Transcendental submanifolds of Rn

In this paper we show how the restriction of the complex algebraic cycles to real part of a complex algebraic set îs related to the real algebraic cycles of the real part As a corollary we give examples of smooth submanifolds of a Euchdean space which can not be isotoped to real parts of complex nonsingular subvaneties in the corresponding projective space Algebraic numbers are dense in R. The problem of whether closed smooth sub¬ manifolds M c: R" can be approximated by nonsingular algebraic subsets could be viewed as a possible higher dimensional version of this property. By adapting a stronger version of the notion of nonsingularity (complexifîcation is nonsingular) the results of this paper show that this is not the case. By identifying R" c RP" we prove: THEOREM. There are closed smooth submanifolds McR" which can not be isotoped to the real parts of nonsingular complex algebraic subvarieties of CP". Furthermore, we can choose M to be nonsingular real algebraic subsets of R". Now a brief history: Seifert showed that if M ci R" has trivial normal bundle then it can be isotoped to a nonsingular component of an algebraic subset of R" ([S]). Nash proved that in gênerai M can be isotoped to a nonsingular sheet of an algebraic subset of R"; but the sheets might intersect each other ([N]). In [AK4] it was shown that M can be isotoped to a nonsingular component of an algebraic subset of Rw, i.e. thèse sheets can be separated. Whether M can be isotoped to (not just to a component of) a nonsingular algebraic subset of Rw still remains open. We should emphasize that stably the answers of thèse problems are known. For example, Nash already proved that M can be isotoped to a nonsingular component of an algebraic set inside of a larger Euclidean space R" x Rfc; and later Tognoli showed that in a larger Euclidean space the extra components of the algebraic set can be removed ([T]). In [AK4] and [AK5] it was shown that any McR" can be isotoped to a nonsingular algebraic subset of R" x R; more generally M can be isotoped to a nonsingular algebraic subset of R" if and only if M is cobordant through immersions to an algebraic subset of R". Transcendental submanifolds of R" 309 We obtain the above results as a corollary to our main theorem which says that the restriction of complex algebraic cycles of the complexification of a nonsingular algebraic set consists of the cup products of the real algebraic cycles. Another corollary is that the Ponryagin classes of the tangent and the normal bundle of a nonsingular algebraic set in R" are represented by real algebraic subsets, a fact which was known only for the Steifel-Whitney classes. 1. Gysin homomorphism Let / : Mm -> Nn be a map. The Gysin homomorphism /+ : H*(M) -+H*+k(N) is defined by the commutative diagram: H*(M) -^ H* + k(N) where k n — m and the vertical maps are the Poincaré duality isomorphisms. Gysin homomorphism satisfies the following well known properties: LEMMA 1. The Gysin homomorphism satisfies the following properties: (a)/,(/•(«) w»)= « w/+(p). (b) Given a commuting diagram :