An extension theorem for separately meromorphic 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 $$X:=\bigcup _{j=1}^N{A}_1\times ...\times A_{j-1}\times D_j\times A_{j+1}\times ...\times A_N.$$ Let $M\subset X$ be relatively closed. For any $j\in \{1,...,N\}$ let ${\Sigma }_j$ be the set of all $(z^{\prime },z^{\prime \prime })\in (A_1\times ...\times A_{j-1})\times (A_{j+1}\times ...\times A_N)$ such that 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 $multline\ast X^{\prime }:={\bigcup }_{j=1}^N\{(z^{\prime },z_j,z^{\prime \prime })\in (A_1\times ...\times A_{j-1})\times D_j\times (A_{j+1}\times ...\times A_N):(z^{\prime },z^{\prime \prime })\notin {\Sigma }_j\}$. Then there exists a relatively closed pluripolar subset $\tilde{M}\subset \tilde{X}$ of the `envelope of holomorphy' $\tilde{X}$ of $X$ such that: $\bullet$ $\tilde{M}\cap X^{\prime }\subset M$, $\bullet$ every function $f$ separately meromorphic on $X\setminus M$ extends to a (uniquely determined) function $\tilde{f}$ meromorphic on $\tilde{X}\setminus \tilde{M}$, $\bullet$ if $f$ is separately holomorphic on $X\setminus M$, then $\tilde{f}$ is holomorphic on $\tilde{X}\setminus \tilde{M}$, and $\bullet$ $\tilde{M}$ is singular with respect to the family of all functions $\tilde{f}$. \noindent In the case where N=2, $M=\varnothing$, the above result may be strengthened.

11 pages