Relative topology of real algebraic varieties in their complexifications

Yildiray Ozan

Vol. 217 (2004), No. 2, 291–302

DOI: 10.2140/pjm.2004.217.291

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.

Milestones

Received: 9 April 2002

Published: 1 December 2004