Abstract
We discuss algebraic vector bundles on smooth -schemes contractible from the standpoint of -homotopy theory; when , the smooth manifolds are contractible as topological spaces. The integral algebraic K-theory and integral motivic cohomology of such schemes are those of . 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 -contractible strictly quasi-affine scheme that admits a -torsor whose total space is affine, for a unipotent group, possesses a nontrivial vector bundle. Indeed, we produce explicit arbitrary-dimensional families of nonisomorphic -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 , 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 -contractible
Citation
Aravind Asok. Brent Doran. "Vector bundles on contractible smooth schemes." Duke Math. J. 143 (3) 513 - 530, 15 June 2008. https://doi.org/10.1215/00127094-2008-027
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