A Real Shafarevich Conjecture for Universal Covers

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 $X(\mathbb{R})\mathrm{\ne }\mathrm{\varnothing }$, and dianalytic holomorphically convex when $X(\mathbb{R})=\mathrm{\varnothing }$. We prove this refined conjecture in two main cases: when $X$ is a curve, and when the fundamental group of $X$ is nilpotent.