Uniformization of some weight 3 variations of Hodge structure, Anosov representations, and Lyapunov exponents

TL;DR

This paper develops uniformizations for certain weight 3 variations of Hodge structure, proves a conjecture on Lyapunov exponents, and classifies related hypergeometric equations, with implications for monodromy and dynamical properties.

Contribution

It introduces new uniformization techniques for weight 3 VHS, proves a conjecture on Lyapunov exponents, and classifies hypergeometric equations satisfying these conditions.

Findings

Established a conjecture on Lyapunov exponents for VHS.

Proved monodromy representations are log-Anosov.

Classified hypergeometric differential equations including mirror quintic.

Abstract

We develop a class of uniformizations for certain weight 3 variations of Hodge structure (VHS). The analytic properties of the VHS are used to establish a conjecture of Eskin, Kontsevich, M\"oller, and Zorich on Lyapunov exponents. Additionally, we prove that the monodromy representations are log-Anosov, a dynamical property that has a number of global consequences for the VHS. We establish a strong Torelli theorem for the VHS and describe appropriate domains of discontinuity. Additionally, we classify the hypergeometric differential equations that satisfy our assumptions. We obtain several multi-parameter families of equations, which include the mirror quintic as well as the six other thin cases of Doran--Morgan and Brav--Thomas.