Math @ Duke

Publications [#243382] of Robert Bryant
search www.ams.org.Papers Published
 Bryant, RL, Bochnerkähler metrics,
Journal of the American Mathematical Society, vol. 14 no. 3
(July, 2001),
pp. 623715 [MR2002i:53096], [math.DG/0003099], [doi]
(last updated on 2019/08/21)
Abstract: A Kahler metric is said to be BochnerKahler 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 BochnerKahler metrics in
complex
dimension n has real dimension n+1 and a recipe for
an
explicit formula for any BochnerKahler metric is given.
It is shown that any BochnerKahler metric in complex
dimension n has local (real) cohomogeneity at most~n.
The
BochnerKahler 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 BochnerKahler
metric.
The fundamental technique is to construct a
canonical
infinitesimal torus action on a BochnerKahler metric
whose
associated momentum mapping has the orbits of its
symmetry
pseudogroupoid as fibers.


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