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Curvature and the c-projective mobility of Kähler metrics with hamiltonian 2-forms

Part of: Compact analytic spaces Local differential geometry Global differential geometry Classical differential geometry

Published online by Cambridge University Press:  26 April 2016

David M. J. Calderbank ,
Vladimir S. Matveev  and
Stefan Rosemann
David M. J. Calderbank
Affiliation:
Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK email D.M.J.Calderbank@bath.ac.uk
Vladimir S. Matveev
Affiliation:
Institute of Mathematics, FSU Jena, 07737 Jena, Germany email vladimir.matveev@uni-jena.de
Stefan Rosemann
Affiliation:
Institute of Mathematics, FSU Jena, 07737 Jena, Germany email stefan.rosemann@uni-jena.de
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Abstract

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The mobility of a Kähler metric is the dimension of the space of metrics with which it is c-projectively equivalent. The mobility is at least two if and only if the Kähler metric admits a nontrivial hamiltonian 2-form. After summarizing this relationship, we present necessary conditions for a Kähler metric to have mobility at least three: its curvature must have nontrivial nullity at every point. Using the local classification of Kähler metrics with hamiltonian 2-forms, we describe explicitly the Kähler metrics with mobility at least three and hence show that the nullity condition on the curvature is also sufficient, up to some degenerate exceptions. In an appendix, we explain how the classification may be related, generically, to the holonomy of a complex cone metric.

Keywords

Kähler metricprojective geometrycurvature

MSC classification

Primary: 53B35: Hermitian and Kählerian structures
Secondary: 53C55: Hermitian and Kählerian manifolds 53B10: Projective connections 53A20: Projective differential geometry 32J27: Compact Kähler manifolds 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 8 , August 2016 , pp. 1555 - 1575
Copyright
© The Authors 2016 

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