Polyhedral Kähler manifolds

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 $ℂP^2$ with singularities at complex line arrangements.