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arXiv · 2110.07533

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

Simion Filip
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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.

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Simion Filip. 2021-10-14. Uniformization of some weight 3 variations of Hodge structure, Anosov representations, and Lyapunov exponents. https://arxiv.org/abs/2110.07533

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