论文标题
加权$ l^2 $ holomorthic功能在紧凑型kähler歧管上的球纤维束上
Weighted $L^2$ Holomorphic functions on ball-fiber bundles over compact Kähler manifolds
论文作者
论文摘要
令$ \ widetilde {m} $为复杂的歧管，$γ$为$ \ text {aut}（\ widetilde {m}）$的无扭转的cococompact晶格。令$ρ\colonγ\至su（n，1）$为表示形式，$ m：= \ widetilde m/γ$为$ n $二维紧凑型复杂的复合歧管，它承认$ \ imath $ \ imath $ to $ \ imth $σ：在本文中，我们研究了纤维束上的加权$ l^2 $ holomorphic函数之间的关系：= m \times_ρ\ mathbb b^n $与靠背bundle $ \ imath $ \ imath^{ - 1} { - 1}（s^mt^*_ =_σ）$ $ m $ m $ m $ $ m $ $ $ $ $ $ mathbb b^n $与holomorphic部分。特别是，$ a^2_α（ω）$对于任何$α> -1 $具有无限尺寸，如果$ n <n $，则$ a^2 _ { - 1}（ω）$也具有相同的属性。 As an application, if $Γ$ is a torsion-free cocompact lattice in $SU(n,1)$, $n\geq 2$, and $ρ\colon Γ\to SU(N,1)$ is a maximal representation, then for any $α>-1$, $A^2_α(\mathbb B^n\times_ρ \mathbb B^N)$ has infinite dimension.如果$ n <n $，则$ a _ { - 1}^2（\ mathbb b^n \times_ρ\ mathbb b^n）$也具有相同的属性。
Let $\widetilde{M}$ be a complex manifold and $Γ$ be a torsion-free cocompact lattice of $\text{Aut}(\widetilde{M})$. Let $ρ\colonΓ\to SU(N,1)$ be a representation and $M:=\widetilde M/Γ$ be an $n$-dimensional compact complex manifold which admits a holomorphic embedding $\imath$ into $Σ:=\mathbb B^N/ρ(Γ)$. In this paper, we investigate a relation between weighted $L^2$ holomorphic functions on the fiber bundle $Ω:=M\times_ρ\mathbb B^N$ and the holomorphic sections of the pull-back bundle $\imath^{-1}(S^mT^*_Σ)$ over $M$. In particular, $A^2_α(Ω)$ has infinite dimension for any $α>-1$ and if $n<N$, then $A^2_{-1}(Ω)$ also has the same property. As an application, if $Γ$ is a torsion-free cocompact lattice in $SU(n,1)$, $n\geq 2$, and $ρ\colon Γ\to SU(N,1)$ is a maximal representation, then for any $α>-1$, $A^2_α(\mathbb B^n\times_ρ \mathbb B^N)$ has infinite dimension. If $n<N$, then $A_{-1}^2(\mathbb B^n\times_ρ \mathbb B^N)$ also has the same property.
