Big Picard theorems and algebraic hyperbolicity for varieties admitting a variation of Hodge structures
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866917388005408768 |
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| author | Deng, Ya |
| author_facet | Deng, Ya |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2001_04426 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Big Picard theorems and algebraic hyperbolicity for varieties admitting a variation of Hodge structures Deng, Ya Algebraic Geometry Complex Variables 32H25, 14D07, 32Q45 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. |
| title | Big Picard theorems and algebraic hyperbolicity for varieties admitting a variation of Hodge structures |
| topic | Algebraic Geometry Complex Variables 32H25, 14D07, 32Q45 |
| url | https://arxiv.org/abs/2001.04426 |