Vector bundles on contractible smooth schemes

TL;DR

This paper investigates algebraic vector bundles on contractible smooth schemes, revealing that non-affine cases can host non-trivial bundles despite topological contractibility, challenging classical classification assumptions.

Contribution

It demonstrates the existence of non-trivial algebraic vector bundles on certain non-affine, A^1-contractible schemes, providing explicit examples and moduli of such bundles.

Findings

Non-affine A^1-contractible schemes can have non-trivial vector bundles.

Explicit families of non-isomorphic vector bundles of arbitrary large rank are constructed.

Algebraic K-theory cannot distinguish these non-trivial vector bundles.

Abstract

We discuss algebraic vector bundles on smooth k-schemes X contractible from the standpoint of A^1-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 that of Spec k. One might 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 non-affine case this is false: we show that (essentially) every smooth A^1-contractible strictly quasi-affine scheme that admits a U-torsor whose total space is affine, for U a unipotent group, possesses a non-trivial vector bundle. Indeed we produce explicit arbitrary dimensional families of non-isomorphic such schemes, with each scheme in the family equipped with "as many" (i.e., arbitrary dimensional moduli of) non-isomorphic 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 A^1-contractible.