An extension theorem for separately holomorphic functions with pluripolar singularities

Abstract

Let $D_j\subset {\mathbb{C}}^{n_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluriregular set, $j=1,...,N$. Put $$X:=\bigcup _{j=1}^N{A}_1\times ...\times A_{j-1}\times D_j\times A_{j+1}\times ...\times A_N\subset {\mathbb{C}}^{n_1}\times ...\times {\mathbb{C}}^{n_N}={\mathbb{C}}^n.$$ Let $U\subset {\mathbb{C}}^n$ be an open neighborhood of $X$ and let $M\subset U$ be a relatively closed subset of $U$. For $j\in \{1,...,N\}$ let ${\Sigma }_j$ be the set of all $(z^{\prime },z^{\prime \prime })\in (A_1\times ...\times A_{j-1})\times (A_{j+1}\times ...\times A_N)$ for which the fiber $M_{(z^{\prime },\cdot ,z^{\prime \prime })}:=\{z_j\in {\mathbb{C}}^{n_j}(z^{\prime },z_j,z^{\prime \prime })\in M\}$ is not pluripolar. Assume that ${\Sigma }_1,...,{\Sigma }_N$ are pluripolar. Put $$X^{\prime }:=\bigcup _{j=1}^N\{(z^{\prime },z_j,z^{\prime \prime })\in (A_1\times ...\times A_{j-1})\times D_j\times (A_{j+1}\times ...\times A_N)(z^{\prime },z^{\prime \prime })\notin {\Sigma }_j\}.$$ Then there exists a relatively closed pluripolar subset $\hat{M}\subset \hat{X}$ of the `envelope of holomorphy' $\hat{X}\subset {\mathbb{C}}^n$ of $X$ such that: $\hat{M}\cap X^{\prime }\subset M$, for every function $f$ separately holomorphic on $X\setminus M$ there exists exactly one function $\hat{f}$ holomorphic on $\hat{X}\setminus \hat{M}$ with $\hat{f}=f$ on $X^{\prime }\setminus M$, and $\hat{M}$ is singular with respect to the family of all functions $\hat{f}$. Some special cases were previously studied in \cite{Jar-Pfl 2001c}.

19 pages