ANNALES POLONICI MATHEMATICI 80 (2003) An extension theorem for separately holomorphic functions with analytic singularities by Marek Jarnicki (Kraków) and Peter Pflug (Oldenburg) Dedicated to Professor Józef Siciak in honour of his 70th birthday Abstract. Let D j ⊂ C k j be a pseudoconvex domain and let A j ⊂ D j be a locally pluriregular set, j =1,...,N . Put X := N  j=1 A 1 × ... × A j-1 × D j × A j+1 × ... × A N ⊂ C k 1 +...+k N . Let U be an open connected neighborhood of X and let M  U be an analytic subset. Then there exists an analytic subset  M of the “envelope of holomorphy”  X of X with  M ∩ X ⊂ M such that for every function f separately holomorphic on X \ M there exists an  f holomorphic on  X \  M with  f | X\M = f . The result generalizes special cases which were studied in [ ¨ Okt 1998], [ ¨ Okt 1999], [Sic 2001], and [Jar-Pfl 2001]. 1. Introduction. Main theorem. Let N ∈ N, N ≥ 2, and let ∅ = A j ⊂ D j ⊂ C k j , where D j is a domain, j =1,...,N . We define an N -fold cross X := X(A 1 ,...,A N ; D 1 ,...,D N ) (1) := N  j=1 A 1 × ... × A j−1 × D j × A j+1 × ... × A N ⊂ C k 1 +...+k N . Observe that X is connected. Let Ω ⊂ C n be an open set and let A ⊂ Ω. Put h A,Ω := sup{u : u ∈ PSH(Ω),u ≤ 1 on Ω,u ≤ 0 on A}, 2000 Mathematics Subject Classification : 32D15, 32D10. Key words and phrases : separately holomorphic, pluriregular, holomorphic extension. Research of M. Jarnicki partially supported by the KBN grant No. 5 P03A 033 21. Research of P. Pflug partially supported by the Nieders¨ achsisches Ministerium f¨ ur Wissenschaft und Kultur, Az. 15.3 – 50 113(55) PL. [143]