Vector bundles and regulous maps

Marcin Bilski,Anna Valette,Wojciech Kucharz,Guillaume Valette

doi:10.1007/s00209-012-1141-6

Abstract

Let $$X$$ be a compact nonsingular affine real algebraic variety. We prove that every pre-algebraic vector bundle on $$X$$ becomes algebraic after finitely many blowing ups. Using this theorem, we then prove that the Stiefel-Whitney classes of any pre-algebraic $$\mathbb{R }$$ -vector bundle on $$X$$ are algebraic. We also derive that the Chern classes of any pre-algebraic $$\mathbb{C }$$ -vector bundles and the Pontryagin classes of any pre-algebraic $$\mathbb{R }$$ -vector bundle are blow- $$\mathbb{C }$$ -algebraic. We also provide several results on line bundles on $$X$$ .

Highlights

The language of real algebraic geometry, as in [5], is used throughout this paper
A real algebraic variety is a locally ringed space that can be covered by finitely many open sets, each of which, together with the restriction of the structure sheaf, is an affine real algebraic variety, cf. [5, Definition 3.2.11]
All topological notions relatin! g to real algebraic varieties refer to the Euclidean topology

Summary

Introduction

The language of real algebraic geometry, as in [5], is used throughout this paper. an affine real algebraic variety is a locally ringed space isomorphic to an algebraic subset of Rn, for some n, endowed with the Zariski topology and the sheaf of real-valued regular functions. Theorem 1.1 Let ξ be a pre-algebraic F-vector bundle on an affine real algebraic variety X. 2. Corollary 1.2 With notation as in Theorem 1.1, if the variety X is nonsingular, there exists a Zariski closed subset Z of X such that codimX Z ≥ 2 and the restriction ξ|X\Z is an algebraic F-vector bundle on X \Z. Corollary 1.2 immediately implies the following: Corollary 1.3 If X is a nonsingular affine real algebraic variety of dimension 1, every pre-algebraic F-vector bundle on X is algebraic. Proposition 1.4 For any integer n > d(F), there exist a nonsingular affine real algebraic variety X and a topological F-line bundle ξ on X such that X is diffeomorphic to Tn and ξ is not isomorphic to any pre-algebraic F-line bundle on X. Regulous maps are essential in the present paper (see Sect. 2 for the definition)

Sections of pre-algebraic vector bundles

Stiefel-Whitney classes of pre-algebraic vector bundles

A construction of pre-algebraic vector bundles

Chern classes and Pontryagin classes of pre-algebraic vector bundles