Start from a connected closed Riemann surface $\Sigma_g,$ obtained as the (symmetric) covering of an open and/or unoriented surface $\Sigma,$ namely $\Sigma=\Sigma_g/\Omega,$ where $\Omega$ is an antiholomorphic involution.

We have on $\Sigma_g$ $2g$ one cycles $\delta_i$ and we can choose them such that $\delta_i$ intersects positively $\delta_{i+g}$ and no other (this gives the so-called A and B-cycles); we have a basis of $g$ holomorphic one-forms $\omega_k.$ Construct a $g \times 2g$ matrix of complex numbers

$$ \begin{pmatrix} \int_{\delta_1} \omega_1 & \cdots & \int_{\delta_{2g}} \omega_1 \\ \vdots & & \vdots\\ \int_{\delta_1} \omega_g & \cdots & \int_{\delta_{2g}} \omega_g \end{pmatrix}. $$

One can show (see e.g. Griffiths-Harris p. 228) that the $g$ period vectors $\Pi_i=\left( \int_{\delta_1} \omega_i , \ldots , \int_{\delta_{2g}} \omega_i \right)$ are linearly independent over $\mathbb{R},$ and so their linear combinations with integer coefficients span a lattice $\Lambda$ inside $\mathbb{C}^{g}.$ Define the Jacobian of $\Sigma_g$ as $Jac(\Sigma_g)=\mathbb{C}^{g}/\Lambda.$

My QUESTION is: how does $\Omega$ act on (or lifts to) (the cohomology of) $Jac(\Sigma_g)$ ? Namely, if we pick coordinates $z_i$ over the torus, does it amount, inside cohomology, to something like $\mathrm{d}z_i \to \mathrm{d}\overline{z}_i$ ? Is there some good reference about this story?

Bonus QUESTION: How does $\Omega$ lift to the moduli space $\mathcal{M}_{g,n}$ ?


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