On a problem of Piatetski-Shapiro and Shafarevich

TL;DR

This paper revisits classical work on algebraic uniformization, providing partial answers to whether certain coverings characterize algebraic varieties with bounded symmetric domain universal coverings.

Contribution

It offers a partial solution to a longstanding problem linking proalgebraic coverings to bounded symmetric domains in algebraic varieties.

Findings

Partial solution to the problem of characterizing varieties with bounded symmetric domain universal coverings.

Clarifies the relationship between proalgebraic quasi-homogeneous coverings and algebraic varieties of general type.

Extends classical results by Piatetski-Shapiro and Shafarevich on uniformization theory.

Abstract

We revisit a classical paper by Piatetski-Shapiro and Shafarevich on algebraic approach to uniformization and provide a partial solution of the problem, namely, whether the existence of proalgebraic quasi-homogeneous coverings of general type is the characteristic property of algebraic varieties whose universal coverings are bounded symmetric domains.