Increasing hyperbolicity of varieties supporting a variation of Hodge structures with level structures
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 X can be made more and more positive by going to finite covers of X defined using level structures of a local system defined on a Zariski-dense open subset.