Subvarieties of quotients of bounded symmetric domains

Abstract

We present a new criterion for the complex hyperbolicity of a non-compact quotient X of a bounded symmetric domain. For each \(p \ge 1\), this criterion gives a precise condition under which the subvarieties \(V \subset X\) with \(\dim V \ge p\) are of general type, and X is p-measure hyperbolic. Then, we give several applications related to ball quotients, or to the Siegel moduli space of principally polarized abelian varieties. In the case of smooth compactifications of ball quotients, we obtain a conditional upper bound on the dimension on the exceptional locus, assuming the Green–Griffiths–Lang conjecture holds true. As another example of application, we give effective lower bounds for levels l so that the moduli spaces of genus g curves with l-level structures are of general type.