Cross theorem

Abstract

Let $D,G\subset \mathbb{C}$ be domains, let $A\subset D$, $B\subset G$ be locally regular sets, and let $X:=(D\times B)\cup (A\times G)$. Assume that $A$ is a Borel set. Let $M$ be a proper analytic subset of an open neighborhood of $X$. Then there exists a pure 1-dimensional analytic subset $\hat{M}$ of the envelope of holomorphy $\hat{X}$ of $X$ such that any function separately holomorphic on $X\setminus M$ extends to a holomorphic function on $\hat{X}\setminus \hat{M}$. The result generalizes special cases which were studied in \cite{Ökt 1998}, \cite{Ökt 1999a}, and \cite{Sic 2000}.

8 pages