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