Complexification and cohomology in real algebraic geometry

Let X be a compact nonsingular real algebraic variety and let (V, j) be a nonsingular projective complexification of X. We discuss the subgroups j*( H*(V; Z )) of the cohomology ring H*(X; Z ) and conclude that the subgroup H*(X; Z )* = j*(H*( V; Z )) does not depend on the choice of the complexification. We also answer the question whether H*(-, Z )* is functorial and give a nontrivial example of the subgroup H*(X; Z )* when X is an abelian variety.;We define the subgroups H*C-alg X;Z ---of algebraic cohomology classes and H*( X; Z )inv---cohomology classes invariant with respect to a homomorphism induced by a translation in X (only when X is an abelian variety) and examine the relation between them. We define KC-alg (X) and KC (X)inv---the objects equivalent to H*C-alg (X; Z ) and H*(X; Z )inv in K-theory and find the condition sufficient for the equality KC-alg (X) = KC (X)inv.