The generalized Serre question

• Applicant: Dr. Tariq Syed
• Subject Area: Mathematics
• Term: from 2021 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 461453992

• Final Report Year: 2024

Final Report Abstract

Answering a question raised by Jean-Pierre Serre, Daniel Quillen and Andrei Suslin independently proved in 1976 that algebraic vector bundles over affine spaces are trivial. The affine spaces over the complex numbers are the primordial examples of topologically contractible smooth affine complex varieties. The generalized Serre question asks whether vector bundles over topologically contractible smooth affine complex varieties are always trivial. While it is known that the generalized Serre question has a positive answer in dimensions 1 and 2, the question remains open in higher dimensions. General results on the classification of vector bundles over smooth affine varieties imply that all vector bundles over a topologically contractible smooth affine complex variety of dimension 3 are trivial if and only if the Chow groups of the variety are trivial. As a consequence of his results on stably trivial vector bundles of rank 2 and on symplectic orbits of unimodular rows, Tariq Syed proved that all vector bundles over a topologically contractible smooth affine complex variety of dimension 4 are trivial if the Chow groups and a specific Hermitian K-theory group of this variety are trivial. The main achievement of this project is a systematic study of concrete examples of topologically contractible smooth affine complex varieties: Such examples can very often be interpreted as cyclic coverings and the main result of this project shows that the Chow groups of such examples become trivial after tensoring with the rational numbers. In the special case of so-called bicyclic coverings it is shown that under suitable assumptions the Chow groups are trivial even without tensoring with the rational numbers. The specific Hermitian K-theory group of interest in dimension 4 is also proven to be trivial for bicyclic coverings of dimension 4 under suitable hypotheses. All these results seem to hint at a positive answer to the generalized Serre question in dimensions 3 and 4. Furthermore, previous results on symplectic orbits of unimodular rows were further strengthened; this served as an inspiration to prove a symplectic version of Suslin’s famous n!-theorem on unimodular rows, which is another major achievement of this project. Finally, other achievements of this project are results on the link between unimodular rows and Spin-orbits of unit vectors as well as the generalization of Suslin matrices, which have played an important role in K-theory, homotopy theory and the study of stably trivial vector bundles in the last decades.

Publications

• A note on Suslin matrices and Clifford algebras
  Tariq Syed
  
• A symplectic version of Suslin’s n!-theorem
  Tariq Syed
  
• Symplectic orbits of unimodular rows. Journal of Algebra, 646, 478-493.
  Syed, Tariq
  
• Some remarks on spin-orbits of unit vectors. Journal of Pure and Applied Algebra, 229(1), 107802.
  Syed, Tariq
  
• Motivic cohomology of cyclic coverings. Advances in Mathematics, 487, 110767.
  Syed, Tariq