (308). Since Λ^{α} is an eigenvector of D_{α}^{β} with eigenvalue ρ, we can furthermore assume without loss of generality that Λ^{α} is proportional to the first vector of our fixed local frame. We restrict our considerations from now on to U.

Differentiating the equation D_{α}^{β}Λ^{α} = ρΛ^{β}, and substituting (131) and ∇_{α}ρ = −Λ_{α}, we obtain

(ρδ_{γ}^{β} − D_{γ}^{β})∇_{α}Λ_{β} = −\frac{1}{2}(g_{αγ}Λ_{β}Λ^{β} − Λ_{α}Λ_{γ} − J_{α}^{β}Λ_{β}J_{γ}^{ϵ}Λ_{ϵ}).  (311)

Since D_{α}^{β} commutes with ∇_{α}Λ^{β} by Proposition 5.13, the block diagonal form of D_{α}^{β} implies that ∇_{α}Λ^{β} has the same block diagonal form. Equation (311) therefore shows that the second and the third block of ∇_{α}Λ^{α} are proportional to the identity with coefficient of proportionality −\frac{Λ_{β}Λ^{β}}{2(ρ−1)} and −\frac{Λ_{β}Λ^{β}}{2ρ} respectively. The condition (309) therefore reduces to the following three equation

(∇_{α}Λ_{β})Λ^{β} = (−μ + 2Bρ)Λ_{α} Λ_{β}Λ^{β} = 2(μ − 2B)(ρ − 1) Λ_{β}Λ^{β} = 2ρμ,  (312)

which can be equivalently rewritten as

(∇_{α}Λ_{β})Λ^{β} = 2B(2ρ − 1)Λ_{α} Λ_{α}Λ^{α} = −4B(ρ − 1)ρ μ = −2B(ρ − 1).  (313)

First note that,

Λ_{α}V^{α} = \frac{d}{dt}_{|t=0}ϕ_{t}^{∗}λ = −\frac{d}{dt}_{|t=0} \frac{ρe^{at}}{ρe^{at} + (1 − ρ)e^{bt}} = (b − a)ρ(1 − ρ),  (314)

which is nowhere vanishing on U. It follows that

ϕ_{t}^{∗}(Λ_{α}V^{α}) = \frac{(b − a)ρ(1 − ρ)e^{at}e^{bt}}{(ρe^{at} + (1 − ρ)e^{bt})^{2}} .  (315)

Let us write P for the orthogonal projection of T U to the line subbundle spanned by Λ^{α}. We have already noticed that V preserves this projection, which implies in particular that L_{V} P (V) = 0. Note also that by (314) the vector field P (V) is nowhere vanishing on U. Hence, we deduce from (302) that

ϕ_{t}^{∗}(g(P (V), P (V))) = g(G_{t}P (V), P (V)) = \frac{g(P (V), P (V))}{(ρe^{at} + (1 − ρ)e^{bt})^{2}e^{(am+bm̃)t}} .  (316)

Since by assumption (m + 1)a + (m̃ + 1)b = 0, equations (315) and (316) show that the function

t ↦ ϕ_{t}^{∗} (\frac{g(Λ, V)}{g(P (V), P (V))}) = \frac{(b − a)ρ(1 − ρ)}{g(P (V), P (V))}  (317)

is constant. Since g(Λ, V) = g(Λ, P (V)), we therefore obtain

ϕ_{t}^{∗}(Λ) = \frac{ρ(1 − ρ)(b − a)}{g(P (V), P (V))}ϕ_{t}^{∗}(P (V)),

and hence (315) implies

ϕ_{t}^{∗}(Λ_{α}Λ^{α}) = \frac{(b − a)^{2}ρ^{2}(1 − ρ)^{2}e^{at}e^{bt}}{(ρe^{at} + (1 − ρ)e^{bt})^{2}g(P (V), P (V))}.  (318)

Differentiating identity (318) gives

(∇_{α}Λ_{β}Λ^{β})V^{α} = \frac{d}{dt}_{|t=0}Φ_{t}^{∗}(Λ_{β}Λ^{β}) = \frac{(b − a)^{3}ρ^{2}(1 − ρ)^{2}(2ρ − 1)}{g(P (V), P (V))}.  (319)

Since (∇_{α}Λ_{β})Λ^{α} = \frac{1}{2}∇_{α}(Λ_{β}Λ^{β}), we can rewrite the first condition of (313) as

∇_{α}(Λ_{β}Λ^{β}) = 4B(2ρ − 1)Λ_{α},

and we conclude from (319) and (314) that contracting the above identity with V^{α} yields

B = \frac{1}{4} \frac{(b − a)^{2}ρ(1 − ρ)}{g(P (V), P (V))}.  (320)

By (318) the second condition of (313) reads

\frac{(b − a)^{2}ρ^{2}(1 − ρ)^{2}}{g(P (V), P (V))} = 4B(1 − ρ)ρ,  (321)

which is of course equivalent to (320). Since the third condition of (313), given by μ = −2B(ρ − 1), simply defines μ in terms of ρ and B, we conclude that there exist functions μ and B on U such that (309) is satisfied. Note that by (320) the function B is also positive as required. It remains to show that B is constant (in a sufficiently small neighbourhood of x_{0}), which implies in particular that μ = −2B(ρ−1) is smooth and that (310) is satisfied.

