CURVATURE AND THE C-PROJECTIVE MOBILITY OF KÄHLER METRICS WITH HAMILTONIAN 2-FORMS

DAVID M.J. CALDERBANK, VLADIMIR S. MATVEEV, AND STEFAN ROSEMANN

Abstract. The mobility of a Kähler metric is the dimension of the space of metrics with which it is c-projectively equivalent. The mobility is at least two if and only if the Kähler metric admits a nontrivial hamiltonian 2-form. After summarizing this relationship, we present necessary conditions for a Kähler metric to have mobility at least three: its curvature must have nontrivial nullity at every point. Using the local classification of Kähler metrics with hamiltonian 2-forms, we describe explicitly the Kähler metrics with mobility at least three and hence show that the nullity condition on the curvature is also sufficient, up to some degenerate exceptions. In an Appendix, we explain how the classification may be related, generically, to the holonomy of a complex cone metric.

Introduction

This article weaves together two threads in Kähler geometry which have been running in parallel for 40–60 years with remarkably little interaction, given their common themes.

The first thread concerns a notion of projective equivalence between Kähler metrics. The classical notion is too strong when applied to Kähler metrics: if two metrics that are hermitian with respect to the same almost complex structure have the same geodesics, they have the same Levi-Civita connection. In 1954, Otsuki and Tashiro [27] introduced a complex, but non-holomorphic, version of projective equivalence, which acquired the unfortunate name of “holomorphically projective” or “h-projective” equivalence in the literature. We prefer the term “c-projective”, which is intended to suggest “complex projective”, without implying that the geometry is holomorphic.

Definition 1. Let (M, J) be a complex manifold of real dimension 2m ≥ 4. Then two J-hermitian Kähler metrics g, g̃ on M, with Levi-Civita connections ∇, ˜∇, are called cprojectively equivalent if there is a 1-form Φ such that

(1)  ˜∇_{X}Y − ∇_{X}Y = Φ(X)Y + Φ(Y)X − Φ(JX)JY − Φ(JY)JX

for all vector fields X, Y.

This notion has been extensively studied by Russian and Japanese schools (see [25] for a list of references up to 1998). One common theme has been the relationship between special curvature properties of a Kähler metric and the existence of metrics c-projectively equivalent to it (e.g. [16]).

The second thread concerns the explicit construction of “optimal” Kähler metrics on complex manifolds, generalizing the constant curvature metrics used in the uniformization of Riemann surfaces. The idea to seek such metrics goes back to Calabi’s famous conjectures in the 1950s (e.g., [8]), but the problem was attacked primarily using analytical methods until the late 1970s. Then Calabi provided fresh impetus by introducing the notion of an extremal Kähler metric and constructing explicit examples on total spaces of complex projective line bundles [9, 10]. Calabi’s construction has been refined and extended considerably by many authors (see e.g., [1, 20]), providing a rich supply of Kähler metrics with special curvature properties (such as extremal Kähler metrics). These generalizations have in common that they introduce first order structure to simplify the second (and higher) order partial differential equations that describe curvature. A single source for this structure was identified in [2], where it was observed that Calabi’s construction and its generalizations reflect the presence of a nontrivial solution to an overdetermined linear differential equation, called a hamiltonian 2-form.

Definition 2. Let (M, g, J, ω) be a Kähler manifold of real dimension 2m ≥ 4. Then a (real) J-invariant 2-form ϕ on M is hamiltonian if

(2)  ∇_{X}ϕ = \frac{1}{2}(d tr_{ω} ϕ ∧ JX^{♭} − Jd tr_{ω} ϕ ∧ X^{♭})

for all vector fields X, where X^{♭} = g(X, ·), JX^{♭} = −X^{♭} ∘ J = (JX)^{♭}, and tr_{ω} ϕ = g(ω, ϕ) is the trace of ϕ with respect to the Kähler form ω.

Kähler manifolds with hamiltonian 2-forms are classified locally in [2] and globally in [3], with applications to extremal Kähler metrics in [4].

The origins of the present article are somewhat serendipitous. In April 2011, the first author was asked to referee the article [21] by the second and third authors, which proves that the only compact c-projective manifold with a one parameter subgroup of “essential” symmetries is complex projective space. This drew the first author’s attention to the “main equation” of c-projective equivalence (equation (4) below), which is manifestly equivalent to the equation for hamiltonian 2-forms (see Remark 1).

As noted in the published version of [21], this equivalence has two main ramifications. First, the organizing principle observed in [2] to underpin explicit constructions of Kähler metrics coincides with the notion of a c-projectively equivalent metric, a topic studied independently for many years previously. Secondly, the classification results in [2, 3] solve open problems in the theory of c-projective equivalence, as well as providing new examples.

Our interest here is in a third ramification: although the methodologies employed in the theories of c-projective equivalence and hamiltonian 2-forms have a large overlap (e.g., as both depend upon the theory of overdetermined PDEs of finite type), they have quite different flavours which might be combined with profit to prove new results. This article is a first attempt to exploit both theories in this way.

We focus on the mobility D(g, J) of a Kähler metric g on (M, J), which is the dimension of the space Sol(g, J) of solutions of equation (4)—or equivalently equation (2). Since the identity map Id (corresponding to the Kähler form ω) is always a solution, D(g, J) ≥ 1, and the presence of an independent solution (or a nontrivial hamiltonian 2-form) means equivalently that D(g, J) ≥ 2.

Our plan is to study the case D(g, J) ≥ 3, using [14, Theorem 5], quoted as Theorem 1 below, which states that any such Kähler metric g is C_{ℂ}(B) (for some B ∈ ℝ) in the sense of Definition 4 (unless all solutions of (4) are parallel). The converse is not true: it is straightforward to construct C_{ℂ}(B) metrics with mobility 2 (e.g., using the cone construction described in the appendix—see §A.4). In Theorems 2 and 3 we establish necessary and sufficient conditions for a Kähler metric to be C_{ℂ}(B), and then, in Theorem 6, describe the additional conditions such that a C_{ℂ}(B) metric g has mobility D(g, J) ≥ 3.

Whereas Theorem 2 draws upon curvature conditions from the theory of c-projective equivalence, Theorem 3 uses hamiltonian 2-form methods. It follows, in Corollary 2, that an extremal Kähler metric with mobility ≥ 3 must have constant scalar curvature.

Our results are closely related to the cone construction of [22], cf. [5, 25], discussed in Appendix A. More precisely, for C_{ℂ}(B) metrics with B < 0 (and we may assume B = −1 by rescaling), this construction gives an explicit isomorphism between Sol(g, J) and the space of parallel hermitian endomorphisms on a complex cone (M̂, ĝ, Ĵ) over (M, g, J), which we summarize in §A.1. The cone is a Kähler manifold of dimension dim_{ℂ} M + 1 and (M, g, J) may be recovered from it by taking a Kähler quotient. It is known, at least since Eisenhart [12], that the existence of a parallel hermitian endomorphism Â on M̂ is (locally) equivalent to a decomposition of M̂ into a direct product of Kähler manifolds.

In §A.2, we derive a formula for the Kähler quotient metric g in terms of radial and angular coordinates on M̂ coming from the decomposition of M̂ induced by Â. In §A.3, we (partially) rederive the local classification formula (9) for g relative to A ∈ Sol(g, J) corresponding to Â; this yields another proof of (one direction of) Theorem 3 by a direct calculation—see Proposition 2. In §A.4 we use the cone construction to give an alternative proof of Theorem 6 for a C_{ℂ}(−1) metric.

1. C-projective equivalence and hamiltonian 2-forms

1.1. C-projective equivalence and C_{ℂ}(B) metrics. Let (M, J) be a complex manifold of real dimension 2m ≥ 4. For J-hermitian metrics g, g̃ on M, we introduce the nondegenerate (g, J)-hermitian (i.e., g-symmetric, J-complex-linear) endomorphism

(3)  A(g, g̃) := (\frac{det g̃}{det g})^{\frac{1}{2(m+1)}} g̃^{−1}g,

where we view g, g̃: T M → T^{∗}M as bundle isomorphisms. A fundamental observation by Domashev and Mikeš [24] is that g and g̃ are c-projectively equivalent if and only if there is a vector field Λ such that A = A(g, g̃) satisfies the “main equation”

∇_{X}A = X^{♭} ⊗ Λ + Λ^{♭} ⊗ X + JX^{♭} ⊗ JΛ + JΛ^{♭} ⊗ JX. (4)

Conversely, a nondegenerate solution A of (4) determines a Kähler metric

(5)  g̃ = (det A)^{− \frac{1}{2}} gA^{−1}

(obtained by solving (3) with respect to g̃) c-projectively equivalent to g. Since Id is always a solution of (4), we can add a multiple of Id to any solution A to obtain (at least locally) a solution which is nondegenerate. In this sense, the solutions A of (4) are (locally, generically) in bijection with Kähler metrics g̃ that are c-projectively equivalent to g.

Definition 3. The space of hermitian endomorphisms A satisfying (4) will be denoted by Sol(g, J). The mobility^{1} D(g, J) of (M, g, J) is the dimension of Sol(g, J).

