TAILIEUCHUNG - Đề tài " A preparation theorem for codimension-one foliations "

Dedicated to C´sar Camacho for his 60th birthday e Abstract After gluing foliated complex manifolds, we derive a preparation-like theorem for singularities of codimension-one foliations and planar vector fields (in the real or complex setting). Without computation, we retrieve and improve results of Levinson-Moser for functions, Dufour-Zhitomirskii for nondegenerate ˙ codimension-one foliations (proving in turn the analyticity), Str´zyna-Zoladek o˙ ´ for non degenerate planar vector fields and Bruno-Ecalle for saddle-node foliations in the plane. . | Annals of Mathematics A preparation theorem for codimension-one foliations By Frank Loray Annals of Mathematics 163 2006 709-722 A preparation theorem for codimension-one foliations By Frank LoRAy Dedicated to César Camacho for his 60th birthday Abstract After gluing foliated complex manifolds we derive a preparation-like theorem for singularities of codimension-one foliations and planar vector fields in the real or complex setting . Without computation we retrieve and improve results of Levinson-Moser for functions Dufour-Zhitomirskii for nondegenerate codimension-one foliations proving in turn the analyticity Strozyna-Zoladek for non degenerate planar vector fields and Bruno-Ecalle for saddle-node foliations in the plane. Introduction We denote by z w the variable of Cn 1 z z1 . zn for n 1. Recall that a germ of non-identically vanishing holomorphic 1-form 0 f1 z w dz1 fn z w dzn g z w dw f1 . fn g G C z w defines a codimension-1 singular foliation F regular outside the zero-set of 0 if and only if it satisfies the Frobenius integrability condition 0 A d0 0. Maybe after division of coefficients of 0 by a common factor the zero-set of 0 has codimension-2 and the foliation F extends as a regular foliation outside this sharp singular set. Our main result is Theorem 1. Let 0 and F be as above and assume that g 0 w vanishes at the order k G N at 0. Then up to analytic change of the w-coordinate w ộ z w the foliation F is also defined by a 1-form 0 P1 z w dz1 Pn z w dzn Q z w dw for w-polynomials P1 . Pn Q G C z w of degree k Q monic. The preliminary version 9 of this work was written during a visit at . Barcelona we thank Marcel Nicolau and the . for hospitality. 710 FRANK LORAY In new coordinates given by Theorem 1 the singular foliation F extends analytically along some infinite cylinder z r X C where C C u to stands for the Riemann sphere . To prove this theorem we just do the converse. Given a germ of foliation we force its endless analytic continuation

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