Quasi-algebraic geometry of curves I. Riemann-Roch theorem and Jacobian

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We discuss an analogue of Riemann-Roch theorem for curves with an infinite number of handles. We represent such a curve X by its Shottki model, which is an open subset U of CP¹ with infinite union of circles as a boundary. An appropriate bundle on X is ω¹/² ⊗ L, L being a bundle with (say) constants as gluing conditions on the circles. An admissible section of an appropriate bundle on X is a holomorphic half-form on U with given gluing conditions and H¹/²-smoothness condition. We study the restrictions on the mutual position of the circles and the gluing constants which guarantee the finite dimension of the space of appropriate sections of admissible bundles, and make the Riemann-Roch theorem hold. The resulting Jacobian variety is described as an infinite-dimension analogue of a torus.