Complex surfaces with CAT(0) metrics

TL;DR

This paper investigates complex surfaces with CAT(0) metrics, constructing such metrics on CP^2 and related orbifolds, and explores their geometric properties and implications for K(pi,1) spaces.

Contribution

It introduces new constructions of CAT(0) metrics on complex surfaces, including CP^2 with orbifold structures and quotients of complex balls, advancing understanding of their geometric and topological properties.

Findings

Constructed CAT(0) metrics on CP^2 with orbifold structures.

Provided criteria for Sasakian 3-manifolds to be globally CAT(1).

Showed certain Kummer coverings of CP^2 are of type K(pi,1).

Abstract

We study complex surfaces with locally CAT(0) polyhedral Kahler metrics and construct such metrics on CP^2 with various orbifold structures. In particular, in relation to questions of Gromov and Davis-Moussong we construct such metrics on a compact quotient of the two-dimensional unite complex ball. In the course of the proof of these results we give criteria for Sasakian 3-manifolds to be globally CAT(1). We show further that for certain Kummer coverings of CP^2 of sufficiently high degree their desingularizations are of type K(pi,1).