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, $j=1,\dots ,N$. Put \[ X:=\bigcup _{j=1}^N A_1\dots A_{j-1}\times D_j\times A_{j+1}\dots A_N \subset \mathbb {C}^{n_1}\dots \mathbb {C}^{n_N}=\mathbb {C}^n.\] 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 $j\in \{1,\dots ,N\}$ let $\Sigma _j$ be the set of all $(z’,z'')\in (A_1\dots A_{j-1}) \times (A_{j+1}\dots A_N)$ for which the fiber $M_{(z’,\cdot ,z'')}:=\{z_j\in \mathbb {C}^{n_j}: (z’,z_j,z'')\in M\}$ is not pluripolar. Assume that $\Sigma _1,\dots ,\Sigma _N$ are pluripolar. Put \begin{multline*} X’:=\bigcup _{j=1}^N\{(z’,z_j,z'')\in (A_1\dots A_{j-1})\times D_j \times (A_{j+1}\dots A_N): (z’,z'')\notin \Sigma _j\}. \end{multline*} Then there exists a relatively closed pluripolar subset $\widehat {M}\subset \widehat X$ of the “envelope of holomorphy” $\widehat {X}\subset \mathbb {C}^n$ of $X$ such that: $\bullet$ $\widehat M\cap X’\subset M$, $\bullet$ for every function $f$ separately holomorphic on $X\setminus M$ there exists exactly one function $\widehat f$ holomorphic on $\widehat X\setminus \widehat M$ with $\widehat f=f$ on $X’\setminus M$, and $\bullet$ $\widehat M$ is singular with respect to the family of all functions $\widehat f$.