$\mathbb{A}^1$-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
- Publication:
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arXiv e-prints
- Pub Date:
- October 2021
- arXiv:
- arXiv:2110.05799
- Bibcode:
- 2021arXiv211005799H
- Keywords:
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- Mathematics - Algebraic Geometry
- E-Print:
- 8 Pages. Comments welcome