Bochner-Kahler metrics

Abstract

A Kahler metric is said to be Bochner-Kahler if its Bochner curvature vanishes. This is a nontrivial condition when the complex dimension of the underlying manifold is at least 2. In this article it will be shown that, in a certain well-defined sense, the space of Bochner-Kahler metrics in complex dimension n has real dimension n+1 and a recipe for an explicit formula for any Bochner-Kahler metric will be given. It is shown that any Bochner-Kahler metric in complex dimension n has local (real) cohomogeneity at most n. The Bochner-Kahler metrics that can be `analytically continued' to a complete metric, free of singularities, are identified. In particular, it is shown that the only compact Bochner-Kahler manifolds are the discrete quotients of the known symmetric examples. However, there are compact Bochner-Kahler orbifolds that are not locally symmetric. In fact, every weighted projective space carries a Bochner-Kahler metric. The fundamental technique is to construct a canonical infinitesimal torus action on a Bochner-Kahler metric whose associated momentum mapping has the orbits of its symmetry pseudo-groupoid as fibers.