15 research outputs found
The Sard conjecture on Martinet surfaces
Given a totally nonholonomic distribution of rank two on a three-dimensional
manifold we investigate the size of the set of points that can be reached by
singular horizontal paths starting from a same point. In this setting, the Sard
conjecture states that that set should be a subset of the so-called Martinet
surface of 2-dimensional Hausdorff measure zero. We prove that the conjecture
holds in the case where the Martinet surface is smooth. Moreover, we address
the case of singular real-analytic Martinet surfaces and show that the result
holds true under an assumption of non-transversality of the distribution on the
singular set of the Martinet surface. Our methods rely on the control of the
divergence of vector fields generating the trace of the distribution on the
Martinet surface and some techniques of resolution of singularities.Comment: 4 figure
Inner geometry of complex surfaces: a valuative approach
Given a complex analytic germ in , the standard
Hermitian metric of induces a natural arc-length metric on , called the inner metric. We study the inner metric structure of the germ
of an isolated complex surface singularity by means of an infinite
family of numerical analytic invariants, called inner rates. Our main result is
a formula for the Laplacian of the inner rate function on a space of
valuations, the non-archimedean link of . We deduce in particular that
the global data consisting of the topology of , together with the
configuration of a generic hyperplane section and of the polar curve of a
generic plane projection of , completely determine all the inner rates
on , and hence the local metric structure of the germ. Several other
applications of our formula are discussed in the paper.Comment: Proposition 5.3 strengthened, exposition improved, some typos
corrected, references updated. 42 pages and 10 figures. To appear in Geometry
& Topolog
On Lipschitz Normally Embedded complex surface germs
We undertake a systematic study of Lipschitz Normally Embedded normal complex
surface germs. We prove in particular that the topological type of such a germ
determines the combinatorics of its minimal resolution which factors through
the blowup of its maximal ideal and through its Nash transform, as well as the
polar curve and the discriminant curve of a generic plane projection, thus
generalizing results of Spivakovsky and Bondil that were known for minimal
surface singularities. In the appendix, we give a new example of a Lipschitz
Normally Embedded surface singularity.Comment: v3: Section 8 has been removed and will be posted separately in
expanded and vastly improved form; the title had been edited accordingly. 31
pages, 2 figure
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
Résolution des singularités dans un espace feuilleté
Let M be an analytic manifold over the real or complex field, J be a coherent and everywhere non-zero ideal sheaf over M, E be a reduced SNC divisor and Θ an involutive singular distribution everywhere tangent to E. The main objective of this work is to obtain a resolution of singularities for the ideal sheaf J that preserves some ``good" properties of the singular distribution Θ. More precisely, the R-monomial property : the existence of local monomial first integrals. This problem arises naturally when we study the ``interaction" between a variety and a foliation and, thus, is also related with the problem of monomialization of maps and of ``quasi-smooth" resolution of families of ideal sheaves.-The first result is a global resolution if the ideal sheaf J is invariant by the singular distribution Θ;-The second result is a global resolution if the the singular distribution Θ has leaf dimension 1;-The third result is a local uniformization if the the singular distribution Θ has leaf dimension 2;We also present two applications of the previous results. The first application concerns the resolution of singularities in families, either of ideal sheaves or vector fields. For the second application, we apply the results to a dynamical system problem motivated by a question of Mattei.Considérons une variété régulière analytique M sur le corps réel ou complexe, un faisceau d'idéaux J défini sur M, un diviseur à croisement normaux simples E et une distribution singulière involutive Θ tangent à E.L'objectif principal de ce travail est d'obtenir une résolution des singularités du faisceau d'idéaux J qui préserve certaines ``bonnes" propriétés de la distribution singulière Θ. Plus précisément, la propriété de R-monomialité : l'existence d'intégrales premières monomiales. Ce problème est naturel dans le contexte où on doit étudier l'interaction d'une variété et d'un feuilletage et, donc, est aussi reliée au problème de la monomilisation des applications et de résolution ``quasi-lisse" des familles d'idéaux.-Le premier résultat donne une résolution globale si le faisceau d'idéaux J est invariant par la distribution singulière;-Le deuxième résultat donne une résolution globale si la distribution singulière Θ est de dimension 1 ;-Le troisième résultat donne une uniformisation locale si la distribution singulière Θ est de dimension 2.On présente aussi deux utilisations des résultats précédents. La première application concerne la résolution des singularités en famille analytique, soit pour une famille d'idéaux, soit pour une famille de champs de vecteurs. Pour la deuxième, on applique les résultats à un problème de système dynamique, motivé par une question de Mattei
Résolution des singularités dans un espace feuilleté
Considérons une variété régulière analytique M sur le corps réel ou complexe, un faisceau d'idéaux J défini sur M, un diviseur à croisement normaux simples E et une distribution singulière involutive tangent à E.L'objectif principal de ce travail est d'obtenir une résolution des singularités du faisceau d'idéaux J qui préserve certaines bonnes" propriétés de la distribution singulière . Plus précisément, la propriété de R-monomialité : l'existence d'intégrales premières monomiales. Ce problème est naturel dans le contexte où on doit étudier l'interaction d'une variété et d'un feuilletage et, donc, est aussi reliée au problème de la monomilisation des applications et de résolution quasi-lisse" des familles d'idéaux.- Le premier résultat donne une résolution globale si le faisceau d'idéaux J est invariant par la distribution singulière;- Le deuxième résultat donne une résolution globale si la distribution singulière est de dimension 1 ;- Le troisième résultat donne une uniformisation locale si la distribution singulière est de dimension 2.On présente aussi deux utilisations des résultats précédents. La première application concerne la résolution des singularités en famille analytique, soit pour une famille d'idéaux, soit pour une famille de champs de vecteurs. Pour la deuxième, on applique les résultats à un problème de système dynamique, motivé par une question de Mattei.Let M be an analytic manifold over the real or complex field, J be a coherent and everywhere non-zero ideal sheaf over M, E be a reduced SNC divisor and an involutive singular distribution everywhere tangent to E. The main objective of this work is to obtain a resolution of singularities for the ideal sheaf J that preserves some good" properties of the singular distribution . More precisely, the R-monomial property : the existence of local monomial first integrals. This problem arises naturally when we study the interaction" between a variety and a foliation and, thus, is also related with the problem of monomialization of maps and of quasi-smooth" resolution of families of ideal sheaves.- The first result is a global resolution if the ideal sheaf J is invariant by the singular distribution ;- The second result is a global resolution if the the singular distribution has leaf dimension 1;- The third result is a local uniformization if the the singular distribution has leaf dimension 2;We also present two applications of the previous results. The first application concerns the resolution of singularities in families, either of ideal sheaves or vector fields. For the second application, we apply the results to a dynamical system problem motivated by a question of Mattei.MULHOUSE-SCD Sciences (682242102) / SudocSudocFranceF
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 is a
subanalytic set, whose complement has codimension two in .Comment: 50 page