A general version of the Hartogs extension theorem for separately holomorphic mappings between complex analytic spaces

Abstract

Using recent development in Poletsky theory of discs, we prove the following result: Let $X,$ $Y$ be two complex manifolds, let $Z$ be a complex analytic space which possesses the Hartogs extension property, let $A$ (resp. $B$) be a non locally pluripolar subset of $X$ (resp. $Y$). We show that every separately holomorphic mapping $f: W:=(A\times Y) \cup (X\times B)\longrightarrow Z$ extends to a holomorphic mapping $\hat{f}$ on $\hat{W}:=\left\lbrace(z,w)\in X\times Y:\ \widetilde{\omega}(z,A,X)+\widetilde{\omega}(w,B,Y)<1 \right\rbrace$ such that $\hat{f}=f$ on $W\cap \hat{W},$ where $\widetilde{\omega}(\cdot,A,X)$ (resp. $\widetilde{\omega}(\cdot,B,Y))$ is the plurisubharmonic measure of $A$ (resp. $B$) relative to $X$ (resp. $Y$). Generalizations of this result for an $N$-fold cross are also given.