ADJUNCTION FORMULA, POINCARE RESIDUE AND HOLOMORPHIC DIFFERENTIALS ON RIEMANN SURFACES

There is still a big gap between knowing that a Riemann surface of genus g has g holomorphic differential forms and being able to find them explicitly. The aim of this paper is to show how to construct holomorphic differential forms on compact Riemann surfaces. As known, the dimension of the space H-1(D, C) of holomorphic differentials of a compact Riemann surface D is equal to its genus, dim H-1(D, C) = g(D) = g. When the Riemann surface is concretely described, we show that one can usually present a basis of holomorphic differentials explicitly. We apply the method to the case of relatively complicated Riemann surfaces.