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 CP² with singularities at complex line arrangements.

Keywords

polyhedral metric, Kobayashi–Hitchin correspondence, line arrangement

Mathematical Subject Classification

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