On uniformization of compact Kahler manifolds

TL;DR

This paper proves a uniformization theorem for compact Kähler manifolds with certain fundamental group properties, showing their universal covers are bounded domains in complex affine space.

Contribution

It establishes a uniformization result for compact Kähler manifolds with large, residually finite, nonamenable fundamental groups, linking their universal covers to bounded domains.

Findings

Universal cover is a bounded domain in complex affine space.

Conditions on the fundamental group are crucial for uniformization.

Provides a geometric characterization of certain Kähler manifolds.

Abstract

Theorem (uniformization). Let X be a compact Kahler manifold of dimension n with large, residually finite and nonamenable fundamental group. Then its universal covering is a bounded domain in the n-dimensional affine space.