Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Topological Hyperbolicity of Moduli spaces of Elliptic Surfaces

Xin Lü, Ruiran Sun, Kang Zuo

Abstract

We introduce the notion of topological hyperbolicity to characterize the largeness of the topological fundamental group of a complex variety. Inspired by the Shafarevich conjecture, we propose to study the topological hyperbolicity of moduli spaces of polarized manifolds. We provide two pieces of supporting evidence: first, we show that moduli spaces where the infinitesimal Torelli theorem holds are very close to being topologically hyperbolic. Second, we establish a weak form of topological hyperbolicity for moduli spaces of elliptic surfaces of Kodaira dimension one without multiple fibers, where the infinitesimal Torelli theorem generally does not hold.

Topological Hyperbolicity of Moduli spaces of Elliptic Surfaces

Abstract

We introduce the notion of topological hyperbolicity to characterize the largeness of the topological fundamental group of a complex variety. Inspired by the Shafarevich conjecture, we propose to study the topological hyperbolicity of moduli spaces of polarized manifolds. We provide two pieces of supporting evidence: first, we show that moduli spaces where the infinitesimal Torelli theorem holds are very close to being topologically hyperbolic. Second, we establish a weak form of topological hyperbolicity for moduli spaces of elliptic surfaces of Kodaira dimension one without multiple fibers, where the infinitesimal Torelli theorem generally does not hold.

Paper Structure

This paper contains 5 sections, 13 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Variation of Hodge structures and topological hyperbolicity
  3. Topological hyperbolicity of moduli spaces of elliptic surfaces
  4. Infinitesimal Torelli Theorem
  5. Relative Iitaka fibration and the proof of \ref{['main-thm2']}

Key Result

Theorem 1.1

Let $Y$ be a smooth projective curve of genus $g$ over an algebraically closed field $k$ of characteristic $0$ and $\Delta \subset Y$ a finite subset. Fix an integer $q \geq 2$. Let $f:\, X \to Y$ be a non-isotrivial family of curves of genus $q$, and the resitiction $f:\, X \setminus f^{-1}(\Delta)

Theorems & Definitions (21)

  • Theorem 1.1: Shafarevich Hyperbolicity Conjecture
  • Definition 1.2
  • Theorem 1
  • Theorem 2
  • Proposition 2.1
  • Theorem 2.2: Theorem 1.2 in BT-Ax
  • proof : Proof of \ref{['main-thm1']}
  • Lemma 2.3
  • proof
  • Lemma 2.4: Lemma 2 in Mil68
  • ...and 11 more