Algebraic Geometry Seminar
Morgan Opie
UCLA
Some classification and finiteness results for rank 2 vector bundles on smooth affine fourfolds
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.
Monday April 21, 2025 at 3:00 PM in 636 SEO