An extension theorem for separately meromorphic functions with pluripolar singularities

2002/09/30 by Marek Jarnicki, Peter Pflug
Mathematics · #math.CV #msc:32D15 #msc:32D10

paper · pdf

published as Kyushu J. Math. 57 (2003), 291-302. · 11 pages

arxiv created 2002/10/16 · arxiv updated 2009/11/30

Abstract

Let D_j⊂\mathbb C^n_j be a pseudoconvex domain and let A_j⊂ D_j be a locally pluriregular set, j=1,...,N. Put X:=\bigcup_j=1^N A₁×...× A_j-1× D_j× A_j+1×...× A_N. Let M⊂ X be relatively closed. For any j∈\1,...,N\ let Σ_j be the set of all (z',z'')∈(A₁×...× A_j-1)×(A_j+1×...× A_N) such that the fiber M_(z',⋅,z''):=\z_j∈\mathbb C^n_j: (z',z_j,z'')∈ M\ is not pluripolar. Assume that Σ₁,...,Σ_N are pluripolar. Put multline* X':=\bigcup_j=1^N\(z',z_j,z'')∈(A₁×...× A_j-1)× D_j ×(A_j+1×...× A_N): (z',z'')∉Σ_j\. Then there exists a relatively closed pluripolar subset \widetilde M⊂\widetilde X of the `envelope of holomorphy' \widetilde X of X such that: \bullet \widetilde M∩ X'⊂ M, \bullet every function f separately meromorphic on X∖ M extends to a (uniquely determined) function \widetilde f meromorphic on \widetilde X∖\widetilde M, \bullet if f is separately holomorphic on X∖ M, then \widetilde f is holomorphic on \widetilde X∖\widetilde M, and \bullet \widetilde M is singular with respect to the family of all functions \widetilde f. \noindent In the case where N=2, M=\varnothing, the above result may be strengthened.

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