论文标题
$ l^2 $ HOLOMORTHIC功能的对称差异和喷气机扩展
Symmetric differentials and jets extension of $L^2$ holomorphic functions
论文作者
论文摘要
令$σ= \ Mathbb b^n/γ$成为一个复杂的双曲空间b^n $ - 纤维捆绑包上$σ$。本文的目的是研究$σ$的对称差异与加权$ l^2 $ holomorphic函数之间的关系。如果在$ω$上存在一个全体形态功能，并且它在$ω$中的最大紧凑型复合物上最多消失，则存在$ K+1 $ $σ$的对称差异。使用此属性，我们表明$σ$始终具有任何$ n \ geq n+2 $的对称差异$ n $。此外，如果$σ$紧凑，对于$σ$的每个对称差异，我们在$ω$上构建了一个加权$ l^2 $ holomorphic函数。我们还表明，当$ h^0（σ，s^{m}t_σ^*）$ 0 <m \ leq n+1 $时，$ω$上的任何有限的全态函数都是恒定的。
Let $Σ= \mathbb B^n/Γ$ be a complex hyperbolic space with discrete subgroup $Γ$ of the automorphism group of the unit ball $\mathbb B^n$ and $Ω$ be a quotient of $\mathbb B^n \times\mathbb B^n$ under the diagonal action of $Γ$ which is a holomorphic $\mathbb B^n$-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 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 $Σ$ always has a symmetric differential of degree $N$ for any $N \geq n+2$. 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 when $H^0 (Σ, S^{m} T_Σ^* )=0$ for every $0 < m \leq n+1$.