The formula (320) for B shows that B is proportional with a constant coefficient to (317). Hence, we have ∇_{V} B = 0. To show that the derivative of B also vanishes along vector field transversal to V consider the 2n − 1 dimensional submanifold of U given by the level set of ρ

M_{y} := {x ∈ U : ρ(x) = ρ(y)},  (322)

where y is some arbitrary point in U. Since the derivative of ρ is nontrivial along the c-projective vector field V by (314), V is transversal to M_{y}. We claim that the derivative of B at y vanishes along all vectors in T_{y}M_{y}. Indeed, note that in view of (320) this is equivalent to the vanishing of the derivative of g(P (V), P (V)) at y along tangent vectors of M_{y}. Since Λ and P (V) are proportional to each other by definition, we have

g(P (V), P (V)) = \frac{(g(P (V), Λ))^{2}}{g(Λ, Λ)} .

It is thus sufficient to show that at y the derivative of g(P (V), Λ), respectively g(Λ, Λ), vanishes along tangent vectors of M_{y}. By (314) this follows immediately for the derivative of g(P (V), Λ) = g(V, Λ). Now consider g(Λ, Λ) and let W ∈ T_{y}M_{y}. Then we compute

W^{α}∇_{α}(Λ_{β}Λ^{β}) = 2W^{α}(∇_{α}Λ_{β})Λ^{β}
= −2μW^{α}Λ_{α} + 4BW^{α}D_{αβ}Λ^{β} = −2μW^{α}Λ_{α} + 4BρW^{α}Λ_{α}.

Since Λ_{α} = −∇_{α}ρ, we see that W^{α}Λ_{α} vanishes and consequently so does W^{α}∇_{α}(Λ_{β}Λ^{β}). Hence, the derivative of B vanishes at y ∈ U. Since y ∈ U was an arbitrary point, we conclude that ∇B is identically zero on U, which completes the proof. □

We can now complete the proof of Theorem 7.6. Under the assumption that (307) holds, Lemma 7.8 shows that in an open neighbourhood of any regular point there exists a positive constant B (which a priori may depend on the neighbourhood) and a function μ such that the triple (D, Λ, μ) satisfies, in addition to the mobility equation, the equations (309) and (310). Since the set of regular points is open and dense, Theorem 6.7 implies that B is actually the same constant at all regular points and that the equations (309) and (310) hold on M for some smooth function μ. Theorem 6.7 also implies that the function λ = −\frac{1}{2}D_{α}^{α} satisfies the equivalent conditions of Proposition 6.22 on M for a positive constant B. Since Λ_{α} = ∇_{α}λ is not identically zero, Tanno’s result [97] completes the proof.

8. Outlook

There has been considerable activity in c-projective geometry since we began work on this article in 2013. Despite this, there remain many open questions and opportunity for further work. In this final section, we survey some of the developments and opportunities which we have not discussed already in the article.

8.1. Metrisability and symmetry. One of the main focuses of this article has been metrisable c-projective structures and their mobility. However, in later sections, we restricted attention to integrable complex structures (the torsion-free case). It would be interesting to extend more of the theory to non-integrable structures with a view to applications in quasi-Kähler geometry, including 4-dimensional almost Kähler geometry—here some partial results have been obtained in [1].

Even in the integrable case, however, a basic question remains wide open: when is a c-projective structure metrisable? One would like to provide a complete c-projectively invariant obstruction, analogous to the obstruction found for the 2-dimensional real projective case in [21].

Additional questions concern the symmetry algebra 𝔠𝔭𝔯𝔬𝔧(J, [∇]) of infinitesimal automorphisms of an almost c-projective 2n-manifold (M, J, [∇]). As with any parabolic geometry [29, 36], the prolongation of the infinitesimal automorphism equation (see Section 3.4, Proposition 3.9) shows that 𝔠𝔭𝔯𝔬𝔧(J, [∇]) is finite dimensional, with its dimension bounded above—in this case, by 2(n + 1)^{2} − 2. This bound is attained only in the c-projectively flat case, and, as shown in [65], in the non-flat case, the dimension of 𝔠𝔭𝔯𝔬𝔧(J, [∇]) is at most 2n^{2}−2n+4+2δ_{3,n} (the so-called “submaximal dimension”). The determination of the possible dimensions of 𝔠𝔭𝔯𝔬𝔧(J, [∇]) remains an open question.

