Reflexive symmetric differentials and quotients of bounded symmetric domains
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866916086554820608 |
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| author | Patel, Aryaman |
| author_facet | Patel, Aryaman |
| contents | For each classical irreducible bounded symmetric domain $\mathcal{D}$, Klingler has computed the minimum number $m_{\mathcal{D}}$ such that any smooth projective quotient $X=\mathcal{D}/Γ$, for $Γ\in\textrm{Aut}^0(\mathcal{D})$, satisfies $H^0(X,\mathrm{Sym}^iΩ^1_X)=0$ for $0<i<m_{\mathcal{D}}$. In this article, we extend Klingler's result to the case when $X$ is normal and projective. This, together with a normal version of Arapura's result about the relationship between the vanishing of global symmetric differentials on $X$ and the rigidity of finite dimensional representations of $π_1(X)$, gives rigidity statements for representations of $π_1(X)$ and $π_1(X_{reg})$ in a low dimensional range, when $X$ is a normal projective quotient of a bounded symmetric domain. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_16814 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Reflexive symmetric differentials and quotients of bounded symmetric domains Patel, Aryaman Algebraic Geometry 14E20, 32Q05, 32Q30, 14J60 For each classical irreducible bounded symmetric domain $\mathcal{D}$, Klingler has computed the minimum number $m_{\mathcal{D}}$ such that any smooth projective quotient $X=\mathcal{D}/Γ$, for $Γ\in\textrm{Aut}^0(\mathcal{D})$, satisfies $H^0(X,\mathrm{Sym}^iΩ^1_X)=0$ for $0<i<m_{\mathcal{D}}$. In this article, we extend Klingler's result to the case when $X$ is normal and projective. This, together with a normal version of Arapura's result about the relationship between the vanishing of global symmetric differentials on $X$ and the rigidity of finite dimensional representations of $π_1(X)$, gives rigidity statements for representations of $π_1(X)$ and $π_1(X_{reg})$ in a low dimensional range, when $X$ is a normal projective quotient of a bounded symmetric domain. |
| title | Reflexive symmetric differentials and quotients of bounded symmetric domains |
| topic | Algebraic Geometry 14E20, 32Q05, 32Q30, 14J60 |
| url | https://arxiv.org/abs/2311.16814 |