On effective existence of symmetric differentials of complex hyperbolic space forms

Abstract

For a noncompact complex hyperbolic space form of finite volume $$X=\mathbb {B}^n/\varGamma $$ , we consider the problem of producing symmetric differentials vanishing at infinity on the Mumford compactification $$\overline{X}$$ of X similar to the case of producing cusp forms on hyperbolic Riemann surfaces. We introduce a natural geometric measurement which measures the size of the infinity $$\overline{X}-X$$ called canonical radius of a cusp of $$\varGamma $$ . The main result in the article is that there is a constant $$r^*=r^*(n)$$ depending only on the dimension, so that if the canonical radii of all cusps of $$\varGamma $$ are larger than $$r^*$$ , then there exist symmetric differentials of $$\overline{X}$$ vanishing at infinity. As a corollary, we show that the cotangent bundle $$T_{\overline{X}}$$ is ample modulo the infinity if moreover the injectivity radius in the interior of $$\overline{X}$$ is larger than some constant $$d^*=d^*(n)$$ which depends only on the dimension.