On algebraic vector bundles of rank $2$ over smooth affine fourfolds

Abstract: The classification of algebraic vector bundles of rank 2 over smooth affine fourfolds is a notoriously difficult problem. Isomorphism classes of such vector bundles are not uniquely determined by their Chern classes, in contrast to the situation in lower dimensions. Given a smooth affine fourfold over an algebraically closed field of characteristic not equal to $2$ or $3$, we study cohomological criteria for finiteness of the fibers of the Chern class map for rank $2$ bundles. As a consequence, we give a cohomological classification of such bundles in a number of cases. For example, if $d\leq 4$, there are precisely $d^2$ non-isomorphic algebraic vector bundles over the complement of a smooth hypersurface of degree $d$ in $\mathbb P^4_{\mathbb C}$.