Multiple products of meromorphic functions

TL;DR

This paper constructs parametric extensions of coboundary operators for meromorphic functions on higher genus Riemann surfaces, using Schottky uniformization to explore algebraic and geometric structures in infinite-dimensional Lie algebra modules.

Contribution

It introduces a novel geometric approach to extend coboundary operators for meromorphic functions on complex surfaces, linking Lie algebra modules with Riemann surface uniformization.

Findings

New family of coboundary operator extensions constructed

Application of Schottky uniformization to higher genus surfaces

Enhanced understanding of meromorphic function complexes

Abstract

Let $\mathfrak g$ be an infinite-dimensional Lie algebra, and $G$ be the algebraic completion of a $\mathfrak g$-module. Using the geometric model of Schottky uniformization of Riemann sphere to obtain a higher genus Riemann surface, we construct a family of parametric extensions of coboundary operators for the double complexes of meromorphic functions depending on elements of $G$ that possess prescribed analytic properties.