On the Algebro-Geometric Analysis of Meromorphic (1,0)-forms

TL;DR

This paper explores the algebro-geometric properties of meromorphic (1,0)-forms on the Riemann sphere, establishing foundational results about meromorphic functions, divisors, and the algebraic structure of the surface.

Contribution

It provides a detailed analysis of meromorphic forms on the Riemann sphere, linking complex analysis with algebraic geometry and offering a group-theoretic approach to classify simply connected Riemann surfaces.

Findings

Every non-constant meromorphic function on the sphere has equal zeros and poles.

Principal divisors on the sphere have degree zero, consistent with the Riemann-Roch theorem.

The surface can be characterized as a non-singular projective algebraic variety.

Abstract

In this paper, we analyze the theory of meromorphic $(1,0)$-forms $\omega\in\mathcal{M}\Omega^{(1,0)}(\mathbb{CP}^1).$ Hence, we show that on a compact Riemann surface of genus $g=0,$ isomorphic to $\mathbb{CP}^1,$ every non-constant meromorphic function $f:X\to\mathbb{CP}^1$ has as many zeros as poles, where each is counted according to multiplicities. Such an analysis gives rise to the following result. Invoking the Riemann-Roch theorem for a compact Riemann $X$ with canonical divisor $K,$ it follows that $deg(f)=0$ for any principal divisor $(f):=D$ on $X.$ More precisely, $\ell(D)-\ell(K-D)=deg(D)+1=1$ or $\ell(D)-\ell(K-D)-1=0.$ Furthermore, for a diffeomorphism $\eta:X\to\mathbb{CP}^1$ of a certain kind, a multistep program is implemented to show $X$ is a compact algebraic variety of dimension one, i.e. a non-singular projective variety. Hence, we adopt a group-theoretic approach and provide a useful heuristic, that is, a set of technical conditions to facilitate the algebro-geometric analysis of simply connected Riemann surfaces $X.$