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Vector bundles over real algebraic surfaces and threefolds

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Vector bundles over real algebraic surfaces and threefolds COMPOSITIO MATHEMATICA WOJCIECHKUCHARZ Vector bundles over real algebraic surfaces and threefolds Compositio Mathematica, tome 60, no 2 (1986), p. 209-225. <http://www.numdam.org/item?id=CM_1986__60_2_209_0> © Foundation Compositio Mathematica, 1986, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 209 VECTOR BUNDLES OVER REAL ALGEBRAIC SURFACES AND THREEFOLDS Wojciech Kucharz Compositio Mathematica 60: 209-225 (1986) © Martinus Nijhoff Publishers, Dordrecht - Printed in the Netherlands Introduction Algebraic vector subbundles of product vector bundles, called strongly algebraic vector bundles, over affine real algebraic varieties have re- markable properties reflecting an interplay between algebra, geometry, and topology. The theory of strongly algebraic vector bundles has attracted the attention of several mathematicians. However, only in some special cases is it understood when a given continuous vector bundle over a real algebraic variety is C0 isomorphic to a strongly algebraic one (cf. [3], [4], [5], [10], [11], [12], [19], [23], [26]). In this paper necessary and sufficient conditions are given for a continuous vector bundle over a compact affine nonsingular real alge- braic surface or threefold to be C° isomorphic to a strongly algebraic vector bundle. The result is used to compare algebraic and topological K-theory. The paper is organized as follows. The main results are formulated in Section 1 and proved in Section 4. The

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