Remark 1. Obviously, two metrics g, g̃ are affinely equivalent ( ˜∇ = ∇) if and only if the endomorphism A = A(g, g̃) is parallel. By (4), if the metrics are c-projectively equivalent, they are affinely equivalent if and only if the vector field Λ is identically zero.

Taking the trace on both sides of equation (4), shows that

Λ = \frac{1}{4} grad_{g} tr A, (6)

hence (4) is a linear PDE system on A, which is equivalent to equation (2) for a hamiltonian 2-form ϕ by writing g(AX, Y) = ϕ(X, JY).

In [2, 21], the nonconstant eigenvalues ξ_{1}, … ξ_{ℓ} of A, considered as functions on M, are shown to be continuous, and smooth on a dense open subset M^{0}. Moreover, their (complex) multiplicity on this subset is one. Thus we can express Λ on M^{0} as

Λ = \frac{1}{2} ∑_{i=1}^{ℓ} grad_{g} ξ_{i}. (7)

1In the classical c-projective literature, this is known as the “degree of mobility”.

For each nonconstant eigenvalue ξ_{i} of A, grad_{g} ξ_{i} lies in the corresponding eigenspace (see [2, 21]). Hence the vanishing of Λ is equivalent to all eigenvalues of the endomorphism A (considered as functions on the manifold) being constant.

An important standard result in c-projective geometry is the fact that JΛ is Killing.

Lemma 1. Let (M, g, J) be a Kähler manifold of real dimension 2m ≥ 4. Then for any A ∈ Sol(g, J), the corresponding vector field Λ is holomorphic, and JΛ is a Killing vector field—equivalently ∇Λ is (g, J)-hermitian.

Proof. This is well known: see [24, Eq. (13)], [2, Proposition 3] and [14, Corollary 3]. □

As the introduction explains, our study builds on the following theorem.

Theorem 1. [14] Let (M, g, J) be a connected Kähler manifold of real dimension 2m ≥ 4 and mobility D(g, J) ≥ 3. Then there is a unique B ∈ ℝ such that for every A ∈ Sol(g, J), with corresponding vector field Λ, there is a function μ such that the system

∇_{X}A = X^{♭} ⊗ Λ + Λ^{♭} ⊗ X + JX^{♭} ⊗ JΛ + JΛ^{♭} ⊗ JX,
∇Λ = μId + BA,
∇μ = 2BΛ^{♭}

(8)

holds at every point of M.

Remark 2. If for A ∈ Sol(g, J), A ≠ const · Id, with corresponding vector field Λ, there exists a function μ such that (A, Λ, μ) solves (8) for a certain constant B, then this holds for any other element Ã ∈ Sol(g, J). This is clear if Ã is a linear combination of Id and A and follows from Theorem 1 if Id, A, Ã are linearly independent.

Definition 4. Let B be a real number. A Kähler metric (g, J) is called^{2} C_{ℂ}(B) if it admits a solution (A, Λ, μ) to the system (8) with Λ not identically zero.

Remark 3. In Definition 4 we require B to be a constant. If B is initially assumed to be a function, it turns out that this function must be (locally) constant provided there exists at almost every point a nonzero vector contained in the B-nullity of the curvature, see Definition 5 and Theorem 2 below.

Remark 4. Neither equation (2) nor equation (4) provide the most natural formulation of c-projective equivalence and mobility because they treat the metrics g and g̃ asymmetrically. This can be remedied by observing that the defining equation (1) for c-projective equivalence is really an equivalence relation between complex affine connections (connections ∇ on T M with ∇J = 0). A c-projective structure on a complex manifold (M, J) is a c-projective equivalence class of such complex affine connections. Equation (4) can be rewritten without reference to a background metric g replacing A with the metric h on T^{∗}M defined by h(α, β) = g(α ∘ A, β). Then equation (4) becomes

∇_{X}h = X ⊗ Λ + Λ ⊗ X + JX ⊗ JΛ + JΛ ⊗ JX

(for all vector fields X) and this equation for h depends only on the c-projective class of ∇ provided that h is viewed as a section of L^{∗} ⊗ S^{2}T M, where L^{⊗(m+1)} = ∧^{2m}T M.

This viewpoint is developed in detail in the forthcoming survey [11] on c-projective geometry—see also [29]. For the present article, we shall always have in mind a background metric, and so we do not pursue this reformulation any further.

2Here “C_{ℂ}” suggests constant/curvature/cone and complex/c-projective, and replaces the term “K_{n}(B)”, often used in the classical c-projective literature, in which K_{n} denotes a Kähler n-manifold.

1.2. The classification of hamiltonian 2-forms. According to [2], a Kähler metric (g, J, ω) admitting a hamiltonian 2-form—or equivalently an A ∈ Sol(g, J)—is locally a bundle over a product of Kähler 2m_{η}-manifolds indexed by the constant eigenvalues η of A (m_{η} being the multiplicity of η), whose “orthotoric” fibres are totally geodesic with the nonconstant eigenvalues ξ_{1}, … ξ_{ℓ} of A as coordinates. On a dense open set, we may write

(9)  g = \underbrace{∑_{η} p_{nc}(η)g_{η}}_{base metric} + \underbrace{∑_{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}}_{fibre metric},

(10)  ω = ∑_{η} p_{nc}(η)ω_{η} + ∑_{r=1}^{ℓ} dσ_{r} ∧ θ_{r},  with dθ_{r} = ∑_{η}(−1)^{r}η^{ℓ−r}ω_{η},

where p_{nc}(t) = ∏_{i=1}^{ℓ}(t − ξ_{i}), σ_{r} is the rth elementary symmetric function of {ξ_{1}, … ξ_{ℓ}}, σ_{r−1}(ξ̂_{j}) is the (r − 1)st such function of {ξ_{k} : k ≠ j}, Δ_{j} = ∏_{k≠j}(ξ_{j} − ξ_{k}), and

(11)  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{Δ_{j}}{Θ_{j}(ξ_{j})}ξ_{j}^{ℓ−r} dξ_{j}.

For any metric of this form,

ϕ : = ∑_{η} η p_{nc}(η)ω_{η} + ∑_{j=1}^{ℓ} ξ_{j} dξ_{j} ∧ (∑_{r=1}^{ℓ} σ_{r−1}(ξ̂_{j})θ_{r})

= ∑_{η} η p_{nc}(η)ω_{η} + ∑_{r=1}^{ℓ}(σ_{r}dσ_{1} − dσ_{r+1}) ∧ θ_{r}

is a hamiltonian 2-form. The extension of this local classification to pseudo-riemannian metrics is subject of the forthcoming paper [6].

Curvature properties of the metric g in (9) are also computed in [2], to which we refer for details and explanations. Let p_{c}(t) = ∏_{η}(t − η)^{m_{η}} be the (monic) polynomial whose roots are the constant eigenvalues η of ϕ, counted with multiplicity.

(1) g is Bochner-flat if and only if the functions Θ_{j}(t) are equal, given by a polynomial Θ(t) of degree ≤ ℓ + 2, with Θ(η) = 0 for all constant eigenvalues η, and the base metrics g_{η} have constant holomorphic sectional curvature (CHSC), given by −Θ′(η). The metric g is itself CHSC if and only if in addition deg Θ(t) ≤ ℓ + 1.

(2) g is weakly Bochner-flat if and only if the functions (p_{c}Θ_{j})′(t)/p_{c}(t) are equal, given by a polynomial Ψ(t) of degree ≤ ℓ+1, and the base metrics g_{η} are Kähler–Einstein, with \frac{1}{m_{η}} Scal_{g_{η}} = −Ψ(η). The metric g is itself Kähler–Einstein if and only if in addition deg Ψ(t) ≤ ℓ.

In particular (applying (1) fibrewise, using the case that there are no constant eigenvalues), the orthotoric fibres have CHSC if and only if the functions Θ_{j}(t) are equal to a common polynomial of degree ≤ ℓ + 1.

It will also be useful to recall from [2] that there is a “Gray–O’Neill” formula [15, 26] for the Levi-Civita connection of g in terms of the fibre and base metrics, where the Gray–O’Neill tensor of the horizontal distribution is given by

(12)  2C(X, Y) = ∑_{r=1}^{ℓ}(Ω_{r}(X, Y)JΛ_{r} − Ω_{r}(JX, Y)Λ_{r})

for Ω_{r} = ∑_{η}(−1)^{r}η^{ℓ−r}ω_{η} and Λ_{r} = grad_{g} σ_{r}.

2. Curvature nullity and the extended system

Let R ∈ Ω^{2}(M, 𝔤𝔩(T M)) denote the curvature of the Kähler manifold (M, g, J),

R(X, Y)Z = (∇_{X}∇_{Y} − ∇_{Y} ∇_{X} − ∇_{[X,Y]})Z,

and let

K(X, Y) = \frac{1}{4}(Y^{♭} ⊗ X − X^{♭} ⊗ Y + JY^{♭} ⊗ JX − JX^{♭} ⊗ JY + 2g(X, JY)J) (13)

be the algebraic curvature tensor of constant holomorphic sectional curvature.

Lemma 2. Let (M, g, J) be a Kähler manifold of real dimension 2m ≥ 4. Then every A ∈ Sol(g, J) satisfies the identity

[R(X, Y), A] = −4[K(X, Y), ∇Λ] (14)

at every point for all tangent vectors X, Y.

