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

Nguyên, Viêt-Anh

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 2, pp. 219-254.

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 $f:W:=(A\times Y)\cup (X\times B)\to Z$ extends to a holomorphic mapping $\widehat{f}$ on $\widehat{W}:=\left\{(z,w)\in X\times Y:\tilde{\omega }(z,A,X)+\tilde{\omega }(w,B,Y)<1\right\}$ such that $\widehat{f}=f$ on $W\cap \widehat{W},$ where $\tilde{\omega }(·,A,X)$ (resp. $\tilde{\omega }(·,B,Y))$ 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.

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