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
$$ \varrho_k\colon H_{\operatorname{alg}}^k(X(\mathbb R),\mathbb Z/2)\to H_{\operatorname{alg}}^{2k}(X(\mathbb C),\mathbb Z/2). $$
is considered. This homomorphism agrees with the cycle mappings. Estimates for the dimension of the kernel of this homomorphism are given.