Relative Topology of Real Algebraic Varieties in Their Complexifications

We investigate, for a given smooth closed manifold M , the existence of an algebraic model X for M (i.e., a nonsingular real algebraic variety diffeomorphic to M) such that some nonsingular projective complexification 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 étale cohomology, we see that mod p Steenrod powers give an obstruction for the induced map on cohomology, i∗ : H(XC,Zp) → H(X,Zp), to be onto, if we require that X is defined over rational numbers.