a real variety and $\pi: X_{\C} \to X$ is its complexification then $G$ acts on $X_{\C}(\C)$ and induces a homeomorphism $$X(\R) \xrightarrow{\iso} X_{\C}(\C)^G.$$

In particular if $X$ is a projective real variety then by [@F:ac Proposition 1.1] $\mcal{C}_q(X_{\C}) = \mcal{C}_{r}(X)_{\C}$ and so we have the isomorphism of topological monoids $$\mcal{C}_{r}(X)(\R) \xrightarrow{\iso} \mcal{C}_{r}(X_{\C})(\C)^{G} .$$

Let $U$ be a quasiprojective real variety. Then $$\mcal{Z}_{r}(U) \xrightarrow{\iso} \mcal{Z}_{r}(U_{\C})^{G}$$ is an isomorphism of topological abelian groups. In particular the group $\mcal{Z}_{r}(U)$ is independent of projectivization $U\subseteq \overline{U}$.

For $U$ projective this follows immediately from Proposition \[appcycagr\]. The quasi-projective case now follows by a comparison of short exact sequences of topological abelian groups $$\xymatrix{
0 \ar[r] & \mcal{Z}_{r}(\overline{U}\backslash U) \ar[r]\ar[d] & \mcal{Z}_{r}(\overline{U}) \ar[r]\ar[d] & \mcal{Z}_{r}(U) \ar[r]\ar[d] & 0 \\
0 \ar[r] & \mcal{Z}_{r}(\overline{U}_{\C}\backslash U_{\C})^{G} \ar[r] & \mcal{Z}_{r}(\overline{U}_{\C})^{G} \ar[r] & \mcal{Z}_{r}(U_{\C})^{G} \ar[r] & 0,
}$$ where the exactness of the bottom row is a consequence of the Lemma \[ccom\].

\[addqp\] Let $U$ be a quasi-projective real variety. Since $+:\mcal{Z}_{k}(U_{\C})\times\mcal{Z}_{k}(U_{\C}) \to \mcal{Z}_{k}(U_{\C})$ is closed we see by taking $G$-fixed points that $+:\mcal{Z}_{k}(U) \times \mcal{Z}_{k}(U) \to \mcal{Z}_{k}(U)$ is a closed map for any real variety $U$.

\[ccom\] Let $Y\subseteq X$ be a closed subvariety of a real projective variety. 