Yohan Brunebarbe: Hyperbolicity of moduli spaces of abelian varieties with a level structure

For any positive integers g and n, let \(A_g(n)\) be the moduli space of principally polarized abelian varieties with a level-n structure (it is a smooth quasi-projective variety for \(n>2\)). Building on works of Nadel and Noguchi, Hwang and To have shown that the minimal genus of a curve contained in \(A_g(n)\) grows with n. We will explain a generalization of this result dealing with subvarieties of any dimension. In particular, we show that all subvarieties of \(A_g(n)\) are of general type when \(n > 6g\). Similar results are true more generally for quotients of bounded symmetric domains by lattices.