Relative topology of real algebraic varieties in their complexifications

Abstract

We investigate, for a given smooth closed manifoldM, the existence of an algebraic model X forM (i.e., a nonsingular real algebraic variety di.eomorphic to M) such that some non-singular projective complexi.cation i : X ?? XC of X admits a retraction r : XC ?? X. If such an X exists, we show that M must be formal in the sense of Sullivan's minimal models, and that all rational Massey products on M are trivial. We also study the homomorphism on cohomology induced by I for algebraic models X of M. Using ?etale cohomology, we see that mod p Steenrod powers give an obstruction for the induced map on cohomology, i. : Hk(XC,Zp) ?? Hk(X, Zp), to be onto, if we require that X is de.ned over rational numbers.