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A Real Shafarevich Conjecture for Universal Covers

Rodolfo Aguilar, Cristhian Garay

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.

A Real Shafarevich Conjecture for Universal Covers

Abstract

The classical Shafarevich conjecture predicts that the universal cover of a complex smooth projective variety 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 of . Specifically, we conjecture that the universal cover is real holomorphically convex when , and dianalytic holomorphically convex when . We prove this refined conjecture in two main cases: when is a curve, and when the fundamental group of is nilpotent.
Paper Structure (13 sections, 7 theorems, 14 equations)

This paper contains 13 sections, 7 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Real setting
  4. Plurisubharmonicity
  5. Real structures and Dianalytic spaces
  6. Real Shafarevich conjecture
  7. Metric geometry of semisimple isometries
  8. Real curves
  9. Non-empty real locus
  10. Empty real locus
  11. Nilpotent covers
  12. Non-empty real locus
  13. Empty real locus

Key Result

Proposition 2.1

Let $X$ be a complex manifold equipped with a real structure, that is, an antiholomorphic involution $\sigma: X \to X$. Let $\pi: \tilde{X} \to X$ be the universal covering map. If the real locus $X(\mathbb{R}) = \{x \in X \mid \sigma(x) = x\}$ is non-empty, then $\tilde{X}$ admits a canonical real

Theorems & Definitions (24)

  • Conjecture 1.1
  • Conjecture 1.2
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • ...and 14 more