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arXiv 2201.10915math.CV
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紧致Kähler流形上球纤维丛的加权$L^2$全纯函数

Weighted $L^2$ Holomorphic functions on ball-fiber bundles over compact Kähler manifolds

Seungjae Lee, Aeryeong Seo

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AI总结：

本文研究紧致Kähler流形上球纤维丛的加权$L^2$全纯函数与拉回对称幂余切丛全纯截面的关系，证明在$\alpha>-1$及$n<N$条件下相关函数空间具有无穷维，并应用于极大表示情形。

摘要：

设$\widetilde{M}$为复流形，$\Gamma$为$\text{Aut}(\widetilde{M})$的无挠余紧格。设$\rho\colon\Gamma\to SU(N,1)$为表示，$M:=\widetilde M/\Gamma$为$n$维紧致复流形，且存在到$\Sigma:=\mathbb B^N/\rho(\Gamma)$的全纯嵌入$\imath$。本文研究纤维丛$\Omega:=M\times_\rho\mathbb B^N$上的加权$L^2$全纯函数与$M$上拉回丛$\imath^{-1}(S^mT^*_\Sigma)$的全纯截面之间的关系。特别地，对任意$\alpha>-1$，$A^2_\alpha(\Omega)$具有无穷维；且若$n<N$，则$A^2_{-1}(\Omega)$也具有同样性质。作为一个应用，若$\Gamma$为$SU(n,1)$（$n\geq 2$）中的无挠余紧格，且$\rho\colon \Gamma\to SU(N,1)$为极大表示，则对任意$\alpha>-1$，$A^2_\alpha(\mathbb B^n\times_\rho \mathbb B^N)$具有无穷维。若$n<N$，则$A_{-1}^2(\mathbb B^n\times_\rho \mathbb B^N)$也具有同样性质。

英文摘要：

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.

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