Une nouvelle version du théorème d'extension de Hartogs pour les applications séparément holomorphes entre espaces analytiques. (A new version of Hartogs' extension theorem for the separately holomorphic mappings between analytic spaces) (Q2724133)

Let \(X, Y, Z\) be complex analytic spaces, \(E\subset K\subset X\) and \(F\subset L\subset Y\) be non-pluripolar sets, \(W:=(K\times F)\cup (E\times L)\). A mapping \(f:W\to Z\) is said to be separately holomorphic if for any \(x\in E\), the mapping \(f(x,\cdot):L\to Z\) extends to a holomorphic mapping of a neighbourhood of \(L\), and similarly with respect to the second variable.NEWLINENEWLINENEWLINEThe main result of the paper is as follows. For any separately holomorphic mapping of a ``cross'' \(W\) with Borel \(K\) and \(L\) of type \({\mathcal F}_\sigma\), there exist Borel sets \(E'\subset E\) and \(F'\subset F\) such that \(E\setminus E'\) and \(F\setminus F'\) are pluripolar and the mapping \(f\) extends to a holomorphic mapping of a neighbourhood of \((K\times F')\cup (E'\times L)\). In particular, the set of regular points of \(W\) where \(f\) has no holomorphic extension is double pluripolar (i.e., its projections to \(X\) and \(Y\) are pluripolar).NEWLINENEWLINENEWLINEUnder certain additional assumptions on the ``crosses'', more precise results are deduced. This gives generalizations for a number of known results on the subject as well.