HAMILTONIAN 2-FORMS IN KÄHLER GEOMETRY, I GENERAL THEORY

VESTISLAV APOSTOLOV, DAVID M. J. CALDERBANK, AND PAUL GAUDUCHON

Abstract. We introduce the notion of a hamiltonian 2-form on a Kähler manifold and obtain a complete local classification. This notion appears to play a pivotal role in several aspects of Kähler geometry. In particular, on any Kähler manifold with co-closed Bochner tensor, the (suitably normalized) Ricci form is hamiltonian, and this leads to an explicit description of these Kähler metrics, which we call weakly Bochner-flat. Hamiltonian 2-forms also arise on conformally Einstein Kähler manifolds and provide an Ansatz for extremal Kähler metrics unifying and extending many previous constructions.

In a previous paper [3], while investigating Kähler 4-manifolds whose antiselfdual Weyl tensor is co-closed, we happened upon a remarkable linear differential equation for (1, 1)-forms ϕ on a Kähler manifold. This equation states (in any dimension)

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

for all vector fields X, where (g, J, ω) is the Kähler structure with Levi-Civita connection ∇. A hamiltonian 2-form is a (nontrivial) solution ϕ of (1).

Hamiltonian 2-forms underpin many explicit constructions in Kähler geometry. They arise in particular on Bochner-flat Kähler manifolds and on Kähler manifolds (of dimension greater than four) which are conformally Einstein, both of which have been classified recently, respectively by Bryant [5], and Derdziński and Maschler [8]. In this paper we obtain an explicit local classification of all Kähler metrics with a hamiltonian 2-form, which provides a unifying framework for these works, and at the same time extends Bryant’s local classification to the much larger class of Kähler manifolds with co-closed Bochner tensor, called weakly Bochner-flat.

The key feature of hamiltonian 2-forms ϕ on Kähler 2m-manifolds M—and the reason for the name—is that if σ_{1}, … σ_{m} are the elementary symmetric functions of the m eigenvalues of ϕ (viewed as a hermitian operator via the Kähler form ω), then the hamiltonian vector fields K_{r} = J grad_{g} σ_{r} are Killing. Further, the Poisson brackets {σ_{r}, σ_{s}} are all zero, so that the vector fields K_{1}, … K_{m} commute.

If K_{1}, … K_{m} are linearly independent, then the Kähler metric is toric. However, not every toric Kähler metric arises in this way: the hamiltonian property also implies that the eigenvalues of ϕ have orthogonal gradients. We say that a toric manifold is orthotoric if there is a momentum map (σ_{1}, … σ_{m}) for the torus action (with respect to some basis of the Lie algebra) such that the gradients of the roots of the polynomial ∑_{r=0}^{m}(−1)^{r}σ_{r}t^{m−r} are orthogonal—here σ_{0} = 1.

Of course K_{1}, … K_{m} need not be independent; then on an open set where the span is ℓ-dimensional, there is a local hamiltonian ℓ-torus action by isometries, so the Kähler metric on M may be described (locally) by the Pedersen–Poon construction [20], as a fibration, with 2ℓ-dimensional toric fibres, over a 2(m−ℓ)-dimensional complex manifold S equipped with a family of Kähler quotient metrics parameterized by the momentum map of the local ℓ-torus action.

Date: November 2002.

The first author was supported in part by FCAR grant NC-7264, and by NSERC grant OGP0023879, the second author by the Leverhulme Trust and the William Gordon Seggie Brown Trust. All three authors are members of EDGE, Research Training Network HPRN-CT-2000-00101, supported by the European Human Potential Programme.

The hamiltonian property of ϕ has further implications for the geometry of this fibration and of the base S. We say that a hamiltonian ℓ-torus action is rigid if the metric on the orbits depends only on the momentum map, and that the base S is semisimple if the Kähler quotient metrics are simultaneously diagonalizable and have common Levi-Civita connection. The rigidity condition has its origins in work of Calabi on Kähler metrics on holomorphic bundles [6, 7] and has a number of formulations: it means, for instance, that the local fibration of M over S is totally geodesic, or equivalently, that M is associated, locally, to a principal ℓ-torus bundle with connection over S. When ℓ = 1 semisimplicity is closely related to the ‘σ-constancy’ of Hwang–Singer [12]. Both rigidity and semisimplicity have explicit descriptions as special cases of the Pedersen–Poon construction.

Our main result shows that any Kähler manifold with a hamiltonian 2-form induces a (local) semisimple rigid ℓ-torus action (ℓ ≤ m) with orthotoric fibres, and that conversely, such an explicit Kähler metric has a hamiltonian 2-form. From this we deduce a local classification of weakly Bochner-flat Kähler metrics, rederiving in particular Bryant’s classification of Bochner-flat Kähler metrics. We also obtain a new proof of the Derdziński–Maschler classification of conformally Einstein Kähler metrics in higher dimensions, and an Ansatz for extremal Kähler metrics—including all constant scalar curvature and Kähler–Einstein metrics with a hamiltonian 2form—which unifies and extends many constructions in the literature.

The structure of the paper is as follows. In section 1, after reviewing some background material, we explain how equation (1) arises on weakly Bochner-flat Kähler manifolds and on conformally Einstein Kähler manifolds. Thus motivated, we begin the study of hamiltonian 2-forms in section 2, where we derive the existence of the hamiltonian Killing vector fields, and show that the equation for hamiltonian 2-forms is an overdetermined equation of finite type which is completely integrable on manifolds of constant holomorphic sectional curvature.

In section 3 we study (isometric) hamiltonian torus actions in general. This section is almost entirely independent of the first two, although our analysis is motivated by the special properties of hamiltonian 2-forms. We first show that the Pedersen–Poon construction [20] has a natural and essentially coordinate-free description in terms of a potential G, which is a fibrewise Legendre transform of a Kähler potential, and is required to satisfy only open conditions—such ‘dual potentials’ appeared first in the toric case, in work of Guillemin [11] and Abreu [1]. We also describe the invariant pluriharmonic functions and compute the Ricci form.

In subsections 3.2 and 3.3 we introduce rigid and semisimple hamiltonian torus actions respectively. In the case of circle actions (in particular) these conditions originally arose as an Ansatz for the construction of extremal Kähler metrics and Kähler–Einstein metrics [6, 7, 12, 14, 20, 23, 24, 25]: for semisimple rigid actions in momentum coordinates, the Ricci form is linear in the matrix of inner products of the Killing vector fields generating the action. Our approach provides a natural interpretation, particularly for the rigidity condition.

Subsection 3.4 is devoted to orthotoric Kähler metrics: in four dimensions, these were introduced in [3] and explicitly classified; here we extend the definition and classification to all dimensions. The decisive feature of orthotoric 2m-manifolds is that they depend effectively on m functions of 1 variable, rather than the 1 function of m variables (the dual potential G) that governs toric Kähler metrics in general. This means that curvature conditions are (functional) ordinary differential equations, rather than partial differential equations.

The central results of this paper can be found in section 4, where we bring the work of sections 2 and 3 together. We first prove that on a connected Kähler 2mmanifold with hamiltonian 2-form ϕ and associated Killing vector fields K_{1}, … K_{m}, there is an integer ℓ, with 0 ≤ ℓ ≤ m, such that the span of K_{1}, … K_{m} is everywhere at most ℓ-dimensional, but on a dense open set K_{1}, … K_{ℓ} are linearly independent. We show that ℓ roots of the momentum polynomial p(t) := (−1)^{m} pf(ϕ − tω) are functionally independent, the remainder being constant. (Here pf ψ = \frac{1}{m!}∗(ψ^{∧m}) stands for the pfaffian of a 2-form ψ.) We call ℓ the order of ϕ and prove that Kähler manifolds admitting a hamiltonian 2-form of order ℓ are exactly those admitting a local hamiltonian ℓ-torus action such that

• the fibres are orthotoric;

• the action is rigid;

• the base is semisimple, with relative eigenvalues of a special form. In section 5 we study the curvature of our explicit metrics and hence obtain

classifications of extremal Kähler metrics with a hamiltonian 2-form, of weakly Bochner-flat Kähler metrics, and of Bochner-flat Kähler metrics [5].

To summarize, we have the following result.

Theorem. Let (M, g, J, ω) be a connected Kähler 2m-manifold with a hamiltonian 2-form ϕ of order ℓ. Then there are functions F_{1}, … F_{ℓ} of one variable such that on a dense open subset M^{0} of M, the Kähler structure may be written

g = ∑_{ξ} p_{nc}(ξ)g_{ξ} + ∑_{j=1}^{ℓ} \frac{p′(ξ_{j})}{F_{j}(ξ_{j})}dξ_{j}^{2} + ∑_{j=1}^{ℓ} \frac{F_{j}(ξ_{j})}{p′(ξ_{j})}(∑_{r=1}^{ℓ} σ_{r−1}(ξ̂_{j})θ_{r})^{2},

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

Jdξ_{j} = \frac{F_{j}(ξ_{j})}{p′(ξ_{j})} ∑_{r=1}^{ℓ} σ_{r−1}(ξ̂_{j}) θ_{r},  Jθ_{r} = (−1)^{r} ∑_{j=1}^{ℓ} \frac{p_{c}(ξ_{j})}{F_{j}(ξ_{j})}ξ_{j}^{ℓ−r} dξ_{j}.

Any Kähler metric of this form admits a hamiltonian 2-form of order ℓ, namely

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

In these expressions:

• σ_{r} is the rth elementary symmetric function of the non-constant roots ξ_{1}, … ξ_{ℓ} of the momentum polynomial p(t)—so σ_{ℓ+1} = 0—and σ_{r−1}(ξ̂_{j}) is the (r − 1)st elementary symmetric functions of the ℓ − 1 roots {ξ_{k} : k ≠ j};

• summation over ξ denotes the sum over the different constant roots of the momentum polynomial and (g_{ξ}, ω_{ξ}) is a positive or negative definite Kähler metric on a manifold S_{ξ} of the same (real) dimension 2m_{ξ} as the ξ-eigenspace of −J ∘ ϕ;

• p(t) = p_{nc}(t)p_{c}(t), where p_{nc}(t) = ∏_{j=1}^{ℓ}(t − ξ_{j}) and p_{c}(t) = ∏_{ξ}(t − ξ)^{m_{ξ}}; also p′(t) is the t-derivative of p(t), so that p′(ξ_{j}) = p_{c}(ξ_{j}) ∏_{k≠j}(ξ_{j} − ξ_{k}).

Now define polynomials p̌_{c}(t) = ∏_{ξ}(t−ξ)^{m_{ξ}−1}, and p̂_{c}(t) = ∏_{ξ}(t−ξ)^{m_{ξ}+1}. Then we have the following special cases.

(i) (g, J, ω) is an extremal Kähler metric if • for all j, F′′_{j}(t) = p̌_{c}(t)(∑m̌ ; r=0 a_{r}t^{m̌−r}), where a_{0}, … am̌ are arbitrary constants

(independent of j) and m̌ = ℓ + ∑_{ξ} 1 = m − ∑_{ξ}(m_{ξ} − 1);

• for all ξ, ±(g_{ξ}, ω_{ξ}) has Scal_{±g_{ξ}} = ∓(∑m̌ ; r=0 a_{r}ξ^{m̌−r})/ ∏_{η≠ξ}(ξ − η).

The scalar curvature is constant if and only if a_{0} = 0 and zero if and only if also a_{1} = 0. An extremal Kähler metric with a hamiltonian 2-form arises in this way if the gradient of the scalar curvature is tangent to the fibration defined by the 2-form. (ii) (g, J, ω) is weakly Bochner-flat if

• for all j, F′_{j}(t) = p_{c}(t)(∑_{r=−1}^{ℓ} b_{r}t^{ℓ−r}), where b_{−1}, … b_{ℓ} are arbitrary constants (independent of j);

• for all ξ, ±(g_{ξ}, ω_{ξ}) is Kähler–Einstein with ‘Kähler–Einstein constant’

\frac{1}{m_{ξ}} Scal_{±g_{ξ}} = ∓ ∑_{r=−1}^{ℓ} b_{r}ξ^{ℓ−r}.

(g, J, ω) is Kähler–Einstein if and only if b_{−1} = 0 and Ricci-flat if and only if also b_{0} = 0. Any weakly Bochner-flat Kähler metric arises in this way. (iii) (g, J, ω) is Bochner-flat if

• for all j, F_{j}(t) = p̂_{c}(t)(∑m̂ ; r=−2 c_{r}t^{m̂−r}), where c_{−2}, … cm̂ are arbitrary constants (independent of j) and m̂ = ℓ − ∑_{ξ} 1 = m − ∑_{ξ}(m_{ξ} + 1);

• for all ξ, ±(g_{ξ}, ω_{ξ}) has constant holomorphic sectional curvature

\frac{1}{m_{ξ}(m_{ξ} + 1)}Scal_{±g_{ξ}} = ∓( ∑_{r=−2}^{m̂} c_{r}ξ^{m̂−r}) ∏_{η≠ξ}(ξ − η).

(g, J, ω) has constant holomorphic sectional curvature if and only if c_{−2} = 0 and is flat if and only if also c_{−1} = 0. Any Bochner-flat Kähler metric arises in this way.

This theorem follows from Theorems 1 and 2, and Propositions 15, 16 and 17 in the text below. We end by discussing hamiltonian 2-forms of order 1, and the classification of conformally Einstein Kähler metrics [8]. There are also two appendices. In Appendix A, we relate hamiltonian 2-forms to conformal Killing forms, recently studied by Moroianu and Semmelmann [18, 22]. In Appendix B, we collect some Vandermonde identities, which we have used freely in the paper.

We thank Uwe Semmelmann for discussing conformal Killing forms with us, and Christina Tønnesen-Friedman for her helpful comments and interest in this work.

Contents

1. The curvature of a Kähler manifold 5 1.1. Riemannian curvature 5 1.2. The Bochner tensor of a Kähler manifold 6 1.3. The differential Bianchi identity in Kähler geometry 7 1.4. Conformally Einstein Kähler metrics 8 2. Hamiltonian 2-forms 9 2.1. Hamiltonian 2-forms and Killing vector fields 9 2.2. The connection on 2-jets of hamiltonian 2-forms 11 2.3. The differential system in the weakly Bochner-flat case 12 2.4. Complex projective, hyperbolic and euclidean space 13 3. Hamiltonian torus actions 14 3.1. The Pedersen–Poon construction 14 3.2. Rigid hamiltonian torus actions 17 3.3. Semisimple Kähler quotients 19 3.4. Orthotoric Kähler metrics 19 4. Classification of hamiltonian 2-forms 21 4.1. Rough classification of hamiltonian 2-forms 21 4.2. Explicit description of the metric 23 4.3. The hamiltonian 2-form 26 5. The curvature of Kähler manifolds with a hamiltonian 2-form 28 5.1. The Ricci potential and scalar curvature 28 5.2. Extremal Kähler metrics 29 5.3. Weakly Bochner-flat Kähler metrics 30 5.4. Bochner-flat Kähler metrics 31 5.5. The Calabi-type case 34 5.6. Strongly conformally Einstein Kähler metrics 35 Appendix A. Conformal Killing forms and hamiltonian 2-forms 36 Appendix B. Vandermonde matrices 38 B.1. The inverse of a Vandermonde matrix 38 B.2. The determinant of a Vandermonde matrix 38 B.3. Vandermonde identities 39 References 40

1. The curvature of a Kähler manifold

In this section we review some background material in order to fix notation, and to present the notions of Bochner-flat, weakly Bochner-flat, and conformally Einstein Kähler metrics. Our conventions mainly follow [4].

1.1. Riemannian curvature. The curvature R of a n-dimensional riemannian manifold (M, g) is defined by

R_{X,Y} Z = ∇_{[X,Y]}Z − [∇_{X}, ∇_{Y}]Z,

for all vector fields X, Y, Z, where ∇ denotes the Levi-Civita connection. It is a 2-form with values in the adjoint bundle AM (the bundle of skew endomorphisms of the tangent bundle T M) and satisfies the algebraic Bianchi identity: R_{X,Y} Z + R_{Y,Z}X +R_{Z,X}Y = 0. Via the metric g, AM can be identified with the bundle Λ^{2}M of 2-forms and R can be viewed as a section of Λ^{2}M ⊗ Λ^{2}M. Then, the algebraic Bianchi identity is equivalent to the following two conditions:

(i) R belongs the the symmetric part, S^{2}Λ^{2}M, of Λ^{2}M ⊗ Λ^{2}M; (ii) R belongs to the kernel of the linear map, β, from S^{2}Λ^{2}M to Λ^{4}M determined by the wedge product. RM := ker β ⊆ S^{2}Λ^{2}M is called the bundle of (abstract) curvature tensors.

The Ricci contraction is the linear map c from RM to the bundle SM of symmetric bilinear forms of M sending R to the bilinear form Ric defined by Ric_{X,Y} = tr(Z → R_{X,Z}Y). We thus obtain an orthogonal decomposition:

RM = c^{∗}(SM) ⊕ WM,

where WM, called the bundle of (abstract) Weyl tensors of (M, g), denotes the kernel of c in RM. Accordingly, the curvature R splits as R = c^{∗}(h) + W, where W is the Weyl tensor of (M, g), whereas h satisfies cc^{∗}(h) = Ric. For n ≥ 3, c is surjective; its adjoint c^{∗} is then injective and h is determined by

h = \frac{Scal}{2n(n − 1)}g + \frac{Ric_{0}}{n − 2},

where Scal is the scalar curvature of g, i.e., the trace of Ric with respect to g, and Ric_{0} denotes the traceless part of Ric (so that Ric = \frac{1}{n}Scal g + Ric_{0}), which is a section of S_{0}M, the bundle of symmetric traceless bilinear forms. For n = 2, c^{∗} has kernel S_{0}M so that Ric_{0} = 0 and the tracefree part of h is undetermined.

Finally, the curvature R, viewed as a symmetric endomorphism of Λ^{2}M using g, splits into three orthogonal pieces as follows:

(2)  R = \frac{Scal}{n(n − 1)}Id|_{Λ^{2}M} + \frac{1}{n − 2}\{Ric_{0}, ·\} + W,

where {Ric_{0}, ·} acts on ψ ∈ Λ^{2}M to give the anticommutator {Ric_{0}, ψ} := Ric_{0} ∘ ψ +ψ ∘Ric_{0} of Ric_{0} and ψ, which are viewed, via g, as endomorphisms, respectively symmetric and skew, of T M. When n = 2, the second term is zero.

Each piece of (2) is an element of RM. The corresponding subbundle of RM is associated to an irreducible representation of the orthogonal group O(n), respectively the trivial representation, the Cartan product ℝ^{n} ⊙ ℝ^{n}, the Cartan product 𝔬(n) ⊙ 𝔬(n), where 𝔬(n) ≅ Λ^{2}ℝ^{n} denotes the Lie algebra of O(n). (The Cartan product of two irreducible representations with dominant weights λ_{1} and λ_{2} is the irreducible sub-representation of the tensor product with dominant weight λ_{1} +λ_{2}.)

1.2. The Bochner tensor of a Kähler manifold. Let (M, g, J, ω) be a Kähler manifold of dimension n = 2m. By definition, J is an orthogonal complex structure which is parallel with respect to the Levi-Civita connection ∇. The Kähler form ω is defined by ω(X, Y) = g(JX, Y). The Ricci form ρ and its primitive part ρ_{0} are defined in a similar way: ρ(X, Y) = Ric(JX, Y) and ρ_{0}(X, Y) = Ric_{0}(JX, Y). The Ricci tensor is J-invariant, and so ρ and ρ_{0} are 2-forms. We denote by

Λ^{2}M = Λ^{J,+}M ⊕ Λ^{J,−}M,

the (orthogonal) decomposition of Λ^{2}M into its J-invariant part, Λ^{J,+}M, and its J-anti-invariant part, Λ^{J,−}M. The riemannian curvature R has values in Λ^{J,+}M and therefore acts trivially on Λ^{J,−}M. More generally, we call an element of RM kählerian if it acts trivially on Λ^{J,−}M. The set of abstract kählerian curvature tensors is a vector subbundle of RM, denoted by KM; thus KM is the kernel of the linear map from S^{2}Λ^{J,+}M to Λ^{4}M determined by the wedge product.

The curvature tensor R of a Kähler manifold (M, g, J) is a section of KM, but in general none of its components in (2) is. Indeed, the first component of R in (2) is only an element of KM if it is zero or n = 2, while the second component is only an element of KM if it is zero or n = 4. We define the Bochner tensor W^{K} to be the orthogonal projection of the third component, the Weyl tensor W, onto WM ∩ KM. We thus obtain a new decomposition of the curvature R inside KM.

