Symmetric differentials on complex hyperbolic manifolds with cusps

Abstract

Let $(X,D)$ be a logarithmic pair, and let $h$ be a smooth metric on $T_{X \setminus D}$. We give a simple criterion on the curvature of $h$ for the bigness of $\Omega_X(\operatorname{log} D)$ or $\Omega_X$. As an application, we obtain a metric proof of the bigness of $\Omega_X(\operatorname{log} D)$ on any toroidal compactification of a bounded symmetric domain. Then, we use this singular metric approach to study the bigness and the nefness of $\Omega_X$ in the more specific case of the ball. We obtain effective ramification orders for a cover $X^\prime \to X$, étale outside the boundary, to have all its subvarieties with big cotangent bundle. We also prove that $\Omega_{X^\prime}$ is nef if the ramification is high enough. Moreover, the ramification orders we obtain do not depend on the dimension of the ball quotient we consider.