Ya Deng - Big Picard theorems and algebraic hyperbolicity for varieties admitting a variation of Hodge structures

epiga:8393 - Épijournal de Géométrie Algébrique, April 24, 2023, Volume 7 - https://doi.org/10.46298/epiga.2023.volume7.8393
Big Picard theorems and algebraic hyperbolicity for varieties admitting a variation of Hodge structuresArticle

Authors: Ya Deng

    In this paper, we study various hyperbolicity properties for a quasi-compact Kähler manifold $U$ which admits a complex polarized variation of Hodge structures so that each fiber of the period map is zero-dimensional. In the first part, we prove that $U$ is algebraically hyperbolic and that the generalized big Picard theorem holds for $U$. In the second part, we prove that there is a finite étale cover $\tilde{U}$ of $U$ from a quasi-projective manifold $\tilde{U}$ such that any projective compactification $X$ of $\tilde{U}$ is Picard hyperbolic modulo the boundary $X-\tilde{U}$, and any irreducible subvariety of $X$ not contained in $X-\tilde{U}$ is of general type. This result coarsely incorporates previous works by Nadel, Rousseau, Brunebarbe and Cadorel on the hyperbolicity of compactifications of quotients of bounded symmetric domains by torsion-free lattices.


    Volume: Volume 7
    Published on: April 24, 2023
    Accepted on: January 14, 2023
    Submitted on: August 26, 2021
    Keywords: Mathematics - Algebraic Geometry,Mathematics - Complex Variables,32H25, 14D07, 32Q45

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