Propri\'et\'es topologiques des polyn\^omes de deux variables complexes, et automorphismes alg\'ebriques de l'espace C2

Abstract

Propri\'et\'es topologiques des polyn\^omes de deux variables complexes, et automorphismes alg\'ebriques de l'espace $C^{2}$ By Masakazu SUZUKI ( Re\cau le 5 sept. , 1972) Introduction.Les recherches r\'ecentes, dues \'a T. Nishino et H. Sait\^o1), sur les fonctions enti\'eres de deux variables complexes, indiquent que les singularit\'es2) de ces fonctions sont soumises \'a certaines restrictions, caus\'ees par la topologie de l'espase $C^{2}$ . A. Gutwirth, M. Nagata et I. Wakabayashi ont aussi rencontr\'e un probl\'eme de ce caract\'ere3).Or, pour les (bonnes) fonctions r\'eelles, c'\'etait l'id\'ee fondamentale, due \'a M. Morse4), de d\'eterminer le type topologique de l'espace o\'u est d\'efinie la fonction consid\'er\'ee (et celui des sous-vari\'et\'es niveaux), d'apr\'es les types de singularit\'es de la fonction.Dans le Pr\'esent m\'emoire, en appliquant cette id\'ee aux fonctions poly-n\^omes de deux variables complexes, nous trouverons deux esp\'eces de restrictions auxquelles sont soumises les singularit\'es de ces fonctions; l'une est l'ind\'ependance, \'a un certain sens, des l-cycles homologiques engendr\'es aux points singuliers de la fonction consid\'er\'ee (voir les \S \S 2, 3), et l'autre concerne l'existence de la singularit\'e au point \'a l'infini (le \S 4). Nos r\'esultats donnent aussi une solution au probl\'eme cit\'e plus haut, qui est reli\'e aux automorphismes alg\

Zeroes of complete polynomial vector fields

“…We also classify the complete polynomial vector fields with rational first integral and, using Andersen's result in [2], those with one zero p which is not of Poincaré-Dulac type, when it is nondicritical and at least two of the separatrices through it are algebraic at infinity. When p is dicritical with no rational first integral, or nondicritical with just one separatrix algebraic at infinity, the induced foliation F X is, as in Brunella's result [4], P -complete where P can be written in a simple form due to [17] and [16] (Proposition 2.1 and Theorem 2.2).…”

“…Thus, if one Σ i is transcendental at infinity, it follows from [4] that there is a nonconstant (primitive) polynomial P : C 2 → C of type C or C * such that F X is P -complete. The set of points where F X is not transverse to P is an algebraic curve S ⊂ P −1 (Q), so p i ∈ S. If P is of type C, since by [16] there is ϕ ∈ Aut[C 2 ] such that P • ϕ(z 1 , z 2 ) = z 1 , one sees as above that p i ∈ {z 1 = λ}, for i = 1, 2, again a contradiction.…”

“…• Assume that the generic orbit of X is C. Suppose that {H = 0} C, so that according to Abhyankar-Moh-Suzuki's Theorem [16], there exists…”

The ball embedding property of the open unit disc

“…Indeed, such an embedding F is non-straightenable in the sense that there exists no holomorphic automorphism ϕ of C n for which ϕ•F (C) = C×{0} ⊂ C n . This result was extended to the case n = 2 by Forstnerič, Globevnik and Rosay [7] and shows that the well-known result by Abhyankar-Moh [1] and Suzuki [12]-stating that any polynomial embedding of C into C 2 can be made into the canonical inclusion by composition with a polynomial automorphism of C 2 -does not hold true in the holomorphic setting. The ideas and methods of Rosay and Rudin were generalised by Buzzard and Fornaess, who constructed a proper holomorphic embedding F :…”

Number of singular points of an annulus in \mathbb{C}^2

“…One of the most important results in this domain is the Abhyankar-Moh-Suzuki theorem ( [1,9]) stating that any algebraic curve in C 2 that is diffeomorphic to a disk is in fact algebraically isomorphic to a line. Another one, due to M. Zaidenberg and V. Lin [11], says that any curve homeomorphic to a disk is algebraically equivalent to a curve of the type x p = y q for p, q coprime.…”