A Riemann Roch Theorem for infinite genus Riemann surfaces

Abstract

In this article we prove a Riemann Roch Theorem for a class of holomorphic line bundles over Riemann surfaces of infinite genus. The theorem shows that the space of holomorphic sections satisfying a pointwise asymptotic growth condition has finite dimension and it provides a formula for this dimension. The gluing functions describing the surface and the transition functions defining the line bundle have to satisfy some asymptotic bounds. The theorem applies to holomorphic line bundles associated to divisors of infinite degree that assign one point to every handle on the surface. Applications of this Riemann Roch Theorem to the description of the Kadomcev Petviashvilli flow were provided in the author’s doctoral thesis.