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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Bochner-Kähler metrics
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by Robert L. Bryant
J. Amer. Math. Soc. 14 (2001), 623-715
Published electronically: March 20, 2001


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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Bibliographic Information
  • Robert L. Bryant
  • Affiliation: Department of Mathematics, Duke University, P.O. Box 90320, Durham, North Carolina 27708-0320
  • MR Author ID: 42675
  • Email:
  • 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.
  • © Copyright 2001 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 14 (2001), 623-715
  • MSC (2000): Primary 53B35; Secondary 53C55
  • DOI:
  • MathSciNet review: 1824987