A generalized Cousin problem for subvarieties of the bidisk

Abstract

We study a generalized Cousin problem that arises in connection with the study of varieties in the unit polydisk in C² on which every analytic function h admits a decomposition of the form h(z,w) = ϕ(z) + ψ(w), where ϕ,ψ are analytic on the disk. A one dimensional analytic variety has this property whenever the coordinate projections are one-to-one and onto a neighborhood of the boundary of the coordinate disks. As an interesting consequence of our method, it is shown that a holomorphic mapping of an annulus into the unit disk that is a diffeomorphism of the outer boundary of the annulus onto the boundary of the disk must be one-to-one on the outer half of the annulus.