The symmetry algebra 𝔠𝔭𝔯𝔬𝔧(J, [∇]) acts (by Lie derivative) on the space of solutions to the metrisability equation, and for any nondegenerate solution, corresponding to a compatible metric g, it has subalgebras 𝔦𝔰𝔬𝔪(J, g) ⊆ 𝔞𝔣𝔣(J, g) of holomorphic Killing fields and infinitesimal complex affine transformations. Thus the presence of compatible metrics constrains 𝔠𝔭𝔯𝔬𝔧(J, [∇]) further, the Yano–Obata theorems being global examples of this. Even locally, if (M, J, g) admits an essential c-projective vector field, i.e., an element X ∈ 𝔠𝔭𝔯𝔬𝔧(J, [∇]) \ 𝔦𝔰𝔬𝔪(J, g), then g must have mobility ≥ 2, and if X ∉ 𝔞𝔣𝔣(J, g), then there are metrics c-projectively, but not affinely, equivalent to g. For example, nontrivial c-projective vector fields with higher order zeros are essential (because a Killing vector field is determined locally uniquely by its 1-jet at a point), and such strongly essential local flows exist only on c-projectively flat geometries [79].

Constraints on the possible dimensions of 𝔠𝔭𝔯𝔬𝔧(J, [∇])/𝔦𝔰𝔬𝔪(J, g) for Kähler manifolds are given in [78], and an explicit classification of 4-dimensional (pseudo-)Kähler metrics admitting essential c-projective vector fields is given in [14].

An open question here is whether locally nonlinearizable c-projective vector fields exist on nonflat c-projective geometries.

8.2. Applications in Kähler geometry. The original motivation for c-projective geometry [90] was to extend methods of projective geometry to Kähler metrics (and this is one reason why we have concentrated so much on the metrisability equation). Thus one expects ideas from c-projective geometry to be useful in Kähler geometry, and indeed important concepts in Kähler geometry have c-projective origins: for instance, Hamiltonians for Killing vector fields form the kernel of the c-projective Hessian.

Apostolov et al. [2, 3, 4, 5] use c-projectively equivalent metrics (in the guise of Hamiltonian 2-forms) to study extremal Kähler metrics, where the scalar curvature of a Kähler metric lies in the kernel of its c-projective Hessian, and it would be natural to consider extremal quasi-Kähler metrics in the same light. Kähler–Ricci solitons (and generalisations) admitting c-projectively equivalent metrics have also been studied in special cases [5, 67, 71], but the picture is far from complete.

A more recent development is the work of Čap and Gover [32], which extends previous work on projective compactification of Einstein metrics [31] to Kähler (and quasi-Kähler) metrics using c-projective geometry.

Let us now touch on prospective applications in the theory of finite dimensional integrable systems. As we have seen in Section 5, the Killing equations for Hermitian symmetric Killing tensors are c-projectively invariant. This suggests to study these equations from a c-projective viewpoint. We expect that in this way one may find interesting new examples of integrable systems, in particular on closed Kähler or Hermitian manifolds (note that only few such examples are known). Moreover, the analogy between metric projective and c-projective geometries suggests that ideas and constructions from the theory of integrable geodesic flows on n-dimensional Riemannian manifolds could be used in the construction and description of integrable geodesics flows on 2n-dimensional Kähler manifolds. For Killing tensors of valence two this approach is very close to the one in [59], and we expect similar applications for Killing tensors of higher valence.

8.3. Projective parabolic geometries. We noted in the introduction that there are many analogies between methods and results in projective and c-projective geometry, and we have followed the literature in exploiting this observation. We have already noted a partial explanation for the similarities: both are Cartan geometries modelled on flag varieties G/P, one a complex version of the other. However, the c-projective metrisability equation for compatible (pseudo-)Kähler metrics is not the complexification of the corresponding projective metrisability equation. Instead, both are first BGG operators for a G-representation with a 1-dimensional P -subrepresentation. These representations determine projective embeddings of the model G/P, namely, the Veronese embedding of ℝℙ^{n} as rank one symmetric matrices in the projective space of S^{2}ℝ^{n+1}, and the analogous projective embedding of ℂℙ^{n} using rank one Hermitian matrices.

Symmetric and Hermitian matrices are examples of Jordan algebras, and this relation with projective geometry is well known (see e.g. [11]), which suggests to define a projective parabolic geometry as one in which the model has a projective embedding into a suitable Jordan algebra. Apart from projective and c-projective geometry, the examples include quaternionic geometry, conformal geometry, and the geometry associated to the Cayley plane over the octonions. In his PhD thesis [47], G. Frost has shown the much of the metrisability theory of these geometries can be developed in a unified framework. Further, in addition to being analogous, projective parabolic geometries are closely interrelated. For instance, S. Armstrong [6] uses cone constructions to realise quaternionic and c-projective geometry as holonomy reductions of projective Cartan connections, while the generalised Feix–Kaledin construction [17] shows how to build quaternionic structures from c-projective structures, modelled on the standard embedding of ℂℙ^{n} in ℍℙ^{n}.

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Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK

E-mail address: D.M.J.Calderbank@bath.ac.uk

School of Mathematical Sciences, University of Adelaide, SA 5005 Australia

E-mail address: meastwoo@member.ams.org

Institute of Mathematics, FSU Jena, 07737 Jena, Germany

E-mail address: vladimir.matveev@uni-jena.de

Mathematical Institute, Charles University, Sokolovská 83, Praha, Czech Republic

E-mail address: kath.neusser@gmail.com