Obstructions to algebraizing topological vector bundles

TL;DR

This paper investigates whether algebraic Chern classes ensure that complex topological vector bundles on smooth complex varieties are algebraic, revealing that in dimensions four and higher, additional obstructions prevent algebraization.

Contribution

The authors introduce a new Steenrod operation-based obstruction to algebraizability, demonstrating its non-triviality and showing that algebraic Chern classes alone are insufficient in higher dimensions.

Findings

Algebraicity of Chern classes guarantees algebraizability in dimensions ≤3.

In dimensions ≥4, algebraic Chern classes do not ensure algebraizability.

A new obstruction using Steenrod operations is constructed and shown to be non-trivial.

Abstract

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, i.e., 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 $\leq 3$, it is known that algebraicity of Chern classes of a vector bundle guarantees algebraizability of the vector bundle. In contrast, we show in dimension $\geq 4$ 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 non-trivial in general.