On Boundary Regularity of Holomorphic Correspondences

Abstract

Abstract. Let D be an arbitrary domain in C n , n > 1, and M ⊂ ∂D be an open piece of the boundary. Suppose that M is connected and ∂D is smooth real-analytic of finite type (in the sense of D'Angelo) in a neighborhood ofM . Let f : D → C n be a holomorphic correspondence such that the cluster set cl f (M ) is contained in a smooth closed real-algebraic hypersurface M ′ in C n of finite type. It is shown that if f extends continuously to some open peace of M , then f extends as a holomorphic correspondence across M . As an application, we prove that any proper holomorphic correspondence from a bounded domain D in C n with smooth real-analytic boundary onto a bounded domain D ′ in C n with smooth real-algebraic boundary extends as a holomorphic correspondence to a neighborhood ofD.