Note on the behavior spaces on open Riemann surfaces and its applications to multiplicative differentials (Q1321790)

To obtain a generalization of the Riemann-Roch theorem on an arbitrary Riemann surfaces. Shiba and Başkan defined the \(S\)-behavior-spaces and the \(B\)-behavior spaces in their works. But the general existence of these spaces are not known yet. In this work the author, under some restrictive conditions, shows the existence of \(B\)-spaces and \(S\)-spaces and then applies these to the Abelian integral theory. In the last part of the work, the author contracts a sequence of the \(S\)-spaces and gives the conditions under which the limit of this sequence is a \(S\)-space. Moreover the author also gives a formulation of a duality theorem (Riemann-Roch type theorem) for multiplicative differentials which is, in case that the surface is symmetric closed, different from Prym-Weyl's theorem.