On complex vector bundles on projective threefold
type: Item
instance of: scholarly article
title: On complex vector bundles on projective threefold
zbMATH Open document ID: 0626.14016
DOI: 10.1007/BF01388917
author: Constantin Bănică
author: Mihai Putinar
published in: Inventiones Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143461
review text: The subject of this paper is the determination of those topological vector bundles on a given complex projective manifold that admit an algebraic structure. This question is answered almost completely for three-dimensional projective manifolds. A necessary condition is of course that the Chern classes of the given topological bundle be algebraic. It is proved in this paper that for rank \(\geq 3\), the algebraicity of the Chern classes is also sufficient for the existence of an algebraic structure. For bundles of rank 2, there is the weaker result that there exists an algebraic rank-2 vector bundle with the same Chern classes as the given topological bundle. A classification of topological vector bundles (of ranks 2 and 3) in terms of Chern classes is also given. A rank-3 bundle is determined by its Chern classes, and there is in integrality condition on these Chern classes which is necessary and sufficient for the existence of a 3-bundle having these classes. For rank-2 bundles there is the same integrality condition, but in this case the Chern classes determine the bundle only up to a finite number of possibilities. The extra data are a generalized Atiyah-Rees invariant \(\alpha\) and another, similar invariant called \(\beta\). - Applied to projective rational threefolds, these theorems imply that every topological vector bundle admits algebraic structures.
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 55R25
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 57R20
zbMATH DE Number: 4017070
zbMATH Keywords: topological vector bundle
zbMATH Keywords: algebraic vector bundle
zbMATH Keywords: algebraicity of the Chern classes
zbMATH Keywords: projective rational threefolds
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4017070