R = \frac{Scal}{2m(m + 1)}(Id|_{Λ^{J,+}M} + ω ⊗ ω)

+ \frac{1}{m + 2} (\{Ric_{0}, ·\}|_{Λ^{J,+}M} + ρ_{0} ⊗ ω + ω ⊗ ρ_{0})

+ W^{K}.

(3)

Here |_{Λ^{J,+}M} has to be interpreted as the orthogonal projection ψ ↦ ψ^{J,+} of Λ^{2}M onto its J-invariant part Λ^{J,+}M, and ρ_{0} ⊗ ω acts on ψ ∈ Λ^{2}M to give ⟨ρ_{0}, ψ⟩ω, where the inner product on 2-forms is normalized so that ⟨ω, ω⟩ = m = n/2.

The three pieces of R appearing in (3) are sections of subbundles of KM associated to irreducible representations of the unitary group U(m), viewed as a subgroup of O(n), namely: the trivial representation, the Cartan product ℂ^{m} ⊙ ℂ^{m}, the Cartan product 𝔰𝔲(m) ⊙ 𝔰𝔲(m) respectively, 𝔰𝔲(m) being the Lie algebra of SU(m).

If Scal is a positive constant, then the first component of R in (3) agrees with the curvature of the complex projective space ℂP^{m} with the Fubini–Study metric of holomorphic sectional curvature equal to \frac{Scal}{m(m+1)}.

The second component in (3) agrees with the second component in (2) when n = 4, since then {Ric_{0}, ψ^{J,−}} = [Jψ^{J,−}, ρ_{0}] = 0, whereas {Ric_{0}, ψ^{J,+}} = ⟨ψ, ρ_{0}⟩ω+ ⟨ψ, ω⟩ρ_{0}. The four dimensional case is also special because the Weyl tensor W splits into selfdual and antiselfdual parts as W = W^{+} + W^{−}, and on a Kähler 4-manifold the selfdual part is identified with the scalar curvature by

W^{+} = \frac{Scal}{12} (\frac{3}{2}ω ⊗ ω − Id|_{Λ^{J,+}M}).

Bringing together W^{+} and the scalar part of R in (2), we deduce that W^{K} = W^{−}.

In higher dimensions, the Weyl tensor W of a Kähler manifold splits into three pieces: one is the Bochner tensor W^{K}, while the other two are identified with Ric_{0} and Scal. In other words, on a Kähler manifold of dimension n ≥ 6 the information given by the riemannian curvature is already contained in the Weyl tensor; in particular, for n ≥ 6, a locally conformally flat Kähler metric is flat.

1.3. The differential Bianchi identity in Kähler geometry. The differential Bianchi identity

(4)  ∇_{X}R_{Y,Z} + ∇_{Y} R_{Z,X} + ∇_{Z}R_{X,Y} = 0,

easily implies the following one, known as the Matsushima identity:

(5)  (δR)_{JX} = −∇_{X}ρ.

(We specialize (4) by X = e_{j}, Y = Je_{j}, where {e_{j}} is a local, J-adapted, orthonormal frame and we observe that ρ = \frac{1}{2} ∑_{j=1}^{n} R_{e_{j},Je_{j}}; here we define (δR)_{JX} := − ∑_{j=1}^{n} ∇_{e_{i}}R_{e_{i},JX}.) The Matsushima identity immediately implies that the Ricci tensor of a Kähler manifold is parallel if and only if the curvature is co-closed, as a 2-form with values in Λ^{2}M. The Ricci form ρ may also be expressed as ρ(X, Y) = \frac{1}{2} ∑_{j=1}^{n}⟨R_{X,Y} e_{j}, Je_{j}⟩, and so it is closed by (4). Hence from (3) and (5), we infer the following expression for the codifferential of the Bochner tensor:

(δW^{K})_{JX} = −\frac{m}{m + 2} ∇_{X}ρ_{0} − \frac{1}{2(m + 1)(m + 2)} dScal(X) ω

+ \frac{m}{4(m + 1)(m + 2)} (dScal ∧ JX − d^{c}Scal ∧ X).

(6)

Here d^{c} = J ∘ d, X is any vector field, and we identify vector fields and 1-forms via g. In view of this identity, we introduce a normalized Ricci form ρ̃ defined by

ρ̃ = ρ_{0} + \frac{Scal}{2m(m + 1)} ω.

Then, identity (6) reduces to

(7)  \frac{m + 2}{m}(δW^{K})_{JX} = −∇_{X} ρ̃ + \frac{1}{2}(ds ∧ JX − d^{c}s ∧ X),

where the normalized scalar curvature s = \frac{Scal}{2(m+1)} is the trace of ρ̃ with respect to ω: s = ⟨ρ̃, ω⟩.

Definition 1. A Kähler manifold (M, g, J) is called Bochner-flat (or Bochner– Kähler) if the Bochner tensor vanishes, W^{K} = 0, and weakly Bochner-flat if the Bochner tensor is co-closed, δW^{K} = 0.

By (7), a Kähler manifold is weakly Bochner-flat if and only if it satisfies the following weak Einstein condition:

(8)  ∇_{X} ρ̃ = \frac{1}{2}(ds ∧ JX − d^{c}s ∧ X).

1.4. Conformally Einstein Kähler metrics. A Kähler manifold (M, g, J, ω) of dimension n = 2m ≥ 4 is said to be conformally Einstein if there is a nonvanishing function τ such that g̃ := τ^{−2}g is an Einstein metric, i.e., Ricg̃ ; 0 = 0. A straightforward and standard computation of the conformal change of the Ricci tensor shows that g is conformally Einstein with conformal factor τ if and only if

(9)  2(m − 1)∇Jdτ = −τρ + λω

for some function λ—the trace of this equation then determines that

λ = −\frac{m − 1}{m}Δτ + \frac{1}{2m}Scal τ,

where Δτ = − tr_{g} ∇dτ = −⟨dd^{c}τ, ω⟩.

We recall that a hamiltonian vector field K = J grad_{g} f is Killing if and only if it preserves J, if and only if the hessian ∇df is J-invariant, if and only if ∇Jdf = \frac{1}{2}dd^{c}f, in which case f is said to be a Killing potential.

Clearly equation (9) implies that 2(m − 1)∇Jdτ = (m − 1)dd^{c}τ, so that by differentiating (9), we obtain dτ ∧ ρ − dλ ∧ ω = 0 and hence dτ ∧ dλ ∧ ω = 0. We shall say that g is strongly conformally Einstein if dτ ∧ dλ = 0; this is automatic if n ≥ 6 since the wedge product with ω is then injective on 2-forms.

The fact that conformally Einstein Kähler metrics are strongly conformally Einstein in 6 or more dimensions was first observed by Derdziński and Maschler [8], who used this to obtain an explicit description of such metrics. A key step is essentially equivalent to the following.

Lemma 1. [8] Suppose that g is strongly conformally Einstein, with conformal factor τ. Then on the open set where dτ is nonzero,

(10)  2∇Jdτ = p ω + q dτ ∧ d^{c}τ

for some functions p, q with dp ∧ dτ = 0.

Proof. On the open set where dτ is nonzero, we may write dλ = λ_{τ} dτ, so that dτ ∧ (ρ − λ_{τ} ω) = 0. It follows that the J-invariant 2-form ρ − λ_{τ} ω is equal to fdτ ∧ d^{c}τ for some function f. Therefore:

2(m − 1)∇Jdτ = (λ − τλ_{τ})ω − τf dτ ∧ d^{c}τ

and clearly d(λ − τλ_{τ}) ∧ dτ = 0. □

Remark 1. More generally, the conclusions of this lemma hold on the open set where dτ, ξ_{τ} ≠ 0 if 2∇Jdτ = ξ(τ)ρ + η(τ)ω, with essentially the same proof [8].

In order to interpret the work of Derdziński and Maschler in the present work, we reformulate equation (10). We first note that if τ is any function satisfying (10), for some functions p, q with dp ∧ dτ = 0, then in fact we have

d(|dτ|^{2}) ∧ dτ = 0,  dq ∧ dτ = 0,  and  p = \frac{a}{aτ + b}|dτ|^{2}

for some constants a and b not both zero. Indeed, contracting (10) with Jdτ we obtain d(|dτ|^{2}) = (p + q|dτ|^{2})dτ which gives the first two observations. Hence dd^{c}τ = f|dτ|^{2}ω + q dτ ∧ d^{c}τ, where df ∧ dτ = 0. The exterior derivative of this equation gives |dτ|^{2}(df + f^{2}dτ) ∧ ω = 0, so that f = a/(aτ + b).

Lemma 2. A Killing potential τ satisfies the equation

dd^{c}τ = \frac{a}{aτ + b}|dτ|^{2}ω + q dτ ∧ d^{c}τ

(for some function q and constants a, b not both zero) if and only if the 2-form χ := (aτ + b)dτ ∧ d^{c}τ/|dτ|^{2} satisfies

(11)  ∇_{X}χ = \frac{a}{2}(dτ ∧ JX − d^{c}τ ∧ X).

Proof. If χ = (aτ + b)dτ ∧ d^{c}τ/|dτ|^{2}, then

∇_{X}χ = a dτ(X)\frac{dτ ∧ d^{c}τ}{|dτ|^{2}} − (aτ + b)⟨ι_{X}dd^{c}τ, d^{c}τ⟩\frac{dτ ∧ d^{c}τ}{|dτ|^{4}}

+ (aτ + b)\frac{ι_{JX}dd^{c}τ ∧ d^{c}τ + dτ ∧ ι_{X}dd^{c}τ}{2|dτ|^{2}} .

This can only equal \frac{a}{2}(dτ ∧JX −d^{c}τ ∧X), if dd^{c}τ is of the form f|dτ|^{2}ω +q dτ ∧d^{c}τ for some functions f, q, in which case we obtain

∇_{X}χ = (a − (aτ + b)f)dτ(X)\frac{dτ ∧ d^{c}τ}{|dτ|^{2}} + \frac{1}{2}(aτ + b)f (dτ ∧ JX − d^{c}τ ∧ X).

The result is now immediate. □

2. Hamiltonian 2-forms

In this section we introduce the notion of a hamiltonian 2-form and develop the most basic general properties and the simplest examples.

2.1. Hamiltonian 2-forms and Killing vector fields. The definition of hamiltonian 2-forms is motivated both by weakly Bochner-flat Kähler manifolds and by strongly conformally Einstein Kähler manifolds. The reason for the terminology will shortly become apparent.

Definition 2. Let ϕ be any (real) J-invariant 2-form on the Kähler manifold (M, g, J, ω). We say ϕ is hamiltonian if there is a function σ on M such that

(12)  ∇_{X}ϕ = \frac{1}{2}(dσ ∧ JX − d^{c}σ ∧ X)

for any vector field X. When M is a Riemann surface, we require in addition that σ is a Killing potential.

It follows immediately from the definition that dσ = d tr ϕ, where tr ϕ = ⟨ϕ, ω⟩ is the trace of ϕ with respect to ω, so without loss of generality, we may take σ = tr ϕ. The defining equation for hamiltonian 2-forms is therefore linear. Note that, for a general hamiltonian 2-form ϕ, A = ϕ + σω is closed, dA = 0.

Example 1. On any Kähler manifold, any J-invariant parallel 2-form is hamiltonian. In particular, a constant multiple of the Kähler form ω is hamiltonian. It follows that if ϕ is hamiltonian, then so is ϕ_{t} := ϕ − tω for any constant t.

Example 2. We shall be particularly interested in the hamiltonian 2-forms arising from the following immediate consequence of equation (8).

Proposition 1. A Kähler manifold of dimension 2m ≥ 4 is weakly Bochner-flat if and only if the normalized Ricci form ρ̃ = ρ_{0} + \frac{1}{m}s ω = ρ − s ω is hamiltonian.

Example 3. In view of equation (11), we also have the following result.

Proposition 2. On any strongly conformally Einstein Kähler manifold of dimension 2m ≥ 4, with conformal factor τ, there are constants a, b not both zero, such that χ = (aτ +b)dτ ∧Jdτ/|dτ|^{2} is hamiltonian on the open set where dτ is nonzero.

The equation for hamiltonian 2-forms is overdetermined. By differentiating and skew-symmetrizing (12), we get

\begin{aligned}R_{X,Y} · ϕ = [R_{X,Y} , ϕ] & = \frac{1}{2}(∇_{Y} dσ ∧ JX − ∇_{X}dσ ∧ JY \\ & − J∇_{Y} dσ ∧ X + J∇_{X}dσ ∧ Y). \end{aligned} ; (13)

This formula underlies most of the basic theory of hamiltonian 2-forms. In particular, we shall use it to explain the use of the term “hamiltonian” in this context.

To do that, we first recall that the pfaffian of a 2-form ϕ is defined by

(14)  pf ϕ = \frac{1}{m!}∗(ϕ ∧ ⋯ ∧ ϕ),

where ∗ denotes the Hodge star operator. The normalization is chosen so that pf ω = 1 and thus ϕ∧⋯∧ϕ = (pf ϕ)ω ∧⋯∧ω. We let ϕ_{t} = ϕ−tω as in Example 1 above, and (following Bryant [5]) define the momentum polynomial of ϕ to be

(15)  p(t) := (−1)^{m} pf ϕ_{t} = t^{m} − (tr ϕ) t^{m−1} + ⋯ + (−1)^{m} pf ϕ.

Proposition 3. If ϕ is a hamiltonian 2-form, then the functions p(t) on M (for each t ∈ ℝ) are Poisson-commuting hamiltonians for Killing vector fields K(t) := J grad_{g} p(t) which preserve ϕ. In particular, the vector fields K(t) all commute.

Proof. We first prove that K := J grad_{g} σ is Killing, i.e., ∇dσ is J-invariant. Since R_{X,Y} is J-invariant in X and Y, equation (13) implies that

S(X) ∧ JY − JS(X) ∧ Y − S(Y) ∧ JX + JS(Y) ∧ X = 0, (16) S(X) = ∇_{X}dσ + J∇_{JX}dσ. where

Contracting (16) with a vector field Z and taking the trace over Y and Z yields 2(1 − m)JS(X) = 0 and hence ∇dσ is J-invariant—by definition when m = 1.

We now show that the other hamiltonian vector fields are Killing. To do this we differentiate pf ϕ_{t}, using the fact that ϕ_{t} is hamiltonian with tr ϕ_{t} = tr ϕ − mt and hence d tr ϕ_{t} = dσ. Therefore, from (12) and (14), we get

(17)  d pf ϕ_{t} = \frac{1}{(m − 1)!}∗(Jdσ ∧ ϕ_{t} ∧ ⋯ ∧ ϕ_{t}).

Using (12) again, we then obtain

\begin{aligned}∇_{X}d pf ϕ_{t} & = \frac{1}{(m − 1)!}∗(∇_{X}Jdσ ∧ ϕ_{t} ∧ ⋯ ∧ ϕ_{t}) \\ & + \frac{1}{2 (m − 2)!}∗(JX ∧ dσ ∧ Jdσ ∧ ϕ_{t} ∧ ⋯ ∧ ϕ_{t}). \end{aligned} ; (18)

The second term on the right hand side is automatically J-invariant, while the first one is also J-invariant since K is Killing. Hence J grad_{g} p(t) is Killing for all t.

It remains to prove that the Killing vector fields preserve ϕ and that their momentum maps Poisson-commute. Contracting equation (17) with Jdσ, we deduce that ⟨Jd pf ϕ_{t}, dσ⟩ = 0 and hence K(t) preserves σ for all t. It follows that L_{K(t)}ϕ = L_{K(t)}(ϕ_{t} + σω) = d ι_{K(t)}(ϕ_{t} + σω), since ϕ_{t} + σω is closed.

Now equation (17) also implies that

ϕ_{t}(grad_{g}(pf ϕ_{t}), ·) = \frac{1}{m!} ∑_{j=1}^{2m}⟨Jdσ, ∗ι_{e_{j}}(ϕ_{t} ∧ ⋯ ∧ ϕ_{t})⟩ε_{j} = (pf ϕ_{t})Jdσ

(using a local frame e_{j} with dual frame ε_{j}) so that

(19)  ϕ_{t}(J grad_{g} p(t), ·) = −p(t)dσ.

Hence ι_{K(t)}(ϕ_{t} + σω) = −d(σ p(t)) is closed and so K(t) preserves ϕ. It follows that K(t) preserves p(s) for all s, t ∈ ℝ and {p(s), p(t)} = ⟨J K(s), K(t)⟩ = d(p(s))(K(t)) = 0 for all s, t; thus p(s) and p(t) Poisson-commute. □

Obviously p(t) is a Killing potential for all t if and only if its coefficients are all Killing potentials.

2.2. The connection on 2-jets of hamiltonian 2-forms. We have noted already that the equation for hamiltonian 2-forms is overdetermined. In fact it has finite type, i.e., the space of local solutions is finite dimensional, being given by the parallel sections for a connection on the 2-jets of hamiltonian 2-forms.

Proposition 4. If ϕ is a hamiltonian 2-form then

∇ϕ + \frac{1}{2}(K ∧ Id + JK ∧ J) = 0 (20)

∇K + \frac{1}{2m}(2u ω − J\{ρ, ϕ\} − 2R(ϕ)) = 0 (21)

du + ρ(K) = 0. (22)

(Here, as elsewhere, we identify vectors with 1-forms and bilinear forms with endomorphisms using g, and we recall that {·, ·} denotes the anticommutator.)

Thus (ϕ, K, u)—with K = J grad_{g} σ, u = \frac{1}{2}Δσ and σ = tr ϕ—is parallel with respect to a natural covariant derivative D on Λ^{J,+}M ⊕ T M ⊕ M×ℝ.

The integrability condition F^{D} · (ϕ, K, u) = 0 is equivalent to the equations

m[R(ψ), ϕ] − [R(ϕ) + \frac{1}{2}J\{ρ, ϕ\}, ψ^{J,+}] = 0 (23)

(m + 1)R_{K,X} − ρ(K, X)ω + \frac{1}{2}J\{ρ, K ∧ X^{J,+}\} − \frac{1}{2}J\{∇_{X}ρ, ϕ\} − ∇_{X}R(ϕ) = 0 (24) −m∇_{K}ρ + [R(ϕ), ρ] = 0. (25)

for any 2-form ψ and vector field X. Note that (23) with ψ = ω gives [ρ, ϕ] = 0.

Proof. Equation (20) is immediate by definition. Contracting (13) with a vector field Z and taking the trace over Y and Z gives ∑_{j=1}^{2m} R_{X,e_{j}}ϕ(e_{j}) + ϕ(Ric(X)) = −\frac{1}{2}(Δσ)JX − mJ∇_{X}dσ (for a local orthonormal frame e_{j}) and (21) is the Jinvariant part of this (as ∇dσ is J-invariant). Since K is a Killing vector field, ∇_{X}∇K = R_{K,X} [15], and (22) is obtained by contracting this with ω.

The first integrability condition (23) follows from (13) by substituting for ∇K = J∇dσ. Differentiating (21) using ∇_{X}∇K = R_{K,X} and the equations for ∇_{X}ϕ and du(X) gives (24). Finally, from equation (22), 0 = d(ρ(K)) = L_{K}ρ = ∇_{K}ρ − [∇K, ρ], which yields (25) by substituting for ∇K.

The three components of F^{D}·(ϕ, K, u) are the left hand sides of (23)–(25) divided by m, after applying the isomorphism alt: T^{∗}M ⊗ Λ^{2}M → Λ^{2}M ⊗ T M to (24). □

Remark 2. It follows that hamiltonian 2-forms enjoy the properties of parallel sections, such as unique continuation, extendibility to submanifolds of codimension at least two, and an upper bound, here equal to m^{2} + 2m + 1, on the dimension of space of hamiltonian 2-forms.

