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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
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[8] Rothstein, W.: Dernier manuscrit (non publié)
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