Algebraic deRham cohomology of log-Riemann surfaces of finite type

TL;DR

This paper develops an algebraic de Rham cohomology theory for log-Riemann surfaces of finite type, establishing a finite-dimensional cohomology group and a nondegenerate period pairing involving exponential singularities.

Contribution

It introduces a new cohomology framework for log-Riemann surfaces with exponential singularities and proves its nondegeneracy and finite-dimensionality, extending classical theories.

Findings

The dimension of the cohomology group is finite and explicitly computed.

A nondegenerate period pairing is established for differentials with exponential singularities.

The cohomology dimension depends on genus, ramification points, and exponential singularities.

Abstract

Log-Riemann surfaces of finite type are Riemann surfaces with finitely generated fundamental group equipped with a local diffeomorphism to C such that the surface has finitely many infinite order ramification points. We define and prove nondegeneracy of a period pairing for log-Riemann surfaces of finite type, given by pairing differentials with finitely many exponential singularities, of the form g exp(\int R_0) dz (where g, R_0 are meromorphic functions on a compact Riemann surface, with R_0 fixed) with closed curves and curves joining infinite order ramification points. As a consequence we show that the dimension of a cohomology group (given by differentials with exponential singularities of fixed type, modulo differentials of functions with exponential singularities of the same fixed type) is finite, equal to (2g + #R + (n-2)), where g is the genus of the compact Riemann surface, R is the set of infinite order ramification points, and n the number of exponential singularities.