Proof. Equation (14) is well known in the theory of c-projectively equivalent metrics, see for example [24, 25]. To prove it, consider the identity

[R(X, Y), A] = ∇_{X}(∇A)_{Y} − ∇_{Y} (∇A)_{X} (15)

which holds for any endomorphism A ∈ Γ(𝔤𝔩(T M)). Assuming that A ∈ Sol(g, J), we can replace the covariant derivatives of A in (15) with (4), to derive an integrability condition for (4). A straightforward calculation yields the desired equation (14). We note that we have to use that ∇Λ commutes with J, see Lemma 1. □

Definition 5. For p ∈ M and B ∈ ℝ, the B-nullity space of the curvature R at p is the linear space

(16)  N(B)_{p} = {Z ∈ T_{p}M : N_{B}(X, Y)Z = 0 ∀X, Y ∈ T_{p}M},

where N_{B}(X, Y) = R(X, Y) + 4BK(X, Y).

Remark 5. Since g(N_{B}(·, ·)·, ·) is a section of S^{2}(∧^{2}T^{∗}M), N(B)_{p} is the set of Z ∈ T_{p}M whose contraction into any entry of g(N_{B}(·, ·)·, ·) is zero. Note also that N(B)_{p} is Jinvariant, i.e., a complex linear subspace of T_{p}M.

Remark 6. The real number B in the definition of the nullity is unique: if Z ∈ N(B)_{p} and Z′ ∈ N(B′)_{p} are nonzero vectors, then B = B′. To see this, we replace X by Z′ in the nullity condition for Z, and apply the nullity condition for Z′ to obtain (B−B′)K(Z, Z′) = 0. Hence, B = B′ or K(Z, Z′) = 0. The last equation implies Z′ is a multiple of Z. Thus (B − B′)K(X, Y)Z = 0 for all vectors X, Y, which, for Z nonzero, forces B = B′.

However, B may depend on the point p, and (of course) the metric g.

Proposition 1. Let (M, g, J) be a Kähler manifold of real dimension 2m ≥ 4, and let A ∈ Sol(g, J) with corresponding vector field Λ. Then for any functions B, μ, we have

(17)  [K(X, Y), ∇Λ − BA − μId] + \frac{1}{4}[N_{B}(X, Y), A] = 0

and, if B and μ are smooth,

(18) ∇_{X}(∇Λ − BA − μId) + JN_{B}(X, JΛ) + (∇_{X}μ − 2Bg(Λ, X))Id + dB(X)A = 0.

Proof. Equation (17) is immediate from Lemma 2(14). Recall from Lemma 1 that JΛ is a Killing vector field, and hence ∇_{X}∇Λ = −J∇_{X}∇JΛ = −JR(X, JΛ) (by the standard formula ∇_{X}∇K = R(X, K), X ∈ T M, which holds for any Killing vector field K, see [18]). Equation (18) follows from this by expanding ∇_{X}(∇Λ − BA − μId) and substituting for ∇_{X}A from equation (4). □

Lemma 3. Let Q be a hermitian endomorphism and Z a nonzero tangent vector at p ∈ M such that [K(X, Z), Q] = 0 for all X ∈ T_{p}M. Then Q is a multiple of the identity.

Proof. We may assume Q is tracefree and prove it vanishes. By definition (13) of K,

(19)  [Z^{♭} ⊗ X − X^{♭} ⊗ Z + JZ^{♭} ⊗ JX − JX^{♭} ⊗ JZ, Q] = 0.

Let e_{1}, … e_{2m} be an orthonormal frame of T_{p}M. We take a trace by applying (19) to e_{i} with X = e_{i} and summing over i. Since Q and Q ∘ J = J ∘ Q are trace-free, and Q is hermitian, we obtain (with summation understood)

0 = g(Z, Qe_{i})e_{i} − g(Z, e_{i})Qe_{i} + g(e_{i}, e_{i})QZ + g(JZ, Qe_{i})Je_{i} − g(JZ, e_{i})QJe_{i} = 2mQZ.

Thus QZ = 0, which we substitute into (19) to obtain:

Z^{♭} ⊗ QX + (QX)^{♭} ⊗ Z + JZ^{♭} ⊗ Q(JX) + Q(JX)^{♭} ⊗ JZ = 0.

For any Y ∈ span{Z, JZ}^{⊥} this yields (using that Q is hermitian)

0 = g(QX, Y)Z + g(Q(JX), Y)JZ = g(X, QY)Z + g(JX, QY)JZ.

Since Z ≠ 0, Q vanishes on span{Z, JZ}^{⊥}. But Q vanishes on span{Z, JZ}, so Q = 0. □

Theorem 2. Let (M, g, J) be a connected Kähler manifold of real dimension 2m ≥ 4. Then for any A ∈ Sol(g, J) with corresponding vector field Λ such that A is not parallel (equivalently, Λ ≠ 0) the following statements are equivalent:

(1) there is a constant B such that [N_{B}(X, Y), A] = 0 for all vector fields X, Y;

(2) there is a constant B and a smooth function μ such that ∇Λ = BA + μId;

(3) there is a constant B and a smooth function μ such that A satisfies the extended system (8);

(4) there is a constant B such that Λ is in the B-nullity space N(B)_{p} at every p ∈ M— equivalently N_{B}(X, JΛ) = 0 for all X ∈ T M;

(5) at every point p of a dense subset, there is a real number B = B(p) such that the B-nullity space N(B)_{p} is nonzero;

(6) there is a constant B such that for any open subset U of M and any eigenvalue ξ of A smoothly defined on U, grad_{g} ξ is in the B-nullity of the curvature on U.

If for given B, these conditions hold for some non-parallel A ∈ Sol(g, J), then they hold for all A ∈ Sol(g, J) (with the same constant B). In particular, the metric g is C_{ℂ}(B).

Proof. (1)⇔(2) by equation (17): if ∇Λ−BA commutes with K(X, Y) for all X, Y ∈ T_{p}M, then it commutes with all skew-hermitian endomorphisms of T_{p}M and is hence a multiple of the identity at p.

(2)⇔(3) by equation (18), which reduces to

g(N_{B}(X, JΛ)Y, Z) = (∇_{X}μ − 2Bg(Λ, X))g(JY, Z)

for all X, Y, Z: the left hand side satisfies the Bianchi identity in X, Y, Z while the right hand side does not (for n > 1), so they must vanish independently.

(3)⇒(4) by equation (18) again: the extended system (8) implies N_{B}(X, JΛ) = 0.

(4)⇒(5) is immediate: if Λ is not identically zero, it is nonzero on an open dense subset, because JΛ is a Killing vector field by Lemma 1.

(5)⇒(3). Given a nonzero Z ∈ N(B)_{p}, substitute Y = Z and μ = 0 in equation (17) to obtain [K(X, Z), ∇Λ − BA] = 0. Hence by Lemma 3 there is a scalar μ = μ(p) such that ∇Λ − BA = μ Id at p. This holds at every point of a dense subset for functions μ, B defined on this subset. Moreover, A is not proportional to the identity at every point of a dense open set (this is straightforward to show using (4)—for a proof see [14, Lemma 4]). Then on a neighbourhood U of any point in this dense open set, B and μ are smooth functions (being solutions of an inhomogeneous linear system of maximal rank with smooth coefficients). We need to show that B is constant and τ := dμ − 2Bg(Λ, ·) is identically zero on U. For this, suppose that a nonzero vector Z is in the B-nullity of the curvature and insert ∇Λ = μId + BA into (18) to obtain

(20)  JN_{B}(X, JΛ) + τ(X)Id + dB(X)A = 0,

and hence, by applying this identity to Z, τ(X)Z + dB(X)AZ = 0. If Z is not an eigenvector of A, we have τ(X) = dB(X) = 0 for all X ∈ T U which is what we wanted to show. We may thus assume AZ = ξZ for some function ξ, so that τ = −ξdB and

N_{B}(X, Λ) = dB(JX)(A − ξId)J

= ((A − ξId)X)^{♭} ⊗ (dB)^{♯} − dB ⊗ (A − ξId)X, (21)

where α^{♯} denotes the metric dual of a 1-form α, and the second line follows from the Bianchi symmetry satisfied by g(N_{B}(X, Λ)Y, W) = g(N_{B}(Y, W)X, Λ). Comparing the second and third lines, it follows that A−ξId has complex rank ≤ 1. It remains to show the following.

Lemma 4. Suppose A ∈ Sol(g, J) and that on an open subset U: A is not parallel with exactly two (distinct) eigenvalues, both smooth, and Z is an eigenvector of A in the Bnullity of g for smooth B. Then dB = 0 on U.

Given this lemma, whose proof give below, we obtain also τ = −ξdB = 0 on U, and hence the system (8) holds in a neighbourhood of every point of an open dense subset for a (local) constant B and a smooth function μ. On the other hand, it was proven in [14, §2.5] that the constants B are the same for each such neighbourhood. Taking the trace of the second equation in (8), we obtain 2mμ = tr ∇Λ − B tr(A), so that the functions μ coincide on overlaps and patch together to a globally defined function. Hence the system (8) holds everywhere on M for a constant B and a smooth function μ.

