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Polyhedral Kähler manifolds
Dmitri Panov
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Geometry & Topology 13 (2009)
2205–2252
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Abstract
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In this article we introduce the notion of polyhedral Kähler manifolds, even
dimensional polyhedral manifolds with unitary holonomy. We concentrate on the
4–dimensional case, prove that such manifolds are smooth complex surfaces and
classify the singularities of the metric. The singularities form a divisor and the
residues of the flat connection on the complement of the divisor give us a system
of cohomological equations. A parabolic version of the Kobayshi–Hitchin
correspondence of T Mochizuki permits us to characterize polyhedral Kähler
metrics of nonnegative curvature on CP2 with singularities at complex line
arrangements.
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Keywords
polyhedral metric,
Kobayashi–Hitchin correspondence, line arrangement
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Mathematical Subject Classification
Primary: 53C56
Secondary: 32Q15, 53C55
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Publication
Received: 29 January 2009
Revised: 5 May 2009
Accepted: 26 April 2009
Published: 26 May 2009
Proposed: Dmitri Burago
Seconded: Simon Donaldson, Jim Bryan
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