On complex vector bundles on projective threefold
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.
scientific article; zbMATH DE number 4017070
scientific article