A characterization of complex quasi-projective manifolds uniformized by unit balls

TL;DR

This paper provides a precise criterion for when complex quasi-projective manifolds are uniformized by complex unit balls, extending Simpson's earlier uniformization theorem and offering new insights into their geometric structure.

Contribution

It establishes a necessary and sufficient condition for such manifolds to be uniformized by complex unit balls, generalizing Simpson's uniformization theorem.

Findings

Provides a complete characterization of uniformization by complex unit balls.

Generalizes Simpson's uniformization theorem.

Derives several new geometric byproducts.

Abstract

In 1988 Simpson extended the Donaldson-Uhlenbeck-Yau theorem to the context of Higgs bundles, and as an application he proved a uniformization theorem which characterizes complex projective manifolds and quasi-projective curves whose universal coverings are complex unit balls. In this paper we give a necessary and sufficient condition for quasi-projective manifolds to be uniformized by complex unit balls. This generalizes the uniformization theorem by Simpson. Several byproducts are also obtained in this paper.