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Recent developments in the theory of separately holomorphic mappings

100%
Nguyên V.-A.
Colloquium Mathematicum
|
2009
|
tom 117
|
nr 2
175-206
EN
We describe a part of the recent developments in the theory of separately holomorphic mappings between complex analytic spaces. Our description focuses on works using the technique of holomorphic discs.
2
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A boundary cross theorem for separately holomorphic functions

63%
Pflug P., Nguyên V.-A.
Annales Polonici Mathematici
|
2004
|
tom 84
|
nr 3
237-271
EN
Let D ⊂ ℂⁿ and $G ⊂ ℂ^m$ be pseudoconvex domains, let A (resp. B) be an open subset of the boundary ∂D (resp. ∂G) and let X be the 2-fold cross ((D∪A)×B)∪(A×(B∪G)). Suppose in addition that the domain D (resp. G) is locally 𝓒² smooth on A (resp. B). We shall determine the "envelope of holomorphy" X̂ of X in the sense that any function continuous on X and separately holomorphic on (A×G)∪(D×B) extends to a function continuous on X̂ and holomorphic on the interior of X̂. A generalization of this result to N-fold crosses is also given.
3
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Boundary cross theorem in dimension 1

63%
Pflug P., Nguyên V.-A.
Annales Polonici Mathematici
|
2007
|
tom 90
|
nr 2
149-192
EN
Let X, Y be two complex manifolds of dimension 1 which are countable at infinity, let D ⊂ X, G ⊂ Y be two open sets, let A (resp. B) be a subset of ∂D (resp. ∂G), and let W be the 2-fold cross ((D∪A)×B) ∪ (A×(B∪G)). Suppose in addition that D (resp. G) is Jordan-curve-like on A (resp. B) and that A and B are of positive length. We determine the "envelope of holomorphy" Ŵ of W in the sense that any function locally bounded on W, measurable on A × B, and separately holomorphic on (A × G) ∪ (D × B) "extends" to a function holomorphic on the interior of Ŵ.
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