Extension of proper holomorphic mappings past the boundary
The authors prove that a proper holomorphic map \(f: \Omega\to D\) between bounded pseudoconvex domains in \({\mathbb{C}}^ n\) with real analytic boundaries extends to be holomorphic in a neighbourhood of \({\bar \Omega}\) if either (1) \(\Omega\) and D are defined by real polynomials or (2) D satisfies ''condition Q'' (i.e. the Bergman projection associated to D maps \(C_ 0^{\infty}(D)\) into \(H(\bar D)\), for instance if D is strictly pseudoconvex, if D is a complete Reinhardt domain, or if the \({\bar \partial}\)-Neumann problem for D is globally analytic hypoelliptic). Main tools are extension of holomorphic correspondences and a transformation rule for the Bergman projection under correspondences. Consequence: If D is a simply connected strictly pseudoconvex domain in \({\mathbb{C}}^ n\) with real analytic boundary, an irreducible proper holomorphic self correspondence of D is an automorphism of D.
scientific article; zbMATH DE number 3933466