\({\bar\partial}\)-problem on a family of weakly pseudoconvex manifolds
type: Item
instance of: scholarly article
title: \({\bar\partial}\)-problem on a family of weakly pseudoconvex manifolds
zbMATH Open document ID: 0599.32017
author: Hideaki Kazama
author: Kwang Ho Shon
published in: Proceedings of the Japan Academy. Series A
publication date: 1986-01-01T00:00:00Z
review text: Let \(E\in Pic^ 0(T^ n)\) be a holomorphic line bundle on the complex torus \(T^ n\) with Chern class zero. The authors study the cohomology group \(H^ p(E,0)\) of the structure sheaf 0 of E. Let d(, ) be the invariant distance on \(Pic^ 0(T^ n)\), then the main result is the following criterion: (1) If \(d(I,E^{\ell})=0\) for some \(\ell \geq 1\), then \(H^ p(E,0)\) is an infinite-dimensional Hausdorff space (1\(\leq p\leq n);\) (2) If there exists \(a>0\) such that \(\exp (-a\ell)\leq d(I,E^{\ell})\) for any \(\ell \geq 1\), then dim \(H^ p(E,0)=\left( \begin{matrix} n\\ p\end{matrix} \right)\) (1\(\leq p\leq n);\) (3) If \(d(I,E^{\ell})\neq 0\) for any \(\ell \geq 1\) and \(\lim \inf_{\ell \to \infty}\exp (a\ell)d(I,E^{\ell})=0\) for any \(a>0\), then \(H^ p(E,0)\) is not Hausdorff (1\(\leq p\leq n).\) The complete proof will appear elsewhere.
Mathematics Subject Classification ID: 32W05
Mathematics Subject Classification ID: 32L10
Mathematics Subject Classification ID: 32T99
zbMATH DE Number: 3966677
zbMATH Keywords: \({\bar \partial }\)-problem
zbMATH Keywords: Picard variety
zbMATH Keywords: weakly pseudoconvex manifold
zbMATH Keywords: holomorphic line bundle
zbMATH Keywords: complex torus
zbMATH Keywords: cohomology group
reviewed by: Chun-Te Lee
MaRDI profile type: Publication
full work available at URL: pjaa.62.19
OpenAlex ID: W2020151469
cites work: Bemerkenswerte pseudokonvexe Mannigfaltigkeiten
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cites work: \({\bar\partial}\)-cohomology of (H,C)-groups
cites work: Complex abelian Lie groups with finite-dimensional cohomology groups
cites work: Q4052463
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cites work: Line bundles on toroidal groups.
cites work: Two remarks concerning toroidal groups
DOI: 10.3792/PJAA.62.19
description: scientific article; zbMATH DE number 3966677