#### Volume 13, issue 4 (2009)

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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 $ℂ\phantom{\rule{0.3em}{0ex}}{P}^{2}$ with singularities at complex line arrangements.

##### Keywords
polyhedral metric, Kobayashi–Hitchin correspondence, line arrangement
##### Mathematical Subject Classification 2000
Primary: 53C56
Secondary: 32Q15, 53C55