Hi all,

Given a variety $X$ over the real numbers, we can consider the singular cohomology of the space $X(\mathbb{C})$, with coefficients in $\mathbb{Q}$, say. The action of complex conjugation on $X(\mathbb{C})$ induces an action on these cohomology groups. How do you compute/describe this action in concrete examples? For example, what happens in the case of $H^1$ of a curve of genus $g$, or even in the supposedly trivial case where $X = Spec(\mathbb{R})$? Other examples would also be interesting, or general hints on how to understand the action.