We now expand the curvature R using (3), which may be rewritten

(26)  R(ψ) = W^{K}(ψ) − J{ρ̂, ψ^{J,+}} + ⟨ρ̂, ψ⟩ω + (tr ψ)ρ̂

for any 2-form ψ, where ρ̂ = \frac{1}{m+2}(ρ̃ + \frac{1}{2}s ω) = \frac{1}{m+2}(ρ − \frac{1}{2}s ω). Then (21) reads

\begin{aligned}& ∇K & = \frac{1}{2}J\{ρ̂, ϕ\} + \frac{1}{m}(W^{K}(ϕ) + (tr ϕ)ρ̂ − (tr ρ̂)ϕ + (⟨ρ̂, ϕ⟩ − u)ω). \\ (27) \end{aligned}

Using [ρ, ϕ] = 0, equation (23) implies

(28)  [W^{K}(ψ), ϕ] = \frac{1}{m}[W^{K}(ϕ), ψ^{J,+}] + J(ρ̂_{0} ∘ ψ^{J,+} ∘ ϕ_{0} − ϕ_{0} ∘ ψ^{J,+} ∘ ρ̂_{0})

(ρ̂_{0} = \frac{ρ_{0}}{m+2}). Equation (24) is complicated when fully expanded. Instead we use the fact that the J-invariant part of K ∧ X is −∇_{X}ϕ to obtain

R_{K,X} = W_{K,X}^{K} + J{ρ̂, ∇_{X}ϕ} − ⟨ρ̂, ∇_{X}ϕ⟩ω − dσ(X)ρ̂

and du(X) = −ρ(K, X) = (m + 2)⟨ρ̂, ∇_{X}ϕ⟩ + dσ(X) tr ρ̂. Substituting these into the covariant derivative of (27) (using ∇_{X}∇K = R_{K,X} as before) we have

\begin{aligned}W_{K,X}^{K} − \frac{1}{m}∇_{X}(W^{K}(ϕ)) & = \frac{1}{2}J\{∇_{X} ρ̂_{0}, ϕ_{0}\} + \frac{1}{m}⟨∇_{X} ρ̂_{0}, ϕ_{0}⟩ω − \frac{1}{m}ds(X)ϕ_{0} \\ & − \frac{1}{2}J\{ρ̂_{0}, ∇_{X}ϕ_{0}\} − \frac{1}{m}⟨ρ̂_{0}, ∇_{X}ϕ_{0}⟩ω + \frac{1}{m}dσ(X)ρ_{0} \end{aligned} ; (29)

The important point we shall need later is that the right hand side vanishes if ρ̃ is a constant linear combination of ϕ and ω.

2.3. The differential system in the weakly Bochner-flat case. On a weakly Bochner-flat Kähler manifold, the normalized Ricci form ρ̃ is hamiltonian. We also want to study hamiltonian 2-forms on Kähler–Einstein manifolds. These cases can be considered together by supposing that (g, J, ω) is a weakly Bochner-flat Kähler metric with a hamiltonian 2-form ϕ such that ρ̃ is a constant linear combination of ϕ and ω. We set ρ̃ = (m + 2)aϕ + bω, and find that (27) may be written

∇K = \frac{1}{m}W^{K}(ϕ) + a J ∘ ϕ^{2} − (aσ + b)ϕ + \frac{1}{m}(a(σ^{2} + ⟨ϕ, ϕ⟩) + bσ − \frac{1}{2}Δσ)ω.

Let us put τ_{0} = −2a, τ_{1} = −2(aσ+b) and τ_{2} = \frac{2}{m}(a(σ^{2}+⟨ϕ, ϕ⟩)+bσ− \frac{1}{2}Δσ). Then, as d⟨ϕ, ϕ⟩ = −2ϕ(K), we obtain the following formulation of the system (20)–(22):

\begin{aligned}∇ϕ & = −\frac{1}{2}(K ∧ Id + JK ∧ J), \\ ∇K & = \frac{1}{m}W^{K}(ϕ), −\frac{1}{2}τ_{0} J ∘ ϕ^{2} + \frac{1}{2}τ_{1} ϕ + \frac{1}{2}τ_{2} ω, \\ dτ_{2} & = −τ_{0} ϕ(K) − τ_{1} JK,  dτ_{1} = −τ_{0} JK,  dτ_{0} = 0. \end{aligned} ; (30)

When W^{K}(ϕ) = 0 this system yields an invariant polynomial—in particular for τ_{0} ≠ 0 and W^{K} = 0, such a polynomial was found by Bryant [5] and is the basis for his classification of Bochner-flat Kähler metrics.

Proposition 5. Let (ϕ, K, τ_{0}, τ_{1}, τ_{2}) be a solution of (30) with W^{K}(ϕ) = 0 and define a polynomial

(31)  F_{c}(t) := (τ_{0}t^{2} + τ_{1}t + τ_{2})p(t) − ⟨K, K(t)⟩.

Then F_{c}(t) has constant coefficients. (Recall that K(t) = J grad_{g} p(t), where p(t) = (−1)^{m} pf ϕ_{t} and ϕ_{t} = ϕ − tω.)

Proof. Equation (17) implies that ⟨K, JX⟩p(t) = ⟨ϕ_{t}(X), K(t)⟩. Hence differentiating (τ_{0}t^{2} + τ_{1}t + τ_{2})p(t) along a vector field X, using the system (30), gives

⟨K, τ_{0}t JX⟩p(t) + ⟨K, τ_{0} ϕ(X) + τ_{1} JX⟩p(t) + (τ_{0}t^{2} + τ_{1}t + τ_{2})⟨JX, K(t)⟩
= ⟨(τ_{0}t ϕ_{t} − τ_{0} ϕ_{t} ∘ ϕ ∘ J + τ_{1} ϕ_{t} + τ_{0}t^{2} J + τ_{1}t J + τ_{2} J)(X), K(t)⟩
= ⟨(−τ_{0} J ∘ ϕ^{2} + τ_{1} ϕ + τ_{2} J)(X), K(t)⟩ = 2⟨∇_{X}K, K(t)⟩.

Now (18) gives ⟨K, ∇_{X}(K(t))⟩ = ⟨∇_{X}K, K(t)⟩, which proves the proposition. □

Following Bryant, we refer to F_{c} as the characteristic polynomial of (g, J, ω, ϕ).

2.4. Complex projective, hyperbolic and euclidean space. A Kähler metric g has constant holomorphic sectional curvature if and only if it is Bochner-flat and Kähler–Einstein (and we require constant scalar curvature when m = 1). It follows from (23)–(25) that the connection D is flat in this case; hence on any simply connected domain, the space of hamiltonian 2-forms has dimension (m+1)^{2}. Conversely, if D is flat, then ρ_{0} = 0 (as [ρ_{0}, ϕ] = 0 for all ϕ ∈ Λ^{J,+}M); now (28) implies that W^{K} = 0 (since [W^{K}(ϕ), ψ] = 0 for all ϕ, ψ ∈ Λ^{J,+}M); finally (29) gives ds = 0 (even if m = 1), so g has constant holomorphic sectional curvature.

Hamiltonian 2-forms on constant holomorphic sectional curvature manifolds correspond to solutions of (30) with W^{K} = 0, τ_{0} = 0 and τ_{1} = −2s/m. We first consider the case that s is nonzero, i.e., up to scale, the Kähler metric is the Fubini–Study metric of complex projective space, or the Bergman metric of complex hyperbolic space. If we put τ_{2} = 2sτ/m, the system (30) becomes

\begin{aligned}∇ϕ & = −\frac{1}{2}(K ∧ Id + JK ∧ J) \\ ∇K & = −\frac{s}{m}(ϕ − τω) \\ dτ & = JK. \end{aligned} ; (32)

The last two equations show that τ is a Killing potential for −K, and that the hamiltonian 2-form ϕ is completely determined by τ. Furthermore, the Kostant identity ∇_{X}∇K = R_{K,X} shows that any Killing potential defines a hamiltonian 2-form in this way. Hence there is a bijection ϕ ↦ \frac{1}{m}σ − \frac{1}{2s}Δσ (with inverse τ ↦ \frac{m}{2s}dd^{c}τ + τω) from the space hamiltonian 2-forms to the space of Killing potentials, which may be identified with the unitary Lie algebra 𝔲(m+1) or 𝔲(m, 1), using the Poisson bracket. We remark, though we shall not make use of this, that the Lie bracket, Killing form and (monic, degree m + 1) characteristic polynomial can be computed and turn out to be given by

[(ϕ, K, τ), (ϕ̂, K̂, τ̂)] = ([ϕ, ϕ̂] + \frac{m}{s}K∧K̂^{J,+}, ιK̂(ϕ − τω) − ι_{K}(ϕ̂ − τ̂ω), \frac{m}{s}ω(K, K̂))
⟨(ϕ, K, τ), (ϕ̂, K̂, τ̂)⟩ = ⟨ϕ, ϕ̂⟩ + ττ̂ + \frac{m}{s}⟨K, K̂⟩
bP_{(ϕ,K,τ)}(t) = −\frac{m}{2s}F_{c}(t) = (t − τ)p(t) + \frac{m}{2s}⟨K, K(t)⟩.

On a flat Kähler manifold (e.g., on complex euclidean space ℂ^{m}) the system (30) reduces to

\begin{aligned}∇ϕ & = −\frac{1}{2}(K ∧ Id + JK ∧ J) \\ ∇K & = \frac{1}{2}κω. \end{aligned} ; (33)

with κ constant. Thus inside the space of hamiltonian 2-forms we have the parallel 2-forms (K = 0); modulo such parallel 2-forms, we then have the parallel vector fields (κ = 0); then finally, the space of hamiltonian 2-forms on ℂ^{m}, modulo those with K parallel, is one dimensional, a representative element being dt ∧ d^{c}t, where t is the distance squared to the origin. The characteristic polynomial is now

F_{c}(t) = κp(t) + \frac{1}{2}⟨K, K(t)⟩

which has degree m if κ ≠ 0, degree m−1 if κ = 0 and K ≠ 0, and is zero if K = 0.

We shall obtain an explicit description of the hamiltonian 2-forms on complex projective, hyperbolic and euclidean space, with a given characteristic polynomial F_{c}(t), in section 5.4 below.

3. Hamiltonian torus actions

We have seen that on a Kähler 2m-manifold with a hamiltonian 2-form, there is a family of Poisson-commuting hamiltonian Killing vector fields K(t) = J grad_{g} p(t). Since p(t) is a monic polynomial of degree m, the span of the K(t) is at most mdimensional. If they are not all zero, then on an open set where the span has rank ℓ, 1 ≤ ℓ ≤ m, the K(t) generate a local action of an ℓ-dimensional torus.

We next study hamiltonian ℓ-torus actions in general. Our discussion is independent of the theory of hamiltonian 2-forms, but is strongly motivated by it. Roughly speaking, there are three aspects to the description of such torus actions: first, the toric geometry of the fibres of the complexified action; second, the geometry of the base of this action, the local Kähler quotient; third, the way the fibre and base geometries fit together. In full generality, these structures are quite difficult to handle. However, there is a class of toric manifolds, called orthotoric, of Kähler quotients, called semisimple, and of fibrations, called rigid, which are more amenable to computation. It will turn out that the hamiltonian torus actions induced by hamiltonian 2-forms are always rigid with semisimple base and orthotoric fibres.

3.1. The Pedersen–Poon construction.

Definition 3. A local (isometric) hamiltonian ℓ-torus action on a Kähler 2mmanifold (M, g, J, ω) is an ℓ-dimensional family of holomorphic Killing vector fields K ∈ C^{∞}(M, T M) ⊗ ℝ^{ℓ∗} which are linearly independent on a dense open set M^{0} and isotropic in the sense that ω(K, K) = 0. The last condition means that every component of JK is orthogonal to every component of K. It follows that ℓ ≤ m—if equality holds, we say that (M, g, J, ω) is a toric Kähler manifold.

For clarity, we write K_{r} = K(e_{r}) (r = 1, … ℓ) for the components of K with respect to a basis e_{r} of ℝ^{ℓ}—this could also be interpreted as an abstract index notation. In this subsection and the next two (only) we adopt the summation convention, i.e., repeated indices imply contraction.

Since L_{K_{r}}ω = 0 for all r and ω is closed, we have d(ι_{K_{r}}ω) = 0 and ι_{[K_{r},K_{s}]}ω = −d(ω(K_{r}, K_{s})) = 0. Furthermore, since L_{K_{r}}J = 0 for all r and J is integrable, we have L_{JK_{r}}J = 0 and [JK_{r}, JK_{s}] = J[JK_{r}, K_{s}] = 0.

Remark 3. Definition 3 can be extended to almost hermitian manifolds, but if ω is not closed, we assume a priori that [K_{r}, K_{s}] = 0 for all r, s, while if J is not integrable, we assume that [JK_{r}, JK_{s}] is in the span of JK for all r, s.

To obtain a local description of these metrics, valid near any point in M^{0}, we may assume that K generates a free ℓ-torus action, so that M is a principal ℓ-torus bundle over a (2m − ℓ)-dimensional manifold B, and that the foliation generated by K, JK descends to a fibration of B over a 2(m − ℓ)-dimensional manifold S.

Since J is integrable and K-invariant, the components of JK are holomorphic vector fields, so that S is a complex manifold.

Further, since ω is closed and K-invariant, we may locally write ι_{K}ω = −dσ where σ: M → ℝ^{ℓ∗} is a K-invariant momentum map for the torus action. Thus we may locally identify B with S × U, where U is an open subset of ℝ^{ℓ∗} and σ is given by projection to U. S is then the Kähler quotient of M: it is a complex manifold equipped with a family of compatible Kähler structures parameterized by U.

It is useful to split the exterior derivative on B into horizontal and vertical parts:

dα = d_{h}α + dσ_{r} ∧ L_{∂/∂σ_{r}}α.

(Note L_{∂/∂σ_{r}} commutes with d_{h}. We write ∂α/∂σ_{r} as a shorthand for L_{∂/∂σ_{r}}α.)

Let θ ∈ C^{∞}(M, Λ^{1}M) ⊗ ℝ^{ℓ} be the connection 1-forms dual to K and let Jθ = −θ∘J; thus θ_{r}(K_{s}) = δ_{rs} and θ_{r} vanishes on the horizontal distribution of M → B . (We may locally write θ_{r} = dt_{r} +α_{r} where t: M → ℝ^{ℓ} and α_{r}(K_{s}) = 0 = α_{r}(JK_{s}).)

The two families of 1-forms Jθ and ι_{K}ω span the same ℓ-dimensional space. Hence we may write

Jθ_{r} = G_{rs}ι_{K_{s}}ω  and  ι_{K_{r}}ω = H_{rs}Jθ_{s},

where G_{rs} and H_{rs} are mutually inverse and K-invariant. Since ⟨K_{r}, K_{s}⟩ = ω(K_{r}, JK_{s}) = H_{rs}, we deduce that G_{rs} and H_{rs} are symmetric and positive definite.

Proposition 6. [20] Let (S, J) be a complex 2(m − ℓ)-manifold, let B = S × U with U open in ℝ^{ℓ∗}, and let M be a principal ℓ-torus bundle over B. Denote the components of the projection σ: B → ℝ^{ℓ∗} by σ_{r}. Now suppose that:

(i) (h, ω_{h}) is family of compatible Kähler metrics on the level surfaces of σ in B; (ii) G_{rs} is a symmetric positive definite matrix of functions on B, with inverse matrix H_{rs}, satisfying the equations

(34)  \frac{∂G_{rs}}{∂σ_{t}} = \frac{∂G_{rt}}{∂σ_{s}}  and  d_{h}d_{h}^{c}G_{rs} + \frac{∂^{2}ω_{h}}{∂σ_{r}∂σ_{s}} = 0;

(iii) θ: M → ℝ^{ℓ} is the connection 1-form of a principal connection on M over B whose curvature satisfies the equation

(35)  dθ_{r} = \frac{∂ω_{h}}{∂σ_{r}} + d_{h}^{c}G_{rs} ∧ dσ_{s}.

Then the almost hermitian structure

\begin{aligned}g & = h + G_{rs}dσ_{r}dσ_{s} + H_{rs}θ_{r}θ_{s} \\ ω & = ω_{h} + dσ_{r} ∧ θ_{r} \\ Jθ_{r} & = −G_{rs}dσ_{s} \end{aligned} ; (36)

on M is Kähler with a free hamiltonian ℓ-torus action and Kähler quotient S.

Any Kähler manifold with a local hamiltonian ℓ-torus action arises locally in this way on the dense open set M^{0} where the Killing vector fields are independent.

Proof. We have seen already that any Kähler structure with a local hamiltonian ℓ-torus action can be written in the form (36), where (h, J, ω_{h}) is Kähler for each fixed σ and dθ_{r}(JK_{s}, JK_{t}) = −θ_{r}([JK_{s}, JK_{t}]) = 0. Now under these conditions, ζ_{r} = G_{rs}dσ_{s} +iθ_{r} generate the (1, 0)-forms on the fibres, so (g, J, ω) is Kähler if and only if the ζ_{r} generate a differential ideal modulo horizontal forms and ω is closed. Since dω = (∂ω_{h}/∂σ_{r}−dθ_{r})∧dσ_{r}, and dζ_{r} = d_{h}G_{rs}∧dσ_{s}−(∂G_{rs}/∂σ_{t})dσ_{s}∧dσ_{t}+idθ_{r}, it follows easily that (g, J, ω) is Kähler if and only if G_{rs} satisfies (34) and θ_{r} satisfies (35): the second part of (34) follows from the integrability of (35). □

Remark 4. It is crucial here that the local torus action is hamiltonian in the strong sense that the components of σ Poisson-commute, i.e., dσ_{r}(K_{s}) = 0. This condition is often missed in the literature, since if K_{r} and K_{s} commute, dσ_{r}(K_{s}) is constant, and on a compact manifold σ_{r} must have a critical point, so the constant is zero. However, we are not assuming compactness: indeed, the above local description is only valid on an open set where σ_{r} has no critical points!

Proposition 6 shows that a Kähler metric with a local hamiltonian ℓ-torus action may be specified by essentially free data. Indeed (35) is integrable by virtue of (34). To solve the latter, observe that the first part implies we can write G_{rs} = ∂u_{r}/∂σ_{s}, and since G_{rs} is symmetric, u_{r} = ∂G/∂σ_{r} for some function G on B such that the d_{h}-closed J-invariant 2-form ω_{h} + d_{h}d_{h}^{c}G depends affinely on σ. Since we can add a d_{h}d_{h}^{c} potential for this 2-form to G without altering G_{rs}, we can assume (locally) that for each fixed σ, ω_{h} + d_{h}d_{h}^{c}G = 0. Thus G determines ω_{h} and G_{rs} and is now subject only to the open condition that these are positive definite.

In fact G is a fibrewise Legendre transform of a Kähler potential, generalizing work of Guillemin in the toric case [11] (see also [1]). Observe first that d^{c}u_{r} = d_{h}^{c}u_{r} + θ_{r}, and so dd^{c}u_{r} = ∂ω_{h}/∂σ_{r} + d_{h}d_{h}^{c}u_{r} = 0, i.e., u_{r} is pluriharmonic. Now dd^{c}(σ_{r}u_{r} − G) = d(σ_{r} ∧ d^{c}u_{r} − d_{h}^{c}G) = dσ_{r} ∧ θ_{r} − d_{h}d_{h}^{c}G = ω, so H := σ_{r}u_{r} − G is a Kähler potential. Since G_{rs} is nondegenerate, the u_{r} also form a coordinate system on each fibre of B over S, and we let ∂/∂u_{r} = H_{rs} ∂/∂σ_{s} be the coordinate vector fields tangent to the fibres, so that H_{rs} = ∂σ_{s}/∂u_{r} and σ_{s} = ∂H/∂u_{r}. If we locally set d^{c}u_{r} = dt_{r} then u + it: M → ℂ^{ℓ} is holomorphic and θ_{r} = dt_{r} + α_{r} with α_{r} = −d_{h}^{c}u_{r}. This is the fibrewise Legendre dual coordinate system to (σ, t): M → ℝ^{ℓ∗} × ℝ^{ℓ}, and we refer to G as a dual potential.

It is convenient to introduce a fixed (σ-independent) volume form vol_{S} on S and write vol_{ω_{h}} = Q vol_{S}. Observe in particular that

(37)  ⟨ω_{h}, \frac{∂ω_{h}}{∂σ_{r}} ⟩_{h} = ⟨ω_{h}^{−1}, \frac{∂ω_{h}}{∂σ_{r}} ⟩ = \frac{1}{Q} \frac{∂Q}{∂σ_{r}} .

Proposition 7. Let (M, g, J, ω) be Kähler with a local hamiltonian ℓ-torus action.

(i) Let f be any invariant function on M. Then

dd^{c}f = d_{h}d_{h}^{c}f + \frac{∂f}{∂u_{r}} \frac{∂ω_{h}}{∂σ_{r}}

+ d_{h}(\frac{∂f}{∂u_{r}} ) ∧ θ_{r} + d_{h}^{c}(\frac{∂f}{∂u_{r}} ) ∧ Jθ_{r} (38)

+ \frac{∂}{∂σ_{r}} (\frac{∂f}{∂u_{s}} )dσ_{r} ∧ θ_{s},

(39)  Δf = Δ_{h}f − \frac{1}{Q} \frac{∂}{∂σ_{r}} (Q\frac{∂f}{∂u_{r}} ).

