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

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

1997-10-10

arxiv.org/abs/alg-geom/9710013v1

Mathematics

Algebraic Geometry

99 pages, AmsLaTeX

Scientific paper

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 CP^{1} with infinite union of circles as a boundary. An appropriate bundle on X is \omega^{1/2} \otimes 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 H^{1/2}-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.

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