Open Access

# Adjunction formula, Poincaré residue and holomorphic differentials on Riemann surfaces

### Abstract

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 $H^1(\mathcal{D}, \mathbb{C})$ of holomorphic differentials of a compact Riemann surface $\mathcal{D}$ is equal to its genus, $\dim H^1(\mathcal{D}, \mathbb{C})=g(\mathcal{D})=g$. 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.

Key words: Adjunction formula, Poincaré residue, Riemann surfaces, holomorphic differentials.

### Article Information

 Title Adjunction formula, Poincaré residue and holomorphic differentials on Riemann surfaces Source Methods Funct. Anal. Topology, Vol. 24 (2018), no. 1, 41-52 Milestones Received 08/02/2017 Copyright The Author(s) 2018 (CC BY-SA)

### Authors Information

A. Lesfari
Department of Mathematics, Faculty of Sciences, University of Choua¨ıb Doukkali, B.P. 20, El Jadida, Morocco

### Citation Example

A. Lesfari, Adjunction formula, Poincaré residue and holomorphic differentials on Riemann surfaces, Methods Funct. Anal. Topology 24 (2018), no. 1, 41-52.

### BibTex

@article {MFAT1024,
AUTHOR = {A. Lesfari},
TITLE = {Adjunction formula, Poincaré residue and holomorphic differentials on Riemann surfaces},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {24},
YEAR = {2018},
NUMBER = {1},
PAGES = {41-52},
ISSN = {1029-3531},
URL = {http://mfat.imath.kiev.ua/article/?id=1024},
}

Coming Soon.