An extension theorem for separately holomorphic functions with singularities

Abstract

Let $D_j\subset {\mathbb{C}}^{k_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluripolar set, $j=1,...,N$. Put$$X:=\bigcup _{j=1}^N{A}_1\times ...\times A_{j-1}\times D_j\times A_{j+1}\times ...\times A_N\subset {\mathbb{C}}^{k_1+...+k_N}.$$Let $U$ be an open connected neighborhood of $X$ and let $MU$ be an analytic subset. Then there exists an analytic subset $\hat{M}$ of the `envelope of holomorphy' $\hat{X}$ of $X$ with $\hat{M}\cap X\subset M$ such that for every function $f$ separately holomorphic on $X\setminus M$ there exists an $\hat{f}$ holomorphic on $\hat{X}\setminus \hat{M}$ with $\hat{f}{|}_{X\setminus M}=f$. The result generalizes special cases which were studied in \cite{Ökt 1998}, \cite{Ökt 1999}, \cite{Sic 2000}, and \cite{Jar-Pfl 2001}.

20 pages; This a new version of the paper (including "An extension theorem for separately holomorphic functions with singularities, II")