Bor de la fibre de Milnor d'une singularité Nexton-non-dégénérée de surface complexe

We give in this work an explicit combinatorial algorithm for the description of the Milnor fiber of a Newton non degenerate surface singularity as a graph manifold. This is based on a previous work by the author describing a general method for the computation of the boundary of the Milnor fiber of any reduced non isolated singularity of complex surface.Nous donnons dans ce travail un algorithme explicite permettant la description de la fibre de Milnor d'une singularité de surface Newton non dégénérée. Ce travail est basé sur un travail précédent de l'auteur proposant une méthode générale pour le calcul du bord de la fibre de Milnor d'une singularité réduite non-isolée de surface complex

On the Nash points of subanalytic sets

Based on a recently developed rank Theorem for Eisenstein power series, we provide new proofs of the following two results of W. Pawlucki: I) The non regular locus of a complex or real analytic map is an analytic set. II) The set of semianalytic or Nash points of a subanalytic set X is a subanalytic set, whose complement has codimension two in X.Comment: Important: Our original pre-print arXiv:2205.03079 had two set of distinct results. We have divided that pre-print in two. This paper contains the second set of results ; v2 of the original submission contains the first set of results. We have divided our pre-prin

A proof of A. Gabrielov’s rank theorem

This article contains a complete proof of Gabrielov's rank Theorem, a fundamental result in the study of analytic map germs. Inspired by the works of Gabrielov and Tougeron, we develop formal-geometric techniques which clarify the difficult parts of the original proof. These techniques are of independent interest, and we illustrate this by adding a new (very short) proof of the Abhyankar-Jung Theorem. We include, furthermore, new extensions of the rank Theorem (concerning the Zariski main Theorem and elimination theory) to commutative algebra.Comment: 61 page

Bor de la fibre de Milnor d'une singularité Nexton-non-dégénérée de surface complexe

We give in this work an explicit combinatorial algorithm for the description of the Milnor fiber of a Newton non degenerate surface singularity as a graph manifold. This is based on a previous work by the author describing a general method for the computation of the boundary of the Milnor fiber of any reduced non isolated singularity of complex surface.Nous donnons dans ce travail un algorithme explicite permettant la description de la fibre de Milnor d'une singularité de surface Newton non dégénérée. Ce travail est basé sur un travail précédent de l'auteur proposant une méthode générale pour le calcul du bord de la fibre de Milnor d'une singularité réduite non-isolée de surface complex

Topologie des lissages de singularités non-isolées de surfaces complexes

This thesis is dedicated to the study of the topology of smoothings of non-isolated singularities of complex surfaces. The question is to describe the topology of the manifold, called \textbf{Milnor fiber}, which appears during this process of smoothing. Considering the great difficulty of a description of the whole of this topology, many researches have focused on the study of the \textbf{boundary} of the Milnor fiber. In the case of isolated singularities, it is known since the work of Mumford (1961) that this boundary is a graph manifold, isomorphic to the link of the singularity. Different results (Michel \& Pichon 2003, 2014, Némethi \& Szil\'ard 2012) have then proved that, in the case of reduced non-isolated singularities \ajo[of surfaces], the boundary of the Milnor fiber is again a graph manifold, while restraining to the case of a smooth total space of smoothing. Fern\'andez de Bobadilla \& Menegon-Neto (2014) have widened the context, considering non-reduced surfaces, and allowing the total space to have an isolated singularity. In this work, we pursue the extension of this result to a larger context, allowing the total space to present non-isolated singularities, while restraining ourselves to the study of reduced surface singularities. Our proof is inspired by the one of Némethi and Szilard, and allows us furthermore to provide a method for the computation of \hsout{this manifold} \ajo[the boundary of the Milnor fiber]. This makes possible the actual computation of a large number of examples, representing a step forward in the quest for the comprehension of the manifolds that can actually appear as boundaries of Milnor fibers.We apply in particular the method to Newton non-degenerate singularities defined on $3$-dimensional toric germs. This is a generalization of a theorem of Oka (1986), expressing the boundary of the Milnor fiber in terms of the Newton polyhedron of the singularity.Cette thèse s'intéresse à la topologie des lissages des singularités non-isolées de sufaces complexes. La question est celle de la description de la topologie de la variété, appelée \textbf{fibre de Milnor}, qui survient lors de ce procédé de lissage. Devant la difficulté de décrire la totalité de cette topologie, beaucoup de recherches se sont concentrées sur le \textbf{bord} de la fibre de Milnor. Dans le cas des singularités isolées, il est connu depuis les travaux de Mumford (1961), que ce bord est une variété graphée, isomorphe au bord de la singularité. Différents résultats (Michel \& Pichon 2003, 2014, Némethi \& Szil\'ard 2012) ont par la suite prouvé que dans le cas des singularités réduites non-isolées \ajo[de surfaces], si l'espace total du lissage est lui-même lisse, le bord de la fibre de Milnor est encore une variété graphée. Fern\'andez de Bobadilla \& Menegon-Neto (2014) ont quant à eux élargi le contexte, considérant le cas d'une surface non réduite dans un espace total à singularité isolée. Dans ce travail, on poursuit l'extension de ce résultat à un plus large contexte, autorisant l'espace total du lissage à présenter des singularités non-isolées, tout en imposant à la surface d'être réduite. Notre preuve s'inspire de celle de Némethi et Szilard, permettant comme chez eux de produire une méthode pour le calcul \hsout{de cette variété} \ajo[du bord de la fibre de Milnor]. Ceci rend praticable le calcul effectif d'une grande quantité d'exemples, représentant un progrès dans la quête de la compréhension des variétés pouvant apparaître comme bords de fibres de Milnor.Nous appliquons en particulier la méthode aux singularités Newton-non-dégénérées définies sur des germes toriques tridimensionnels quelconques. Nous généralisons de cette manière un théorème de Oka (1986), en exprimant le bord de la fibre de Milnor en termes du polyèdre de Newton de la singularité

