## Duke Mathematical Journal

### Vector bundles on contractible smooth schemes

#### Abstract

We discuss algebraic vector bundles on smooth $k$-schemes $X$ contractible from the standpoint of ${\mathbb A}^1$-homotopy theory; when $k = {\mathbb C}$, the smooth manifolds $X({\mathbb C})$ are contractible as topological spaces. The integral algebraic K-theory and integral motivic cohomology of such schemes are those of $\mathrm{Spec} k$. One may hope that, furthermore, and in analogy with the classification of topological vector bundles on manifolds, algebraic vector bundles on such schemes are all isomorphic to trivial bundles; this is almost certainly true when the scheme is affine. However, in the nonaffine case, this is false: we show that (essentially) every smooth ${\mathbb A}^1$-contractible strictly quasi-affine scheme that admits a $U$-torsor whose total space is affine, for $U$ a unipotent group, possesses a nontrivial vector bundle. Indeed, we produce explicit arbitrary-dimensional families of nonisomorphic ${\mathbb A}^1$-contractible schemes, with each scheme in the family equipped with “as many” (i.e., arbitrary-dimensional moduli of) nonisomorphic vector bundles, of every sufficiently large rank $n$, as one desires; neither the schemes nor the vector bundles on them are distinguishable by algebraic K-theory. We also discuss the triviality of vector bundles for certain smooth complex affine varieties whose underlying complex manifolds are contractible but that are not necessarily ${\mathbb A}^1$-contractible

#### Article information

Source
Duke Math. J., Volume 143, Number 3 (2008), 513-530.

Dates
First available in Project Euclid: 3 June 2008

https://projecteuclid.org/euclid.dmj/1212500465

Digital Object Identifier
doi:10.1215/00127094-2008-027

Mathematical Reviews number (MathSciNet)
MR2423761

Zentralblatt MATH identifier
1167.14025

#### Citation

Asok, Aravind; Doran, Brent. Vector bundles on contractible smooth schemes. Duke Math. J. 143 (2008), no. 3, 513--530. doi:10.1215/00127094-2008-027. https://projecteuclid.org/euclid.dmj/1212500465

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