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

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, Möller, 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.

improved exposition; 103 pages, 3 tables, 5 figures

References in corpus (3)

• Variations of Hodge structures for hypergeometric differential operators and parabolic Higgs bundles
• Relatively dominated representations
• Degenerations of Hodge structure