Global deformations of \(\mathbb{P}^2\)-bundles over \(\mathbb{P}^1\)
type: Item
instance of: scholarly article
title: Global deformations of \(\mathbb{P}^2\)-bundles over \(\mathbb{P}^1\)
zbMATH Open document ID: 0933.14005
author: Iku Nakamura
published in: Journal of Mathematics of Kyoto University
publication date: 1999-09-02T00:00:00Z
review text: Every holomorphic vector bundle over \(\mathbb{P}^1\) is a direct sum of line bundles. Hence the analytic classification of holomorphic \(\mathbb{P}^2\)-bundles over \(\mathbb{P}^1\) may be considered known. Not obvious is when a \(\mathbb{P}^2\)-bundle may be deformed to another \(\mathbb{P}^2\)-bundle. In this paper the author studies the complex analytic global deformation of \(\mathbb{P}^2\)-bundles on \(\mathbb{P}^1\). The author proves the following results. Theorem 1. The set of all \(\mathbb{P}^2\)-bundles over \(\mathbb{P}^1\) with first Chern class divisible by 3 is closed and transitive under global deformations. Theorem 2. The set of all \(\mathbb{P}^2\)-bundles over \(\mathbb{P}^1\) whose first Chern class is not divisible by 3 and which are not homeomorphic to \(\mathbb{P}^1\times \mathbb{P}^2\) is closed and transitive under global deformations. Theorem 3. The union of the set of all \(\mathbb{P}^2\)-bundles over \(\mathbb{P}^1\) homeomorphic to \(\mathbb{P}^1\times\mathbb{P}^2\) and the set of all \(\mathbb{P}^1\)-bundles over \(\mathbb{P}^2\) homeomorphic to \(\mathbb{P}^1 \times\mathbb{P}^2\) is closed and transitive under global deformations. Furthermore, the author shows that there are infinitely many non-isomorphic \(\mathbb{P}^1\)-bundles over \(\mathbb{P}^2\) homeomorphic to \(\mathbb{P}^1\times \mathbb{P}^2\) which arise from topologically trivial unstable rank two vector bundles over \(\mathbb{P}^2\).
reviewed by: E. Ballico
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 32G08
Mathematics Subject Classification ID: 14H60
Mathematics Subject Classification ID: 14D15
Mathematics Subject Classification ID: 55R40
zbMATH DE Number: 1295040
zbMATH Keywords: global deformation
zbMATH Keywords: first Chern class
zbMATH Keywords: unstable vector bundle
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250518158
OpenAlex ID: W1954668670
DOI: 10.1215/KJM/1250518158
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description: scientific article; zbMATH DE number 1295040