Cohomology of multipoint connections on complex curves

TL;DR

This paper develops a cohomology theory for multipoint connections on complex curves using generalized holomorphic connections, with explicit examples involving higher genus elliptic functions and functional equations.

Contribution

It introduces a novel cohomology framework for multipoint connections on complex curves based on generalized holomorphic connections.

Findings

Cohomology expressed via higher genus elliptic functions

Explicit solutions obtained through analytic continuation

Functional equations related to complex curve connections

Abstract

Assuming complex functions defined on complex curves satisfy recursion relations with respect to number of parameters, we express the corresponding cohomology theory via generalizations of holomorphic connections. In examples provided, the cohomology is explicitly found in terms of higher genus counterparts of elliptic functions as analytic continuations of solutions for functional equations.