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Analyticité séparée et prolongement analytique. (Separate analyticity and analytic extension). (French) Zbl 0701.32003
This work deals with the old and always interesting question of separate analyticity.
Definition. A subset E of a polydisc $$\Delta^ n\subset {\mathbb{C}}^ n$$ is said to be separately analytic if for each hyperplane H parallel the axes $$E\cap H$$ is empty or it is analytic in $$\Delta^ n$$, with all irreducible components of dimension $$\geq 1.$$
Definition. A function f: $$U\to X$$ mapping a domain $$U\subset {\mathbb{C}}^ n$$ into a complex reduced analytic space X is said to be separately analytic if for each hyperplane H parallel to axes the restriction of f to $$U\cap H$$ is analytic.
The author generalizes a theorem of Rothstein by getting rid of additional assumptions and obtains the following:
Theorem 1.4. Each separately analytic subset of $$\Delta^ n$$ is analytic in $$\Delta^ n.$$
The question is, as always, more delicate for functions. Here the author gets:
Proposition 1.8. A separately analytic mapping from a domain $$U\subset {\mathbb{C}}^ n$$ to a complex analytic variety, f: $$U\to M$$, such that $$\dim U\geq \dim M+2,$$ is analytic.
The methods of proof, inspired partly by the work of H. Alexander, B. A. Taylor and J. L. Ullman [Invent. Math. 16, 335-341 (1972; Zbl 0238.32007)] and Rothstein [the author and W. Rothstein, Lect. Notes Math. 683, 179-202 (1978; Zbl 0409.32010)] are very much likeable (especially to the reviewer) for being direct, clear and geometrical despite the fact that the proofs are not easy.
Reviewer: Z.Denkowska

MSC:
 32A10 Holomorphic functions of several complex variables 32B15 Analytic subsets of affine space
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References:
 [1] Alexander, H., Taylor, B.A., Ullman, J.L: Areas of projections of analytic sets. Invent. Math.16, 335-341 (1972) · Zbl 0238.32007 [2] Dloussky, G.: Analyticité séparée et prolongements analytiques (d’après le dernier manuscrit de W. Rothstein) Variétés analytiques compactes. Colloque Nice 1977 Y. Hervier, A. Hirschowitz (eds.) (Lecture Notes in Mathematics, Vol. 683). Berlin Heidelberg New York: Springer 1978 [3] Dloussky, G.: Structure de surfaces de Kato. Mém. Soc. Math. Fr.14 (1984) · Zbl 0543.32012 [4] Hirschowitz, A.: Les deux types de méromorphie diffèrent J. Reine Angew. Math.313, 157-160 (1980) · Zbl 0412.32006 [5] Remmert, R., Stein, K.: Über die wesentlichen Singularitäten analytischer Mengen. Math. Ann.126, 263-306 (1953) · Zbl 0051.06303 [6] Rothstein, W.: Zur Theorie der analytischen Mengen. Math. Ann.174, 8-32 (1967) · Zbl 0172.37805 [7] Rothstein, W.: Das Maximumprinzip und die Singularitäten analytischer Mengen. Invent. Math.6, 163-184 (1968) · Zbl 0164.38201 [8] Rothstein, W.: Dernier manuscrit (non publié) [9] Stein, K.: Meromorphic mappings. L’enseignement mathématique14, 29-46 (1968) · Zbl 0165.40502 [10] Stoll, W.: Über meromorphe Abbildungen komplexer Räume. I, II. Math. Ann.136, 201-239, 393-429 (1958) · Zbl 0096.06202
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