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

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

arXiv comment10 pages. Minor corrections and modifications. Results unchanged. Final Version. Accepted for publication in Journal of Inst. of Math of Jussieu