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1
Ruiz, Camila Mariana. "Do número de Milnor ao número de Milnor de Lê". Universidade de São Paulo, 2011. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-10082011-150046/.
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Neste trabalho,apresentamos um breve compêndio sobre o estudo topológico das fibras de Milnor. Abordamoso caso clássico, estudado por J. Milnor, e a generalização apresentada por Lê D. T. para o caso de germes de funções analíticas definidas em variedades singulares. Nestas duas situações, os resultados principais tratam de germes de funções com singularidades isoladasIn this work, we present a brief compendium about the topological study of J. Milnor fibers. We address the classic case, studied by Milnor, and the generalization presented by Lê D. T. for the case of germs of analytic functions defined on singular varieties. In both situations, the main results deal with germs of functions with isolated singularities
2
Ribeiro, Maico Felipe Silva. "Singular Milnor Fibrations". Universidade de São Paulo, 2018. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-06072018-115031/.
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In this work we present the most recent developments in the direction of local fibrations structures of analytic singularities. Using techniques and tools from stratification theory we prove structural theorems in the stratified sense, which will be called singular Milnor tube fibration and Milnor-Hamm sphere fibration. In addition, we present algorithms with the purpose of creating a large number of examples in this new setting and compare our results obtained with the current ones found in the literature. Our results generalize all previous result in both cases: in the classical and in the stratified ones.Neste trabalho apresentamos os mais recentes desenvolvimentos na direção de estruturas de fibrações locais de singularidades analíticas. Usando técnicas e ferramentas da teoria de estratificação, provamos alguns teoremas estruturais no sentido estratificado, os quais serão chamados fibração singular de Milnor sobre o tubo e fibração de Milnor-Hamm sobre a esfera. Além disso, apresentamos algoritmos com o intuito de criar uma ampla variedade de exemplos e comparamos nossos resultados com os atuais encontrados na literatura. Nossos resultados generalizam todos os previamente existentes tanto no caso clássico, quanto no sentido estratificado.
3
Keating, Ailsa Macgregor. "Symplectic properties of Milnor fibres". Thesis, Massachusetts Institute of Technology, 2014. http://hdl.handle.net/1721.1/90186.
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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2014.69
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 121-123).
We present two results relating to the symplectic geometry of the Milnor fibres of isolated affine hypersurface singularities. First, given two Lagrangian spheres in an exact symplectic manifold, we find conditions under which the Dehn twists about them generate a free non-abelian subgroup of the symplectic mapping class group. This extends a result of Ishida for Riemann surfaces. The proof generalises the categorical version of Seidel's long exact sequence to arbitrary powers of a fixed Dehn twist. We also show that the Milnor fibre of any isolated degenerate hypersurface singularity contains such pairs of spheres. In the second half of this thesis, we study exact Lagrangian tori in Milnor fibres. The Milnor fibre of any isolated hypersurface singularity contains many exact Lagrangian spheres: the vanishing cycles associated to a Morsification of the singularity. Moreover, for simple singularities, it is known that the only possible exact Lagrangians are spheres. We construct exact Lagrangian tori in the Milnor fibres of all non-simple singularities of real dimension four. This gives examples of Milnor fibres whose Fukaya categories are not generated by vanishing cycles. Also, this allows progress towards mirror symmetry for unimodal singularities, which are one level of complexity up from the simple ones.
by Ailsa Macgregor Keating.
Ph. D.
4
Kerz, Moritz. "Milnor K-theory of local rings". kostenfrei, 2008. http://www.opus-bayern.de/uni-regensburg/volltexte/2008/991/.
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5
Feld, Niels. "Faisceaux et modules de Milnor-Witt". Thesis, Université Grenoble Alpes, 2021. http://www.theses.fr/2021GRALM001.