Topology of smoothings of non-isolated singularities of complex surfaces

Cette thèse s’intéresse à la topologie des lissages des singularités non-isoléesde surfaces complexes. La question est celle de la description de la topologie de la variété,appelée fibre de Milnor, qui survient lors de ce procédé de lissage. Devant la difficulté dedécrire la totalité de cette topologie, beaucoup de recherches se sont concentrées sur le bordde la fibre de Milnor. Dans le cas des singularités isolées, il est connu depuis les travaux deMumford (1961), que ce bord est une variété graphée, isomorphe au bord de la singularité.Différents résultats (Michel & Pichon 2003, 2014, Némethi & Szilárd 2012) ont par lasuite prouvé que dans le cas des singularités réduites non-isolées, le bord de la fibre de Milnorest encore une variété graphée, en imposant à l’espace total du lissage d’être lui-mêmelisse. Fernández de Bobadilla & Menegon-Neto (2014) ont quant à eux élargi le contexte,considérant le cas d’une surface non réduite dans un espace total à singularité isolée. Dansce travail, on poursuit l’extension de ce résultat à un plus large contexte, autorisant l’espacetotal du lissage à présenter des singularités non-isolées, tout en imposant à la surface d’êtreréduite. Notre preuve s’inspire de celle de Némethi et Szilard, permettant comme chez euxde produire une méthode pour le calcul de cette variété. Ceci rend praticable le calcul effectifd’une grande quantité d’exemples, représentant un progrès dans la quête de la compréhensiondes variétés pouvant apparaître comme bords de fibres de Milnor.Nous appliquons en particulier la méthode aux singularités Newton-non-dégénéréesdéfinies sur des germes toriques tridimensionnels quelconques. Nous généralisons de cettemanière un théorème de Oka (1986), en exprimant le bord de la fibre de Milnor en termesdu polyèdre de Newton de la singularité.This thesis is dedicated to the study of the topology of smoothings of non-isolated singularities of complex surfaces. The question is to describe the topology of themanifold, called Milnor fiber, which appears during this process of smoothing. Consideringthe great difficulty of a description of the whole of this topology, many researches havefocused on the study of the boundary of the Milnor fiber. In the case of isolated singularities,it is known since the work of Mumford (1961) that this boundary is a graph manifold,isomorphic to the link of the singularity.Different results (Michel & Pichon 2003, 2014, Némethi & Szilárd 2012) have then provedthat, in the case of reduced non-isolated singularities, the boundary of the Milnor fiber isagain a graph manifold, while restraining to the case of a smooth total space of smoothing.Fernández de Bobadilla & Menegon-Neto (2014) have widened the context, consideringnon-reduced surfaces, and allowing the total space to have an isolated singularity. In thiswork, we pursue the extension of this result to a larger context, allowing the total spaceto present non-isolated singularities, while restraining ourselves to the study of reducedsurface singularities. Our proof is inspired by the one of Némethi and Szilard, and allows usfurthermore to provide a method for the computation of this manifold. This makes possiblethe actual computation of a large number of examples, representing a step forward in thequest for the comprehension of the manifolds that can actually appear as boundaries ofMilnor fibers.We apply in particular the method to Newton non-degenerate singularities defined on3-dimensional toric germs. This is a generalization of a theorem of Oka (1986), expressingthe boundary of the Milnor fiber in terms of the Newton polyhedron of the singularity

Topology of non-isolated singularities of complex surfaces

Milnor fibers play a crucial role in the study of the topology of a singularity of surface. They correspond to the different possible smoothings of this singularity. A description of this fiber is known in some particular cases, but in general it is not, even for isolated singularities. However, the study of its boundary has been an active field of research in the last decades. In different settings, this boundary has been proven to be a graph manifold. (Mumford, 1961, for isolated singularities, Michel-Pichon, 2003, 2014, for a smoothing of a reduced surface with smooth total space, N\'emethi-Szilard, 2012, with the same hypothesis, Bobadilla-Menegon Neto, 2014, for a non-reduced surface and a total space with isolated singularity). I will explain how the constructive proof provided by N\'emethi and Szilard can be adapted to prove, constructively, the same result for a smoothing of a reduced surface with any total space. This allows the hope for a characterization of the manifolds bounding Milnor fibers of surface singularities. Furthermore, I provide a simple algorithm for computing the boundary of the Milnor fiber, in the case of a surface defined by a generic function on a toric germ.Non UBCUnreviewedAuthor affiliation: Université Lille 1Researche

On rank Theorems and the Nash points of subanalytic sets

We prove a generalization of Gabrielov's rank theorem for families of rings of power series which we call W-temperate. Examples include the family of complex analytic functions and of Eisenstein series. Then the rank theorem for Eisenstein series allows us to give new proofs of the following two results of W. Pawlucki: I) The non regular locus of a complex or real analytic map is an analytic set. II) The set of semianalytic or Nash points of a subanalytic set $X$ is a subanalytic set, whose complement has codimension two in $X$.Comment: 50 page