$\mathbb{A}^1$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 AsokMorel. We also give an explicit example of a variety which is $\mathbb{A}^1$hcobordant 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 AsokKebekusWendt
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.05799
 Bibcode:
 2021arXiv211005799H
 Keywords:

 Mathematics  Algebraic Geometry
 EPrint:
 8 Pages. Comments welcome