It follows that dd^{c}f = 0 if and only if f = A_{r}u_{r} + B where the A_{r} are constant, and B is a pluriharmonic function on S. (ii) Suppose that κ_{h} is a Ricci potential for ω_{h} for each fixed σ, i.e., d_{h}d_{h}^{c}κ_{h} = ρ_{h} where ρ_{h} is the Ricci form of ω_{h}. Then κ = κ_{h} + \frac{1}{2} log det G_{rs} is a Ricci potential for g, and we have

(40)  d_{h}d_{h}^{c}κ = ρ_{h} + \frac{1}{2}d_{h}(H_{rs}d_{h}^{c}G_{rs}),
(41)  \frac{∂κ}{∂u_{r}} = −\frac{1}{2Q} \frac{∂}{∂σ_{t}} (QH_{rt}).

Proof. (i) Expanding d into horizontal and vertical parts, we get

dd^{c}f = d_{h}d_{h}^{c}f + \frac{∂f}{∂u_{r}} dθ_{r} + dσ_{r} ∧ \frac{∂}{∂σ_{r}} d_{h}^{c}f + d_{h} \frac{∂f}{∂u_{r}} ∧ θ_{r} + \frac{∂}{∂σ_{s}} \frac{∂f}{∂u_{r}} dσ_{s} ∧ θ_{r}.

The second term in the equation (35) for dθ_{r} combines with the third term in the above equation to give d_{h}^{c}(∂f/∂u_{r})∧Jθ_{r}. The formula for the laplacian (39) follows by contracting with ω, using (37).

Now dd^{c}f = 0 if and only if the three lines on the right hand side of (38) are separately zero. Hence ∂f/∂u_{r} must be constant, i.e., f = A_{r}u_{r} + B with A_{r} constant and ∂B/∂u_{r} = 0, and so dd^{c}f = 0 if and only if dd^{c}B = d_{h}d_{h}^{c}B = 0. (ii) A Ricci potential has the form −\frac{1}{2} log(vol_{ω} / vol_{J}) where vol_{ω} = \frac{1}{m!}ω ∧ ⋯ ∧ ω and vol_{J} is a holomorphic volume form. We first observe (see [20, 24]) that if dz_{μ} is a local frame of holomorphic (1, 0)-forms on S, then there are functions B_{rμ} such that ∑_{μ} B_{rμ}dz_{μ} + G_{rs}dσ_{s} + iθ_{r}, together with dz_{μ}, form a local holomorphic frame of M. Since ω = ω_{h} + dσ_{r} ∧ θ_{r}, the formula for κ is immediate.

Equation (40) follows easily using the fact that for any matrix valued function A, d log det A = tr A^{−1}dA. For (41) we also note that κ_{h} = −\frac{1}{2} log(Q vol_{S} / vol_{J_{h}}), where vol_{J_{h}} is a (σ-independent) holomorphic volume form on S, and so

\frac{∂κ}{∂u_{r}} = H_{rt} \frac{∂κ}{∂σ_{t}} = \frac{1}{2}H_{rt}(H_{pq} \frac{∂G_{pq}}{∂σ_{t}} − \frac{1}{Q} \frac{∂Q}{∂σ_{t}} ) = \frac{1}{2}(H_{rt} \frac{∂G_{tp}}{∂σ_{q}} H_{pq} − \frac{1}{Q} \frac{∂Q}{∂σ_{t}} H_{rt})

= −\frac{1}{2}(\frac{∂H_{rt}}{∂σ_{t}} + \frac{1}{Q} \frac{∂Q}{∂σ_{t}} H_{rt}),

where we use the symmetry of ∂G_{pq}/∂σ_{t} in p, q, t. □

This result provides conditions for M to be Kähler–Einstein, using the fact that invariant Ricci and Kähler potentials then differ by an invariant pluriharmonic function. More generally, substituting (40) and (41) into (38) and (39) gives the Ricci form and scalar curvature. These expressions are rather complicated in general. However, if we suppose that d_{h}G_{rs} = 0 and d_{h}Q = 0 then

ρ = ρ_{h} − \frac{1}{2Q} \frac{∂(QH_{rt})}{∂σ_{t}}ω_{h} − \frac{∂}{∂σ_{s}} (\frac{1}{2Q} \frac{∂(QH_{rt})}{∂σ_{t}})dσ_{s} ∧ θ_{r},
Scal = Scal_{h} − \frac{1}{Q} \frac{∂^{2}}{∂σ_{r}∂σ_{s}} (QH_{rs}).

Note that these expressions depend linearly in H_{rs}: this fact was emphasised by Abreu [1] in the toric case, and by Hwang–Singer [12] in the case of circle symmetry: we have just combined their arguments. We shall see the significance of the conditions d_{h}G_{rs} = 0 and d_{h}Q = 0 shortly.

We remark that when m = 2, ℓ = 1, the Pedersen–Poon construction reduces to LeBrun’s construction [16]:

g = we^{u}(dx^{2} + dy^{2}) + w dz^{2} + w^{−1}θ^{2}, ω = we^{u} dx ∧ dy + dz ∧ θ,

where w_{xx} + w_{yy} + (we^{u})_{zz} = 0, which is the integrability condition for dθ = w_{x} dy ∧ dz − w_{y} dx ∧ dz + (we^{u})_{z} dx ∧ dy. Here d_{h}w = w_{x} dx + w_{y}dy. Note κ = −\frac{1}{2}u is a Ricci potential for g.

3.2. Rigid hamiltonian torus actions. Kähler manifolds with a hamiltonian ℓ-torus action are too complicated, in their fullest generality, for constructing interesting Kähler metrics. Indeed most applications, including those in [20], use only Kähler metrics in the following subclass.

Proposition 8. Suppose the Kähler manifold (M, g, J, ω) has a local (isometric) hamiltonian ℓ-torus action K = J grad_{g} σ, for σ: M → ℝ^{ℓ∗}, and let F be the foliation generated by K_{r}, JK_{r} (r = 1, … ℓ). Then on the open dense set M^{0}, where the action is locally free, the following are equivalent:

(i) the leaves of F are totally geodesic; (ii) the connection θ: T M^{0} → ℝ^{ℓ}, with ker θ = (span K)^{⊥}, is JK-invariant; (iii) ⟨K_{r}, K_{s}⟩ is constant on the level surfaces of σ for all r, s; (iv) the family of Kähler forms ω_{h} = ω − dσ_{r} ∧ θ_{r} on the local leaf space of F depends affinely on σ and the linear part pulls back to the curvature of θ;

(v) there is a (local) K-invariant Kähler potential of the form H = H_{0}+H_{σ} where H_{0} is constant on the leaves of F and H_{σ} is constant on the level surfaces of σ.

Proof. The conditions (i)–(iii) are all equivalent to the fact that ⟨∇_{K_{r}}K_{s}, X⟩ = 0 for all r, s and all X orthogonal to F. Indeed, since J is parallel and K_{r} is holomorphic this says that F is totally geodesic. On the other hand, ⟨L_{JK_{r}}(JX), K_{s}⟩ = ⟨JL_{JK_{r}}(X), K_{s}⟩ = ⟨J∇_{JK_{r}}X + ∇_{X}K_{r}, K_{s}⟩ = −2⟨∇_{K_{r}}K_{s}, X⟩, so it also says that the connection θ is JK_{r}-invariant. Finally, it means that ∂_{X}⟨K_{r}, K_{s}⟩ = −2⟨∇_{K_{r}}K_{s}, X⟩ = 0 for all X orthogonal to F.

To establish the equivalence of the local conditions (iii)–(v), we use the Pedersen– Poon construction, Proposition 6. (iii) means that d_{h}H_{rs} = 0, or equivalently d_{h}G_{rs} = 0, which by (35) is equivalent to dθ_{r} = ∂ω_{h}/∂σ_{r}; this is (iv), since (34) then shows that ω_{h} is affine in σ. (v) gives that H_{rs} is the hessian of H_{σ} which implies (iii). Conversely (iii) implies that the dual potential G is an affine function of σ, so that d_{h}H = σ_{r}(∂/∂σ_{r})(d_{h}G) − d_{h}G is independent of σ; then it has a local σ-independent d_{h}-potential H_{0}, and d_{h}H_{σ} = 0 where H_{σ} = H − H_{0}. □

If M is given by the Pedersen–Poon construction (as it is locally), then (i) means that the fibres of M → S (the complex orbits) are totally geodesic, (ii) that M → B is the pullback of a principal bundle with connection over S, and (iii) that the metric on the fibres of M → B (the torus orbits) depends only on the momentum map; the condition (iv) on the Kähler quotient is a kind of rigid Duistermaat–Heckman property (it holds in cohomology by [9]), while (v) generalizes Calabi’s Ansatz [6] for Kähler metrics on holomorphic bundles.

Definition 4. A local hamiltonian ℓ-torus action K = J grad_{g} σ on a Kähler manifold will be called rigid if ⟨K_{r}, K_{s}⟩ is constant on the level surfaces of σ.

Proposition 9. Suppose that M arises from the Pedersen–Poon construction for a rigid hamiltonian ℓ-torus action, and let ∇^{∥} and ∇^{H} be respectively the Levi-Civita connection on the fibres of M over S, and the Levi-Civita connection on the level surfaces of σ in B, lifted to the horizontal distribution of M → S. Let X, Y be horizontal vector fields and U, V be vertical vector fields. Then

(42)  ∇_{X}Y =  ∇_{X}^{H}Y  − C(X, Y)
(43)  ∇_{X}U =  ⟨C(X, ·), U⟩  + [X, U]^{∥}
(44)  ∇_{U}X =  [U, X]^{H} + ⟨C(X, ·), U⟩
(45)  ∇_{U}V =  ∇_{U}^{∥}V,

where ^{H} and ^{∥} denote the horizontal and vertical components, and the O’Neill tensor C is given by

(46)  2C(X, Y) = Ω_{r}(X, Y)K_{r} + Ω_{r}(JX, Y)JK_{r}.

Proof. These observations all follow from the Koszul formula

\begin{aligned}2⟨∇_{X}Y, Z⟩ & = ∂_{X} ⟨Y, Z⟩ + ∂_{Y} ⟨X, Z⟩ − ∂_{Z} ⟨X, Y ⟩ \\ & + ⟨[X, Y], Z⟩ − ⟨[X, Z], Y ⟩ − ⟨[Y, Z], X⟩. \end{aligned} ; (47)

The contraction of (42) with a horizontal vector field Z is immediate because M → B is a riemannian submersion. For the vertical component, (47) gives

2⟨∇_{X}Y, Z⟩ = (L_{Z}g)(X, Y) + ⟨[X, Y], Z⟩.

(with Z vertical). Taking Z = ∂/∂σ_{r} and Z = K_{r} we obtain (42) with C given by (46), since [X, Y]^{∥} = −Ω_{r}(X, Y)K_{r}, JK_{r} = − grad_{g} σ_{r} and

\frac{∂g}{∂σ_{r}} (X, Y) = −\frac{∂ω_{h}}{∂σ_{r}} (JX, Y) = −Ω_{r}(JX, Y).

The remaining three equations are much easier: ⟨∇_{X}U, Y ⟩ = −⟨∇_{X}Y, U⟩, ∇_{U}X − ∇_{X}U = [U, X] and ⟨∇_{U}X, V ⟩ = −⟨∇_{U}V, X⟩, so we only need to check ⟨∇_{X}U, V ⟩ = ⟨[X, U], V ⟩ and (45). These follow immediately because the metric on the fibres is constant along horizontal curves and the fibres are totally geodesic. □

3.3. Semisimple Kähler quotients.

Definition 5. A complex manifold (S, J) with a family of Kähler metrics (h, ω_{h}) (with parameter σ) is semisimple if there is a Kähler form Ω_{S} on S with respect to which the ω_{h} are simultaneously diagonalizable and parallel. A local hamiltonian torus action is semisimple if its local Kähler quotient is.

We can of course take Ω_{S} to be ω_{h} for some fixed σ, but it will be convenient later to make a different choice.

Proposition 10. If (S, J) is semisimple then (S, Ω_{S}) is a locally a Kähler product of (S_{a}, ω_{a}) (a = 1, … N, N ≥ 1) such that ω_{h} = ∑_{a=1}^{N} c_{a}(σ)ω_{a}, where c_{a}(σ) is constant on S. The Levi-Civita connection of ω_{h} is independent of σ, being equal to the Levi-Civita connection of Ω_{S}.

Proof. As ω_{h} is a parallel (1, 1)-form on S, the (J-invariant, simultaneous) eigendistributions of ω_{h} are parallel, and S splits as a local Kähler product by the deRham theorem. The Levi-Civita connections of ω_{h} and Ω_{S} agree, because on each factor S_{a} of the local Kähler product they are related by a constant multiple. □

In the case of local hamiltonian ℓ-torus actions, the semisimplicity condition implies in particular that the quantity Q defined by vol_{ω_{h}} = Q vol_{S}, where vol_{S} is the volume form of Ω_{S}, is constant on S, being given by

(48)  Q = ∏_{a=1}^{N} c_{a}(σ)^{m_{a}},  where  dim S_{a} = 2m_{a}.

If the action is also rigid, ω_{h} depends affinely on σ ∈ ℝ^{ℓ∗}, so we can write ω_{h} = Ω_{0} + ⟨σ, Ω⟩ where Ω_{0} and Ω are closed J-invariant (σ-independent) 2-forms on S, the latter with values in ℝ^{ℓ}. Letting Ω_{r} denote the components of Ω, we have Ω_{r} = ∑_{a=1}^{N} c_{ar}ω_{a} for r = 0, … ℓ, where c_{a}(σ) = c_{a0} + c_{ar}σ_{r}.

3.4. Orthotoric Kähler metrics.

Definition 6. A Kähler 2m-manifold (M, g, J, ω) is orthotoric if it is equipped with m Poisson-commuting functions σ_{1}, … σ_{m} such that K_{r} = Jgrad_{g}σ_{r} are Killing vector fields and, on a dense open set M^{0}, the roots ξ_{j} of ∑_{r=0}^{m}(−1)^{r}σ_{r}t^{ℓ−r} (σ_{0} = 1) are smoothly defined, with linearly independent, orthogonal gradients.

Note that an orthotoric Kähler manifold is toric, and any toric Riemann surface is orthotoric. For a higher dimensional toric manifold it is hard to detect whether it is orthotoric, since the condition depends on a choice of basis for Lie algebra of the torus. Because of this choice, we abandon the summation convention.

The exterior derivative of the identity

∏_{k=1}^{m}(t − ξ_{k}) = ∑_{r=0}^{m}(−1)^{r}σ_{r}t^{m−r}

at t = ξ_{j} yields

(49)  dξ_{j} = \frac{1}{Δ_{j}} ∑_{r=0}^{m}(−1)^{r−1}ξ_{j}^{m−r}dσ_{r},

where Δ_{j} = ∏_{k≠j}(ξ_{j} − ξ_{k}). This is inverse to the identity

(50)  dσ_{r} = ∑_{j=1}^{m} σ_{r−1}(ξ̂_{j})dξ_{j},

where σ_{r−1}(ξ̂_{j}) denote the elementary symmetric functions of the m − 1 functions ξ_{k} with ξ_{j} deleted (with the convention that σ_{0} = 1). Hence the coordinate systems given by ξ_{j} and σ_{r} are related by the Vandermonde matrix and its inverse. We have collected some Vandermonde identities that we need in Appendix B.

Proposition 11. Let (M, g, J, ω) be an orthotoric Kähler 2m-manifold. Then, on any simply connected domain U in M^{0}, there are m functions t_{r}, each determined up to an additive constant, and m functions Θ_{j} of one variable, such that {ξ_{1}, … ξ_{m}, t_{1}, … t_{m}} form a coordinate system with respect to which the Kähler structure may be written

\begin{aligned}g & = ∑_{j=1}^{m} \frac{Δ_{j}}{Θ_{j}(ξ_{j})} dξ_{j}^{2} + ∑_{j=1}^{m} \frac{Θ_{j}(ξ_{j})}{Δ_{j}}(∑_{r=1}^{m} σ_{r−1}(ξ̂_{j}) dt_{r})^{2}, \\ ω & = ∑_{j=1}^{m} dξ_{j} ∧ (∑_{r=1}^{m} σ_{r−1}(ξ̂_{j})dt_{r}) = ∑_{r=1}^{m} dσ_{r} ∧ dt_{r}, \\ Jdξ_{j} & = \frac{Θ_{j}(ξ_{j})}{Δ_{j}} ∑_{r=1}^{m} σ_{r−1}(ξ̂_{j}) dt_{r}, Jdt_{r} = (−1)^{r} ∑_{j=1}^{m} \frac{ξ_{j}^{m−r}}{Θ_{j}(ξ_{j})} dξ_{j}. \end{aligned} ; (51)

Conversely, for any m real functions Θ_{j} of one variable, the almost-hermitian structure defined by (51) is Kähler and orthotoric with dual potential

(52)  G = − ∑_{j=1}^{m} ∫ ^{ξ_{j}} \frac{∏_{k}(t − ξ_{k})}{Θ_{j}(t)}dt = − ∑_{j=1}^{m} ∫ ^{ξ_{j}} \frac{∑_{r=0}^{m}(−1)^{r}σ_{r}t^{m−r}}{Θ_{j}(t)} dt

and Kähler potential

(53)  H = ∑_{j=1}^{m} ∫ ^{ξ_{j}} \frac{t^{m}}{Θ_{j}(t)}dt.

Proof. We apply Proposition 6 to obtain the local expression. The condition that the ξ_{j} have orthogonal gradients means that

(54)  H_{rs} = ∑_{j=1}^{m} σ_{r−1}(ξ̂_{j})σ_{s−1}(ξ̂_{j})|dξ_{j}|^{2}

and hence

(55)  G_{rs} = ∑_{j=1}^{m} \frac{(−1)^{r+s}ξ_{j}^{m−r}ξ_{j}^{m−s}}{Δ_{j}^{2}|dξ_{j}|^{2}} .

We set Θ_{j} = Δ_{j}|dξ_{j}|^{2} so that

(56)  ∑_{s=1}^{m} G_{rs}dσ_{s} = ∑_{j=1}^{m} \frac{(−1)^{r−1}ξ_{j}^{m−r}}{Θ_{j}} dξ_{j}.

If G_{rs} is the hessian of a function G, this must be closed, i.e.,

ξ_{j}^{m−r} \frac{∂Θ_{j}}{∂ξ_{k}} = ξ_{k}^{m−r} \frac{∂Θ_{k}}{∂ξ_{j}} ,

for all j, k, r = 1, … m. Multiplying this by (−1)^{r}dσ_{r} and using (49) to sum over r (which amounts to inverting the above Vandermonde system) shows that Δ_{j}∂Θ_{j}/∂ξ_{k} vanishes for j ≠ k; it follows that each Θ_{j} only depends on ξ_{j}. Changing coordinates from Proposition 6 yields (51).

Conversely if Θ_{j} is a function only of ξ_{j} then (56) is equal to du_{r} where

(57)  u_{r} = − ∑_{j=1}^{m} ∫ ^{ξ_{j}} \frac{(−1)^{r}t^{m−r}}{Θ_{j}(t)} dt.

Since the integrand in (52) vanishes when t = ξ_{j}, the derivative of G with respect to σ_{r} is u_{r}, so its hessian is G_{rs}; ∑_{r=1}^{m} u_{r}σ_{r} − G then gives (53). Since the dξ_{j} are evidently pairwise orthogonal, the structure is orthotoric. □

We end with an alternative characterization of orthotoric Kähler metrics.

Proposition 12. A toric Kähler structure (g, J, ω) is orthotoric if and only if there is a momentum map (σ_{1}, … σ_{m}) such that ∑_{r=1}^{m}(σ_{1}σ_{r}−σ_{r+1})du_{r} is a closed 1-form, where u_{r} = ∂G/∂σ_{r} for a dual potential G.