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On généralise la théorie des modules de cycles de Rost en utilisant la K-théorie de Milnor-Witt au lieu de la K-théorie de Milnor. On obtient un cadre (quadratique) pour étudier certains complexes de cycles et leurs groupes de (co)homologie.De plus, on démontre que le coe ur de la catégorie homotopique stable de Morel-Voevodsky au-dessus d'un corps parfait (équipé de sa t-structure homotopique) est équivalente à la catégorie des modules de cycles de Milnor-Witt.Finalement, on explore une conjecture de Morel concernant les transferts de Bass-Tate définis sur la contraction d'un faisceau homotopique et démontre que la conjecture est vraie à coefficients rationnels. On étudie aussi les relations entre faisceaux homotopiques (contractés), faisceaux homotopiques avec transferts généralisés et MW-faisceaux homotopiques, et démontre une équivalence de catégories. Comme applications, on décrit l'image essentielle du foncteur canonique qui oublie les MW-transferts et utilise ces résultats pour discuter de la conjecture de conservativité en A1-homotopie due à Bachmann et YakersonWe generalize Rost's theory of cycle modules using the Milnor-Witt K-theory instead of the classical Milnor K-theory. We obtain a (quadratic) setting to study general cycle complexes and their (co)homology groups.Moreover, we prove that the heart of Morel-Voevodsky stable homotopy category over a perfect field (equipped with its homotopy t-structure) is equivalent to the category of Milnor-Witt cycle modules.Finally, we explore a conjecture of Morel about the Bass-Tate transfers defined on the contraction of a homotopy sheaf and prove that the conjecture is true with rational coefficients. We also study the relations between (contracted) homotopy sheaves, sheaves with generalized transfers and MW-homotopy sheaves, and prove an equivalence of categories. As applications, we describe the essential image of the canonical functor that forgets MW-transfers and use theses results to discuss the conservativity conjecture in A1-homotopy due to Bachmann and Yakerson
6
Santana, Hellen Monção de Carvalho. "Número de Milnor associado a curvas reduzidas /". São José do Rio Preto, 2016. http://hdl.handle.net/11449/136361.
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Orientador: Michelle Ferreira Zanchetta MorgadoBanca: Bruna Oréfice Okamoto
Banca: Parham Salehyan
Resumo: O objetivo deste trabalho é estudar curvas reduzidas. Associado a elas, Buchweitz e Greuel definem um número, chamado número de Milnor de curvas reduzidas, pois no caso de curvas planas este coincide com o número de Milnor definido por Milnor. Este número é obtido através de um importante objeto algébrico: o módulo dual de Grothendieck. Com o intuito de facilitar a obtenção deste número, mostraremos que ele está relacionado com outro número, chamado delta, mais fácil de ser calculado. Por fim, mostraremos que, de maneira análoga, Nuño-Ballesteros e Tomazella definem um número associado a germes de função finita definidos em curvas reduzidas. Este número está relacionado com o grau deste germe e com o número de Milnor da curva reduzida associada
Abstract: The aim of this work is to study reduced curves. Associate to them, Buchweitz and Greuel define a number, called Milnor number once that in the case of plane curves, this number coincides to the Milnor number defined by Milnor. This number is obtained through an important algebraic object: dual module of Grothendieck. In order to make it easier to obtain this number, we will prove that it is related to another number, called delta, easier to be computed. At last, we prove that, in the same way, Nuño-Ballesteros and Tomazella define a number associate to finite function germs defined over reduced curves. This number is related to the degree of this germ and to the Milnor number of the reduced curve associated to it
Mestre
7
Martins, Rafaella de Souza. "Sobre a topologia das fibrações de Milnor". Universidade de São Paulo, 2018. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-25072018-104835/.
