ON BOUNDARY REGULARITY OF HOLOMORPHIC CORRESPONDENCES

Title & Authors
ON BOUNDARY REGULARITY OF HOLOMORPHIC CORRESPONDENCES
Ourimi, Nabil;

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