An extension theorem for separately holomorphic functions with pluripolar singularities

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Let Dⱼ⊂ Cⁿʲ be a pseudoconvex domain and let Aⱼ⊂ Dⱼ be a locally pluriregular set, j=1,...,N. Put X:=ⱼ₌₁ⁿ A₁×...× Aⱼ₋₁× Dⱼ× Aⱼ₊₁×...× Aₙ⊂ Cⁿ¹×...× Cⁿⁿ= Cⁿ. Let U⊂ Cⁿ be an open neighborhood of X and let M⊂ U be a relatively closed subset of U. For j∈{1,...,N} let Σⱼ be the set of all (z',z'')∈(A₁×...× Aⱼ₋₁) ×(Aⱼ₊₁×...× Aₙ) for which the fiber M₍z',·,z''₎:={zⱼ∈ Cⁿʲ (z',zⱼ,z'')∈ M} is not pluripolar. Assume that Σ₁,...,Σₙ are pluripolar. Put X':=ⱼ₌₁ⁿ{(z',zⱼ,z'')∈(A₁×...× Aⱼ₋₁)× Dⱼ ×(Aⱼ₊₁×...× Aₙ) (z',z'')∉Σⱼ}. Then there exists a relatively closed pluripolar subset M⊂ X of the `envelope of holomorphy' X⊂ Cⁿ of X such that: M∩ X'⊂ M, for every function f separately holomorphic on X M there exists exactly one function f holomorphic on X M with f=f on X' M, and M is singular with respect to the family of all functions f. Some special cases were previously studied in Jar-Pfl 2001c.