Stratified-algebraic vector bundles

TL;DR

This paper studies stratified-algebraic vector bundles on real algebraic varieties, revealing their unique properties that differ from purely algebraic or topological bundles, with implications for understanding vector bundle structures.

Contribution

It introduces the concept of stratified-algebraic vector bundles and explores their distinctive properties compared to algebraic and topological bundles.

Findings

Stratified-algebraic vector bundles have unique properties not shared by purely algebraic or topological bundles.

Every algebraic vector bundle is a stratified-algebraic vector bundle.

Stratified-algebraic vector bundles exhibit surprising and distinct characteristics.

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 on X is called a stratified-algebraic vector bundle if, roughly speaking, there exists a stratification of X such that the restriction of the bundle to each stratum is an algebraic vector bundle. 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.