(1–5)⇒(6). Since N_{B}(X, Y)Λ = 0, equation (7) implies

(22)  0 = ∑_{i=1}^{l} N_{B}(X, Y) grad_{g} ξ_{i},

where ξ_{1}, … ξ_{l} are the eigenvalues of A. It was shown in [2, Proposition 14] and [21, Proposition 1] that the gradient grad_{g} ξ_{i} is contained in the eigenspace of A corresponding to ξ_{i}. Since [N_{B}(X, Y), A] = 0, N_{B}(X, Y) leaves the eigenspaces of A invariant. Then wherever N_{B}(X, Y) grad_{g} ξ_{i} is nonzero, it is an eigenvector of A corresponding to the eigenvalue ξ_{i} and (22) shows that N_{B}(X, Y) grad_{g} ξ_{i} = 0.

(6)⇒(5). It was shown in [2, Proposition 14] that every nonconstant eigenvalue ξ of A ∈ Sol(g, J) has nonvanishing differential on an open and dense subset.

The final observation of the theorem follows because condition (5) is independent of A ∈ Sol(g, J), and if A is ∇-parallel (i.e., the corresponding Λ is zero), then equation (17) and Lemma 3 imply that A is a multiple of the identity or B = 0. □

Proof of Lemma 4. Since A is nonparallel (i.e., Λ ≠ 0), it has at least one nonconstant eigenvalue. We consider first the case that A has one nonconstant eigenvalue ξ and one constant eigenvalue, which we may assume to be zero. The ξ-eigenspace is therefore spanned by Λ, and if this is in the nullity, then (20) implies dB = 0. Thus we may assume AZ = 0, hence dμ = 2Bg(Λ, ·) = B dξ, so that μ and B are functions of ξ. For any X with AX = 0 we have

ξμX = −(A − ξ Id)μX = −(A − ξ Id)∇_{X}Λ = (∇_{X}A − dξ(X))Λ = g(Λ, Λ)X

since ∇Λ = μ Id + BA, dξ(X) = 0 and ∇A is given by (4). Hence ξμ = g(Λ, Λ).

On the other hand, using the Gray–O’Neill formulae [2, 15] or the explicit form

g = −ξg_{0} + \frac{dξ^{2}}{Θ(ξ)} + Θ(ξ)θ^{2}

of the metric, we obtain that

(23)  −4BK(X, Y)Z = R(X, Y)Z = R_{0}(X, Y)Z − \frac{4g(Λ, Λ)}{ξ^{2}}K(X, Y)Z,

for X, Y in the zero eigenspace of A, where R_{0} denotes the curvature of g_{0} (the Kähler quotient by JΛ), lifted to the zero eigenspace. Thus R_{0}(X, Y)Z = 4(Bξ − μ)K_{0}(X, Y)Z, where K_{0} = −K/ξ is the algebraic constant holomorphic sectional curvature tensor of g_{0}. Taking the trace over X (on the zero eigenspace), Ric_{0}(Z) = 2(m − 1)(Bξ − μ)Z and so Bξ −μ is independent of ξ, hence constant. This combines with dμ = B dξ to give dB = 0 as required.

Now we turn to the case when A has two nonconstant eigenvalues ξ_{1} and ξ_{2}. Note that in this case, M is necessarily real 4-dimensional. Let V_{1} = grad_{g} ξ_{1}, V_{2} = grad_{g} ξ_{2} and suppose that V_{2} is contained in the B-nullity of the curvature R. To compute B and μ, we apply V_{1} and V_{2} to the equation ∇Λ = μ Id + BA to obtain the linear system m_{1} = μ + ξ_{1}B, m_{2} = μ + ξ_{2}B, where m_{1}, m_{2} are the eigenvalues of ∇Λ, i.e., ∇_{V_{1}}Λ = m_{1}V_{1} and ∇_{V_{2}}Λ = m_{2}V_{2}. Hence,

μ = \frac{ξ_{2}m_{1} − ξ_{1}m_{2}}{ξ_{2} − ξ_{1}} , B = \frac{m_{1} − m_{2}}{ξ_{1} − ξ_{2}} .

To calculate m_{1}, m_{2}, we recall that Λ = \frac{1}{2}(V_{1} + V_{2}) and so ∇_{Λ}Λ = \frac{1}{2}(m_{1}V_{1} + m_{2}V_{2}), or, dually, d(g(Λ, Λ)) = m_{1}dξ_{1} + m_{2}dξ_{2}. The classification of hamiltonian 2-forms from §1.2 shows that, in a neighbourhood of almost every point, g takes the form

g = \frac{ξ_{1} − ξ_{2}}{F_{1}(ξ_{1})} dξ_{1}^{2} + \frac{ξ_{2} − ξ_{1}}{F_{2}(ξ_{2})} dξ_{2}^{2} + \frac{F_{1}(ξ_{1})}{ξ_{1} − ξ_{2}} (dt_{1} + ξ_{2}dt_{2})^{2} + \frac{F_{2}(ξ_{2})}{ξ_{2} − ξ_{1}} (dt_{1} + ξ_{1}dt_{2})^{2}

in local coordinates ξ_{1}, ξ_{2}, t_{1}, t_{2}. From this, we obtain g(Λ, Λ) in terms of the functions F_{1}, F_{2}. Calculating d(g(Λ, Λ)) and comparing coefficients, we obtain

B = \frac{m_{1} − m_{2}}{ξ_{1} − ξ_{2}} = \frac{(F′_{1}(ξ_{1}) + F′_{2}(ξ_{2}))(ξ_{1} − ξ_{2}) − 2(F_{1}(ξ_{1}) − F_{2}(ξ_{2}))}{4(ξ_{1} − ξ_{2})^{3}}. (24)

Replacing X in (21) by the vector JV_{2} in the nullity, we see that dB(V_{2}) = 0, i.e., B does not depend on the variable ξ_{2}. Using (24), it is straightforward to show that the condition dB/dξ_{2} = 0 is equivalent to

0 = F′′_{2}(ξ_{2})(ξ_{1} − ξ_{2})^{2} + 2(F′_{1}(ξ_{1}) + 2F′_{2}(ξ_{2}))(ξ_{1} − ξ_{2}) − 6(F_{1}(ξ_{1}) − F_{2}(ξ_{2})). (25)

Taking three derivatives of this equation w.r.t. ξ_{1} yields F_{1}^{(4)}(ξ_{1}) = 0, hence F_{1}(ξ_{1}) is a polynomial of degree ≤ 3. Inserting this condition back into (25), a straightforward calculation shows F_{1} = F_{2}. Inserting these polynomials into (24) shows that B is a constant. This also follows from [2] where it is shown that (g, J) has constant holomorphic sectional curvature (and hence B is constant) if F_{1} = F_{2} is a polynomial of degree ≤ 3. □

Remark 7. Recall from Remark 1 that A not being parallel is necessary for (5). All other conditions are automatically fulfilled for parallel A, in which case we have μ = B = 0.

To relate this result to the local classification of metrics with hamiltonian 2-forms (see §1.2), observe that at each point in a dense open set, the J-linear span of the gradients of the eigenvalues of A is the tangent space to the orthotoric fibres of the metric g.

Corollary 1. A ∈ Sol(g, J) satisfies the extended system (8) for B ∈ ℝ if and only if A is parallel (in which case, we may assume B = 0) or the orthotoric fibres of A are in the B-nullity of g. In particular, since these fibres are totally geodesic, they have constant holomorphic sectional curvature −4B.

By Definition 4, a Kähler metric is C_{ℂ}(B) for a constant B if one of the conditions in Theorem 2 is satisfied for some non-parallel A ∈ Sol(g, J). We next describe the conditions on the parameters in formula (9) under which a Kähler metric is C_{ℂ}(B).

For this, we first observe that the extended system (8) is equivalent to the special case τ_{0} = 0, τ_{1} = −4B, τ_{2} = −4μ of the system [2, §2.3, Equation (30)], with the term “W^{K}(ϕ)” omitted. Hence (the proof of) [2, Proposition 5] applies to show that the polynomial

(26)  F (t) = −4(Bt + μ)p_{A}(t) − g(K, K(t)),

has constant coefficients, where p_{A}(t) is the characteristic polynomial of A, K = J grad_{g} σ_{1} and K(t) = J grad_{g} p_{A}(t) (thus K coincides with the Killing vector field 2JΛ). To interpret this fact geometrically, we next observe that any triple (A, Λ, μ) ∈ 𝔤𝔩(T M)⊕T M ⊕M ×ℝ, with A hermitian, defines a hermitian (bundle) metric on {(σ, ρ) ∈ Hom(T M, ℂ)⊕M ×ℂ : σ(JX) = iσ(X)}, via the expression

(27)  [\begin{matrix}ρ \\ σ \end{matrix}]^{†} [\begin{matrix}μ & Λ \\ Λ & A \end{matrix}] [\begin{matrix}ρ′ \\ σ′ \end{matrix}] := μ\overline{ρ}ρ′ + \overline{σ}(Λ)ρ′ + \overline{ρ}σ′(Λ) + g(\overline{σ} ∘ A, σ′).

When B = −1, this bundle may be identified with the (holomorphic) tangent bundle of the complex cone over (M, g, J) studied in [22], which we discuss in Appendix A. For any B ∈ ℝ, the bundle carries a connection D defined by

(28)  D_{X} [\begin{matrix}ρ \\ σ \end{matrix}] = [\begin{matrix}∇_{X}ρ + σ(X) \\ ∇_{X}σ + Bg(X + iJX, ·)ρ \end{matrix}] .

