Math @ Duke
Publications [#243382] of Robert Bryant
- Bryant, RL, Bochner-Kähler metrics,
Journal of the AMS, vol. 14 no. 3
pp. 623-715 [MR2002i:53096], [math.DG/0003099]
(last updated on 2017/12/18)
A Kahler metric is said to be Bochner-Kahler if its
curvature vanishes. This is a nontrivial condition when
complex dimension of the underlying manifold is at
In this article it will be shown that, in a certain well-
defined sense, the space of Bochner-Kahler metrics in
dimension n has real dimension n+1 and a recipe for
explicit formula for any Bochner-Kahler metric is given.
It is shown that any Bochner-Kahler metric in complex
dimension n has local (real) cohomogeneity at most~n.
Bochner-Kahler metrics that can be `analytically
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
symmetric examples. However, there are compact
orbifolds that are not locally symmetric. In fact, every
weighted projective space carries a Bochner-Kahler
The fundamental technique is to construct a
infinitesimal torus action on a Bochner-Kahler metric
associated momentum mapping has the orbits of its
pseudo-groupoid as fibers.
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