Vector bundles and regulous maps
- PDF / 271,123 Bytes
- 16 Pages / 439.37 x 666.142 pts Page_size
- 63 Downloads / 212 Views
Mathematische Zeitschrift
Vector bundles and regulous maps Marcin Bilski · Wojciech Kucharz · Anna Valette · Guillaume Valette
Received: 24 July 2012 / Accepted: 25 November 2012 © The Author(s) 2013. This article is published with open access at Springerlink.com
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 R-vector bundle on X are algebraic. We also derive that the Chern classes of any pre-algebraic C-vector bundles and the Pontryagin classes of any pre-algebraic R-vector bundle are blowC-algebraic. We also provide several results on line bundles on X . Keywords Real algebraic variety · Pre-algebraic vector bundle · Algebraic vector bundle · Multiblowup · Regulous map. Mathematics Subject Classification (2000)
14P05 · 14P25 · 14P99.
Research partially supported by NCN grants 2011/01/B/ST1/01289, 2011/01/B/ST1/03875. M. Bilski · W. Kucharz (B) · A. Valette Wydział Matematyki i Informatyki Uniwersytetu Jagiello´nskiego, ul. S. Łojasiewicza 6, 30-348 Kraków, Poland e-mail: [email protected] M. Bilski e-mail: [email protected] A. Valette e-mail: [email protected] G. Valette ´ Tomasza 30, Instytut Matematyczny PAN, ul. Sw. 31-027 Kraków, Poland e-mail: [email protected]
123
M. Bilski et al.
1 Introduction The language of real algebraic geometry, as in [5], is used throughout this paper. Thus, 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. 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]. The underlying topology of any real algebraic variety is called the Zariski topology. Each quasi-projective real algebraic variety is affine, cf. [5, Proposition 3.2.10, Theorem 3.4.4]. Morphisms of real algebraic varieties are called regular maps. Each real algebraic variety carries also the Euclidean topology, which is induced by the usual metric on R. Unless explicitly stated otherwise, all topological notions relatin! g to real algebraic varieties refer to the Euclidean topology. Let F stand for R, C or H (the quaternions). Only left F-vector spaces will be considered. When convenient, F will be identified with Rd(F) , where d(F) = dimR F. Let X be an affine real algebraic variety. In the present paper investigated are relationships between pre-algebraic and algebraic F-vector bundles on X , cf. [5, Chapter 12] for the definitions of such bundles. It should be mentioned that pre-algebraic vector bundles and algebraic vector bundles had been called algebraic vector bundles and strongly algebraic vector bundles, respectively, in the literature predating the
Data Loading...
