Duke Mathematical Journal
- Duke Math. J.
- Volume 143, Number 3 (2008), 513-530.
Vector bundles on contractible smooth schemes
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
Duke Math. J., Volume 143, Number 3 (2008), 513-530.
First available in Project Euclid: 3 June 2008
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx]
Secondary: 19E08: $K$-theory of schemes [See also 14C35] 14L24: Geometric invariant theory [See also 13A50]
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