Proof. Let ξ_{j} be the roots of the polynomial ∑_{r=0}^{m}(−1)^{r}σ_{r}t^{m−r}. Then

∑_{r=1}^{m} d(σ_{1}σ_{r} − σ_{r+1}) ∧ du_{r} = ∑_{r,s} G_{rs}(σ_{r} dσ_{1} − dσ_{r+1}) ∧ dσ_{s}

= ∑_{r,s,j,k} G_{rs}(ξ_{j} σ_{r−1}(ξ̂_{j})dξ_{j}) ∧ (σ_{s−1}(ξ̂_{k})dξ_{k}),

which is zero if and only if ∑_{r,s} G_{rs}(ξ_{j} − ξ_{k})σ_{r−1}(ξ̂_{j})σ_{s−1}(ξ̂_{k}) = 0 for all j, k. The left hand side is (ξ_{j} − ξ_{k})⟨∂/∂ξ_{j}, ∂/∂ξ_{k}⟩, so the result follows. □

4. Classification of hamiltonian 2-forms

4.1. Rough classification of hamiltonian 2-forms. On any Kähler manifold, any J-invariant parallel 2-form is hamiltonian. However, in this case the Killing vector fields K(t) are all identically zero. For a general hamiltonian 2-form ϕ, it is important to know how many of the K(t) are linearly independent. To do this, we (temporarily) write p(t) = ∑_{r=0}^{m}(−1)^{r}σ_{r}t^{m−r} and K_{r} = J grad_{g} σ_{r} for the coefficients of the momentum polynomial and corresponding Killing vector fields. Hence K(t) = ∑_{r=1}^{m}(−1)^{r}K_{r}t^{m−r} is a linear combination of K_{1}, … K_{m} for any t.

Proposition 13. Let ϕ be a hamiltonian 2-form on a (connected) Kähler 2mmanifold M. Then there is an integer ℓ, with 0 ≤ ℓ ≤ m such that K_{1} ∧ ⋯ ∧ K_{ℓ} is nonzero on a dense open subset, but dim span{K_{1}, … K_{m}} ≤ ℓ on all of M.

Proof. The coefficient of t^{m−r} in identity (19) gives

(58)  K_{r+1} = ϕ(JK_{r}) + σ_{r}K_{1}.

Suppose for some z ∈ M and 1 < s < m, K_{s} is a linear combination of K_{1}, … K_{s−1} at z. Then K_{s+1} is also a linear combination of K_{1}, … K_{s−1} at z: to see this, use (58) with r = s, then write ϕ(JK_{s}) as a linear combination of ϕ(JK_{r}) with r < s and use (58) again to express these in terms of K_{1}, … K_{s}. Hence, at each z ∈ M, dim span{K_{1}, … K_{m}} is the largest integer ℓ_{z} such that K_{1}, … K_{ℓ_{z}} are linearly independent at z. However, for any integer r, K_{1}, … K_{r} are linearly dependent if and only if the holomorphic r-vector K_{1}^{1,0} ∧ ⋯ ∧ K_{r}^{1,0} is zero. Hence the set where K_{1}, … K_{r} are linearly independent is empty or dense. The result follows. □

The integer ℓ of this proposition will be called the order of ϕ and we let M^{0} be the dense open set where K_{1}, … K_{ℓ} are independent. We shall identify the order of ϕ with the number of non-constant roots of the momentum polynomial p.

Lemma 3. If, on an open subset of M, ϕ(Z, ·) = ξ ω(Z, ·) with Z nonvanishing, then dξ is the orthogonal projection of dσ onto the span of Z and JZ.

Proof. Without loss of generality, we can take Z to be a unit vector field, and hence (∇_{X}ϕ)(Z, JZ) = dξ(X) for all vector fields X. By (12), this becomes

(59)  dξ = dσ(Z)Z + dσ(JZ)JZ,

which is what we wanted to prove. □

The roots of the momentum polynomial are the eigenvalues of −Jϕ = −J ∘ ϕ, viewed as a J-commuting symmetric endomorphism of T M. At each point of M, these eigenvalues are real and there is an orthogonal J-invariant direct sum decomposition of the tangent space into eigenspaces. We count an eigenvalue with multiplicity k if the corresponding eigenspace has real dimension 2k; for the moment, we denote by ξ_{1}, … ξ_{m} the m (not necessarily distinct) eigenvalues of −Jϕ. It follows that, for any t,

p(t) = ∏_{j=1}^{m}(t − ξ_{j}) = t^{m} − σ_{1}t^{m−1} + ⋯ + (−1)^{m}σ_{m},

where σ_{1}, … σ_{m} are the elementary symmetric functions of ξ_{1}, … ξ_{m}. The above lemma, and the independence of K_{1}, … K_{ℓ} on M^{0}, yields a fundamental fact.

Proposition 14. Let ϕ be a hamiltonian 2-form on M. Then the roots ξ_{i} of p(t) and their derivatives dξ_{i} may be defined smoothly on the dense open set M^{0} and the roots extend continuously to M. Furthermore, for i ≠ j, dξ_{i} and dξ_{j} are orthogonal on M. In particular, any repeated root (on an open set) is constant.

If the order of ϕ is ℓ, then there are ℓ non-constant roots and they are functionally independent on the dense open set where K_{1}, … K_{ℓ} are independent.

Proof. The ordered roots ξ_{1} ≤ ⋯ ≤ ξ_{m} are continuous on M. By the maximality of ℓ, wherever ξ_{1}, … ξ_{m} can be smoothly defined, at most ℓ of the dξ_{i} are independent. On the other hand, on any open subset of M^{0} where ξ_{1}, … ξ_{m} are smoothly defined, at least ℓ of the dξ_{i} are independent. It follows that the ξ_{i} and dξ_{i} can be defined smoothly on M^{0}, that precisely ℓ of the ξ_{i} are functionally independent there. Lemma 3 now shows that for i ≠ j, dξ_{i} and dξ_{j} are orthogonal. □

The order expresses the extent to which ϕ constrains the Kähler geometry of M. At one extreme, when ℓ = 0, we have an orthogonal J-invariant decomposition of T M into eigenspaces of −Jϕ, and it is easy to see that this makes M into a Kähler product: if we write ω = ∑_{ξ} ω_{ξ} and ϕ = ∑_{ξ} ξω_{ξ} where ω_{ξ} is the restriction of the Kähler form to the ξ eigenspace, then the closedness of ϕ and ω is equivalent to dω_{ξ} = 0 for each ξ—for instance there could be only one eigenspace (in which case ϕ is a constant multiple of the Kähler form), or there could be m (in which case M is an arbitrary Kähler product of Riemann surfaces). At the other extreme, when ℓ = m, M is toric and Lemma 3 shows in fact that it is orthotoric, so has the explicit form of Proposition 11, determined by m functions of one variable.

4.2. Explicit description of the metric. We now present the general description of Kähler metrics with a hamiltonian 2-form. To do this, it will be convenient to adopt different notation from the previous subsection.

Definition 7. Let p(t) = (−1)^{m} pf(ϕ − tω) be the momentum polynomial of a hamiltonian 2-form ϕ of order ℓ, and let ξ_{1}, … ξ_{ℓ} the non-constant roots of p. We denote by σ_{0}, … σ_{ℓ} the elementary symmetric functions of these non-constant roots, set K_{r} := J grad_{g} σ_{r} for r = 1, … ℓ, and write p(t) = p_{c}(t)p_{nc}(t), where

\begin{aligned}p_{nc}(t) & := ∏_{j=1}^{ℓ}(t − ξ_{j}) = t^{ℓ} − σ_{1}t^{ℓ−1} + ⋯ + (−1)^{ℓ}σ_{ℓ}, \\ p_{c}(t) & := ∏_{ξ}(t − ξ)^{m_{ξ}}, \end{aligned} ; (60)

and the product over ξ denotes the product over the different constant roots of p(t), m_{ξ} being the multiplicity of the root ξ.

If we denote the σ’s of the previous subsection by σ̃_{1}, … σ̃_{m}, then for ℓ = m σ̃_{r} = σ_{r}; otherwise, since the roots of p_{c} are constant, it follows that dσ̃_{1}, … dσ̃_{m} are constant linear combinations of dσ_{1}, … dσ_{ℓ}. Now dσ̃_{1}, … dσ̃_{ℓ} are linearly independent. Hence dσ_{1}, … dσ_{ℓ} are also linearly independent, and σ_{1}, … σ_{ℓ} are constant affine linear combinations of σ̃_{1}, … σ̃_{ℓ}. (Note in particular that dσ_{1} = dσ̃_{1} = dσ.)

Hence K_{1}, … K_{ℓ} generate a local hamiltonian ℓ-torus action and M^{0} is locally a bundle over the Kähler quotient S with toric fibres. The tangent space V to the fibres of M over S is spanned by K_{1}, … K_{ℓ} and JK_{1}, … JK_{ℓ}, while the orthogonal distribution H is the direct sum of the eigendistributions H_{ξ} corresponding to the constant eigenvalues ξ of −Jϕ. We let Ω be the the V-valued 2-form on H defined by Ω(Y, Z) = [Y, Z]^{V}, the orthogonal projection of the Lie bracket onto V.

Lemma 4. (i) For all ξ and r, the distribution H_{ξ} is K_{r} and JK_{r} invariant, and descends to a parallel distribution on S (with respect to each quotient metric). (ii) If Y and Z belong to H_{ξ} and H_{η} for distinct constant eigenvalues ξ ≠ η then Ω(Y, Z) = 0. If instead Y and Z both belong to H_{ξ}, then

(61)  Ω(Y, Z) = \frac{ω(Y, Z)}{p_{nc}(ξ)} ∑_{r=1}^{ℓ}(−1)^{r−1}ξ^{ℓ−r}K_{r}.

Proof. Suppose that ϕ(Z) = ξJZ for a constant root ξ. Then

ϕ(∇_{Y} Z) = ∇_{Y} (ξJZ) − (∇_{Y} ϕ)(Z) = ξJ∇_{Y} Z − \frac{1}{2}ι_{Z}(dσ_{1} ∧ JY − d^{c}σ_{1} ∧ Y)

for any vector field Y, and hence

(62)  2(ϕ − ξJ)(∇_{Y} Z) = ω(Y, Z)dσ_{1} − ⟨Y, Z⟩Jdσ_{1}

since dσ_{1}(Z) = 0 = Jdσ_{1}(Z) (cf. Lemma 3 and equation (59)).

(i) We apply (62) with Z in H_{ξ} and Y orthogonal to H_{ξ} to deduce that ∇_{Y} Z also belongs to H_{ξ}. Now L_{Y} Z = ∇_{Y} Z −∇_{Z}Y, and the first term is in H_{ξ} for Y = K_{r} or JK_{r}. On the other hand, for X orthogonal to H_{ξ}, ⟨∇_{Z}JK_{r}, JX⟩ = ⟨∇_{Z}K_{r}, X⟩ = −⟨∇_{X}K_{r}, Z⟩ = ⟨K_{r}, ∇_{X}Z⟩ = 0, so that ∇_{Z}Y is also in H_{ξ} for Y = K_{r} or JK_{r}. Thus H_{ξ} descends to S, and since the Levi-Civita connections of the Kähler quotient metrics lift to the horizontal (H) part of ∇, this distribution is parallel by (62).

(ii) This again follows from (62): if Y and Z belong to distinct eigenspaces, then ω(Y, Z) = ⟨Y, Z⟩ = 0, so that [Y, Z] is in H; otherwise, if they both belong the the ξ eigenspace, we have

(ϕ − ξJ)([Y, Z]) = ω(Y, Z)dσ_{1} = ω(Y, Z) ∑_{j=1}^{ℓ} dξ_{j} = ω(Y, Z) ∑_{j=1}^{ℓ} \frac{(ϕ − ξJ)Jdξ_{j}}{ξ − ξ_{j}} .

Hence  Ω(Y, Z) = ω(Y, Z) ∑_{j=1}^{ℓ} \frac{Jdξ_{j}^{♯}}{ξ − ξ_{j}} = \frac{ω(Y, Z)}{∏_{k=1}^{ℓ}(ξ − ξ_{k})} ∑_{j=1}^{ℓ}(∏_{k≠j}(ξ − ξ_{k}))Jdξ_{j}^{♯}

from which (61) easily follows, since K_{r} = Jdσ_{r}^{♯} = ∑_{j=1}^{ℓ} σ_{r−1}(ξ̂_{j})Jdξ_{j}^{♯}. □

This lemma, with Propositions 8, 11 and 14, yields our classification.

Theorem 1. Let (M, g, J, ω) be a connected Kähler 2m-manifold with a hamiltonian 2-form ϕ of order ℓ. Then there are functions F_{1}, … F_{ℓ} of one variable such that on a dense open subset M^{0}, the Kähler structure may be written

\begin{aligned}g & = ∑_{ξ} p_{nc}(ξ)g_{ξ} + ∑_{j=1}^{ℓ} \frac{p′(ξ_{j})}{F_{j}(ξ_{j})}dξ_{j}^{2} + ∑_{j=1}^{ℓ} \frac{F_{j}(ξ_{j})}{p′(ξ_{j})}(∑_{r=1}^{ℓ} σ_{r−1}(ξ̂_{j})θ_{r})^{2}, \\ ω & = ∑_{ξ} p_{nc}(ξ)ω_{ξ} + ∑_{r=1}^{ℓ} dσ_{r} ∧ θ_{r},  dθ_{r} = ∑_{ξ}(−1)^{r}ξ^{ℓ−r}ω_{ξ}, \\ Jdξ_{j} & = \frac{F_{j}(ξ_{j})}{p′(ξ_{j})} ∑_{r=1}^{ℓ} σ_{r−1}(ξ̂_{j}) θ_{r},  Jθ_{r} = (−1)^{r} ∑_{j=1}^{ℓ} \frac{p_{c}(ξ_{j})}{F_{j}(ξ_{j})}ξ_{j}^{ℓ−r} dξ_{j}. \end{aligned} ; (63)

where summation over ξ denotes the sum over the different constant roots of p(t), σ_{r−1}(ξ̂_{j}) denote the elementary symmetric functions of the ℓ − 1 functions ξ_{k} with ξ_{j} deleted, p′(t) is the derivative of the momentum polynomial p(t) with respect to t, and ±(g_{ξ}, ω_{ξ}) is a Kähler metric on a manifold S_{ξ} of the same dimension as the ξ-eigenspace of −Jϕ. Dual and Kähler potentials for (g, J, ω) are given by

(64)  G = − ∑_{r=0}^{ℓ} H_{r}σ_{r} − ∑_{j=1}^{ℓ} ∫ ^{ξ_{j}} \frac{p(t)}{F_{j}(t)}dt,

(65)  H = H_{0} + ∑_{j=1}^{ℓ} ∫ ^{ξ_{j}} \frac{p_{c}(t)t^{ℓ}}{F_{j}(t)} dt,

where H_{r} is a (σ-independent) dd^{c} potential for Ω_{r} = ∑_{ξ}(−1)^{r}ξ^{ℓ−r}ω_{ξ}.

Furthermore, in these coordinates, the hamiltonian 2-form may be written

\begin{aligned}ϕ & = ∑_{ξ} ξ p_{nc}(ξ)ω_{ξ} + ∑_{j=1}^{ℓ} ξ_{j}dξ_{j} ∧ (∑_{r=1}^{ℓ} σ_{r−1}(ξ̂_{j})θ_{r}) \\ & = ∑_{ξ} ∑_{r=0}^{ℓ}(−1)^{r}σ_{r}ξ^{ℓ+1−r}ω_{ξ} + ∑_{r=1}^{ℓ}(σ_{r}dσ_{1} − dσ_{r+1}) ∧ θ_{r}. \end{aligned} ; (66)

We also have a local dd^{c} potential for the closed form ϕ + σ_{1}ω:

(67)  ϕ + σ_{1}ω = ∑_{ξ} ξ^{ℓ+1}ω_{ξ} + ∑_{r=1}^{ℓ} d((σ_{1}σ_{r} − σ_{r+1})θ_{r}) = dd^{c}Φ

where

(68)  Φ = −H_{−1} + ∑_{j=1}^{ℓ} ∫ ^{ξ_{j}} \frac{p_{c}(t)t^{ℓ+1}}{F_{j}(t)} dt

and H_{−1} is a (σ-independent) dd^{c} potential for Ω_{−1} = − ∑_{ξ} ξ^{ℓ+1}ω_{ξ}.

Proof. By Lemma 4, the distribution H is preserved by JK_{1}, … JK_{ℓ}; hence, by Proposition 8, the local fibration of M^{0} over S is totally geodesic, and the toric structure on the fibres is constant on the level surfaces of σ. Therefore, the restriction of ϕ to any fibre is hamiltonian (even when ℓ = 1 since the trace of ϕ is a hamiltonian for a Killing vector field tangent to the fibres). By Proposition 14, the fibres are orthotoric, and the functions G_{rs} are independent of the fibre, hence so are the functions Θ_{j} = Δ_{j}|dξ_{j}|^{2} defining the orthotoric structure in Proposition 11.

Again using Lemma 4, for the constant eigenvalues ξ, H_{ξ} descends to a Jinvariant distribution on S, and T S is the direct sum of these distributions. For each fixed value of σ, these distributions are parallel with respect to the Kähler quotient metric (h, ω_{h}) and so S splits locally as a Kähler product of manifolds S_{ξ}. Furthermore, the curvature Ω of H descends to S, so that the 2-form ω_{ξ} = ω/p_{nc}(ξ), appearing in the formula (61), descends to give a Kähler structure on S_{ξ}, after restricting it to the ξ-eigenspace distribution. (Note however, that this Kähler structure will be negative definite if p_{nc}(ξ) is negative.)

Clearly ω_{h} = ∑_{ξ} p_{nc}(ξ)ω_{ξ} = ∑_{r=0}^{ℓ} σ_{r}Ω_{r} with

(69)  Ω_{r} = ∑_{ξ}(−1)^{r}ξ^{ℓ−r}ω_{ξ}

(r = 0, … ℓ). The explicit form of the metric on M^{0} easily follows: we define F_{j}(t) = p_{c}(t)Θ_{j}(t), and observe that p′(ξ_{j}) = p_{c}(ξ_{j})Δ_{j}, since Δ_{j} = ∏_{k≠j}(ξ_{j} − ξ_{k}).

It remains to establish the explicit form of the potentials: observe that

ω = Ω_{0} + ∑_{r=1}^{ℓ} d(σ_{r}θ_{r}) = dd^{c}H_{0} + ∑_{j=1}^{ℓ} dJ(\frac{ξ_{j}^{ℓ} p_{c}(ξ_{j})dξ_{j}}{F_{j}(ξ_{j})} ) = dd^{c}H.

The equation H = ∑_{r=1}^{ℓ} σ_{r}∂G/∂σ_{r} − G determines G up to a linear combination of the σ_{r} with basic coefficients. We also require d_{h}d_{h}^{c}G = −ω_{h} so the functions u_{r} = ∂G/∂σ_{r} are pluriharmonic; G given by (64) has the required properties with

(70)  u_{r} = −H_{r} − ∑_{j=1}^{ℓ} ∫ ^{ξ_{j}} \frac{(−1)^{r}p_{c}(t)t^{ℓ−r}}{F_{j}(t)} dt

for r = 1, … ℓ, where H_{r} is a dd^{c}-potential for Ω_{r} on S.

In the formula for ϕ, we have used the fact that ξ_{j}σ_{r−1}(ξ̂_{j}) = σ_{r} − σ_{r}(ξ̂_{j}). It is then straightforward to check that Φ is a dd^{c} potential for ϕ + σ_{1}ω. □

Remark 5. Formally, we set u_{0} = −H, so that G = ∑_{r=1}^{ℓ} σ_{r}u_{r} − H = ∑_{r=0}^{ℓ} σ_{r}u_{r}. Similarly, we can write u_{−1} = Φ, and in general extend (69)–(70) to all r ≤ ℓ, where dd^{c}H_{r} = Ω_{r}. For r > 0 dd^{c}u_{r} = 0, while for r = −k ≤ 0 we have dd^{c}u_{−k} = (−1)^{k+1}ϕ_{k} where

ϕ_{k} = ∑_{ξ} ξ^{ℓ+k}ω_{ξ} + ∑_{j,r=1}^{ℓ} d(\frac{ξ_{j}^{ℓ+k}σ_{r−1})(ξ̂_{j})}{Δ_{j}}Jdξ_{j})

= ∑_{ξ}(\frac{ξ^{ℓ+k}}{∏_{k=1}^{ℓ}(ξ − ξ_{k})} + ∑_{j=1}^{ℓ} \frac{ξ_{j}^{ℓ+k}}{Δ_{j}(ξ_{j} − ξ)})p_{nc}(ξ)ω_{ξ}

+ ∑_{i,j,r=1}^{ℓ} \frac{∂}{∂ξ_{i}} (\frac{ξ_{j}^{ℓ+k}σ_{r−1}(ξ̂_{j})}{Δ_{j}})dξ_{i} ∧ θ_{r}.

