Algebraic vector bundles over real algebraic varieties

Abstract

By an affine algebraic variety, we mean in this note a locally ringed space (X, Rx) which is isomorphic to a ringed space of the form (V, Ry), where V is a Zariski closed subset in R n and Ry is the sheaf of rings of regular functions on V. Recall that £y 00 is the localization of the ring of polynomial functions on V with respect to the multiplicatively closed subset consisting of functions vanishing nowhere on V [2,15].Let F be one of the fields R, C or H (quaternions). A continuous F-vector bundle £ over X is said to admit an algebraic structure if there exists a finitely generated projective module P over the ring %x{X) (8>R F such that the Fvector bundle over X, associated with P in the standard way, is C° isomorphic toe Our purpose is to study the following PROBLEM. Characterize continuous F-vector bundles over X which admit an algebraic structure. This is an old problem, but despite considerable effort, the situation is well understood only in a few special cases: when X is the unit sphere S n [4,16], when X is the real projective space RF n [5,7] and when dimX < 3 and F = R [8, 9] (cf. also [13] for a short survey).Clearly, RF n with its natural structure of an abstract real algebraic variety is actually an affine variety and every affine real algebraic variety admits a locally closed embedding in some RP n .Let us first consider C-vector bundles. Let X be an affine nonsingular real algebraic variety and assume for a moment that X is embedded in RP n as a locally closed subvariety. Consider RP n as a subset of the complex projective space CP n . Let U be a Zariski neighborhood of X in the set of nonsingular points of the Zariski (complex) closure of X in CP n . Denote by H% [£ n (U,Z) the subgroup of the cohomology group H even (U, Z) generated by the cohomology classes which are Poincaré dual to the homology classes in the Borel-Moore homology group #even(CAZ) represented by the closed irreducible complex algebraic subvarieties of U (cf. [3]). Let #g^l g (X,Z) be the image of H%™{U,Z) via the restriction homomorphism # even (*7,Z) -• if even (X,Z). One easily checks that i^-aig (^ 21) does n°t depend on the choice of U or the choice of the embedding of X in RP n .

Complex cycles on real algebraic models of a smooth manifold

“…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.…”

Symplectic complex bundles over real algebraic four-folds

“…In [4,6] (cf. also Section 2) we have denned the graded subring k>0 of the cohomology ring H even (X, Z).…”

“…by a polynomial G, then V{H) will denote the subvariety of RP n denned by G. THEOREM [6]) that there exists a positive integer ko such that for every integer k greater than ko, one can find a subset 2* of P(n, k) which is a countable union of proper Zariski closed algebraic subvarieties of P(n, k) and has the property that for every …”

Complex vector bundles on real algebraic varieties of small dimension

“…For dimX ^ 3 and F = R a very satisfactory solution is given in [4] (see also [3,12,13] for earlier results). In [1] (see also [7]) most results are first obtained for C-vector bundles and then many of them are extended on to F-vector bundles, F = R or H, by using the realification and quaternionification. The main tool of [1], which will also be used here, is the functor HT^^ (•, Z) from affine real algebraic varieties to graded rings (we recall the definition of H^e^ (•> Z) in the next section).…”