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Bochner-Kähler metrics

Author: Robert L. Bryant
Journal: J. Amer. Math. Soc. 14 (2001), 623-715
MSC (2000): Primary 53B35; Secondary 53C55
Published electronically: March 20, 2001
MathSciNet review: 1824987
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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.

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Additional Information

Robert L. Bryant
Affiliation: Department of Mathematics, Duke University, P.O. Box 90320, Durham, North Carolina 27708-0320

Keywords: K\"ahler metrics, Bochner tensor, momentum map, polytope
Received by editor(s): July 6, 2000
Received by editor(s) in revised form: December 19, 2000
Published electronically: March 20, 2001
Additional Notes: The research for this article was made possible by support from the National Science Foundation through grant DMS-9870164 and from Duke University.
Article copyright: © Copyright 2001 American Mathematical Society