The Jacobian of a nonorientable Klein surface

Using divisors, an analog of the Jacobian for a compact connected nonorientable Klein surface $Y$ is constructed. The Jacobian is identified with the dual of the space of all harmonic real one-forms on $Y$ quotiented by the torsion-free part of the first integral homology of $Y$. Denote by $X$ the double cover of $Y$ given by orientation. The Jacobian of $Y$ is identified with the space of all degree zero holomorphic line bundles $L$ over $X$ with the property that $L$ is isomorphic to $\sigma^*\bar{L}$, where $\sigma$ is the involution of $X$.