This connection induces the extended system (8) in the following sense (cf. [14, §§4.1-4.2] in the case B ≠ 0).

Lemma 5. For any sections (A, Λ, μ) and (σ, ρ) as above, we have

∂_{X}([\begin{matrix}ρ \\ σ \end{matrix}]^{†} [\begin{matrix}μ & Λ \\ Λ & A \end{matrix}] [\begin{matrix}ρ′ \\ σ′ \end{matrix}]) − (D_{X} [\begin{matrix}ρ \\ σ \end{matrix}])^{†} [\begin{matrix}μ & Λ \\ Λ & A \end{matrix}] [\begin{matrix}ρ′ \\ σ′ \end{matrix}] − [\begin{matrix}ρ \\ σ \end{matrix}]^{†} [\begin{matrix}μ & Λ \\ Λ & A \end{matrix}] D_{X} [\begin{matrix}ρ′ \\ σ′ \end{matrix}]

= [\begin{matrix}ρ \\ σ \end{matrix}]^{†} [\begin{matrix}∇_{X}μ − 2Bg(Λ, X) & ∇_{X}Λ − μX − BAX \\ ∇_{X}Λ − μX − BAX & ∇_{X}A − ( \begin{matrix}X^{♭}⊗Λ+Λ^{♭}⊗X \\ +JX^{♭}⊗JΛ+JΛ^{♭}⊗JX) \end{matrix} ) \end{matrix}] [\begin{matrix}ρ′ \\ σ′ \end{matrix}]

The proof is a straightforward computation. Up to a normalization constant, the function F (t) is the (complex) determinant of the hermitian form on Hom(T M, ℂ)⊕M ×ℂ defined by (A − t Id, Λ, μ + Bt), so its roots are the relative eigenvalues of the hermitian forms defined by (A, Λ, μ) and (Id, 0, −B). This gives another proof that F (t) has constant coefficients when (A, Λ, μ) solves (8), and further shows that the relative eigenspaces are D-parallel subbundles of Hom(T M, ℂ) ⊕ M × ℂ.

Theorem 3. Let (g, J, ω) be a Kähler metric with a non-parallel hamiltonian 2-form, given explicitly by (9) on a dense open set. Then g is C_{ℂ}(B) if and only if Θ_{j}(t) = Θ(t), a polynomial of degree ≤ ℓ+1 (independent of j) with leading coefficient −4B, and Θ(η) = 0 for all constant eigenvalues η.

Proof. If g is C_{ℂ}(B) then the (totally geodesic) orthotoric fibres have constant holomorphic sectional curvature (CHSC). The hamiltonian 2-form restricts to a hamiltonian 2-form on each fibre whose characteristic polynomial is p_{A}(t)/p_{c}(t). Applying [2, Proposition 18] fibrewise, we thus have Θ_{j}(t) = Θ(t) := F (t)/p_{c}(t) for all j (where we recall that p_{c}(t) is the monic polynomial whose roots are the constant eigenvalues η of A). It remains to show that any root η of p_{c}(t) is a root of Θ(t), i.e., the multiplicity of η as a root of F (t) is greater than its multiplicity as a root of p_{c}(t). The latter is the dimension of the kernel of A − η Id in Hom(T M, ℂ) which is a subspace U of the relative η-eigenspace, i.e., the kernel of the hermitian form defined by (A − η Id, Λ, μ + Bη). However by (28), U cannot be D-parallel, so the dimension of the relative η-eigenspace is strictly larger, hence so is the multiplicity of η as a root of F (t).

Conversely, if Θ_{j}(t) = Θ(t) as stated, then the orthotoric fibres belong to the B-nullity of g. To see this, observe that the Gray–O’Neill curvature formulae [15, 26] (with Gray– O’Neill tensor (12)) imply that all components of the curvature of g, apart from the purely horizontal part, depend on the base metrics g_{η} in (9) only to first order at each point. Hence, to compute R(X, Y)Z for Z vertical, we may use a metric g̃ which agrees with g at a given point, but where we replace the base metrics g_{η} with metrics g̃_{η} which have CHSC equal to Θ′(η) at that point. By [2, Proposition 17], g̃ has CHSC given by a multiple of B, hence the fibres are in the B-nullity. Consequently the same holds for g. □

Corollary 2. Let (g, J) be a C_{ℂ}(B) Kähler metric (e.g., with D(g, J) ≥ 3) which is weakly Bochner-flat (or is Bochner-flat). Then g is Kähler–Einstein (or has constant holomorphic sectional curvature, respectively).

Recall from [2] that a Kähler metric (g, J) of dimension 2m is orthotoric if it admits a hamiltonian 2-form having m nonconstant eigenvalues ξ_{1}, … ξ_{m} (these metrics are also “Kähler–Liouville”—see [17]).

Corollary 3. Let (g, J) be a C_{ℂ}(B) Kähler metric (e.g., with D(g, J) ≥ 3) which is orthotoric. Then g has constant holomorphic sectional curvature.

Remark 8. An analog of the corollary in real projective geometry, which is also true under more general assumptions, can be found in [7].

Theorem 3 has the following global consequence.

Theorem 4. Let M be a closed connected 2m-orbifold (2m ≥ 4) and suppose (g, J) is a C_{ℂ}(B) Kähler metric on M. Then (M, g, J) is an orbifold quotient of ℂP^{m} with a Fubini–Study metric.

Proof. By assumption, M admits a hamiltonian 2-form of order ℓ ≥ 1. The theory of [3, Section 2], which extends to orbifolds following [19], shows that the universal orbifold cover of M has a blow-up M̂ which is a bundle of (connected) toric orbifolds over an orbifold S which is a complete Kähler product over the constant eigenvalues η of A. Since blow-up does not change the orbifold fundamental group, M̂ is a simply connected orbifold, hence so is S (since the fibres of M̂ → S are connected). Now, since every constant eigenvalue η is a root of the function Θ of Theorem 3, it follows from [3, Proposition 6] (or rather, its proof, extended straightforwardly to orbifolds) that S is a Kähler product of complex projective spaces where the Kähler metric on the factor corresponding to a root η has CHSC −Θ′(η) (see [3, Theorem 5(iv–v)]). As discussed in §1.2(1), these are precisely the conditions (given that Θ is a polynomial of degree ≤ ℓ + 1 vanishing on the constant eigenvalues η) which ensure that the metric on M has CHSC [2]. (This is not a coincidence: the Fubini–Study metric on ℂP^{m} admits hamiltonian 2-forms of any order 0 ≤ ℓ ≤ m.) Since K = JΛ is a nonparallel Killing vector field on M (it is hamiltonian, hence has zeros), the curvature of g must be positive by Bochner’s argument. Hence the universal cover of M is isometric to ℂP^{m} with a Fubini–Study (positive CHSC) metric. □

Corollary 4. Let (M, g, J) be a closed connected Kähler orbifold of dimension 2m ≥ 4 and mobility D(g, J) ≥ 3. Then either M is an orbifold quotient of ℂP^{m} with a Fubini–Study metric, or every Kähler metric c-projectively equivalent to g is affinely equivalent to g.

Remark 9. This corollary is immediate from Theorem 4 and Theorem 1 (i.e., [14, Section 2]): in the manifold case, it is the main result of [14], where it was established for metrics of arbitrary signature. Indeed, on manifolds, the analogue of Theorem 4 for metrics of arbitrary signature was obtained in [14, Remark 12]. Furthermore, the proof in [14] proceeds by first reducing to the case that −Bg is positive definite, and this part of the argument extends straightforwardly to orbifolds. Hence Theorem 4 is actually valid in all signatures.

On the other hand, in the remaining case, where (without loss of generality) B = −1 and g is positive definite, [14, Lemma 8] shows that the extended system (8) yields a nontrivial solution of the kählerian Tanno equation, and so the manifold case of Theorem 4 follows from [28, Theorem 10.1]. In fact, as shown in [13, §3, see (4)], the Tanno equation is equivalent to the extended system in this case, and so Theorem 4 may be regarded as providing a natural generalization of [28, Theorem 10.1] to orbifolds of arbitrary signature. Note that our method of proof for Theorem 4 is very different from [28].

The corollary is a rigidity result for closed connected Kähler orbifolds (M, g, J) which are not quotients of ℂP^{m}, but admit a c-projectively equivalent metric which is not affine equivalent (i.e., a hamiltonian 2-form of order ℓ > 0). This has several consequences. First, as observed in [14], the isometry group of g has codimension ≤ 1 in the group of c-projective transformations of M: this is because the latter group acts on the projectivization of Sol(M, g), with the isometry group of g as a point stabilizer. Secondly, since the hamiltonian 2-form is essentially unique (i.e., A ∈ Sol(M, g) is unique up to a linear combination with the identity solution), the ℓ-torus action it defines must be central.

3. Classification of metrics with c-projective mobility ≥ 3

Let us recall the following result:

Theorem 5 ([3, 14]). Let (M, g, J) be a connected Kähler manifold of real dimension 4. Then D(g, J) ≥ 3 if and only if the holomorphic sectional curvature is constant.

