An extension theorem for separately holomorphic functions with analytic singularities
Tom 80 / 2003
Streszczenie
Let $D_j\subset{\mathbb C}^{k_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluriregular set, $j=1,\dots,N$. Put $$ X:=\bigcup_{j=1}^N A_1\times\dots\times A_{j-1}\times D_j\times A_{j+1}\times\dots\times A_N \subset{\mathbb C}^{k_1+\dots+k_N}. $$ Let $U$ be an open connected neighborhood of $X$ and let $M\varsubsetneq U$ be an analytic subset. Then there exists an analytic subset $\widetilde M$ of the “envelope of holomorphy” $\skew3\widetilde X$ of $X$ with $\widetilde M\cap X\subset M$ such that for every function $f$ separately holomorphic on $X\setminus M$ there exists an $\skew5\widetilde f$ holomorphic on $\skew3\widetilde X\setminus\widetilde M$ with $\skew5\widetilde f\,|_{X\setminus M}=f$. The result generalizes special cases which were studied in \cite{Ökt 1998}, \cite{Ökt 1999}, \cite{Sic 2001}, and \cite{Jar-Pfl 2001}.
