ON BOUNDARY REGULARITY OF HOLOMORPHIC CORRESPONDENCES

Abstract

Let D be an arbitrary domain in <TEX>$\mathbb{C}^n$</TEX>, n > 1, and <TEX>$M{\subset}{\partial}D$</TEX> be an open piece of the boundary. Suppose that M is connected and <TEX>${\partial}D$</TEX> is smooth real-analytic of finite type (in the sense of D'Angelo) in a neighborhood of <TEX>$\bar{M}$</TEX>. Let f : <TEX>$D{\rightarrow}\mathbb{C}^n$</TEX> be a holomorphic correspondence such that the cluster set <TEX>$cl_f$</TEX>(M) is contained in a smooth closed real-algebraic hypersurface M' in <TEX>$\mathbb{C}^n$</TEX> 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 <TEX>$\mathbb{C}^n$</TEX> with smooth real-analytic boundary onto a bounded domain D' in <TEX>$\mathbb{C}^n$</TEX> with smooth real-algebraic boundary extends as a holomorphic correspondence to a neighborhood of <TEX>$\bar{D}$</TEX>.