  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.
