Real Algebraic Realization of Characteristic Classes

Throughout this paper X denotes a real non-singular affine algebraic variety of dimension n. We will give a realization of the characteristic classes (the Stiefel-Whitney classes, the Pontrjagin classes and the Euler classes) of real affine algebraic vector bundles over X by algebraic subvarieties (Theorems 1, 2). For the complex field, Grothendieck [4] showed that the Chern classes of an algebraic vector bundle over a complex non-singular quasi-projective variety are realized by algebraic cycles. Morimoto [6] considered the complex analytic case. We prove Theorems 1, 2 by the method used there. If we work over a real analytic vector bundle, Thorn's transversality theorem shows easily a realization of the characteristic classes by analytic subsets (see Suzuki [10]). Theorem 1 was partially proved in [2], [8], and two different applications of them were given in [2], [9]. In Section 4, Theorems 3, 4 will show that the smoothing of algebraic subvarieties of X of codimension 1 for homological equivalence is always possible. The proof uses an idea in [8]. Given two cohomology classes of X which are realized by algebraic subvarieties, it seems likely that their cup product is realized by an algebraic subvariety. We prove this under some assumptions, applying Theorems 1, 2 (Theorem 5). We must remark that a realization of the cup product by an analytic subset is always possible according to the transversality theorem. Section 6 considers an affine algebraic structure of a topological vector bundle over X. If the rank is 1, and if the Stiefel-Whitney class is realized by an algebraic subvariety, then the bundle has an affine algebraic structure.