## Abstract

Let Σ = B^{n}/Γ be a complex hyperbolic space with discrete subgroup Γ of the automorphism group of the unit ball B^{n}, and Ω be a quotient of B^{n} × B^{n} under the diagonal action of Γ which is a holomorphic B^{n}-fiber bundle over Σ. The goal of this article is to investigate the relation between symmetric differentials of Σ and the weighted L^{2} 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 L^{2} holomorphic function on Ω. We also show that any bounded holomorphic function on Ω is constant.

Original language | English |
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Pages (from-to) | 1239-1272 |

Number of pages | 34 |

Journal | Indiana University Mathematics Journal |

Volume | 72 |

Issue number | 3 |

DOIs | |

State | Published - 2023 |

## Keywords

- Complex hyperbolic space forms
- L holomorphic functions
- symmetric differentials
- ∂̄-equations

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