Chern classes of linear submanifolds with application to spaces of $k$-differentials and ball quotients

Abstract

We provide formulas for the Chern classes of linear submanifolds of the moduli spaces of Abelian differentials and hence for their Euler characteristic. This includes as special case the moduli spaces of $k$-differentials, for which we set up the full intersection theory package and implement it in the SageMath package diffstrata. As an application, we give an algebraic proof of the theorems of Deligne–Mostow and Thurston that suitable compactifications of moduli spaces of $k$-differentials on the $5$-punctured projective line with weights satisfying the INT-condition are quotients of the complex two-ball.