# Vector bundles and regulous maps

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Mathematische Zeitschrift

, Volume 275, Issue 1–2, pp 403–418

First Online: 17 January 2013Received: 24 July 2012Accepted: 25 November 2012

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\.

KeywordsReal algebraic variety Pre-algebraic vector bundle Algebraic vector bundle Multiblowup Regulous map Research partially supported by NCN grants 2011-01-B-ST1-01289, 2011-01-B-ST1-03875.

Autor: Marcin Bilski - Wojciech Kucharz - Anna Valette - Guillaume Valette