Polyhedral Kahler Manifolds

TL;DR

This paper introduces Polyhedral Kahler manifolds, focusing on 4-dimensional cases, classifies their singularities, and characterizes non-negative curvature metrics on complex projective planes using advanced geometric and cohomological tools.

Contribution

It defines Polyhedral Kahler manifolds, proves their smoothness in 4D, and applies a parabolic Kobayashi-Hitchin correspondence to classify certain metrics on CP^2.

Findings

Polyhedral Kahler manifolds are smooth complex surfaces in 4D.

Singularities form a divisor with specific cohomological properties.

Non-negative curvature metrics on CP^2 relate to complex line arrangements.

Abstract

In this article we introduce the notion of Polyhedral Kahler 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. Parabolic version of Kobayshi-Hitchin correspondence of T. Mochizuki permits us to characterize polyhedral Kahler metrics of non-negative curvature on CP^2 with singularities at complex line arrangements.