Using (95) and (97) from Appendix B, this may be written

−Jϕ_{k} = ∑_{s=0}^{k} h_{k−s} (−Jϕ)^{s}

and h_{p} is the pth complete symmetric function in ξ_{1}, … ξ_{ℓ}.

4.3. The hamiltonian 2-form. In order to complete the classification of Kähler metrics with a hamiltonian 2-form, we must show that the explicit metric of Theorem 1 actually admits a hamiltonian 2-form, with no further constraints.

Theorem 2. Let (g, J, ω) be a Kähler structure given explicitly by (63). Then the J-invariant 2-form ϕ defined by (66) is a hamiltonian 2-form of order ℓ.

Proof. Obviously (for the case m = 1), the trace of ϕ is a Killing potential. Hence, in order to show that ϕ is hamiltonian, we must show that

∇A = dσ_{1} ⊗ ω + \frac{1}{2}(dσ_{1} ∧ J − d^{c}σ_{1} ∧ Id)

where A = ϕ + σ_{1}ω. Since A is manifestly closed, we only need to check the equation for (∇_{X}A)(Y, Z), (∇_{X}A)(Y, U), (∇_{U}A)(V, X) and (∇_{U}A)(V, W), where X, Y, Z and U, V, W are arbitrary horizontal and vertical vector fields respectively. Two of these equations are immediate: (∇_{X}A)(Y, Z) = 0 and (∇_{U}A)(V, X) = 0.

We next consider the equation for (∇_{X}A)(Y, U), which reduces, using Proposition 9, to the equation

(71)  2[ϕ, C(X)] = ⟨X, ·⟩ ⊗ Jdσ_{1}^{♯} − ω(X, ·) ⊗ dσ_{1}^{♯}.

Here the left hand side is the commutator of ϕ with

2C(X) = ∑_{r=1}^{ℓ}(Ω_{r}(X) ⊗ Jdσ_{r}^{♯} − Ω_{r}(JX) ⊗ dσ_{r}^{♯}).

Decomposing into the eigenspaces of −Jϕ, we compute that

2[ϕ, C(X)] = − ∑_{r,j=1}^{ℓ} ∑_{ξ}(−1)^{r}ξ^{ℓ−r}σ_{r−1}(ξ̂_{j})(ξ_{j} − ξ)(ω_{ξ}(X) ⊗ dξ_{j}^{♯} + ω_{ξ}(JX) ⊗ Jdξ_{j}^{♯})

= − ∑_{j=1}^{ℓ} ∑_{ξ}(ξ^{ℓ} + ∑_{r=1}^{ℓ}(−1)^{r}ξ^{ℓ−r}(ξ_{j}σ_{r−1}(ξ̂_{j}) + σ_{r}(ξ̂_{j})))

(ω_{ξ}(X) ⊗ dξ_{j}^{♯} + ω_{ξ}(JX) ⊗ Jdξ_{j}^{♯})

= − ∑_{r=0}^{ℓ} ∑_{ξ}(−1)^{r}σ_{r}ξ^{ℓ−r}(ω_{ξ}(X) ⊗ dσ_{1}^{♯} + ω_{ξ}(JX) ⊗ Jdσ_{1}^{♯}).

This proves (71), by the definition of g and ω.

It remains to verify the equation for (∇_{U}A)(V, W). Since the fibres are totally geodesic, this amounts to showing that ϕ is a hamiltonian 2-form on the fibres. Hence it suffices to prove the result in the orthotoric case, when we have ϕ + σ_{1}ω = ∑_{r} d(f_{r}d^{c}u_{r}), with f_{r} = σ_{1}σ_{r} − σ_{r+1}.

To do this we shall use only the fact that ∑_{r} f_{r}du_{r} = dΦ is closed—equivalently A is J-invariant: A(JK_{r}, K_{s}) = df_{r}(JK_{s}) = df_{s}(JK_{r}) = A(JK_{s}, K_{r}), cf. Proposition 12. We define Φ_{rs} := A(JK_{r}, K_{s}) and recall that ⟨K_{r}, K_{s}⟩ = H_{rs}.

Since A is closed and J-invariant, it suffices to check

(72)  (∇_{K_{r}}A)(JK_{s}, K_{t}) = 0
(73)  (∇_{JK_{r}}A)(JK_{s}, K_{t}) = −dσ_{1}(JK_{r})⟨K_{s}, K_{t}⟩
− \frac{1}{2}dσ_{1}(JK_{s})⟨K_{r}, K_{t}⟩ − \frac{1}{2}dσ_{1}(JK_{t})⟨K_{s}, K_{r}⟩.

Equation (72) follows immediately: the left hand side is

K_{r} · (A(JK_{s}, K_{t})) − A(J∇_{K_{r}}K_{s}, K_{t}) − A(JK_{s}, ∇_{K_{r}}K_{t})

and all three terms are zero here, since A(JK_{s}, K_{t}) is K_{r}-invariant, J∇_{K_{r}}K_{s} is a linear combination of the K_{t}’s and A(K_{r}, K_{s}) = 0 for all r, s.

On the other hand, for equation (73) we have

(∇_{JK_{r}}A)(JK_{s}, K_{t}) = JK_{r} · (df_{s}(JK_{t})) + df_{t}(∇_{K_{r}}K_{s}) + df_{s}(∇_{K_{r}}K_{t}).

Now

∇_{K_{r}}K_{s} = −\frac{1}{2} grad⟨K_{r}, K_{s}⟩ = −\frac{1}{2} ∑_{p} \frac{∂H_{rs}}{∂u_{p}} du_{p}^{♯}

and H_{rs} = ∂^{2}H/∂u_{r}∂u_{s}, so (73) holds if and only if

(74) \frac{∂Φ_{st}}{∂u_{r}} − \frac{1}{2} ∑_{p,q} G_{pq}(Φ_{tp} \frac{∂H_{sq}}{∂u_{r}} + Φ_{sp} \frac{∂H_{tq}}{∂u_{r}} ) = H_{1r}H_{st} + \frac{1}{2}(H_{1s}H_{rt} + H_{1t}H_{rs}).

This simplifies once we observe that Φ_{st} = ∑_{p,q} G_{pq}Φ_{tp}H_{sq} = ∑_{p,q} G_{pq}Φ_{sp}H_{tq}, so that the product rule reduces the left hand side to

(75)  \frac{1}{2} ∑_{p,q}(H_{sq} \frac{∂}{∂u_{r}} (G_{pq}Φ_{tp}) + H_{tq} \frac{∂}{∂u_{r}} (G_{pq}Φ_{sp})).

Now we use the fact that f_{s} = σ_{1}σ_{s} − σ_{s+1} and ∂σ_{s}/∂u_{r} = H_{rs} to deduce that

∑_{p} \frac{∂}{∂u_{r}} (G_{pq}Φ_{tp}) = H_{rt}δ_{1q} + H_{1r}δ_{tq}.

Substituting this into (75) yields (74), and hence (73). □

5. The curvature of Kähler manifolds with a hamiltonian 2-form

5.1. The Ricci potential and scalar curvature. In this section we compute the Ricci potential and scalar curvature for any Kähler manifold (M, g, J, ω) given by (63), using the formulae obtained in section 3. In terms of the Vandermonde matrix V_{rj} and its inverse W_{jr} (see Appendix B), we have

(76)  G_{rs} = ∑_{j=1}^{ℓ} \frac{p_{c}(ξ_{j})V_{rj}V_{sj}}{F_{j}(ξ_{j})Δ_{j}} ,  H_{rs} = ∑_{j=1}^{ℓ} \frac{W_{jr}W_{js}F_{j}(ξ_{j})Δ_{j}}{p_{c}(ξ_{j})} ,

and hence, up to a sign, det G_{rs} is ∏_{j=1}^{ℓ} p_{c}(ξ_{j})F_{j}(ξ_{j})^{−1}. Also, the formula ω_{h} = ∑_{ξ} p_{nc}(ξ)ω_{ξ} for the Kähler quotient gives Q = ∏_{ξ} p_{nc}(ξ)^{m_{ξ}} = ± ∏_{j=1}^{ℓ} p_{c}(ξ_{j}), cf. (48). It follows immediately from Proposition 7 that if κ_{ξ} is a Ricci potential for (S_{ξ}, ω_{ξ}), then a Ricci potential for (M, ω) is

(77)  κ = ∑_{ξ} κ_{ξ} − \frac{1}{2} ∑_{j=1}^{ℓ} log |F_{j}(ξ_{j})|.

In order to obtain the scalar curvature from this, we need a formula for the laplacian in the ξ_{j} coordinates.

Lemma 5. For any function f, we have

(78)  Δf = Δ_{h}f − ∑_{j=1}^{ℓ} \frac{1}{Δ_{j} p_{c}(ξ_{j})} \frac{∂}{∂ξ_{j}} (F_{j}(ξ_{j})\frac{∂f}{∂ξ_{j}} ).

Proof. We just need to change coordinates in equation (39). For this observe that

\frac{∂}{∂σ_{r}} = ∑_{k=1}^{ℓ} \frac{V_{rk}}{Δ_{k}} \frac{∂}{∂ξ_{k}}  and  \frac{∂}{∂u_{r}} = ∑_{s=1}^{ℓ} H_{rs} \frac{∂}{∂σ_{s}} = ∑_{j=1}^{ℓ} \frac{W_{jr}F_{j}(ξ_{j})}{p_{c}(ξ_{j})} \frac{∂}{∂ξ_{j}} ,

so that Δf = Δ_{h}f − \frac{1}{Q} ∑_{r,j,k=1}^{ℓ} \frac{V_{rk}}{Δ_{k}} \frac{∂}{∂ξ_{k}} (\frac{QW_{jr}F_{j}(ξ_{j})}{p_{c}(ξ_{j})} \frac{∂f}{∂ξ_{j}} ).

Since Q = ± ∏_{i=1}^{ℓ} p_{c}(ξ_{i}), this agrees with (78) once we observe that

∑_{k,r=1}^{ℓ} \frac{V_{rk}}{Δ_{k}} \frac{∂W_{jr}}{∂ξ_{k}} = − ∑_{r,k=1}^{ℓ} \frac{W_{jr}}{Δ_{k}} \frac{∂V_{rk}}{∂ξ_{k}} = ∑_{r=1}^{ℓ−1} ∑_{k=1}^{ℓ} \frac{W_{jr}}{Δ_{k}} (−1)^{r}(ℓ − r)ξ_{k}^{ℓ−r−1} = 0

by the Vandermonde identity. □

Applying this formula to the Ricci potential, we deduce immediately that

(79)  Scal = ∑_{ξ} \frac{Scal_{g_{ξ}}}{p_{nc}(ξ)} − ∑_{j=1}^{ℓ} \frac{F′′_{j}(ξ_{j})}{Δ_{j} p_{c}(ξ_{j})},

where Scal_{g_{ξ}} is the scalar curvature of the (possibly negative definite) metric g_{ξ}.

Lemma 6. Suppose that Scal depends polynomially on ξ_{1}, … ξ_{ℓ}. Then

• for all j, F′′_{j}(t) = p̌_{c}(t)R(t), where p̌_{c}(t) = ∏_{ξ}(t − ξ)^{m_{ξ}−1} and R(t) is a polynomial independent of j;

• for all ξ, (g_{ξ}, ω_{ξ}) has Scal_{g_{ξ}} = −R(ξ)/ ∏_{η≠ξ}(ξ − η).

We then have

(80) Scal = − ∑_{ξ} \frac{R(ξ)}{∏_{η≠ξ}(ξ − η) ∏_{k}(ξ − ξ_{k})} − ∑_{j=1}^{ℓ} \frac{R(ξ_{j})}{∏_{k≠j}(ξ_{j} − ξ_{k}) ∏_{η}(ξ_{j} − η)},

where m̌ = ℓ + ∑_{ξ} 1 = m − ∑_{ξ}(m_{ξ} − 1).

Furthermore, if Scal has degree ≤ q in each variable ξ_{j}, then R(t) has degree at most m̌+q −1. Hence, for each j, F_{j}(t) is a polynomial of degree at most m+q +1.

Proof. We multiply the formula (79) by Δ_{k} p_{c}(ξ_{k}) to obtain an equality between polynomials in ξ_{k} (on a nonempty open set, hence everywhere):

Δ_{k} p_{c}(ξ_{k}) Scal

= − ∑_{ξ} \frac{Δ_{k} p̌_{c}(ξ_{k}) ∏_{η≠ξ}(ξ_{k} − η)}{∏_{j≠k}(ξ − ξ_{j})}Scal_{g_{ξ}} − F′′_{k}(ξ_{k}) − ∑_{j≠k} \frac{Δ_{k} p_{c}(ξ_{k})}{Δ_{j} p_{c}(ξ_{j})}F′′_{j}(ξ_{j}).

This clearly shows that F′′_{k} is a polynomial with p̌_{c} as a factor. Evaluating at ξ_{k} = ξ_{j} for some fixed j, we obtain F′′_{k}(ξ_{j}) = F′′_{j}(ξ_{j}) for all ξ_{j} (in a nonempty open set, hence everywhere). Dividing through by p̌_{c}(ξ_{k}) we now have

Δ_{k} ∏_{η}(ξ_{k} − η) Scal

= − ∑_{ξ} \frac{Δ_{k} ∏_{η≠ξ}(ξ_{k} − η)}{∏_{j≠k}(ξ − ξ_{j})}Scal_{g_{ξ}} − R(ξ_{k}) − ∑_{j≠k} \frac{Δ_{k} ∏_{η}(ξ_{k} − η)}{Δ_{j} ∏_{η}(ξ_{j} − η)}R(ξ_{j}).

Evaluating at ξ_{k} = ξ gives the formula for Scal_{g_{ξ}}, and it is straightforward to count the degree in ξ_{k}. Dividing by Δ_{k} ∏_{η}(ξ_{k} − η) now gives (80). □

To interpret (80), we adjoin the distinct constant roots to the variables ξ_{1}, … ξ_{ℓ}. If we label these ξ_{1}, … ξ_{ℓ}, ξ_{ℓ+1}, … ξm̌, and let Δ_{j}^{∨} = ∏_{k≠j}(ξ_{j} − ξ_{k}), where the product is over k = 1, … m̌, then the right hand side of (80) is just − ∑m̌ ; j=1 R(ξ_{j})/Δ_{j}^{∨} which is a polynomial of degree at most q in each ξ_{j}, by the Vandermonde identity.

5.2. Extremal Kähler metrics. Recall that a Kähler metric is called extremal if the scalar curvature is a Killing potential [7]. Weakly Bochner-flat Kähler metrics of dimension 2m ≥ 4 are extremal, since the scalar curvature is then the trace of a hamiltonian 2-form. In this section we classify the extremal Kähler metrics with a hamiltonian 2-form such that d_{h}Scal = 0.

Proposition 15. Let (M, g, J, ω) be Kähler with a hamiltonian 2-form ϕ. Then Scal is a hamiltonian for a Killing vector field tangent to the fibres of M over the Kähler quotient S if and only if (g, J, ω) has the explicit form (63) where:

• for all j, F′′_{j}(t) = p̌_{c}(t)(∑m̌ ; r=0 a_{r}t^{m̌−r}), and a_{0}, … am̌ are arbitrary constants (independent of j);

• for all ξ, (g_{ξ}, ω_{ξ}) has Scal_{g_{ξ}} = −(∑m̌ ; r=0 a_{r}ξ^{m̌−r})/ ∏_{η≠ξ}(ξ − η).

The scalar curvature of (g, J, ω) is then given by Scal = −(a_{0}σ̌_{1} + a_{1}), where σ̌_{1} := ∑m̌ ; j=1 ξ_{j} = σ_{1} + ∑_{ξ} ξ, so that Scal is a hamiltonian for −a_{0}K_{1}.

Any constant scalar curvature Kähler metric with a hamiltonian 2-form arises in this way with a_{0} = 0, and is scalar-flat if and only if a_{1} = 0.

Proof. Since Scal is invariant under K_{1}, … K_{ℓ}, it must be a function of σ_{1}, … σ_{ℓ}, and since J grad_{g} Scal commutes with K_{1}, … K_{ℓ} and is in their span at each point, it must in fact be a constant linear combination of K_{1}, … K_{ℓ}, so that Scal is an affine function of σ_{1}, … σ_{ℓ}. Now any such function is a polynomial in ξ_{1}, … ξ_{ℓ} of degree one in each ξ_{j}. Hence we can apply Lemma 6.

Conversely, the Vandermonde identities imply that Scal = −(a_{0}σ̌_{1} + a_{1}), which is a Killing potential for −a_{0}K_{1}, since σ̌_{1} differs from σ_{1} by a constant. □

5.3. Weakly Bochner-flat Kähler metrics. On a weakly Bochner-flat Kähler manifold of dimension 2m ≥ 4, ρ̃ is a hamiltonian 2-form, so we obtain a classification by specializing the work of the previous section to the case ρ̃ = ϕ. In fact we may as well consider more generally the case that ρ̃ = aϕ + bω for constants a, b. Then when a = 0, we will have characterized the Kähler–Einstein metrics among Kähler metrics with a hamiltonian 2-form. Note however, that we have fixed ϕ: it is obviously possible for ρ̃ to be a hamiltonian 2-form without it being equal to aϕ + bω for a given ϕ, but we have nothing to say about this situation.

Proposition 16. Let (M, g, J, ω) be Kähler of dimension 2m ≥ 4 with a hamiltonian 2-form ϕ. Then M is weakly Bochner-flat with ρ̃ a linear combination of ϕ and ω if and only if (g, J, ω) has the explicit form (63) where:

• for all j, F′_{j}(t) = p_{c}(t)(∑_{r=−1}^{ℓ} b_{r}t^{ℓ−r}), and b_{−1}, … b_{ℓ} are arbitrary constants (independent of j);

• for all ξ, (g_{ξ}, ω_{ξ}) is Kähler–Einstein with Kähler–Einstein constant

\frac{1}{m_{ξ}} Scal_{g_{ξ}} = − ∑_{r=−1}^{ℓ} b_{r}ξ^{ℓ−r}.

The Ricci form of (g, J, ω) is then given by ρ = −\frac{1}{2}(b_{−1}(ϕ + σ_{1}ω) + b_{0}ω).

Any Kähler–Einstein metric with a hamiltonian 2-form arises in this way with b_{−1} = 0, and is Ricci-flat if and only if b_{0} = 0.

Proof. −2ρ = b_{−1}(ϕ + σ_{1}ω) + b_{0}ω if and only if dd^{c} potentials for the two sides differ by a pluriharmonic function. This means that −2κ must be of the form − ∑_{r=−1}^{ℓ}(−1)^{r}b_{r}u_{r} where

u_{r} = −H_{r} − ∑_{j=1}^{ℓ} ∫ ^{ξ_{j}} \frac{(−1)^{r}t^{ℓ−r}p_{c}(t)}{F_{j}(t)}dt

for r = −1, … ℓ, the H_{r} being functions on ∏_{ξ} S_{ξ} such that H_{r} is a dd^{c}-potential for Ω_{r} = ∑_{ξ}(−1)^{r}ξ^{ℓ−r}ω_{ξ} when r = 0, … ℓ. (So u_{−1} is a dd^{c} potential for ϕ + σ_{1}ω, u_{0} is a dd^{c} potential for −ω and u_{r} is pluriharmonic for r ≥ 1—see Remark 5.) Now it follows from (77) that

−2κ = −2 ∑_{ξ} κ_{ξ} + ∑_{j=1}^{ℓ} ∫ ^{ξ_{j}} \frac{F′_{j}(t)}{F_{j}(t)}dt.

Comparing this with

∑_{r=−1}^{ℓ}(−1)^{r}b_{r}H_{r} + ∑_{j=1}^{ℓ} ∫ ^{ξ_{j}} \frac{p_{c}(t)(∑_{r=−1}^{ℓ} b_{r}t^{ℓ−r})}{F_{j}(t)}dt,

we obtain the required result. □

5.4. Bochner-flat Kähler metrics. We now rederive Bryant’s classification of Bochner-flat Kähler metrics in the present framework. One interesting feature of our approach is that we can at the same time explicitly classify hamiltonian 2-forms on Bochner-flat Kähler–Einstein manifolds, i.e., on Kähler manifolds of constant holomorphic sectional curvature, cf. section 2.4.

We set p̂_{c}(t) = ∏_{ξ}(t − ξ)^{m_{ξ}+1} and m̂ = ℓ − ∑_{ξ} 1 = m − ∑_{ξ}(m_{ξ} + 1).

