Topological Hyperbolicity of Moduli spaces of Elliptic Surfaces

TL;DR

This paper introduces topological hyperbolicity to measure the largeness of fundamental groups in complex varieties, specifically studying moduli spaces of elliptic surfaces and their topological properties.

Contribution

It proposes the concept of topological hyperbolicity and provides evidence linking it to the infinitesimal Torelli theorem in moduli spaces of elliptic surfaces.

Findings

Moduli spaces with the infinitesimal Torelli theorem are nearly topologically hyperbolic.

A weak form of topological hyperbolicity is established for certain elliptic surface moduli spaces.

The study connects topological properties with classical Torelli-type theorems.

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.