On algebraic vector bundles of rank $2$ over smooth affine fourfolds

TL;DR

This paper investigates the classification of rank 2 algebraic vector bundles over smooth affine fourfolds, revealing that their isomorphism classes are not solely determined by Chern classes and providing cohomological criteria for their classification.

Contribution

It introduces cohomological criteria for classifying rank 2 bundles over smooth affine fourfolds and offers explicit counts of such bundles in specific geometric contexts.

Findings

Finiteness of Chern class fibers is characterized cohomologically.

Explicit classification for bundles over complements of hypersurfaces in projective space.

Number of non-isomorphic bundles over certain fourfolds is precisely determined.

Abstract

The classification of algebraic vector bundles of rank 2 over smooth affine fourfolds is a notoriously difficult problem. Isomorphism classes of such vector bundles are not uniquely determined by their Chern classes, in contrast to the situation in lower dimensions. Given a smooth affine fourfold over an algebraically closed field of characteristic not equal to $2$ or $3$, we study cohomological criteria for finiteness of the fibers of the Chern class map for rank $2$ bundles. As a consequence, we give a cohomological classification of such bundles in a number of cases. For example, if $d\leq 4$, there are precisely $d^2$ non-isomorphic algebraic vector bundles over the complement of a smooth hypersurface of degree $d$ in $\mathbb P^4_{\mathbb C}$.