Proposition 17. [5] Let (M, g, J, ω) be a Kähler manifold of dimension 2m ≥ 4 with a hamiltonian 2-form. Then M is Bochner-flat with ρ̃ a linear combination of ϕ and ω if and only if (g, J, ω) has the explicit form (63) where:

• for all j, F_{j}(t) = p̂_{c}(t)(∑m̂ ; r=−2 c_{r}t^{m̂−r}), and c_{−2}, … cm̂ are arbitrary constants (independent of j);

• for all ξ, (g_{ξ}, ω_{ξ}) has constant holomorphic sectional curvature

\frac{1}{m_{ξ}(m_{ξ} + 1)}Scal_{g_{ξ}} = −( ∑_{r=−2}^{m̂} c_{r}ξ^{m̂−r}) ∏_{η≠ξ}(ξ − η).

The curvature of (g, J, ω) is then given by R = −{J ∘ ρ̂, ·} + ρ̂ ⊗ ω + ω ⊗ ρ̂, where ρ̂ = −\frac{1}{2}(c_{−2}(ϕ + \frac{1}{2} σ̂_{1}ω) + \frac{1}{2}c_{−1}ω) and σ̂_{1} = σ_{1} − ∑_{ξ} ξ.

Any constant holomorphic sectional curvature Kähler metric with a hamiltonian 2-form arises in this way with c_{−2} = 0, and is flat if and only if c_{−1} = 0.

Proof. By Proposition 16, we may assume that (g, J, ω) is weakly Bochner-flat, with F′_{j} a polynomial divisible by p_{c} and S_{ξ} Kähler–Einstein. Since ρ̃ is a linear combination of ϕ and ω, we may use the system (30). Integrating the last three equations gives τ_{0} = C_{−2}, τ_{1} = C_{−2}σ_{1}+C_{−1} and τ_{2} = C_{−2}(σ_{1}^{2}−σ_{2})+C_{−1}σ_{1}+C_{0} for constants C_{−2}, C_{−1} and C_{0}. (We have used the fact that σ−σ_{1} and ⟨ϕ, ϕ⟩+2σ_{2}−σ_{1}^{2} are constants.) Hence

dd^{c}σ_{1} = \frac{2}{m}W^{K}(ϕ) − C_{−2}Jϕ^{2} + (C_{−2}σ_{1} + C_{−1})ϕ + (C_{−2}(σ_{1}^{2} − σ_{2}) + C_{−1}σ_{1} + C_{0})ω

= \frac{2}{m}W^{K}(ϕ) − C_{−2}(Jϕ^{2} − σ_{1}ϕ − (σ_{1}^{2} − σ_{2})ω) + C_{−1}(ϕ + σ_{1}ω) + C_{0}ω.

It follows from Remark 5 that

\frac{2}{m}W^{K}(ϕ) = dd^{c}(σ_{1} + C_{−2}u_{−2} − C_{−1}u_{−1} + C_{0}u_{0}).

(Here we recall that dd^{c}u_{−2} = Jϕ^{2} − σ_{1}ϕ − (σ_{1}^{2} − σ_{2})ω.)

Thus W^{K}(ϕ) is basic if and only if F_{j}(t) = p_{c}(t)(∑_{r=−2}^{ℓ} C_{r}t^{ℓ−r}), where C_{1}, … C_{ℓ} are arbitrary constants. This means that F_{j} is a polynomial divisible by p_{c}. Since F′_{j} is also divisible by p_{c}, W^{K}(ϕ) is basic if and only if F_{j} is a polynomial divisible by p̂_{c}. In particular ∑_{r=−2}^{ℓ} C_{r}ξ^{ℓ−r} = 0 for each constant root ξ, and so

\frac{2}{m}W^{K}(ϕ) = dd^{c}(σ_{1} + ∑_{r=−2}^{ℓ}(−1)^{r}C_{r}u_{r})

= − ∑_{r=−2}^{ℓ}(−1)^{r}C_{r} dd^{c}H_{r} = ∑_{r=−2}^{ℓ} ∑_{ξ} C_{r}ξ^{ℓ−r}ω_{ξ} = 0.

Hence we may suppose that F_{j}(t) = p̂_{c}(t)(∑m̂ ; r=−2 c_{r}t^{m̂−r}) and that W^{K}(ϕ) = 0. Note that c_{−2} = C_{−2} and c_{−1} = C_{−1}+(∑_{ξ} ξ)C_{−2}, so τ_{0} = c_{−2} and τ_{1} = c_{−2}σ̂_{1}+c_{−1}.

Since ρ̃ is an constant linear combination of ϕ and ω, equation (28) implies that [W^{K}(ψ), ϕ] = \frac{1}{m}[W^{K}(ϕ), ψ] = 0. Also equation (29) implies that ι_{K}W^{K}(ψ) = 0.

Now any 2-form commuting with ϕ is the sum of a vertical and a horizontal 2-form (i.e., there is no mixed component); then since −Jϕ has distinct eigenvalues on the fibres of M over S, the vertical component is of the form ∑_{j=1}^{ℓ} μ_{j}dξ_{j} ∧ Jdξ_{j}. If also the contraction with K is zero, we have ∑_{j=1}^{ℓ} μ_{j}dξ_{j}(X)|dξ_{j}|^{2} = 0, for each vector field X, which forces μ_{1} = ⋯ = μ_{r} = 0. We deduce that for any 2-form ψ, W^{K}(ψ) is horizontal, and, since W^{K} is symmetric, W^{K}(ψ) vanishes unless ψ is horizontal.

We next employ the Gray–O’Neill submersion formulae [10, 19] which apply in this situation (the submersion of M over S is not riemannian, so we need the framework of Gray). If X, Y, Z are horizontal vector fields, we have (see Proposition 9)

(R_{X,Y} Z)^{H} = R_{X,Y}^{H} Z − ⟨C(X), C(Y, Z)⟩^{♯} + ⟨C(Y), C(X, Z)⟩^{♯}
+ ⟨C(Z), C(X, Y)⟩^{♯} − ⟨C(Z), C(Y, X)⟩^{♯},
with  2C(X, Y) = ∑_{r=1}^{ℓ}(Ω_{r}(X, Y)K_{r} + Ω_{r}(JX, Y)JK_{r}).

To compute ⟨C(X, X̃), C(Y, Ỹ)⟩ for horizontal vector fields X, X̃, Y, Ỹ, we use the definition of C, and expand Ω_{r} and ⟨K_{r}, K_{s}⟩ to get

⟨C(X, X̃), C(Y, Ỹ)⟩ = \frac{1}{4} ∑_{ξ,η} f_{ξ,η} (ω_{ξ}(X, X̃)ω_{η}(Y, Ỹ) + ω_{ξ}(JX, X̃)ω_{η}(JY, Ỹ)),

where

f_{ξ,η} = ∑_{j=1}^{ℓ} \frac{F_{j}(ξ_{j})}{p_{c}(ξ_{j})Δ_{j}} (∑_{r=1}^{ℓ}(−1)^{r}ξ^{ℓ−r}σ_{r−1}(ξ̂_{j}))(∑_{s=1}^{ℓ}(−1)^{s}η^{ℓ−s}σ_{s−1}(ξ̂_{j}))

= (∏_{k=1}^{ℓ}(ξ − ξ_{k})(η − ξ_{k}))(∑_{j=1}^{ℓ} \frac{F_{j}(ξ_{j})}{p_{c}(ξ_{j})Δ_{j}(ξ_{j} − ξ)(ξ_{j} − η)})

= p_{nc}(ξ)p_{nc}(η)(C_{−2}(σ_{1} + ξ + η) + C_{−1} − δ_{ξ,η} \frac{(∑m̂ ; r=−2 c_{r}ξ^{m̂−r}) ∏_{η≠ξ}(ξ − η)}{∏_{j=1}^{ℓ}(ξ − ξ_{j})}).

For the last line of this calculation, we observe that F_{j}(t)/p_{c}(t) is a polynomial of degree ℓ+2 vanishing when t = ξ; then if ξ ≠ η we can apply the Vandermonde identity with the variables ξ_{1}, … ξ_{ℓ}, ξ, η, whereas if ξ = η, we may use the Vandermonde identity for the polynomial F_{j}(t)/(p_{c}(t)(t − ξ)) = (∑m̂ ; r=−2 c_{r}t^{m̂−r}) ∏_{η≠ξ}(t − η) of degree ℓ + 1, with the variables ξ_{1}, … ξ_{ℓ}, ξ.

We expand the curvature R using (26) and the fact that

ρ̂ = −\frac{1}{2}(τ_{0}ϕ + \frac{1}{2}τ_{1}ω) = −\frac{1}{2}C_{−2}(ϕ + \frac{1}{2}σ_{1}ω) − \frac{1}{4}C_{−1}ω = −\frac{1}{2}c_{−2}(ϕ + \frac{1}{2} σ̂_{1}ω) − \frac{1}{4}c_{−1}ω.

(See section 2.3.) The final ingredient in the computation, from Proposition 16, is the fact that each S_{ξ} is Kähler–Einstein with Kähler–Einstein constant

\frac{1}{m_{ξ}} Scal_{g_{ξ}} = −(m_{ξ} + 1)( ∑_{r=−2}^{m̂} c_{r}ξ^{m̂−r}) ∏_{η≠ξ}(ξ − η).

Putting these ingredients together, bearing in mind that ω|_{S_{ξ}} = p_{nc}(ξ)ω_{ξ}, we find that a remarkable cancellation occurs (cf. [5]) and we obtain

(W_{X,Y}^{K} Z)^{H} = ∑_{ξ} W_{X,Y}^{K,S_{ξ}}Z,

where W^{K,S_{ξ}} denotes the Bochner tensor of S_{ξ} (pulled back to the Kähler product). We deduce that (W_{X,Y}^{K} Z)^{H} = 0 if and only if each S_{ξ} has constant holomorphic sectional curvature given by the stated formula. □

Our proof above is very much inspired by [5, Section 4.5], where Bryant indicates how to obtain an explicit formula for the general Bochner-flat metric, although he stops short of providing the final formula. Our approach has proceeded in reverse, by first finding the general formula, then showing that it is Bochner-flat. This has permitted us to give a proof using standard methods in Kähler geometry, substituting a linear system for the nonlinear system which Bryant integrates using Cartan’s generalization of Lie’s Third Theorem.

Bryant’s remarkable paper also addresses global questions: the compact Bochnerflat Kähler manifolds are necessarily locally symmetric, but Bryant finds compact orbifold examples, and classifies the complete examples (cf. also [2] for the case of Kähler surfaces). In forthcoming work, we shall find that there are many compact weakly Bochner-flat Kähler manifolds (see also [3]).

We now return to the characteristic polynomial of section 2.3. We write F for F_{j} (which is independent of j), and define a minimal polynomial F_{m}(t) := F (t)/p_{c}(t) = (∑m̂ ; r=−2 c_{r}t^{m̂−r}) ∏_{ξ}(t − ξ). When c_{−2} ≠ 0, these polynomials are (up to affine transformation of t) Bryant’s characteristic and reduced characteristic polynomials.

Proposition 18. Let ϕ be a hamiltonian 2-form on a Bochner-flat Kähler manifold (M, g, J, ω) with ρ̃ a linear combination of ϕ and ω, as in Proposition 17. Then F is the characteristic polynomial F_{c} of (g, J, ω, ϕ).

Proof. To compute (τ_{0}t^{2} + τ_{1}t + τ_{2})p(t) − ⟨K, K(t)⟩, observe that

⟨K, K(t)⟩ = − ∑_{j=1}^{ℓ} p_{c}(t)(∏_{k≠j}(t − ξ_{j}))|dξ_{j}|^{2} = p(t) ∑_{j=1}^{ℓ} \frac{F_{m}(ξ_{j})}{(ξ_{j} − t)Δ_{j}} .

Hence, writing F_{m}(t) = ∑_{r=−2}^{ℓ} C_{r}t^{ℓ−r}, Vandermonde identities (with variables ξ_{1}, … ξ_{ℓ}, t) give

⟨K, K(t)⟩ = p(t)(C_{−2}((σ_{1} + t)^{2} − σ_{2} − tσ_{1}) + C_{−1}(σ_{1} + t) + C_{0}) − p(t)\frac{F_{m}(t)}{p_{nc}(t)}

= p(t)(C_{−2}t^{2} + (C_{−2}σ_{1} + C_{−1})t + (C_{−2}(σ_{1}^{2} − σ_{2}) + C_{−1}σ_{1} + C_{0}) − F (t).

Hence (τ_{0}t^{2} + τ_{1}t + τ_{2})p(t) − ⟨K, K(t)⟩ = F (t). □

When c_{−2} = 0 (i.e., τ_{0} = 0), Proposition 17 provides a classification of hamiltonian 2-forms on simply-connected manifolds of constant holomorphic sectional curvature in terms of two polynomials p_{c} and F_{m} respectively of degrees m − ℓ (precisely) and ℓ + 1 (at most), such that every root of p_{c} is a root of F_{m}.

In section 2.4 we showed that hamiltonian 2-forms are then given by parallel sections of a flat connection on a bundle of rank (m + 1)^{2}. In the simply-connected case, or when the Kähler manifold is an open subset of ℂ^{m}, ℂP^{m} or ℂH^{m}, the bundle is trivial, and parallel sections extend globally to ℂ^{m}, ℂP^{m} or ℂH^{m}. For s ≠ 0, we identified the solution space with the Lie algebra 𝔲(m + 1) or 𝔲(m, 1) of Killing potentials, and we gave an explicit description for s = 0.

The positive-definiteness of the explicit metric g implies that F_{m}(ξ_{j})/Δ_{j} must be positive for all j, so F_{m} must have at least ℓ − 1 distinct roots (without loss of generality ξ_{1} < ⋯ < ξ_{ℓ} and F_{m} has a sign change in each interval). The scalar curvature is nonzero when F_{m} has exactly degree ℓ + 1. In the positive case (C_{−1} < 0), it changes sign in (−∞, ξ_{1}) and (ξ_{ℓ}, ∞) and so has ℓ + 1 distinct roots; the roots of p_{c} are thus the multiple roots of the characteristic polynomial F, and the order ℓ of ϕ equals the number of different roots of the characteristic polynomial minus one. (The case of only one root is the trivial case that τ is constant and ϕ = τω.) The negative case is more complicated, corresponding to the fact that 𝔲(m, 1) has non-semisimple elements, and F alone is not enough to classify them—we also need to know the factorization F (t) = F_{m}(t)p_{c}(t), i.e., the minimal polynomial.

5.5. The Calabi-type case. To illustrate the conditions of the previous subsections, and for use in the next subsection, we specialize to the simple, but important case of hamiltonian 2-forms of order one, when the Kähler structure may be written:

\begin{aligned}g & = _{∑ξ}(z − ξ)g_{ξ} + \frac{∏_{ξ}(z − ξ)^{m_{ξ}}}{F (z)} dz^{2} + \frac{F (z)}{∏_{ξ}(z − ξ)^{m_{ξ}}} θ^{2}, \\ ω & = _{∑ξ}(z − ξ)ω_{ξ} + dz ∧ θ,  dθ = _{∑ξ}ω_{ξ}, \\ Jdz & = \frac{F (z)}{∏_{ξ}(z − ξ)^{m_{ξ}}} θ,  Jθ = −\frac{∏_{ξ}(z − ξ)^{m_{ξ}}}{F (z)} dz, \end{aligned}

where 2m_{ξ} = dim S_{ξ} and ±(g_{ξ}, ω_{ξ}) is a Kähler metric on S_{ξ} (compared to (63), we have reversed the sign of (g_{ξ}, ω_{ξ})). Suppose there are N different constant roots ξ. Then Propositions 15, 16 and 17 specialize as follows:

(i) g is extremal when P_{2}(t) := F′′(t)/ ∏_{ξ}(t − ξ)^{m_{ξ}−1} is a polynomial of degree N + 1 and g_{ξ} has constant scalar curvature P_{2}(ξ)/ ∏_{η≠ξ}(ξ − η);

g has constant scalar curvature when P_{2} has degree N; (ii) g is weakly Bochner-flat when P_{1}(t) := F′(t)/ ∏_{ξ}(t − ξ)^{m_{ξ}} is a polynomial of degree 2 and g_{ξ} is Kähler–Einstein with Kähler–Einstein constant P_{1}(ξ);

g is Kähler–Einstein when P_{1} has degree 1; (iii) g is Bochner-flat when P_{0}(t) := F (t)/ ∏_{ξ}(t − ξ)^{m_{ξ}+1} is a polynomial of degree 3 − N and S_{ξ} has constant holomorphic sectional curvature P_{0}(ξ) ∏_{η≠ξ}(η − ξ);

g has constant holomorphic sectional curvature when P_{0} has degree 2 − N. Metrics of this form have been used in many places to provide compact or complete examples on (projective) line bundles over Kähler products: see [14, 25] for Kähler– Einstein metrics, and [7, 12, 23, 24] for extremal Kähler metrics. We shall explore and generalize these global examples in subsequent work.

The case N = 1 is particularly important: the constant roots are all equal, so without loss of generality we may take them all to be zero. The polynomials P_{0}, P_{1}, P_{2} are all quadratic: in case (i), we may write F (t) = c_{−2}t^{m+2} + c_{−1}t^{m+1} + c_{0}t^{m} + bt + a and the scalar curvature of g_{0} is m(m − 1)c_{0}; then case (ii) has b = 0 and Kähler–Einstein constant mc_{0}, while case (iii) has a = b = 0 with holomorphic sectional curvature c_{0}. The Kähler structure is

\begin{aligned}g & = zg_{0} + \frac{z^{m−1}}{F (z)} dz^{2} + \frac{F (z)}{z^{m−1}} θ^{2},  ω = zω_{0} + dz ∧ θ, \\ Jdz & = \frac{F (z)}{z^{m−1}} θ,  Jθ = −\frac{z^{m−1}}{F (z)} dz,  dθ = ω_{0}, \end{aligned} ; (81)

which is the local form of a Kähler metric of Calabi’s type [6] on a line bundle over a Kähler manifold (S, g_{0}, ω_{0}) of dimension 2(m−1)—Calabi’s extremal Kähler metrics [7] were constructed in this way.

We remark that any such metric (81) is conformal to a Kähler metric of the same form, but with an oppositely oriented complex structure. Indeed we may set z̃ = 1/z, g̃ = g/z^{2} and F̃(z̃)/z̃^{m} = F (z)/z^{m}.

5.6. Strongly conformally Einstein Kähler metrics. We return briefly to the strongly conformally Einstein manifolds of Derdziński–Maschler [8] and explain how their classification can be derived in a natural way in our framework.

We recall that if (M, g, J, ω) strongly conformally Einstein with conformal factor τ, then χ = (aτ + b)dτ ∧ d^{c}τ/|dτ|^{2} is a hamiltonian 2-form with trace z = aτ + b. If a = 0 then χ is parallel, and M is a Kähler product of a Riemann surface and a Kähler manifold of dimension 2m−2. We shall concentrate on the more interesting case a ≠ 0. We set a = 1/q and b = −p/q so that τ = qz +p. This allows us to take the limit q → 0, when τ is constant and (M, g, J, ω) is Kähler–Einstein. Since χ is hamiltonian of order one, with all constant roots zero, the Kähler structure has the explicit form (81) given in the previous subsection. This will be conformally Einstein with conformal factor qz + p if and only if

(82)  qdd^{c}z = ξ(z)dd^{c}κ + η(z)ω,

where κ = κ_{0} − \frac{1}{2} log |F (z)|, κ_{0} is a Ricci potential for ω_{0}, ξ(z) = −(qz + p)/(m − 1) and η(z) is an arbitrary function. Such an equation can only hold if ω_{0} is Kähler Einstein, and we let dd^{c}κ_{0} = c ω_{0} so that Scal_{g_{0}} = (m − 1)c. We now compute

dd^{c}z = \frac{F}{z^{m−1}} ω_{0} + \frac{zF′ − (m − 1)F}{z^{m}} dz ∧ θ
dd^{c}κ = (c − \frac{F′}{2z^{m−1}} )ω_{0} − \frac{zF′′ − (m − 1)F′}{2z^{m}} dz ∧ θ.

Substituting in (82), we eliminate η(z) to obtain a differential equation for F (z):

qF (z) − ξ(z)(cz^{m−1} − \frac{1}{2}F′(z)) = η(z) z^{m}

= q(zF′(z) − (m − 1)F (z)) + \frac{1}{2}ξ(z)(zF′′(z) − (m − 1)F′(z)).

