A cohomological classification of vector bundles on smooth affine threefolds

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 ${\mathbb{A}}^1$-homotopy sheaf of the symplectic group. Second, these computations can be used in concert with F. Morel's ${\mathbb{A}}^1$-homotopy classification of vector bundles on smooth affine schemes and obstruction theoretic techniques (stemming from a version of the Postnikov tower in ${\mathbb{A}}^1$-homotopy theory) to reduce the classification results to cohomology vanishing statements. Third, we prove the required vanishing statements.

32 pages; Completely revised and reorganized. Final version (before page proofs) to appear in Duke Math. J

References in corpus (3)

• Hermitian K-theory, derived equivalences and Karoubi's Fundamental Theorem
• Algebraic vector bundles on spheres
• Splitting vector bundles outside the stable range and A^1-homotopy sheaves of punctured affine spaces

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• Finite Chow-Witt correspondences
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• Chow-Witt rings of classifying spaces for symplectic and special linear groups
• Euler class groups and motivic stable cohomotopy
• Generically split octonion algebras and A^1-homotopy theory
• Geometric models for higher Grothendieck-Witt groups in A1-homotopy theory
• The cancellation of projective modules of rank 2 with a trivial determinant
• The Simplicial EHP Sequence in A1-Algebraic Topology
• Enumerating Non-Stable Vector Bundles
• Algebraic vs. topological vector bundles on spheres
• Algebraic vector bundles and $p$-local A^1-homotopy theory
• Splitting criteria for projective modules over polynomial algebras
• Stable motivic pi_1 of low-dimensional fields
• Milnor-Witt motivic cohomology and linear algebraic groups
• A Transfer morphism for Hermitian K-theory of schemes with involution
• On third homology of SL_2 and weak homotopy invariance
• On Tate Milnor-Witt Motives