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Analyticité et algébricité d'applications de Cauchy-Riemann

Abstract : This work concerns the analyticity and the algebraicity of smooth Cauchy-Riemann (CR) mappings between real analytic or real algebraic CR manifolds. There has been recently a renewed activity in this subject, which deals with the extension properties of mappings. Our contribution essentially concerns the study of the non-equidimensional situation and the investigation of the case of higher codimension. In the first part of this thesis, we consider the question of the algebraicity of a local holomorphic mapping f sending a minimal generic real algebraic submanifold M of Cn, n > 1, into a real algebraic subset M' of Cn'. This problem was initiated by the work of Poincaré (1907), and more recently of Webster (1977). The introduction of "characteristic varieties" associated to both the sets M and M' and the mapping f allows us to give two new conditions for the algebraicity of f. In the second part of this thesis, we study the problem of the analyticity of a smooth CR mapping f : M -> M' between a minimal generic real analytic submanifold M of Cn, n>1, and a real analytic subset M' of Cn'. We establish a generalization of the Lewy-Pinchuk reflection principle (1975-77) and we prove that if the characteristic variety if of dimension zero, then f is real analytic. In the third part of this thesis, we deal with the more general situation when the characteristic variety if of arbitrary dimension. We prove that if M' does not contain any complex curves, then f is analytic on a dense open subset of M. More generally, we establish an upper estimate of the partial analyticity of f, which depends on the maximal dimension of local holomorphic foliations contained in M'.
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https://tel.archives-ouvertes.fr/tel-00002244
Contributor : Sylvain Damour <>
Submitted on : Wednesday, January 8, 2003 - 1:48:28 PM
Last modification on : Wednesday, October 10, 2018 - 1:25:44 AM
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Sylvain Damour. Analyticité et algébricité d'applications de Cauchy-Riemann. Mathématiques [math]. Université de Provence - Aix-Marseille I, 2001. Français. ⟨tel-00002244⟩

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