We can reduce the study of algebraic varieties and line bundles over the field of real numbers to the complex case by using the following standard technique. A variety $X$ over $\mathbb{R}$ induces a variety $X_{\mathbb{C}}$ over $\mathbb{C}$ together with an anti-linear involution $\sigma\colon X_{\mathbb{C}}\to X_{\mathbb{C}}$ such that the diagram

commutes, where the arrow below denotes the map induced by complex conjugation. A line bundle $L$ on $X$ determines a line bundle $L_{\mathbb{C}}$ on $X_{\mathbb{C}}$ and an isomorphism $\alpha\colon\sigma^{*}L_{\mathbb{C}}\to L_{\mathbb{C}}$ such that a section $s$ of $L_{\mathbb{C}}$ is real if and only if $\alpha(\sigma^{*}s)=s$. By a metric on $L^{{\text{\rm an}}}$ we will mean a metric $\|\cdot\|$ on $L_{\mathbb{C}}^{{\text{\rm an}}}$ such that the induced map $\sigma^{*}(L_{\mathbb{C}},\|\cdot\|)\to(L_{\mathbb{C}},\|\cdot\|)$ is an isometry.

In this way, the above definitions can be extended to metrized line bundles on varieties over $\mathbb{R}$. For instance, a real smooth metrized line bundle is semipositive if and only if its associated complex smooth metrized line bundle is semipositive. The corresponding signed measure is a measure over $X_{\mathbb{C}}^{{\text{\rm an}}}$ which is invariant under $\sigma$.

In the sequel, every time we have a real variety, we will work with the associated complex variety and quietly ignore the anti-linear involution $\sigma$, because it will play no role in our results.