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Nesta tese abordaremos dois tipos de problemas relacionados aos célebres Teorema da Fibração de Milnor e Teorema da Fibração de Milnor-Lê para o caso real com valores críticos não isolados. Primeiramente, asseguramos fibrações do tipo Milnor-Lê para F : (Xm, 0) → (Yn, 0), germe de aplicação subanalítico com X e Y espaços subanalíticos sobre C \\ {0} uma curva subanalítica conexa em Y e sobre um subespaço analítico suave W ⊂ Y de dimensão p, n ≥ p ≥ 2, sob algumas condições. Em particular, mostramos a existência das fibrações sobre o discriminantes de germe de aplicações subanalíticos, caso esse ainda não estudado na literatura, normalmente o conjunto dos valores críticos são desconsiderados. Finalizando nossa análise da categoria subanalítica, certificamos que existe a fibração de Milnor-Lê para f : (X, 0) →(Rp, 0), com dimensão de X maior que p ≥ 2, subanalítica e X subanalítico com valores críticos não isolados, definindo d-regularidade. Abordamos estes problemas utilizando resultados de campos de vetores rugosos. Em uma segunda etapa apresentamos um novo critério necessário e suficiente para verificar a importante propriedade de transversalidade de um germe de aplicação real f de classe Cl, l ≥ 1. Fazendo uso também de uma recente ferramenta desenvolvida, a D-regularidade, verificamos condições para a existência das fibrações do germe de aplicação Ψ F, X : (Cn, 0) → (C, 0) não holomorfo, dado por Ψ (z, z̄) = Σnj=1 kjtjzj a
8
Neto, Aurelio Menegon. "Números de Milnor e obstrução de Euler". Universidade de São Paulo, 2007. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-14092007-101056/.
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Neste trabalho, definimos a obstrução local de Euler de um espaço analítico complexo singular (X, \'x IND.0\'), denotada por Eu(X, \'x IND.0\'), e a obstrução local de Euler de uma função holomorfa f definida neste espaço, com uma singularidade isolada em \'x IND. 0\', denotada por \'Eu IND. f\' (X, \'x IND.0\'); e apresentamos duas fórmulas para seus respectivos cálculos. Em seguida, através de uma abordagem geométrica, determinamos as relações entre \'Eu IND. f\' (X,\'x IND.0\') e algumas generalizações do número de Milnor para funções em espaços singularesIn this work we define the local Euler obstruction of a complex analytic singularity (X, \'x IND.0\'), denoted Eu(X, \'x IND.0\'), and the local Euler obstruction of a holomorphic function f defined on this space, with an isolated singularity at \'x IND. 0\', denoted \'Eu IND. f\' (X, \'x IND.0\'); and we present two formulas for their respective calculations. Next, using a geometric approach, we determine the relations between \'Eu IND.f\' (X, \'x IND.0\') and several generalizations of the Milnor number for functions on singular spaces
9
Santos, Raimundo Nonato Araújo dos. "Fibrações de Milnor de singularidades analíticas reais". Universidade de São Paulo, 2002. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-17062015-103449/.
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Neste trabalho estudamos a fibração de Milnor associada a singularidades isoladas reais definidas por germes de aplicações f : Rn, 0 → R2, 0 . O principal resultado relaciona a existência da fibração de Milnor com a (c)-regularidade da família de hipersuperfícies com singularidade isolada obtida projetando f sobre a família L-θ de todas as retas pela origem no plano R2. Estudamos também famílias de germes de funções analíticas com singularidades isoladas. O objetivo é encontrar condições suficientes para a trivialidade topológica das famílias e a equivalência das fibrações de Milnor associadas a elas.In this work we study the Milnor\'s fibrations associated to real isolated singularities defined by map-germs f : Rn, 0 → R2, 0. The main result relates the existence of the Milnor\'s fibration with the (c)-regularity of the family of hypersurfaces with isolated singularity obtained by projecting f into the family L-θ of all lines through the origin in the plane R2sup. We also study families of germs of analytic functions with isolated singularities. The aim is to get sufficient condition for the topological triviality of the families and the equivalente of the Milnor fibrations associated to them.
10
Guerrero, Vejarano Darwin Emerson [UNESP]. "Sobre o teorema de fibração de Milnor". Universidade Estadual Paulista (UNESP), 2014. http://hdl.handle.net/11449/122106.