Remark 10. In [3, Proposition 10] and [14, Lemma 7], it was shown that a Kähler manifold of real dimension 4 and of mobility ≥ 3 has constant holomorphic sectional curvature. The fact that every CHSC Kähler manifold of any dimension 2m has mobility (m + 1)^{2} ≥ 3 is a standard result, see for example [2, 25].

By Theorem 1, the condition D(g, J) ≥ 3 implies either that all A ∈ Sol(M, g) are parallel, or that the metric is C_{ℂ}(B), i.e., the equivalent conditions of Theorem 2 hold. Conversely, we now find the metrics satisfying D(g, J) ≥ 3 among those that are C_{ℂ}(B).

Theorem 6. Let (M, g, J) be a connected Kähler manifold of real dimension 2m ≥ 4 which is C_{ℂ}(B). Suppose in addition that there exists A ∈ Sol(g, J) such that

• either the number of nonconstant eigenvalues of A is ≥ 2

• or the number of constant eigenvalues of A is ≥ 3. Then D(g, J) ≥ 3.

Proof. Let us choose A ∈ Sol(g, J) satisfying one of the two conditions on the eigenvalues.

First suppose that the corresponding vector field Λ is identically zero. Then A is covariantly constant and all eigenvalues of A are constant (see Remark 1). The endomorphism Ã = A^{2} is covariantly constant and hence contained in Sol(g, J). It follows that Ã, A and Id are linearly independent and therefore D(g, J) ≥ 3, since otherwise, A would be annihilated by a polynomial with constant coefficients of order two or lower and this contradicts the assumption that the number of constant eigenvalues is at least three. We have proven Theorem 6 under the assumption Λ ≡ 0.

Let us now suppose that Λ is not identically zero.

First case: B = 0. A straightforward computation (using the equations in (8)) shows

Ã = Λ^{♭} ⊗ Λ + JΛ^{♭} ⊗ JΛ

is contained in Sol(g, J), where the corresponding vector field is Λ̃ = μΛ and μ is a constant.

Clearly, Ã is not proportional to Id (since it is multiplication with g(Λ, Λ) on span{Λ, JΛ} and multiplication with 0 on span{Λ, JΛ}^{⊥}). If D(g, J) = 2, we have A = αÃ + βId for certain constants α and β but this contradicts the assumptions on the eigenvalues of A. Theorem 6 is proven in the case B = 0.

Second case: B ≠ 0. Let us multiply the metric with −B, such that the system (8) for the new metric (which we again denote by the symbol g) holds with B = −1. Note that the mobility remains unchanged by this procedure. A straightforward computation (one may also compare [25, p. 1338], [14, equation (88) in the proof of Lemma 10] or the cone construction [22, Theorem 9]—see the appendix below) using the equations in (8) shows

Ã = A^{2} + Λ^{♭} ⊗ Λ + JΛ^{♭} ⊗ JΛ

is contained in Sol(g, J) with corresponding vector field Λ̃ = (A + μId)Λ. Assuming D(g, J) = 2, we obtain (up to rescaling) A = Ã + αId for a certain constant α. Taking the covariant derivative of this equation shows Λ = (A + μId)Λ. Hence, Λ is an eigenvector of A corresponding to the nonconstant eigenvalue 1−μ. Equation (7) (together with the fact that for each nonconstant eigenvalue ξ_{i} of A, grad_{g} ξ_{i} is contained in the corresponding eigenspace) implies that A has exactly one nonconstant eigenvalue. Restricting A = Ã + αId to the orthogonal complement U := span{Λ, JΛ}^{⊥} shows that the restriction A|_{U} is annihilated by a quadratic polynomial. Then the number of nonconstant eigenvalues is at most two. We obtain a contradiction to any of the two conditions on the eigenvalues of A. Hence, D(g, J) ≥ 3 and Theorem 6 is proven. □

Appendix A. Cone construction for C_{ℂ}(−1) metrics

A.1. The cone construction. If g is a C_{ℂ}(−1) Kähler metric then the space Sol(g, J) is isomorphic to the space of solutions (A, Λ, μ) of the PDE system

∇_{X}A = X^{♭} ⊗ Λ + Λ^{♭} ⊗ X + JX^{♭} ⊗ JΛ + JΛ^{♭} ⊗ JX,
∇Λ = μId − A,
∇μ = −2Λ^{♭}.

(29)

The cone construction [22, Theorem 9] (see also the formulae in [25, pp. 1338–1339] for the same statement, though the formula for Â appearing there seems to have a misprint) asserts that the space of solutions (A, Λ, μ) of this system is isomorphic to the space of parallel hermitian endomorphisms Â ∈ End(T M̂) on the cone

M̂ = ℝ_{>0} × ℝ × M, ĝ = dr^{2} + r^{2}(ϕ^{2} + g), Ĵ = \frac{1}{r}∂_{t} ⊗ dr − r∂_{r} ⊗ ϕ + J, (30)

where ϕ = dt − τ and τ is a 1-form on M satisfying dτ = 2ω (ω = g(J·, ·) denotes the Kähler form on M). The construction is local but this is sufficient for our purposes. The correspondence between solutions (A, Λ, μ) of (29) and parallel hermitian endomorphisms Â ∈ End(T M̂) is given by

ĝ(Â·, ·) = μdr^{2} − rdr ⊙ Λ^{♭} + r^{2}(μϕ^{2} + ϕ ⊙ Λ^{♭}(J·) + g(A·, ·)). (31)

Further, we view the manifold N = ℝ×M with metric h = ϕ^{2}+g as naturally embedded into M̂ as the hypersurface N = {r = 1}. The manifold (M, g, J) is recovered from (M̂, ĝ, Ĵ) as the Kähler quotient w.r.t. the action of the hamiltonian Killing vector field K := \frac{1}{2} Ĵgradĝ r^{2} on the level set N, where the function \frac{1}{2}r^{2} serves as the moment map for the (local) hamiltonian S^{1}-action induced by K.

A.2. The Kähler quotient in the presence of a decomposition of the cone into a direct product. By the decomposition theorem for riemannian manifolds [12], the parallel hermitian endomorphisms on a manifold are classified by all the ways the manifold can be decomposed into a direct product of Kähler manifolds. Let (M̂, ĝ, Ĵ) be the cone over a Kähler manifold (M, g, J) given by (30). Suppose ĝ decomposes into a direct product

M = ∏_{i} M_{i}, ĝ = ∑_{i} ĝ_{i}, Ĵ = ∑_{i} Ĵ_{i}. (32)

of Kähler manifolds (M̂_{i}, ĝ_{i}, Ĵ_{i}). Recall that the cone structure on (M̂, ĝ, Ĵ) gives rise to the cone vector field C = r∂_{r} satisfying ˆ∇C = Id. Conversely, a vector field satisfying this equation induces a cone structure by defining the radial coordinate to be

r := \sqrt{ĝ(C, C)}.

The decomposition C = ∑_{i=0}^{ℓ} C_{i} of the cone vector field w.r.t. (32) defines cone vector fields C_{i} on each component (M̂_{i}, ĝ_{i}, Ĵ_{i}) making them into cones over certain Kähler manifolds (M_{i}, g_{i}, J_{i}). Hence, having a decomposition as in (32), we may write

ĝ = ∑_{i=0}^{ℓ} \underbrace{(dr_{i}^{2} + r_{i}^{2}(ϕ_{i}^{2} + g_{i})}_{=ĝ_{i}} . (33)

Here we allow some of the g_{i}’s to be zero, meaning that the corresponding cone (M̂_{i}, ĝ_{i}, Ĵ_{i}) is (complex) 1-dimensional over a base of dimension 0. In particular ĝ_{i} is flat.

As a Kähler riemannian cone, ĝ is of the form (30). Using C = ∑_{i=0}^{ℓ} C_{i} and C_{i} = r_{i}∂_{r_{i}}, we see that

r, K = \frac{1}{2}Ĵgradĝ r^{2}, ∂_{r}, dr and ϕ

relate to the corresponding objects on the components ĝ_{i} of ĝ in (33) by the equations

r^{2} = ∑_{i} r_{i}^{2}, K = ∑_{i} K_{i}, ∂_{r} = \frac{1}{r} ∑_{i} r_{i}∂_{r_{i}}, dr = \frac{1}{r} ∑_{i} r_{i}dr_{i} and ϕ = \frac{1}{r^{2}} ∑_{i} r_{i}^{2}ϕ_{i}. (34)

Next we describe the Kähler quotient of the direct product metric ĝ in (33) w.r.t. the action of the hamiltonian Killing vector field K := \frac{1}{2} Ĵgradĝ r^{2} on the level set r = 1.

Theorem 7. The Kähler quotient metric g of the metric ĝ is given by the formula

g = ∑_{i=0}^{ℓ} dr_{i}^{2} + \frac{1}{2} ∑_{i,j=0}^{ℓ} r_{i}^{2}r_{j}^{2}(ϕ_{i} − ϕ_{j})^{2} + ∑_{i=0}^{ℓ} r_{i}^{2}g_{i}. (35)

Remark 11. The forms ϕ_{i} − ϕ_{j} are basic, i.e., they can be written as the pullback of forms defined on the quotient. Indeed, these forms vanish upon insertion of ∂_{r} and K, they do not depend on r and they are K-invariant (that is, invariant w.r.t. the (local) S^{1}-action).

