Vector bundles on contractible smooth schemes

Abstract

We discuss algebraic vector bundles on smooth k-schemes X contractible from the standpoint of A1-homotopy theory; when k=C, the smooth manifolds X(C) are contractible as topological spaces. The integral algebraic K-theory and integral motivic cohomology of such schemes are those of Speck. 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 A1-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 A1-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 A1-contractible