Shafarevich mappings and period mappings

TL;DR

This paper establishes conditions under which a smooth quasi-projective variety has a holomorphically convex universal cover, focusing on properties of its fundamental group and period mappings related to variations of mixed Hodge structures.

Contribution

It provides new criteria linking the residual nilpotency of the fundamental group and properness of period mappings to the holomorphic convexity of the universal cover.

Findings

Universal cover is holomorphically convex under specified conditions.

Residually nilpotent fundamental group implies convexity.

Proper period mappings are key to the convexity result.

Abstract

We shall show that a smooth, quasi-projective variety $X$ has a holomorphically convex universal covering $\wt X$ when (i) $\pi_1(X)$ is residually nilpotent and (ii) there is an admissable variation of \mhs\ over $X$ whose monodromy representation has a finite kernel, and where in each case a corresponding period mapping is assumed to be proper.