Remark 12. Recall that the metrics g_{i} in (35) are zero if ĝ_{i} = dr_{i}^{2} +r_{i}^{2}(ϕ_{i}^{2} +g_{i}) is (complex) 1-dimensional.

Proof of Theorem 7. Restricted to the level set r = 1, the quotient metric g is given by

g = ĝ − ϕ^{2} = ∑_{i=0}^{ℓ} dr_{i}^{2} + ∑_{i=0}^{ℓ} r_{i}^{2}ϕ_{i}^{2} − ϕ^{2} + ∑_{i=0}^{ℓ} r_{i}^{2}g_{i}. (36)

Using (34), we obtain

∑_{i=0}^{ℓ} r_{i}^{2}ϕ_{i}^{2} − ϕ^{2} = ∑_{i=0}^{ℓ} r_{i}^{2}ϕ_{i}^{2} − ∑_{i,j=0}^{ℓ} r_{i}^{2}r_{j}^{2}ϕ_{i} ⊗ ϕ_{j} = \frac{1}{2} ∑_{i,j=0}^{ℓ} r_{i}^{2}r_{j}^{2}(ϕ_{i} − ϕ_{j})^{2}

which gives us formula (35). □

In what follows, let Â be a parallel hermitian endomorphism for ĝ with distinct eigenvalues C_{0} < ⋯ < C_{ℓ} of multiplicities m_{0}, … m_{ℓ}. Let (33) be the decomposition of ĝ w.r.t. the parallel eigenspace distributions of Â. If we consider Â as a parallel symmetric (0, 2)-tensor field (by lowering one index w.r.t. the metric ĝ), it is given by the formula

Â = ∑_{I=0}^{ℓ} C_{I}(dr_{I}^{2} + r_{I}^{2}(ϕ_{I}^{2} + h_{I})). (37)

Let us relate the (constant) eigenvalues C_{0}, … C_{ℓ} of Â to the (generically nonconstant) eigenvalues ξ_{1}, … ξ_{ℓ} of A ∈ Sol(g, J) corresponding to Â.

Lemma 6. Let Â be given by (37) for numbers C_{0} < ⋯ < C_{ℓ} and let the cone metric ĝ over g be given by (33). Let A ∈ Sol(g, J) correspond to Â via the isomorphism (31). Then the function p_{A}: M̂ × ℝ → ℝ, given by

p_{A}(t) = \frac{1}{r^{2}} ∏_{i=0}^{ℓ}(t − C_{i})^{m_{i}−1} ∑_{i=0}^{ℓ} r_{i}^{2} ∏_{j≠i}(t − C_{j}) (38)

is the characteristic polynomial of A. Moreover, we have

C_{0} ≤ ξ_{1} ≤ C_{1} ≤ ⋯ ≤ ξ_{ℓ} ≤ C_{ℓ}. (39)

where ξ_{i} are the ordered nonconstant eigenvalues of A. In particular, ℓ is the number of nonconstant eigenvalues of A ∈ Sol(g, J) on the base M and the eigenvalues of Â occurring with multiplicity two or higher are the constant eigenvalues of A.

Remark 13. The calculations in the proof of Lemma 6 below are analogous to the derivation of elliptic separation coordinates on the n-sphere, see [23, Section 7].

Proof. Recall that A is the horizontal part of Â. Its action on the horizontal distribution

H = {X ∈ T M̂ : ĝ(X, ∂_{r}) = ĝ(X, Ĵ∂_{r}) = 0}

is then given by

AX = ÂX − ĝ(ÂX, ∂_{r})∂_{r} − ĝ(ÂX, Ĵ∂_{r})Ĵ∂_{r}.

In particular, if ξ is an eigenvalue of A, i.e., AX = ξX for some nonzero X ∈ H, we have (Â − ξId)X = ⟨ÂX, ∂_{r}⟩∂_{r}, where ⟨·, ·⟩ = ĝ − iĝ(Ĵ·, ·) denotes the hermitian inner product associated to ĝ. Thus, ξ is an eigenvalue of A if and only if there exists X ≠ 0 such that

⟨X, ∂_{r}⟩ = 0 and (Â − ξId)X = c∂_{r} for some c ∈ ℂ.

If ξ is not an eigenvalue of Â, this condition is equivalent to ⟨(Â − ξId)^{−1}∂_{r}, ∂_{r}⟩ = 0. Inserting Â given by (37) and ∂_{r} = ∑_{i=0}^{ℓ} \frac{r_{i}}{r} ∂_{r_{i}}, this equation becomes equal to

∑_{i=0}^{ℓ} \frac{r_{i}^{2}}{C_{i} − ξ} = 0. (40)

We obtain that each eigenvalue ξ of A which is not an eigenvalue of Â must be a solution to this equation. For fixed r_{0}, … r_{ℓ}, the function h(ξ) = ∑_{i=0}^{ℓ} \frac{r_{i}^{2}}{C_{i}−ξ} has ℓ + 1 poles at C_{0}, … C_{ℓ} and is monotonously increasing within the intervals (C_{i}, C_{i+1}). Hence, it has ℓ zeros ξ_{1}, … ξ_{ℓ} which are the ℓ nonconstant eigenvalues of A depending on r_{0}, … r_{ℓ}. We have just seen that these eigenvalues have to satisfy the relation (39).

On the other hand, if an eigenvalue C_{i} of Â has multiplicity m_{i} ≥ 2, the corresponding eigenspace must have an m_{i}−1 dimensional intersection with H, hence, C_{i} is also a constant eigenvalue of A of multiplicity m_{i} − 1. The number of eigenvalues of A found so far is

ℓ + ∑_{i=0}^{ℓ}(m_{i} − 1) = −1 + ∑_{i=0}^{ℓ} m_{i} = −1 + dim M̂ = dim M.

Thus, we certainly found all eigenvalues of A.

Multiplying (40) with ∏_{i=0}^{ℓ}(C_{i} − ξ), we obtain ∑_{i=0}^{ℓ} r_{i}^{2} ∏_{j≠i}(C_{j} − ξ) = 0. The left hand side is a polynomial in ξ of degree ℓ and since the nonconstant eigenvalues ξ_{1}, … ξ_{ℓ} are the roots of this polynomial, we obtain

p_{nc}(t) = \frac{1}{r^{2}} ∑_{i=0}^{ℓ} r_{i}^{2} ∏_{j≠i}(t − C_{j}),

where p_{nc}(t) = ∏_{i=1}^{ℓ}(t − ξ_{i}) is the nonconstant part of the characteristic polynomial of A. The characteristic polynomial of A is then given by formula (38). □

Denote by ξ_{1}, … ξ_{ℓ} the nonconstant eigenvalues of A and by η its constant eigenvalues of multiplicity m_{η}. The characteristic polynomial p_{A}(t), expressed in terms of the radial coordinates r_{i}, is given by (38); hence, we obtain the relation

∏_{i=1}^{ℓ}(t − ξ_{i}) = \frac{1}{r^{2}} ∑_{I=0}^{ℓ} r_{I}^{2} ∏_{J≠I}(t − C_{J}), (41)

between the two sets of functions {ξ_{1}, … ξ_{ℓ}} and {r_{0}, … r_{ℓ}}. Inserting t = C_{I} into formula (41), we obtain the functions r_{I} explicitly as functions of the ξ_{i}:

r_{I}^{2} = \frac{∏_{i=1}^{ℓ}(C_{I} − ξ_{i})}{∏_{J≠I}(C_{I} − C_{J})}. (42)

Differentiating yields

2r_{I}dr_{I} = − ∑_{i=1}^{ℓ} \frac{∏_{j≠i}(C_{I} − ξ_{j})}{∏_{J≠I}(C_{I} − C_{J})}dξ_{i}. (43)

A.3. A local description of C_{ℂ}(−1)-metrics. We rederive the part of Theorem 3 stating necessary conditions on the parameters from formula (9) for g being C_{ℂ}(−1).

Proposition 2. Consider a C_{ℂ}(−1) metric g given by formula (9) w.r.t. some A ∈ Sol(g, J) with nonconstant eigenvalues ξ_{1}, … ξ_{ℓ}. Let C_{0} < ⋯ < C_{ℓ} be the distinct eigenvalues of the corresponding parallel hermitian endomorphism Â on the cone. Then Θ_{j}(t) = −4 ∏_{I=0}^{ℓ}(t − C_{I}) for j = 1, … ℓ.

Proof. The part of the metric g in (9) involving the dξ_{i}’s corresponds to the part ∑_{I=0}^{ℓ} dr_{I}^{2} of g in (35). Using (42) and (43), we obtain

4dr_{I}^{2} = \frac{∑_{i_{1},i_{2}=1}^{ℓ} ∏_{j_{1}≠i_{1}}(C_{I} − ξ_{j_{1}}) ∏_{j_{2}≠i_{2}}(C_{I} − ξ_{j_{2}})dξ_{i_{1}} ⊗ dξ_{i_{2}}}{∏_{J≠I}(C_{I} − C_{J}) ∏_{i=1}^{ℓ}(C_{I} − ξ_{i})} . (44)

Let 4 ∑_{I=0}^{ℓ} dr_{I}^{2} =: A_{i_{1}i_{2}}dξ_{i_{1}} ⊗ dξ_{i_{2}}. For i_{1} ≠ i_{2}, (44) implies that

A_{i_{1}i_{2}} = ∑_{I=0}^{ℓ} \frac{∏_{j≠i_{1},i_{2}}(C_{I} − ξ_{j})}{∏_{J≠I}(C_{I} − C_{J})}.

