Algebraic vector bundles on spheres

TL;DR

This paper classifies high-rank vector bundles on certain affine quadrics using ${ m A}^1$-homotopy theory and obstruction methods, providing answers to longstanding questions about unimodular rows and vector bundle completability.

Contribution

It determines the first non-stable ${ m A}^1$-homotopy sheaf of $SL_n$ and classifies vector bundles on split smooth affine quadrics, connecting algebraic and topological perspectives.

Findings

Computed the first non-stable ${ m A}^1$-homotopy sheaf of $SL_n$

Classified vector bundles of rank ≥ d-1 on specific affine quadrics

Provided a criterion for unimodular row completability

Abstract

We determine the first non-stable ${\mathbb A}^1$-homotopy sheaf of $SL_n$. Using techniques of obstruction theory involving the ${\mathbb A}^1$-Postnikov tower, supported by some ideas from the theory of unimodular rows, we classify vector bundles of rank $\geq d-1$ on split smooth affine quadrics of dimension $2d-1$. These computations allow us to answer a question posed by Nori, which gives a criterion for completability of certain unimodular rows. Furthermore, we study compatibility of our computations of ${\mathbb A}^1$-homotopy sheaves with real and complex realization.