On the Algebraic Cohomology of Real Algebraic M-Varieties

For a real algebraic M-variety X, the canonical homomorphism of the algebraic cohomology group of the set of real points into the algebraic cohomology group of the set of complex points $$varrho _k :H_{alg}^k (X(mathbb{R}),mathbb{Z}/2) to H_{alg}^{2k} (X(mathbb{C}),mathbb{Z}/2)$$ is considered. This homomorphism agrees with the cycle mappings. Estimates for the dimension of the kernel of this homomorphism are given.