On Some Holomorphic Cohomological Equations

Abstract

Let Σ be a non compact Riemann surface and \({\gamma :\Sigma \longrightarrow \Sigma}\) an automorphism acting freely and properly such that the quotient M = Σ/γ is a non compact Riemann surface. Using the fact that Σ and M are Stein manifolds, we prove that, for any holomorphic function \({g : \Sigma \longrightarrow {\mathbb C}}\) and any \({\lambda \in {\mathbb C}}\) , there exists a holomorphic function \({f:\Sigma \longrightarrow {\mathbb C}}\) which is a solution of the holomorphic cohomological equation \({f \circ \gamma - \lambda f = g}\) .