Symmetric Differentials and Jets Extension of L² Holomorphic Functions

Let Σ = Bⁿ/Γ be a complex hyperbolic space with discrete subgroup Γ of the automorphism group of the unit ball Bⁿ, and Ω be a quotient of Bⁿ × Bⁿ under the diagonal action of Γ which is a holomorphic Bⁿ-fiber bundle over Σ. The goal of this article is to investigate the relation between symmetric differentials of Σ and the weighted L² holomorphic functions of Ω. If there exists a holomorphic function on Ω and it vanishes up to k-th order but with nonvanishing (k + 1)-th order on the maximal compact complex variety in Ω, then there exists a symmetric differential of degree k+1 on Σ. Using this property, we show that Σ has a symmetric differential of degree N for any N ≥ n + 1 under certain conditions. Moreover, if Σ is compact, for each symmetric differential over Σ we construct a weighted L² holomorphic function on Ω. We also show that any bounded holomorphic function on Ω is constant.