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Let F be the Cartesian product of N closed sets in ℂ. We prove that there exists a function g which is continuous on F and holomorphic on the interior of F such that is complete pluripolar in . Using this result, we show that if D is an analytic polyhedron then there exists a bounded holomorphic function g such that is complete pluripolar in . These results are high-dimensional analogs of the previous ones due to Edlund [Complete pluripolar curves and graphs, Ann. Polon. Math. 84 (2004), 75-86]...
We prove that a Cousin-I open set D of an irreducible projective surface X is locally Stein at every boundary point which lies in . In particular, Cousin-I proper open sets of ℙ² are Stein. We also study K-envelopes of holomorphy of K-complete spaces.
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