Cohomological Obstructions for Mittag-Leffler Problems

TL;DR

This paper surveys cohomological methods used to generalize the Mittag-Leffler Theorem on Riemann surfaces, focusing on local-to-global data passage and solution characterizations.

Contribution

It introduces a cohomological framework for Mittag-Leffler problems, connecting complex analysis with sheaf theory and differential forms.

Findings

Cohomological conditions determine solvability of Mittag-Leffler problems.

Sheaf and differential form techniques unify local and global data analysis.

Characterizations of Riemann surface contexts where solutions exist.

Abstract

This is an extensive survey of the techniques used to formulate generalizations of the Mittag-Leffler Theorem from complex analysis. With the techniques of the theory of differential forms, sheaves and cohomology, we are able to define the notion of a Mittag-Leffler Problem on a Riemann surface as a problem of passage of data from local to global, and discuss characterizations of contexts where these problems have solutions.