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Obstructions to algebraizing topological vector bundles. (English) Zbl 07040512
Summary: Suppose \(X\) is a smooth complex algebraic variety. A necessary condition for a complex topological vector bundle on \(X\) (viewed as a complex manifold) to be algebraic is that all Chern classes must be algebraic cohomology classes, that is, lie in the image of the cycle class map. We analyze the question of whether algebraicity of Chern classes is sufficient to guarantee algebraizability of complex topological vector bundles. For affine varieties of dimension \(\leqslant 3\), it is known that algebraicity of Chern classes of a vector bundle guarantees algebraizability of the vector bundle. In contrast, we show in dimension \(\geqslant 4\) that algebraicity of Chern classes is insufficient to guarantee algebraizability of vector bundles. To do this, we construct a new obstruction to algebraizability using Steenrod operations on Chow groups. By means of an explicit example, we observe that our obstruction is nontrivial in general.
MSC:
14F42 Motivic cohomology; motivic homotopy theory
32L05 Holomorphic bundles and generalizations
55R25 Sphere bundles and vector bundles in algebraic topology
13C10 Projective and free modules and ideals in commutative rings
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