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We will survey recent developments on the construction and classification of algebraic vector bundles on smooth schemes over a field k. in the affine case, w. Algebraic vs. topological vector bundles on spheres aravind asok∗ jean fasel† abstract we study the problem of when a topological vector bundle on a smooth complex afﬁne variety admits an algebraic structure. we prove that all rank 2 topological complex vector bundles on smooth afﬁne quadrics of dimension 11 over the complex numbers. Authors: aravind asok, jean fasel (submitted on 3 apr 2012 ( v1 ), last revised 17 feb 2014 (this version, v5)) abstract: we give a cohomological classification of vector bundles of rank $2$ on a smooth affine threefold over an algebraically closed field having characteristic unequal to $2$. Obstruction theory in topology, to understand algebraic vector bundles on smooth afﬁne schemes. that this can be done, relies on f. morel’s proof of an algebro geometric analog of the classical steenrod representability theorem [ste99] for topological vector bundles on spaces having the ho motopy type of a cw complex. Algebraic vs. topological vector bundles joint with jean fasel and mike hopkins aravind asok (usc) october 27, 2020 aravind asok (usc) algebraic vs. topological vector bundles1 27.

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We give a cohomological classification of vector bundles of rank 2 on a smooth affine threefold over an algebraically closed field having characteristic unequal to 2. as a consequence we deduce that cancellation holds for rank 2 vector bundles on such varieties. the proofs of these results involve three main ingredients. (with a. asok) splitting vector bundles outside the stable range and homotopy theory of punctured affine spaces, j. amer. math. soc. 28 (2014), no.4, 1031 1062. (with b. calmès) groupes classiques, 147 pages, panoramas et synthèses 46 (2015). (with a. asok) a cohomological classification of vector bundles over smooth affine threefolds, duke math. Vector bundles on contractible smooth schemes aravind asok department of mathematics university of washington seattle, wa 98195 [email protected] brent doran school of mathematics institute for advanced study princeton, nj 08540 [email protected] abstract we discuss algebraic vector bundles on smooth k schemes xcontractible from the stand.

Jean Fasel: Algebraic Vector Bundles On Smooth Schemes