$\mathbb{A}^1$-connectivity of moduli of vector bundles on a curve

TL;DR

This paper proves the $A^1$-connectivity of the moduli stack of vector bundles with fixed determinant on a curve, classifies vector bundles and $P^n$-bundles up to $A^1$-equivalence, and provides a counterexample related to $A^1$-h-cobordism.

Contribution

It establishes $A^1$-connectivity of the moduli stack, classifies vector bundles and projective bundles on curves up to $A^1$-equivalence, and constructs a variety counterexample to a previous question.

Findings

Moduli stack of vector bundles with fixed determinant is $A^1$-connected.

Classification of vector bundles on a curve up to $A^1$-concordance.

Existence of a variety $A^1$-h-cobordant to a projective bundle but not isomorphic to one.

Abstract

In this note we prove that the moduli stack of vector bundles on a curve, with a fixed determinant is $\mathbb{A}^1$-connected. We obtain this result by classifying vector bundles on a curve upto $\mathbb{A}^1$-concordance. Consequently we classify$\mathbb{P}^n$- bundles on a curve upto $\mathbb{A}^1$-weak equivalence, extending a result of Asok-Morel. We also give an explicit example of a variety which is $\mathbb{A}^1$-h-cobordant to a projective bundle over $\mathbb{P}^2$ but does not have the structure of a projective bundle over $\mathbb{P}^2$, thus answering a question of Asok-Kebekus-Wendt