Stratified-algebraic vector bundles

Abstract

Abstract We investigate stratified-algebraic vector bundles on a real algebraic variety X. A stratification of X is a finite collection of pairwise disjoint, Zariski locally closed subvarieties whose union is X. A topological vector bundle ξ on X is called a stratified-algebraic vector bundle if, roughly speaking, there exists a stratification {\mathcal{S}} of X such that the restriction of ξ to each stratum S in {\mathcal{S}} is an algebraic vector bundle on S. In particular, every algebraic vector bundle on X is stratified-algebraic. It turns out that stratified-algebraic vector bundles have many surprising properties, which distinguish them from algebraic and topological vector bundles.