Uniformization of complex projective klt varieties by bounded symmetric domains

TL;DR

This paper establishes criteria for complex projective klt varieties to be uniformized by Hermitian symmetric spaces or polydisks, extending classical results with modern Hodge theory and singularity analysis.

Contribution

It provides necessary and sufficient conditions for uniformization of klt varieties by Hermitian symmetric spaces and polydisks, generalizing classical theorems with new singularity considerations.

Findings

Criteria for uniformization by Hermitian symmetric spaces

Conditions for uniformization by polydisks

Extension of classical uniformization results

Abstract

Using classical results from Hodge theory and more contemporary ones valid for complex projective varieties with Kawamata log terminal (klt) singularities, we deduce necessary and sufficient conditions for such varieties to be uniformized by each of the four irreducible Hermitian symmetric spaces of non compact type. We also deduce necessary and sufficient conditions for uniformization by a polydisk, which generalizes a classical result of Simpson.