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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Polyhedral Kähler manifolds

Dmitri Panov

Geometry & Topology 13 (2009) 2205–2252
Abstract

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 P2 with singularities at complex line arrangements.

Keywords
polyhedral metric, Kobayashi–Hitchin correspondence, line arrangement
Mathematical Subject Classification 2000
Primary: 53C56
Secondary: 32Q15, 53C55
References
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
Authors
Dmitri Panov
Department of Mathematics
Imperial College
London, SW7 2AZ
UK
http://www.ma.ic.ac.uk/~dpanov