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${\mathbb A}^1$-CONNECTEDNESS OF MODULI OF VECTOR BUNDLES ON A CURVE
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 23 / Issue 3 / May 2024
- Published online by Cambridge University Press:
- 20 March 2023, pp. 1019-1027
- Print publication:
- May 2024
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- Article
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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 up to 
${\mathbb A}^1$-concordance. Consequently, we classify 
${\mathbb P}^n$-bundles on a curve up to 
${\mathbb A}^1$-weak equivalence, extending a result in [3] 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 [2].