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A fibração de Milnor aparece como a principal ferramenta no estudo local da topologia das singularidades analíticas reais e complexas. Neste trabalho estudaremos o Teorema de Fibração de Milnor e o surpreendente comportamento topológico das Fibras de Milnor. Para tais objetivos, usaremos algumas ferramentas da Geometria Algébrica, Análise Complexa em várias variáeis e um pouco da Teoria de Morse
The Milnor bration appears as the main tool in the topological local study of real and complex analytic singularities. In this work we study the famous Milnor Fibration Theorem, and the surprising topological behavior of the Milnor bres. To reach these objectives we use some tools of classical Algebraic Geometry, Complex Analysis of several variables and also some aspects of Morse Theory
11
Santana, Hellen Monção de Carvalho [UNESP]. "Número de Milnor associado a curvas reduzidas". Universidade Estadual Paulista (UNESP), 2016. http://hdl.handle.net/11449/136361.
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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)
O objetivo deste trabalho é estudar curvas reduzidas. Associado a elas, Buchweitz e Greuel definem um número, chamado número de Milnor de curvas reduzidas, pois no caso de curvas planas este coincide com o número de Milnor definido por Milnor. Este número é obtido através de um importante objeto algébrico: o módulo dual de Grothendieck. Com o intuito de facilitar a obtenção deste número, mostraremos que ele está relacionado com outro número, chamado delta, mais fácil de ser calculado. Por fim, mostraremos que, de maneira análoga, Nuño-Ballesteros e Tomazella definem um número associado a germes de função finita definidos em curvas reduzidas. Este número está relacionado com o grau deste germe e com o número de Milnor da curva reduzida associada.
The aim of this work is to study reduced curves. Associate to them, Buchweitz and Greuel define a number, called Milnor number once that in the case of plane curves, this number coincides to the Milnor number defined by Milnor. This number is obtained through an important algebraic object: dual module of Grothendieck. In order to make it easier to obtain this number, we will prove that it is related to another number, called delta, easier to be computed. At last, we prove that, in the same way, Nuño-Ballesteros and Tomazella define a number associate to finite function germs defined over reduced curves. This number is related to the degree of this germ and to the Milnor number of the reduced curve associated to it.
12
Guerrero, Vejarano Darwin Emerson. "Sobre o teorema de fibração de Milnor /". São José do Rio Preto, 2014. http://hdl.handle.net/11449/122106.
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Orientador: João Carlos Ferreira CostaBanca: Ana Claudia Nabarro
Banca: Luciana de Fátima Martins
Resumo: A fibração de Milnor aparece como a principal ferramenta no estudo local da topologia das singularidades analíticas reais e complexas. Neste trabalho estudaremos o Teorema de "Fibração" de Milnor e o surpreendente comportamento topológico das Fibras de Milnor. Para tais objetivos, usaremos algumas ferramentas da Geometria Algébrica, Análise Complexa em várias variáeis e um pouco da Teoria de Morse
Abstract: The Milnor bration appears as the main tool in the topological local study of real and complex analytic singularities. In this work we study the famous Milnor Fibration Theorem, and the surprising topological behavior of the Milnor bres. To reach these objectives we use some tools of classical Algebraic Geometry, Complex Analysis of several variables and also some aspects of Morse Theory
Mestre
13
AUDOUBERT, BENOIT. "Filtration polaire de la fibre de milnor". Nantes, 2001. http://www.theses.fr/2001NANT2016.
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La these s'inscrit dans le domaine des singularites des espaces analytiques complexes. Classiquement dans le cas des varietes lisses, les invariants de nature topologique tel que la forme d'intersection ou le genre des courbes ont ete largement utilises pour classifier ces varietes. Dans le cas singulier on est conduit a considerer des invariants de nature algebrique (ou analytique) plus elabores comme les varietes polaires locales ou la fibre de milnor locale munie de sa monodromie et de sa filtration polaire. La resolution des singularites fait le lien avec le cas lisse et de ce point de vue la topologie de la fibre de milnor est liee au diviseur a croisements normaux dans une resolution des singularites via une projection naturelle. L'objet de la these est de montrer que la filtration polaire de la fibre de milnor locale, definie par une projection sur un petit disque de c, coincide avec une filtration valuative, dite filtration d'hironaka, construite a l'aide du diviseur a croisements normaux, ce qui permet de relier des invariants associes aux singularites de la projection avec des invariants de la desingularisation.
