2016/02/26 by Kingshook Biswas, Biswas, Kingshook
Mathematics · Physics and Astronomy · #30F30 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1602.08219
openalex publication_date 2016/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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(∫ R0) dz (where g, R0 are meromorphic functions on a compact Riemann surface, with R0 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.