The Jacobian of a Nonorientable Klein Surface

[摘要] 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 $𝜎^*overline{L}$, where 𝜎 is the involution of 𝑋.

[学科分类] 数学（综合）

[关键词] Nonorientable surface;divisor;Jacobian