Project description

This project is concerned with several basic questions about the classification of vector bundles over affine varieties and related topics; this includes, for example, the cancellation and splitting problem of algebraic vector bundles. For a smooth affine variety over an algebraically closed field of dimension at most 3 it is known that its vector bundles are completely classified by their Chern classes and the intersection theory of the underlying variety. However, Kumar has given an example of a stably free non-free module of rank 2 over a 4-dimensional smooth affine variety over an algebraically closed field, which shows that the classification results known in low dimensions cannot be extended to higher dimensions. Motivated by recent results of Fasel-Rao-Swan, we approach this classification problem using classical algebraic $K$-theory and Grothendieck-Witt groups.

Related publications

Tariq Syed, A generalized Vaserstein symbol Annals of K-Theory 4 (2019), no. 4, 671-706

Tariq Syed, The cancellation of projective modules of rank 2 with a trivial determinant, Algebra & Number Theory 15 (2021), no. 1, 109-140

Tariq Syed, Cancellation of vector bundles of rank 3 with trivial Chern classes on smooth affine fourfolds, Preprint (2020) to appear in JPAA arxiv 2010.07690

Tariq Syed, Symplectic orbits of unimodular rows, Preprint (2020) arxiv 2010.06669