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 × Y ) ∪ (X × B) −→ Z extends to a holomorphic mapping f on Ŵ := {(z, w) ∈ X × Y : ω(z, A, X) + ω(w, B, Y ) < 1} such that f = f on W ∩ Ŵ , where ω(·, A, X) (resp. ω(·, 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. Mathematics Subject Classification (2000): 32D15 (primary); 32D10 (secondary).