Vector bundles over real algebraic varieties

Complex Cycles on Real Algebraic Models of a Smooth Manifold

“…Namely, let 3i(X, C) denote the ring of regular functions from X to C (cf. [7,Chapter 12] This theorem can be easily derived from [6] and is recorded here for the sake of completeness, since our more interesting and more precise result below concerns only 77¿alg. Let X be an affine real algebraic variety.…”

“…A topological F-vector bundle over X is said to admit an algebraic structure if it is topologically isomorphic to a strongly algebraic vector bundle over X (cf. [6]). Denote by Kf(X) the Grothendieck group of topological F-vector bundles over X (cf.…”

“…In [6,14] we have defined a contravariant functor Hg£P(-, Z) from affine nonsingular real algebraic varieties to graded rings. If X is an affine nonsingular real algebraic variety, then Hg*^(X, Z) = 0fc>o Hc¿J(X, Z) is a graded subring of the graded ring 77even(X, Z) = @k>0Hlk(X, Z) generated, roughly speaking, by the cohomology classes determined by the complex algebraic cycles on a nonsingular complexification of X (cf.…”

“…[7] and §1 for details). The functor 77¿y¿°(«, Z) has played a crucial role in the study of vector bundles over real algebraic varieties [6,9,14] and real algebraic morphisms [11,12]. Further applications will be discussed in our subsequent papers.…”

“…The study of algebro-geometric properties of the varieties in sf (M) is important for better understanding relations between differential topology and real algebraic geometry (cf. [2,1,6,12,9]). In this paper we investigate the behavior of the groups H^ (X, Z) for X in sf(M).…”