Arakelov inequalities and characterization of totally geodesic ball quotients

I will report on work in progress with Matteo Costantini and Daniel Greb providing an intrinsic numerical characterization of totally geodesic ball quotients inside the moduli space of principally polarized abelian varieties over the complex numbers, obtained in terms of an Arakelov (in)equality associated with the underlying variation of Hodge structure. Our result extends work of Möller, Viehweg, and Zuo by removing some positivity conditions imposed in their statement. Our approach involves first 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 a singular setting.