A cohomological classification of vector bundles on smooth affine threefolds

TL;DR

This paper provides a cohomological classification of rank 2 vector bundles on smooth affine threefolds over algebraically closed fields with characteristic not 2, establishing cancellation properties and using advanced ${ m A}^1$-homotopy techniques.

Contribution

It introduces a cohomological approach to classify rank 2 vector bundles on smooth affine threefolds, utilizing ${ m A}^1$-homotopy sheaves and obstruction theory.

Findings

Classification of rank 2 vector bundles via cohomology

Cancellation property for rank 2 bundles on these varieties

Description of the first non-stable ${ m A}^1$-homotopy sheaf of the symplectic group

Abstract

We give a cohomological classification of vector bundles of rank $2$ on a smooth affine threefold over an algebraically closed field having characteristic unequal to $2$. As a consequence we deduce that cancellation holds for rank $2$ vector bundles on such varieties. The proofs of these results involve three main ingredients. First, we give a description of the first non-stable ${\mathbb A}^1$-homotopy sheaf of the symplectic group. Second, these computations can be used in concert with F. Morel's ${\mathbb A}^1$-homotopy classification of vector bundles on smooth affine schemes and obstruction theoretic techniques (stemming from a version of the Postnikov tower in ${\mathbb A}^1$-homotopy theory) to reduce the classification results to cohomology vanishing statements. Third, we prove the required vanishing statements.