The numerator of each term in this sum is a polynomial of degree ℓ − 2 in C_{I}, hence, applying a Vandermonde identity (see, for instance, the appendix of [2]) in the ℓ + 1 variables C_{0}, … C_{ℓ}, we see that A_{i_{1}i_{2}} = 0 for i_{1} ≠ i_{2}. For the case i = i_{1} = i_{2}, we obtain

A_{ii} = ∑_{I=0}^{ℓ} \frac{∏_{j≠i}(C_{I} − ξ_{j})}{∏_{J≠I}(C_{I} − C_{J})(C_{I} − ξ_{i})}. (45)

The numerator of each term in this sum is a polynomial of degree ℓ − 1 in C_{I}. Applying Vandermonde identities with respect to the ℓ + 2 variables C_{0}, … C_{ℓ}, ξ_{i}, we obtain that

A_{ii} = −\frac{∏_{j≠i}(ξ_{i} − ξ_{j})}{∏_{I=0}^{ℓ}(ξ_{i} − C_{I})}.

Thus we have

∑_{i=0}^{ℓ} dr_{i}^{2} = − ∑_{i=1}^{ℓ} \frac{∏_{j≠i}(ξ_{i} − ξ_{j})}{4 ∏_{I=0}^{ℓ}(ξ_{i} − C_{I})}dξ_{i}^{2}.

Comparing this with (9), we see that Θ_{i}(t) = −4 ∏_{I=0}^{ℓ}(t − C_{I}) as we claimed. □

A.4. C_{ℂ}(−1)-metrics with mobility ≥ 3. The cone construction provides a more geometric explanation why the conditions on the eigenvalues in Theorem 6 imply that the mobility is ≥ 3: since for a C_{ℂ}(−1) metric g the space Sol(g, J) is isomorphic to the space of parallel hermitian endomorphisms on the cone (M̂, ĝ, Ĵ), the decomposition theorem for riemannian manifolds [12] implies that the mobility D(g, J) = dim Sol(g, J) is given by

D(g, J) = f^{2} + i, (46)

where f is the complex dimension of the flat part and i is the number of irreducible (nonflat) components of ĝ (see also [22]). Let C_{0} ≤ ⋯ ≤ C_{n} denote the (not necessarily distinct) eigenvalues of a parallel hermitian endomorphism Â on M̂ and let A be the corresponding element of Sol(g, J). Lemma 6 shows that each repeated eigenvalue C_{i−1} = C_{i} of Â gives rise to a constant eigenvalue of A, while each gap C_{j−1} < C_{j} gives rise to a nonconstant eigenvalue of A taking values in the interval [C_{j−1}, C_{j}]. This explains the assumptions in Theorem 6: if the number of nonconstant eigenvalues of A is ≥ 2 or the number of constant eigenvalues of A is ≥ 3, the number of distinct eigenvalues of Â must be ≥ 3. Now, given a parallel hermitian endomorphism Â on the cone with at least three distinct eigenvalues, the decomposition theorem, together with formula (46) show that the mobility is ≥ 3.

Acknowledgements. We are grateful to Mike Eastwood and Katharina Neusser for pointing out a gap (concerning the constancy of B) in an earlier version of the proof of Theorem 2; this result will also appear in [11] with alternative proofs of the constancy of B. We are likewise indebted to the referee for insightful comments which greatly improved the clarity and scope of the paper. In particular, these observations suggested Theorem 4 and the remarks following it.

We would also like to thank the Mathematical Sciences Institute at the Australian National University for the opportunity to meet at a workshop in March 2013, and the Deutsche Forschungsgemeinschaft (Research training group 1523 — Quantum and Gravitational Fields) and FSU Jena for their financial support.

References

[1] M. Abreu, Kähler geometry of toric varieties and extremal metrics, Int. J. Math. 9 (1998) 641–651.

[2] V. Apostolov, D. M. J. Calderbank and P. Gauduchon, Hamiltonian 2-forms in Kähler geometry. I. General theory, J. Differential Geom. 73 (2006) 359–412.

[3] V. Apostolov, D. M. J. Calderbank, P. Gauduchon and C. Tønnesen-Friedman, Hamiltonian 2-forms in Kähler geometry. II. Global classification, J. Differential Geom. 68 (2004) 277–345.

[4] V. Apostolov, D. M. J. Calderbank, P. Gauduchon and C. Tønnesen-Friedman Hamiltonian 2-forms in Kähler geometry. III Extremal metrics and stability, Invent. Math. 173 (2008) 547–601.

[5] S. Armstrong, Projective holonomy I: principles and properties, II: cones and complete classifications, Ann. Global Anal. and Geom. 33 (2008) 47–69, 137–160.

[6] A. V. Bolsinov, V. S. Matveev and S. Rosemann, Local normal forms for c-projectively equivalent metrics and proof of the Yano–Obata conjecture in arbitrary signature. Proof of the projective Lichnerowicz conjecture for Lorentzian metrics, preprint (2015), arXiv:1510.00275.

[7] A. V. Bolsinov, V. Kiosak and V. S. Matveev, A Fubini theorem for pseudo-Riemannian geodesically equivalent metrics, J. Lond. Math. Soc. (2) 80 (2009) 341–356, MR2545256.

[8] E. Calabi, The space of Kähler metrics, Proc. Internat. Congress Math., Noordhoff, Groningen– Amsterdam, 1954, pp. 206–207.

[9] E. Calabi, Métriques kählériennes et fibrés holomorphes, Ann. Sci. Ecole Norm. Sup. (4) 12 (1979) 269–294.

[10] E. Calabi, Extremal Kähler metrics, Seminar on Differential Geometry, Princeton University Press, 1982.

[11] D. M. J. Calderbank, M. Eastwood, V. S. Matveev and K. Neusser, C-projective geometry, in preparation.

[12] L. P. Eisenhart, Symmetric tensors of the second order whose first covariant derivatives are zero, Trans. Amer. Math. Soc. 25 (1923) 297–306.

[13] A. Fedorova and S. Rosemann, The Tanno theorem for Kählerian metrics with arbitrary signature, Diff. Geom. Appl. 29, suppl. 1 (2011) 71–79.

[14] A. Fedorova, V. Kiosak, V. Matveev and S. Rosemann, The only Kähler manifold with degree of mobility at least 3 is (CP (n), g_{Fubini–Study}), Proc. Lond. Math. Soc. 105 (2012) 153–188.

[15] A. Gray, Pseudo-Riemannian almost product manifolds and submersions, J. Math. Mech. 16 (1967) 715–737.

[16] S. Ishihara and S. Tachibana, A note on holomorphic projective transformations of a Kählerian space with parallel Ricci tensor, Tohoku Math. J. 13 (1961) 193–200.

[17] K. Kiyohara and P. J. Topalov, On Liouville integrability of h-projectively equivalent Kähler metrics, Proc. Amer. Math. Soc. 139 (2011) 231–242.

[18] B. Kostant, Holonomy and the Lie algebra of infinitesimal motions of a Riemann manifold, Trans. Amer. Math. Soc. 80 (1955) 528–542.

[19] E. Lerman and S. Tolman, Hamiltonian torus actions on symplectic orbifolds and toric varieties, Trans. Amer. Math. Soc. 349 (1997) 4201–4230.

[20] A. D. Hwang and M. A. Singer, A momentum construction for circle-invariant Kahler metrics, Trans. Amer. Math. Soc. 354 (2002) 2285–2325.

[21] V. S. Matveev and S. Rosemann, Proof of the Yano–Obata conjecture for h-projective transformations, J. Differential Geom. 92 (2012) 221–261.

[22] V. S. Matveev and S. Rosemann, Conification construction for Kaehler manifolds and its application in c-projective geometry, Adv. Math. 274 (2015) 1–38.

[23] K. Schöbel, The variety of integrable Killing tensors on the 3-sphere, SIGMA 10 (2014), 080.

[24] J. Mikeš and V. V. Domashev, On the theory of holomorphically projective mappings of Kaehlerian spaces, Math. Zametki 23 (1978) 297–303.

[25] J. Mikeš, Holomorphically projective mappings and their generalizations, J. Math. Sci. 89 (1998) 1334–1353.

[26] B. O’Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966) 459–469.

[27] T. Otsuki and Y. Tashiro, On curves in Kählerian spaces, Math. J. Okayama Univ. 4 (1954) 57–78.

[28] S. Tanno, Some Differential Equations On Riemannian Manifolds, J. Math. Soc. Japan 30 (1978) 509–531.

[29] Y. Yoshimatsu, H-projective connections and H-projective transformations, Osaka J. Math. 15 (1978) 435–459.

DMJC: Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK

  E-mail address: D.M.J.Calderbank@bath.ac.uk

VSM, SR: Institute of Mathematics, FSU Jena, 07737 Jena, Germany

  E-mail address: vladimir.matveev@uni-jena.de, stefan.rosemann@uni-jena.de