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 ${f \circ \gamma - \lambda f = g}$ .