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- W4287705066 abstract "Looking at the finite 'etale congruence covers $X(p)$ of a complex algebraic variety $X$ equipped with a variation of integral polarized Hodge structures whose period map is quasi-finite, we show that both the minimal gonality among all curves contained in $X(p)$ and the minimal volume among all subvarieties of $X(p)$ tend to infinity with $p$. This applies for example to Shimura varieties, moduli spaces of curves, moduli spaces of abelian varieties, moduli spaces of Calabi-Yau varieties, and can be made effective in many cases. The proof goes roughly as follows. We first prove a generalization of the Arakelov inequalities valid for any variation of Hodge structures on higher-dimensional algebraic varieties, which implies that the hyperbolicity of the subvarieties of $X$ is controlled by the positivity of a single line bundle. We then show in general that a big line bundle on a normal proper algebraic variety $bar X$ can be made more and more positive by going to finite covers of $bar X$ defined using level structures of a local system defined on a Zariski-dense open subset." @default.
- W4287705066 created "2022-07-26" @default.
- W4287705066 creator A5015288726 @default.
- W4287705066 date "2020-07-25" @default.
- W4287705066 modified "2023-09-30" @default.
- W4287705066 title "Increasing hyperbolicity of varieties supporting a variation of Hodge structures with level structures" @default.
- W4287705066 doi "https://doi.org/10.48550/arxiv.2007.12965" @default.
- W4287705066 hasPublicationYear "2020" @default.
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