A cohomological classification of vector bundles on smooth affine threefolds
By A. Asok, Jean Fasel. 2012
74 citations recorded by the source corpus.
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.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/835b6b2240624184d45e0b6200d8b43cd18b44ad
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Citations in the corpus
- Geometric models for higher Grothendieck-Witt groups in A1-homotopy theory (2013, unknown): “[AF12] and [Zib11] and opens the door to a classification of unstable operations in Grothendieck-Witt theory as done in [Rio10] for K-theory.”
- Affine representability results in 1–homotopy theory, II : Principal bundles and homogeneous spaces (2015, unknown): “For example, if the base k is a perfect field, the A1 –fibration sequence An \ {0} −→ BGLn−1 −→ BGLn was used by F Morel [43, Chapter 8] to develop an obstruction-theoretic approach to answering the question of when a vector bundle over a smooth affine variety splits off a trivial rank 1 summand; th”
- Hermitian K-theory, derived equivalences and Karoubi's fundamental theorem (2012, background): “These groups naturally occur in A(1)-homotopy theory [Mor12], [AF12] and are to oriented Chow groups what algebraic K-theory is to ordinary Chow groups.”