Arakelov inequalities and characterization of totally geodesic ball quotients in $\mathcal{A}_g$

Abstract

We establish an Arakelov inequality for variations of Hodge structures underlying families of principally polarized complex Abelian varieties. In the equality case, this leads to a numerical characterization of certain totally geodesic ball quotients inside the moduli space of Abelian varieties. Our result extends work of Möller-Viehweg-Zuo by removing the strong positivity conditions imposed in their statement. For families over compact base spaces, our proof involves showing that the period map associated with a family of Abelian varieties factors through certain MMP operations and then generalizing the results of Möller, Viehweg, and Zuo to an appropriate singular setting. In the quasiprojective surface case, we implement a new approach that does not pass through Miyaoka-Yau-type uniformisation theorems but uses an argument going back to an idea of Mok, symmetric space theory and semistability considerations instead.