The Jacobian of a nonorientable Klein surface

Details The Jacobian of a nonorientable Klein surface The Jacobian of a nonorientable Klein surface

2003-10-18

arxiv.org/abs/math/0310288v1

Proc. Indian Acad. Sci. (Math. Sci.), Vol. 113, No. 2, May 2003, pp. 139-152

Mathematics

Algebraic Geometry

13 pages

Scientific paper

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 $X$.

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