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

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 $X.$