The linear Shafarevich conjecture for quasiprojective varieties and algebraicity of Shafarevich morphisms

TL;DR

This paper proves a non-proper version of the Shafarevich conjecture for complex algebraic varieties, showing their universal covers are holomorphically convex and establishing algebraic maps related to local systems.

Contribution

It introduces a non-proper version of the Shafarevich conjecture and defines a class of subsets of the Betti stack with specific covering space properties.

Findings

Universal cover of certain algebraic varieties is holomorphically convex.

Existence of algebraic maps contracting subvarieties where local systems are isotrivial.

Extension of Shafarevich conjecture to non-proper varieties.

Abstract

We prove that the universal cover of a normal complex algebraic variety admitting a faithful complex representation of its fundamental group is an analytic Zariski open subset of a holomorphically convex complex space. This is a non-proper version of the Shafarevich conjecture. More generally we define a class of subset of the Betti stack for which the covering space trivializing the corresponding local systems has this property. Secondly, we show that for any complex local system $V$ on a normal complex algebraic variety $X$ there is an algebraic map $f \colon X\to Y$ contracting precisely the subvarieties on which $V$ is isotrivial.