We give a cohomological classification of vector bundles of rank on a smooth affine threefold over an algebraically closed field having characteristic unequal to . As a consequence we deduce that cancellation holds for rank vector bundles on such varieties. The proofs of these results involve three main ingredients. First, we give a description of the first nonstable -homotopy sheaf of the symplectic group. Second, these computations can be used in concert with F. Morel’s -homotopy classification of vector bundles on smooth affine schemes and obstruction theoretic techniques (stemming from a version of the Postnikov tower in -homotopy theory) to reduce the classification results to cohomology vanishing statements. Third, we prove the required vanishing statements.
Aravind Asok. Jean Fasel. "A cohomological classification of vector bundles on smooth affine threefolds." Duke Math. J. 163 (14) 2561 - 2601, 1 November 2014. https://doi.org/10.1215/00127094-2819299