Hamiltonian 2-forms in Kähler geometry, II Global Classification

Abstract

We present a classification of compact Kahler 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 Kahler 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 Kahler manifolds with a rigid hamiltonian torus action are bundles of toric Kahler manifolds. The second idea is a special case of toric geometry, which we call orthotoric. We prove that orthotoric Kahler 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 Kahler–Einstein 4-orbifolds. Combining these two themes, we prove that compact Kahler manifolds with hamiltonian 2-forms are covered by blow-downs of projective bundles over Kahler products, and we describe explicitly how the Kahler metrics with a hamiltonian 2-form are parameterized. We explain how this provides a context for constructing new examples of extremal Kahler 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 Kahler manifolds, since we need it and find the existing literature incomplete.