Homolohy Group and Generalized Riemann -Roch Theorem

Abstract

In this paper we show in particular th a t if X is a connected com plex analytic m anifold w ith fundam ental group F =|= 1, th en the com m utator subgroup [F, F ] determ ines com plete’y the vector space A(X) of holom orphic functions on X. As a consequence, sim ilarly and generally every norm al subgroup D of F such th a t F /D is com m utative determ ines com pletely an ideal of A(X). In this paper we recollect and expand on th e results obtained in [1, 2 ] and deduce thereof th e fundam ental Theorem . The paper is nevertheless self-contained.