A Real Shafarevich Conjecture for Universal Covers

TL;DR

This paper refines the classical Shafarevich conjecture for complex varieties over the reals, introducing real holomorphic convexity notions and proving the conjecture for curves and nilpotent fundamental groups.

Contribution

It proposes a real version of the Shafarevich conjecture, defining new convexity concepts and proving the conjecture in specific cases.

Findings

Universal cover is real holomorphically convex when the real locus is non-empty.

Universal cover is dianalytic holomorphically convex when the real locus is empty.

The conjecture is proven for curves and varieties with nilpotent fundamental groups.

Abstract

The classical Shafarevich conjecture predicts that the universal cover of a complex smooth projective variety $X$ is holomorphically convex. In this paper, we propose a refinement of this conjecture for varieties defined over the reals. In order to do this, we introduce the notions of real holomorphic convexity and transverse holomorphic convexity to capture the geometric differences dictated by the real locus $X(\mathbb{R})$ of $X$. Specifically, we conjecture that the universal cover is real holomorphically convex when $X(\mathbb{R}) \neq \emptyset$, and dianalytic holomorphically convex when $X(\mathbb{R}) = \emptyset$. We prove this refined conjecture in two main cases: when $X$ is a curve, and when the fundamental group of $X$ is nilpotent.