On Bordisms of Real Algebraic $M$-Varieties

Any morphism of nonsingular complete real algebraic varieties $F\colon Y\to X$ determines a holomorphic mapping of the sets of complex points $F_{\mathbb C}\colon Y(\mathbb C)\to X(\mathbb C)$ as well as a differentiable mapping of the sets of real points $F_{\mathbb R}\colon Y(\mathbb R)\to X(\mathbb R)$. These two mappings determine classes of nonoriented bordisms $[F_{\mathbb C}]\in\operatorname{MO}_{2m}(X(\mathbb C))$, $[F_{\mathbb R}]\in\operatorname{MO}_m(X(\mathbb R))$, where $m=\dim Y$. The paper describes relationship between these two classes of bordisms.