A particular integral is 2cz^{m}/m (this is the flat metric), and one solution of the homogeneous equation is (qz − p)(qz + p)^{2m−1}. We can then reduce the equation to first order to find the general homogeneous solution as an integral:

(qz − p)(qz + p)^{2m−1}(a + b ∫ ^{z} \frac{t^{m}}{(qt + p)^{2m}(qt − p)^{2}} dt).

This is in fact a polynomial in z, which may be written

∑_{j=1}^{m} \frac{j}{m}\binom{2m}{m + j}(a_{+}p^{m−j}q^{j−1}z^{m+j} − a_{−}p^{j−1}q^{m−j}z^{m−j})

for constants a_{±}. The polynomial reduces to a multiple of (qz−p)(qz+p)^{2m−1} when q^{m+1}a_{−} = p^{m+1}a_{+}, while for a_{+}a_{−} = 0, but p, q nonzero, it is a hypergeometric function of qz/p, essentially a Gegenbauer polynomial.

In conclusion, then, the Kähler structure (81) is conformally Einstein with conformal factor qz + p if and only if

(83) F (z) = ∑_{j=1}^{m} \frac{j}{m}\binom{2m}{m + j}(a_{+}p^{m−j}q^{j−1}z^{m+j} − a_{−}p^{j−1}q^{m−j}z^{m−j}) + \frac{2c}{m}z^{m}

for constants a_{±}, c with ρ_{0} = c ω_{0}. We have

Scal = \frac{Scal_{g_{0}}}{z} − \frac{F′′(z)}{z^{m−1}} = −a_{+} ∑_{j=1}^{m} \frac{j}{m} \frac{(2m)!}{(m + j − 2)!(m − j)!}p^{m−j}q^{j−1}z^{j−1}

+ a_{−} ∑_{j=1}^{m−2} \frac{j}{m} \frac{(2m)!}{(m + j)!(m − j − 2)!}p^{j−1}q^{m−j}z^{−j−1},

which is polynomial in z if and only if a_{−} = 0, q = 0 or m = 2. It is a Killing potential (i.e., affine in z) if and only if it vanishes or q = 0 or m = 2.

The function Q(z) of Derdziński–Maschler is F (z)/z^{m−1}, and writing τ = qz +p, we recover their solution, except that we have chosen the basis for the solutions in a way that unifies the cases p = 0 and p ≠ 0. We also recall from the previous subsection that g is conformal to a Kähler metric g̃ with the opposite orientation. This new Kähler metric is conformally Einstein with conformal factor τ̃ = q + pz̃. Just as q = 0 corresponds to the case that g is Kähler–Einstein, so p = 0 corresponds to the case that g̃ is Kähler–Einstein.

Appendix A. Conformal Killing forms and hamiltonian 2-forms

We have noted already that for any hamiltonian 2-form ϕ, A = ϕ + σω is closed. One also easily observes that ϕ−mσω is co-closed (divergence-free). In this appendix we relate hamiltonian 2-forms to conformal Killing 2-forms, which have been investigated recently by Moroianu and Semmelmann [18, 22].

Definition 8. A conformal Killing or twistor 2-form on a riemannian manifold of dimension n ≥ 3 is a 2-form ψ satisfying the equation

(84)  ∇_{X}ψ = \frac{1}{n − 1}X ∧ α + \frac{1}{3}ι_{X}β

for all vector fields X, where α is a 1-form, β is a 3-form and ι_{X} denotes contraction.

It follows immediately from this equation that α = −δψ := ∑_{i} ι_{e_{i}}∇_{e_{i}}ψ and that β = dψ. Hence ψ is a conformal Killing 2-form if and only if ∇ϕ, which is a section of Λ^{1}M ⊗ Λ^{2}M, is in the kernel of the natural projections to Λ^{1}M and Λ^{3}M.

The notion of conformal Killing form can be extended to p-forms and is conformally invariant for p-forms of weight 1, i.e., for sections of L^{p+1} ⊗ Λ^{p}M, where L is the density line bundle, i.e., L^{−n} = |Λ^{n}M|.

The following observation is due in part to Sekizawa [21] and Semmelmann [22]. The essential new ingredient is (85).

Proposition 19. Let (M, g, J, ω) be a Kähler manifold of dimension n = 2m ≥ 4.

(i) If ϕ is a hamiltonian 2-form, then the J-invariant 2-form ψ = ϕ − \frac{1}{2}(tr ϕ)ω is a conformal Killing 2-form, with tr ψ = (1 − \frac{m}{2}) tr ϕ. (ii) Conversely, if ψ is a J-invariant conformal Killing 2-form, then

(85)  dψ = −\frac{3}{n − 1}ω ∧ Jδψ

and

(86)  (m − 2)Jδψ = −(2m − 1)d tr ψ.

Hence

∇_{X}ψ = −\frac{1}{2m − 1}(X ∧ δψ + JX ∧ Jδψ − ⟨Jδψ, X⟩ω)

and Jδψ is closed if m > 2, while tr ψ is constant if m = 2.

(iii) If ψ is a J-invariant conformal Killing 2-form and Jδψ = df then ϕ = ψ + \frac{1}{2m−1}f ω is a hamiltonian 2-form.

In particular, for m > 2 the map ϕ ↦ ϕ− \frac{1}{2}(tr ϕ)ω is a bijection from hamiltonian 2forms to J-invariant conformal Killing 2-forms, with inverse ψ ↦ ψ − \frac{1}{m−2}(tr ψ)ω.

Proof. (i) If ϕ is hamiltonian and ψ = ϕ − \frac{1}{2}σω, then

∇_{X}ψ = \frac{1}{2}(dσ ∧ JX − d^{c}σ ∧ X − dσ(X)ω) = \frac{1}{2}(X ∧ d^{c}σ − ι_{X}(ω ∧ dσ)).

Hence ψ is a conformal Killing 2-form with δψ = −\frac{n−1}{2}d^{c}σ and dψ = −\frac{3}{2}ω ∧ dσ.

(ii) Observe that ∇_{X}ψ is J-invariant and so

\frac{1}{3}(dψ(X, JY, Z) + dψ(X, Y, JZ)) = \frac{1}{n − 1}(X ∧ δψ(JY, Z) + X ∧ δψ(Y, JZ))

= −\frac{1}{n − 1}(ω ∧ Jδψ(X, JY, Z) + ω ∧ Jδψ(X, Y, JZ)).

It follows that \frac{1}{3}dψ + \frac{1}{n−1}ω ∧ Jδψ is a (real) J-invariant 3-form, and so it must be zero—for instance one can use the identity

2β(JX, Y, Z) = β(Z, JX, Y) + β(Z, X, JY)
− β(X, JY, Z) − β(X, Y, JZ)
+ β(Y, JZ, X) + β(Y, Z, JX).

Equation (85) follows immediately. Next, the defining equation (84) implies

(n − 1)(∇_{X}ψ(JY) + ∇_{Y} ψ(JX)) = δψ(X)JY + δψ(Y)JX − 2⟨X, Y ⟩Jδψ.

Taking the trace of this formula as a function of Y gives

(n − 1)(2d tr ψ(X) + Jδψ(X)) = 3Jδψ(X)

which is manifestly equivalent to (86).

(iii) This is a simple verification using (85):

∇_{X}ψ − \frac{1}{n − 1}df(X) ω = \frac{1}{n − 1}(−X ∧ Jdf + ι_{X}(ω ∧ df) − df(X)ω)

= −\frac{1}{n − 1}(df ∧ JX − d^{c}f ∧ X).

Note that (86) gives d tr ϕ = d tr ψ − \frac{m}{n−1}df = −\frac{2}{n−1}df. □

We remark that there is also a connection between hamiltonian 2-forms and Killing tensors (cf. [13]), i.e., symmetric 2-tensors S satisfying sym ∇S = 0. Indeed ϕ is hamiltonian if and only if A = ϕ + σω is closed and S = J(ϕ − σω) is a Killing tensor. For the reverse implication, observe that dA = 0 determines ∇_{X}ϕ(Y, Z) + ∇_{Y} ϕ(X, Z) + ∇_{Z}ϕ(X, Y) in terms of the 1-form dσ, while sym ∇S = 0 determines ∇_{X}ϕ(JY, Z) + ∇_{Y} ϕ(JZ, X) + ∇_{Z}ϕ(JX, Y). Replacing JY by Y in the second expression and adding to the first yields a formula for 2∇_{X}ϕ(Y, Z) + ∇_{Y} ϕ(Z, X) − ∇_{JY} ϕ(JZ, X) and the J-invariant part (in Y, Z) is (12).

Appendix B. Vandermonde matrices

B.1. The inverse of a Vandermonde matrix. A Vandermonde matrix is a (m × m)-matrix of the form

V = V (ξ_{1}, … ξ_{m}) = [\begin{matrix}ξ_{1}^{m−1} & ⋯ & ξ_{m}^{m−1} \\ −ξ_{1}^{m−2} & ⋯ & −ξ_{m}^{m−2} \\ ⋮ & ⋯ & ⋮ \\ (−1)^{m−1} & ⋯ & (−1)^{m−1} \end{matrix}] ,

where the ξ_{j}’s are m independent variables; the entries of the Vandermonde matrix V are thus defined by V_{rj} = (−1)^{r−1}ξ_{j}^{m−r}.

We denote by σ_{r} the elementary symmetric functions of the ξ_{j}’s, so that

(87)  ∏_{j=1}^{m}(t − ξ_{j}) = t^{m} − σ_{1}t^{m−1} + ⋯ + (−1)^{m}σ_{m},

for any t. We also define σ_{0} = 1.

Removing the variable ξ_{j} (equivalently, differentiating with respect to ξ_{j}) gives

(88)  ∏_{k≠j}(t − ξ_{k}) = t^{m−1} − σ_{1}(ξ̂_{j})t^{m−1} + … + (−1)^{m−1}σ_{m−1}(ξ̂_{j}),

where the σ_{r}(ξ̂_{j}) are the elementary symmetric functions of the m − 1 variables ξ_{1}, … ξ̂_{j}, … ξ_{m} (ξ_{j} deleted). By putting t = ξ_{i} in the above identity, we get

(89)  ξ_{i}^{m−1} − σ_{1}(ξ̂_{j})ξ_{i}^{m−2} + ⋯ + (−1)^{m−1}σ_{m−1}(ξ̂_{j}) = Δ_{j} δ_{ij},

where δ_{ij} is the Kronecker symbol and Δ_{j} = ∏_{k≠j}(ξ_{j} − ξ_{k}).

This means that the matrix W whose entries are W_{ir} = σ_{r−1}(ξ̂_{i})/Δ_{i}, i.e.,

W = [\begin{matrix}\frac{1}{Δ_{1}} & \frac{σ_{1}(ξ̂_{1})}{Δ_{1}} & ⋯ & \frac{σ_{m−1}(ξ̂_{1})}{Δ_{1}} \\ ⋮ & ⋮ & ⋯ & ⋮ \\ \frac{1}{Δ_{m}} & \frac{σ_{1}(ξ̂_{m})}{Δ_{m}} & ⋯ & \frac{σ_{m−1}(ξ̂_{m})}{Δ_{m}} \end{matrix} ] ,

is a left-inverse of V:

∑_{r=1}^{m} W_{ir}V_{rj} = δ_{ij}.

B.2. The determinant of a Vandermonde matrix. In order to compute the determinant detV of V, we use the fact that W_{11} = 1/Δ_{1} is equal to the determinant of the minor of V_{11} in V divided by det V. Now the minor of V_{11} is clearly −V (ξ_{2}, …, ξ_{m}); we thus get the following induction formula:

det V (ξ_{1}, …, ξ_{m}) = (−1)^{m−1}(ξ_{1} − ξ_{2}) … (ξ_{1} − ξ_{m}) det V (ξ_{2}, …, ξ_{m}),

from which we readily infer that

(90)  det V = (−1)^{m(m−1)/2} ∏_{i<j}(ξ_{i} − ξ_{j}).

Notice that we also have

(91)  (det V)^{2} = (−1)^{m(m−1)/2} ∏_{j=1}^{m} Δ_{j}.

Indeed, both sides are products of elements of the form ξ_{i} − ξ_{j}: for each i < j, we get (ξ_{i} − ξ_{j})^{2} in the left hand side, and (ξ_{i} − ξ_{j})(ξ_{j} − ξ_{i}) = −(ξ_{i} − ξ_{j})^{2} in the right hand side.

B.3. Vandermonde identities. In the ring of m × m matrices, a left inverse is also a right-inverse, so that:

(92)  ∑_{j=1}^{m} V_{sj}W_{jr} = δ_{rs};

we thus obtain the following Vandermonde identity

(93)  ∑_{j=1}^{m} \frac{(−1)^{s−1}ξ_{j}^{m−s}σ_{r−1}(ξ̂_{j})}{Δ_{j}} = δ_{rs},

for any pair r, s = 1, … m. In particular, with r = 1, we have that

(94)  ∑_{j=1}^{m} \frac{ξ_{j}^{m−s}}{Δ_{j}} = δ_{s1}

for s = 1, … m. This identity for s = 1 may be extended to all s ≤ 1 to give

(95)  ∑_{j=1}^{m} \frac{ξ_{j}^{m−1+p}}{Δ_{j}} = h_{p}

for all p ≥ 0, where h_{p} is the pth complete symmetric function of ξ_{1}, … ξ_{m}. By multiplying by t^{p}, for a formal variable t, and summing over p ≥ 0, this equation may be rewritten

∑_{j=1}^{m} \frac{ξ_{j}^{m−1}}{(1 − ξ_{j}t)Δ_{j}} = ∏_{k=1}^{m} \frac{1}{1 − ξ_{k}t}

where the right hand side denotes the (formal) product of geometric series. Hence, to prove (95), it suffices to observe that

∑_{j=1}^{m} ξ_{j}^{m−1} ∏_{k≠j} \frac{1 − ξ_{k}t}{ξ_{j} − ξ_{k}} = 1.

This follows because the left hand side is a polynomial in t, of degree at most m−1, whose value at t = 1/ξ_{j} is equal to 1 for all j = 1, … m. (In fact, this is more or less the Lagrange interpolation formula.)

Similarly, we can extend (93) to obtain

(96)  ∑_{j=1}^{m} \frac{ξ_{j}^{m+k}σ_{r−1}(ξ̂_{j})}{Δ_{j}} = ∑_{s=0}^{k}(−1)^{s}h_{k−s}σ_{r+s}

for all r = 1, … m and all k ≥ 0. Here, by convention, σ_{r+s} = 0 for r + s > m. We reduce (96) to (95) by means of the obvious identity:

ξ_{j}^{m+k}σ_{r−1}(ξ̂_{j}) = ∑_{s=0}^{m−r}(−1)^{s}ξ_{j}^{m−1+k−s}σ_{r+s}.

(Evidently ξ_{j}σ_{r−1}(ξ̂_{j}) = σ_{r} −σ_{r}(ξ̂_{j}).) Substitute this into the left hand side of (96), and note that the summation over s can be made from 0 to k, using the Vandermonde identity (94) to eliminate any extra terms. Now applying (95) for each s yields the right hand side of (96).

There is one further identity we shall need, namely

(97)  \frac{∂}{∂ξ_{i}} (∑_{j=1}^{m} \frac{ξ_{j}^{m+k}σ_{r−1}(ξ̂_{j})}{Δ_{j}}) = σ_{r−1}(ξ̂_{i}) ∑_{s=0}^{k} h_{k−s}ξ_{i}^{s}.

We prove this using (96): multiplying by t^{k} and summing over k, it suffices to show

∑_{s≥0} \frac{∂}{∂ξ_{i}} (\frac{(−1)^{s}σ_{r+s}t^{s}}{∏_{j=1}^{m}(1 − ξ_{j}t)}) = \frac{σ_{r−1}(ξ̂_{i})}{(1 − ξ_{i}t) ∏_{j=1}^{m}(1 − ξ_{j}t)}.

This holds since direct computation of the left hand side gives

∑_{s≥0} \frac{(−1)^{s}σ_{r+s−1}(ξ̂_{i})t^{s} + (−1)^{s}σ_{r+s}(ξ̂_{i})t^{s+1}}{(1 − ξ_{i}t) ∏_{j=1}^{m}(1 − ξ_{j}t)}

using σ_{r+s}(ξ̂_{i}) = σ_{r+s} − ξ_{i}σ_{r+s−1}(ξ̂_{i}). All terms now cancel in pairs except the first one with s = 0.

In fact we shall only make serious use of the identities (96) and (97) for 0 ≤ k ≤ 2. In particular (96) implies

(98)  ∑_{j=1}^{m} \frac{ξ_{j}^{m}}{Δ_{j}} σ_{r−1}(ξ̂_{j}) = σ_{r},

(99)  ∑_{j=1}^{m} \frac{ξ_{j}^{m+1}}{Δ_{j}} σ_{r−1}(ξ̂_{j}) = σ_{1}σ_{r} − σ_{r+1},

(100)  ∑_{j=1}^{m} \frac{ξ_{j}^{m+2}}{Δ_{j}} σ_{r−1}(ξ̂_{j}) = (σ_{1}^{2} − σ_{2})σ_{r} − σ_{1}σ_{r+1} + σ_{r+2}.

References

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

[2] V. Apostolov and P. Gauduchon, Self-dual Einstein Hermitian 4-manifolds, Ann. Scuola Norm. Sup. Pisa (5) 1 (2002) 203–243.

[3] V. Apostolov, D. M. J. Calderbank and P. Gauduchon, The geometry of weakly self-dual Kähler surfaces, Compositio Math. 135 (2003), 279–322.

[4] A. L. Besse, Einstein manifolds, Ergeb. Math. Grenzgeb. 3, Springer-Verlag, Berlin, Heidelberg, New York, 1987.

[5] R. Bryant, Bochner–Kähler metrics, J. Amer. Math. Soc. 14 (2001) 623–715.

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

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

[8] A. Derdziński and G. Maschler, Local types of conformally-Einstein Kähler metrics in higher dimensions, Preprint (2000), available at arXiv:math.DG/0204013.

[9] J. J. Duistermaat and G. J. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982) 259–268.

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

[11] V. Guillemin, Kähler structures on toric varieties, J. Diff. Geom. 40 (1994) 285–309.

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

[13] W. Jelonek, Compact Kähler surfaces with harmonic anti-self-dual Weyl tensor, Diff. Geom. Appl. 16 (2002), 267–276.

[14] N. Koiso and Y. Sakane, Nonhomogeneous Kähler–Einstein metrics on compact complex manifolds, in Curvature and Topology of Riemannian Manifolds (Kataka, 1985), Lecture Notes in Math. 1201, Springer-Verlag, Berlin, 1986, 165–179.

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

[16] C. R. LeBrun, Explicit self-dual metrics on ℂP^{2}# ⋯ #ℂP^{2}, J. Diff. Geom. 34 (1991) 223–253.

[17] M. Matsumoto and S. Tanno, On Kähler spaces with parallel or vanishing Bochner curvature tensor, Tensor N. S. 27 (1973) 291–294.

[18] A. Moroianu and U. Semmelmann, Twistor forms on Kähler manifolds, available at arXiv:math.DG/0204322

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

[20] H. Pedersen and Y. S. Poon, Hamiltonian construction of Kähler–Einstein metrics and Kähler metrics of constant scalar curvature, Comm. Math. Phys. 136 (1991) 309–326.

[21] M. Sekizawa, On conformal Killing tensors of degree 2 in Kählerian spaces, TRU Math. 6 (1970) 1–5.

[22] U. Semmelmann, Conformal Killing forms on Riemannian manifolds, Habilitationschrift, Universität München (2002), available at arXiv:math.DG/0206117

[23] C. Tønnesen-Friedman, Extremal Kähler metrics on minimal ruled surfaces, J. reine angew. Math. 502 (1998) 175–197.

[24] C. Tønnesen-Friedman, Extremal Kahler metrics and Hamiltonian functions II, Glasg. Math. J. 44 (2002) 241–253.

[25] M. Y. Wang, Einstein metrics from symmetry and bundle constructions, Surv. Diff. Geom., VI, Int. Press, Boston, MA, 1999, 287–325.

Vestislav Apostolov, Département de Mathématiques, UQAM, C.P. 8888, Succ. Centre-ville, Montréal (Québec), H3C 3P8, Canada

   E-mail address: apostolo@math.uqam.ca

David M. J. Calderbank, School of Mathematics, University of Edinburgh, King’s Buildings, Mayfield Road, Edinburgh EH9 3JZ, Scotland

   E-mail address: davidmjc@maths.ed.ac.uk

Paul Gauduchon, Centre de Mathématiques, Ecole Polytechnique, UMR 7640 du CNRS, 91128 Palaiseau, France

   E-mail address: pg@math.polytechnique.fr