Abstract

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 → X_C of X admits a retraction r : X_C → 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^k(X_C, Z_p) → H^k(X, Z_p), to be onto, if we require that X is defined over rational numbers.

Authors