Symmetric Differentials and Jets Extension of L-2 Holomorphic Functions

Abstract

Let Sigma = B-n/Gamma be a complex hyperbolic space with discrete subgroup Gamma of the automorphism group of the unit ball B-n, and Omega be a quotient of B-n x B-n under the diagonal action of Gamma which is a holomorphic B-n-fiber bundle over Sigma. The goal of this article is to investigate the relation between symmetric differentials of Sigma and the weighted L-2 holomorphic functions of Omega. If there exists a holomorphic function on Omega and it vanishes up to k-th order but with nonvanishing (k + 1)-th order on the maximal compact complex variety in Omega, then there exists a symmetric differential of degree k + 1 on Sigma. Using this property, we show that Sigma has a symmetric differential of degree N for any N >= n + 1 under certain conditions. Moreover, if Sigma is compact, for each symmetric differential over Sigma we construct a weighted L-2 holomorphic function on Omega. We also show that any bounded holomorphic function on Omega is constant.