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

Let $\widetilde{M}$ be a complex manifold and $\Gamma$ be a torsion-free cocompact lattice of $\text{Aut}(\widetilde{M})$. Let $\rho\colon\Gamma\to SU(N,1)$ be a representation and $M:=\widetilde M/\Gamma$ be an $n$-dimensional compact complex manifold which admits a holomorphic embeddin...

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Published in:	arXiv.org 2023-04
Main Authors:	Lee, Seungjae, Seo, Aeryeong
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Language:	English
Subjects:	Analytic functions Colon Manifolds (mathematics) Mathematical analysis Representations
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description	Let $\widetilde{M}$ be a complex manifold and $\Gamma$ be a torsion-free cocompact lattice of $\text{Aut}(\widetilde{M})$. Let $\rho\colon\Gamma\to SU(N,1)$ be a representation and $M:=\widetilde M/\Gamma$ be an $n$-dimensional compact complex manifold which admits a holomorphic embedding $\imath$ into $\Sigma:=\mathbb B^N/\rho(\Gamma)$. In this paper, we investigate a relation between weighted $L^2$ holomorphic functions on the fiber bundle $\Omega:=M\times_\rho\mathbb B^N$ and the holomorphic sections of the pull-back bundle $\imath^{-1}(S^mT^*_\Sigma)$ over $M$. In particular, $A^2_\alpha(\Omega)$ has infinite dimension for any $\alpha>-1$ and if $n-1$, $A^2_\alpha(\mathbb B^n\times_{\rho} \mathbb B^N)$ has infinite dimension. If \(n
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subjects	Analytic functions Colon Manifolds (mathematics) Mathematical analysis Representations
title	Weighted $L^2$ Holomorphic functions on ball-fiber bundles over compact Kähler manifolds
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Weighted \(L^2\) Holomorphic functions on ball-fiber bundles over compact Kähler manifolds