Polyhedral Kähler manifolds

Panov, Dmitri

doi:10.2140/gt.2009.13.2205

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Polyhedral Kähler manifolds

Dmitri Panov

Geometry & Topology 13 (2009) 2205–2252

DOI: 10.2140/gt.2009.13.2205

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 $ℂ P^{2}$ 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

Volume 13, issue 4 (2009)