Topological Hyperbolicity of Moduli spaces of Elliptic Surfaces

2021/03/23 by Xin Lü, Ruiran Sun, Lü, Xin +3
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2103.12490

openalex publication_date 2021/03/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• On the isotriviality of families of elliptic surfaces

Related

• The Chow rings of moduli spaces of elliptic surfaces over ℙ¹
• The moduli space of rational elliptic surfaces
• The structure of surfaces mapping to the moduli stack of canonically polarized varieties
• Finiteness of pointed maps to moduli spaces of polarized varieties
• Hyperbolicity of coarse moduli spaces and isotriviality for certain families