Polyhedral Kahler Manifolds

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.

References in corpus (4)

• Asymptotic behaviour of tame harmonic bundles and an application to pure twistor $D$-modules
• Kobayashi-Hitchin correspondence for tame harmonic bundles and an application
• Polyhedral approximations of Riemannian manifolds
• A PL-manifold of nonnegative curvature homeomorphic to $S^2\times S^2$ is a direct metric product (Preliminary version)

Cited by in corpus (9)

• Ramification conjecture and Hirzebruch's property of line arrangements
• Singularities of plane complex curves and limits of Kähler metrics with cone singularities. I: Tangent Cones
• Local models for conical Kähler-Einstein metrics
• PIGTIKAL (puzzles in geometry that I know and love)
• Polyhedral Kähler cone metrics on ${\mathbb{C}}^n$ singular at hyperplane arrangements
• Real line arrangements with Hirzebruch property
• A PL-manifold of nonnegative curvature homeomorphic to $S^2\times S^2$ is a direct metric product (Preliminary version)
• Kähler cone-manifolds arising from a projective arrangement
• Asymptotically conical Ricci flat Kähler metrics on ${\mathbb{C}}^2$ with cone singularities along a complex curve