Project Details
Projekt
The generalized Serre question
Applicant
Dr. Tariq Syed
Subject Area
Mathematics
Term
from 2021 to 2024
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 461453992
Answering a question raised by Jean-Pierre Serre, Daniel Quillen and Andrei Suslin independently proved in 1976 that algebraic vector bundles on 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 algebraic vector bundles on topologically contractible smooth affine complex varieties are always trivial.In 1980, Gurjar proved that line bundles on topologically contractible smooth affine complex varieties are always trivial. This immediately implied that vector bundles on topologically contractible smooth affine complex varieties of dimension 1 are always trivial. In 1989, Gurjar and Shastri were able to prove that vector bundles on topologically contractible smooth affine complex varieties of dimension 2 are also trivial.General results on the classification of algebraic vector bundles on smooth affine varieties proven by Kumar-Murthy and Asok-Fasel imply that all vector bundles on a topologically contractible smooth affine complex variety of dimension 3 are trivial if and only if the Chow groups of the variety are trivial. Recent work of Tariq Syed shows that all vector bundles on 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. In higher dimensions, no cohomological criterion for the generalized Serre question is known.The goal of this project is to give an answer to the generalized Serre question in dimensions 3 and 4 and to find cohomological criteria for the generalized Serre question in higher dimensions as well.
DFG Programme
WBP Fellowship
International Connection
Norway, USA
