A cohomological classification of vector bundles on smooth affine threefolds

2012/04/30 by Aravind Asok, Jean Fasel · 1 citation
Mathematics · #math.AG #math.AT #math.KT #msc:13C10 #msc:14F42 #msc:19A13 #msc:19D45 #msc:55S35

paper · pdf · doi:10.1215/00127094-2819299

published as Duke Math. J. 163, no. 14 (2014), 2561-2601 · 32 pages; Completely revised and reorganized. Final version (before page proofs) to appear in Duke Math. J

arxiv created 2014/02/17 · arxiv updated 2015/01/14

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. 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. First, we give a description of the first non-stable \mathbb A¹-homotopy sheaf of the symplectic group. Second, these computations can be used in concert with F. Morel's \mathbb A¹-homotopy classification of vector bundles on smooth affine schemes and obstruction theoretic techniques (stemming from a version of the Postnikov tower in \mathbb A¹-homotopy theory) to reduce the classification results to cohomology vanishing statements. Third, we prove the required vanishing statements.

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