  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.
