Morgan Opie : Some classification and finiteness results for rank 2 vector bundles on smooth affine fourfolds

Posted by Philip Engel , part of the Algebraic Geometry Seminar.

At

April 21, 2025, 3 p.m.

In

636 SEO

Abstract

Given a variety X over a field, it is generally difficult to understand the structure of vector bundles on X. As a first approximation, we might try to understand vector bundles only up to isomorphism. Classical isomorphism invariants of algebraic vector bundles include Chern classes and Euler classes, so we can study the extent to which these invariants determine a vector bundle. The analogous question in topology is a finite one: given a finite-dimensional manifold M, there are only finitely many isomorphism classes of complex rank r topological vector bundles on M with given topological Chern classes. However, such a finiteness result is not, in general, known in algebraic geometry. In this talk, I will discuss conditions under which algebraic characteristic classes determine algebraic rank 2 vector bundles on a given smooth affine fourfold up to finite choices. As a consequence, I will deduce complete isomorphism classification results for certain examples. This is joint work with Thomas Brazelton and Tariq Syed.