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,

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...

International audienceLet $M=P(E)$ be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle $E \to \Sigma$ over a compact complex curve $\Sigma$ of genus $\ge 2$. Building on ideas of Fujiki, we prove that $M$ admits a Kähler metric of constant scalar curvature if and only if $E$ is polystable. We also address the more general existence problem of extremal Kähler metrics on such bundles and prove that the splitting of $E$ as a direct sum of stable subbundles is necessary and sufficient condition for the existence of extremal Kähler metrics in sufficiently small Kähler classes. The methods used to prove the above results apply to a wider class of manifolds, called {\it rigid toric bundles over a semisimple base}, which are fibrations associated to a principal torus bundle over a product of constant scalar curvature Kähler manifolds with fibres isomorphic to a given toric Kähler variety. We discuss various ramifications of our approach to this class of manifolds

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We study the construction and classification of weakly Bochnerflat (WBF) metrics (i.e., Kähler metrics with coclosed Bochner tensor) on compact complex manifolds. A Kähler metric is WBF if and only if its 'normalized' Ricci form is a hamiltonian 2-form: such 2-forms were introduced and studied in previous papers in the series. It follows that WBF Kähler metrics are extremal. We construct many new examples of WBF metrics on projective bundles and obtain a classification of compact WBF Kähler 6-manifolds, extending work by the first three authors on weakly selfdual Kähler 4-manifolds. The constructions are independent of previous papers in the series, but the classification relies on the classification of compact Kähler manifolds with a hamiltonian 2-form [3].