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)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 m r=0 (−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.The hamiltonian property of φ has fu...

This paper concerns the explicit construction of extremal Kähler metrics on total spaces of projective bundles, which have been studied in many places. We present a unified approach, motivated by the theory of hamiltonian 2-forms (as introduced and studied in previous papers in the series) but this paper is largely independent of that theory.We obtain a characterization, on a large family of projective bundles, of those 'admissible' Kähler classes (i.e., the ones compatible with the bundle structure in a way we make precise) which contain an extremal Kähler metric. In many cases, such as on geometrically ruled surfaces, every Kähler class is admissible. In particular, our results complete the classification of extremal Kähler metrics on geometrically ruled surfaces, answering several long-standing questions.We also find that our characterization agrees with a notion of K-stability for admissible Kähler classes. Our examples and nonexistence results therefore provide a fertile testing ground for the rapidly developing theory of stability for projective varieties, and we discuss some of the ramifications. In particular we obtain examples of projective varieties which are destabilized by a non-algebraic degeneration.Ψ * y = y c , Ψ * t = t, and hence Ψ * J = J c .As J c and J are integrable complex structures, Ψ extends to a U (1)-equivariant diffeomorphism of M leaving fixed any point on e 0 ∪e ∞ (since it is fibre preserving).Putω := Ψ * ω. Thenω is a Kähler form on (M, J c ) which (we claim) belongs to the same cohomology class Ω as ω. Indeed, on M 0 we havẽdd c Jc h(y c ) = Ψ * dd c J h(y) =ω − a ω a /x a , so the following implies the claim. Lemma 3. The function h(y c ) − h c (y c ) is smooth on M . Centre-ville,

It is well known that any 4-dimensional hyperkähler metric with two commuting Killing fields may be obtained explicitly, via the Gibbons-Hawking Ansatz, from a harmonic function invariant under a Killing field on R 3 . In this paper, we find all selfdual Einstein metrics of nonzero scalar curvature with two commuting Killing fields. They are given explicitly in terms of a local eigenfunction of the Laplacian on the hyperbolic plane. We discuss the relation of this construction to a class of selfdual spaces found by Joyce, and some Einstein-Weyl spaces found by Ward, and then show that certain 'multipole' hyperbolic eigenfunctions yield explicit formulae for the quaternion-kähler quotients of HP m−1 by an (m − 2)-torus first studied by Galicki and Lawson. As a consequence we are able to place the well-known cohomogeneity one metrics, the quaternion-kähler quotients of HP 2 (and noncompact analogues), and the more recently studied selfdual Einstein Hermitian metrics in a unified framework, and give new complete examples.

We present a classification of compact Kähler manifolds admitting a hamiltonian 2-form (which were classified locally in part I of this work). This involves two components of independent interest.The first is the notion of a rigid hamiltonian torus action. This natural condition, for torus actions on a Kähler manifold, was introduced locally in part I, but such actions turn out to be remarkably well behaved globally, leading to a fairly explicit classification: up to a blow-up, compact Kähler manifolds with a rigid hamiltonian torus action are bundles of toric Kähler manifolds.The second idea is a special case of toric geometry, which we call orthotoric. We prove that orthotoric Kähler manifolds are diffeomorphic to complex projective space, but we extend our analysis to orthotoric orbifolds, where the geometry is much richer. We thus obtain new examples of Kähler-Einstein 4-orbifolds.Combining these two themes, we prove that compact Kähler manifolds with hamiltonian 2-forms are covered by blow-downs of projective bundles over Kähler products, and we describe explicitly how the Kähler metrics with a hamiltonian 2-form are parameterized. We explain how this provides a context for constructing new examples of extremal Kähler metrics-in particular a subclass of such metrics which we call weakly Bochner-flat.We also provide a self-contained treatment of the theory of compact toric Kähler manifolds, since we need it and find the existing literature incomplete.This paper is concerned with the construction of explicit Kähler metrics on compact manifolds, and has several interrelated motivations. The first is the notion of a hamiltonian 2-form, introduced in part I of this series [4].Definition 1. Let φ be any (real) J-invariant 2-form on the Kähler manifold (M, g, J, ω) of dimension 2m. We say φ is hamiltonian iffor any vector field X, where tr φ = φ, ω is the trace with respect to ω. When M is a Riemann surface (m = 1), this equation is vacuous and we require instead that tr φ is a Killing potential, i.e., a hamiltonian for a Killing vector field J grad g tr φ.A second motivation is the notion of a weakly Bochner-flat (WBF) Kähler metric, by which we mean a Kähler metric whose Bochner tensor (which is part of the curvature tensor) is co-closed. By the differential Bianchi identity, this is equivalent (for m ≥ 2) to the condition that ρ + Scal 2(m+1) ω is a hamiltonian 2-form, where ρ is the Ricci form. WBF Kähler metrics are extremal in the sense of Calabi, i.e., the symplectic gradient of the scalar curvature is a Killing vector field, and provide Date: November 3, 2018. We would like to thank C. a class of extremal Kähler metrics which include the Bochner-flat Kähler metrics studied by Bryant [10] and products of Kähler-Einstein metrics. The geometry of WBF Kähler metrics is tightly constrained, because the more specific the normalized Ricci form is, the closer the metric is to being Kähler-Einstein, while the more generic it is, the stronger the consequences of the hamiltonian property.A hamiltonian 2-form φ induces...

Abstract. We give a simple construction of the Bernstein-Gelfand-Gelfand sequences of natural differential operators on a manifold equipped with a parabolic geometry. This method permits us to define the additional structure of a bilinear differential "cup product" on this sequence, satisfying a Leibniz rule up to curvature terms. It is not associative, but is part of an A∞-algebra of multilinear differential operators, which we also obtain explicitly. We illustrate the construction in the case of conformal differential geometry, where the cup product provides a wide-reaching generalization of helicity raising and lowering for conformally invariant field equations.