Bochner-Kähler metrics

Abstract:

A Kähler metric is said to be Bochner-Kähler 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-Kähler metrics in complex dimension $n$ has real dimension $n{+}1$ and a recipe for an explicit formula for any Bochner-Kähler metric will be given. It is shown that any Bochner-Kähler metric in complex dimension $n$ has local (real) cohomogeneity at most $n$. The Bochner-Kähler 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-Kähler manifolds are the discrete quotients of the known symmetric examples. However, there are compact Bochner-Kähler orbifolds that are not locally symmetric. In fact, every weighted projective space carries a Bochner-Kähler metric. The fundamental technique is to construct a canonical infinitesimal torus action on a Bochner-Kähler metric whose associated momentum mapping has the orbits of its symmetry pseudo-groupoid as fibers.