Totally geodesic subvarieties in the moduli space of curves

Abstract

In this paper we study totally geodesic subvarieties $Y\subset {\mathsf{A}}_g$ of the moduli space of principally polarized abelian varieties with respect to the Siegel metric, for $g\ge 4$. We prove that if $Y$ is generically contained in the Torelli locus, then $\dim Y\le (7g-2)/3$.