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Oréfice, Bruna. "O número de Milnor de uma singularidade isolada". Universidade Federal de São Carlos, 2011. https://repositorio.ufscar.br/handle/ufscar/5823.
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Previous issue date: 2011-11-24Financiadora de Estudos e Projetos
Given (X; 0) C (CN; 0) a weighted homogeneous germ of hypersurface with isolated singularity and f : (CN; 0) - C a germ of function finitely determined with respect to X, we show that UBR(f;X) = U(f) + U(X; f), where U(f) and U(X; f) denote the Milnor numbers of f and of the fiber X \ f��1(0), respectively, and UBR(f;X) is the Bruce-Roberts number of f with respect to X. We show that the logarithmic characteristic subvariety, LC(X), is Cohen-Macaulay and we get relations between the Bruce-Roberts number and the Euler obstruction. Given F : (CN; 0) ! Mm;n(C) a holomorphic function germ, let (X; 0) be the isolated determinantal singularity given by X = F-1(Ms m;n(C)) where Ms m;n(C) is the set of the complex matrices with rank less then s, with s an integer number between 0 and minfm; ng such that N < (m - s + 2)(n - s + 2), we will define the vanishing Euler characteristic of (X; 0) and the Milnor number of a holomorphic function germ with an isolated singularity at X, f : (X; 0) - C.
Dados (X; 0) C (CN; 0) um germe de hipersuperfície quase homogêneo com singularidade isolada e f : (CN, 0) - C um germe de função finitamente determinado com respeito a X, mostramos que UBR(f;X) = U(f) + U(X; f), onde U(f) e U(X; f) denotam o número de Milnor de f e da fibra X \ f-1(0), respectivamente, e _BR(f;X) é o número de Bruce-Roberts de f com respeito a X. Mostramos que a variedade logarítmica característica LC(X) é Cohen-Macaulay e obtemos relações entre o número de Bruce-Roberts e a obstrução de Euler. Dado F : (CN; 0) ! Mm;n(C) um germe de função holomorfa, seja (X; 0) a singularidade determinantal isolada dada por X = F-1(Ms m;n(C)) onde Ms m;n(C) é o conjunto das matrizes complexas com posto menor que s, com s um número inteiro entre 0 e minfm; ng tal que N < (m-s+2)(n-s+2), definimos a característica de Euler evanescente de (X; 0) e o número de Milnor de um germe de função holomorfa com uma singularidade isolada em X, f : (X; 0) - C.
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Spohr, Cristina. "Poliedro de Newton e o número de Milnor". Universidade Federal de São Carlos, 2006. https://repositorio.ufscar.br/handle/ufscar/5917.
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Previous issue date: 2006-02-24Financiadora de Estudos e Projetos
In this work, we study the relation between Milnor's number and the Newton's number of a formal series. The former is always greater than or equal to the latter and equality occurs whan the formal series f has non-degenerate newtonian principal part at the origin.
Neste trabalho, estudamos a relação que existe entre o número de Milnor de uma série formal com o número de Newton. O número de Milnor de uma série formal é sempre maior ou igual ao número de Newton e a igualdade entre os números é obtida sempre que a série formal f possui parte principal Newton não-degenerada na origem.
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Zuber, Hugues. "Variétés caractéristiques et non formalité des fibres de Milnor". Phd thesis, Université de Nice Sophia-Antipolis, 2009. http://tel.archives-ouvertes.fr/tel-00440281.
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Le but de cette thèse est l'étude de la fibre de Milnor associée à un complémentaire d'arrangement d'hyperplans. Il est montré par un exemple que cette variété n'est pas toujours formelle, ou même 1-formelle. La formalité est une propriété introduite dans les années 1970 dans le cadre de la théorie de l'homotopie rationnelle. Des avancées récentes ont identifié cette propriété comme critère particulièrement fin pour établir un lien entre variétés caractéristiques et variétés de résonance, associées à l'espace étudié. Ces deux types de variétés sont des invariants dont les définitions présentent beaucoup de points communs, mais dans des espaces différents. Un lien très fort - la variété de résonance est le cône tangent à l'origine de la variété caractéristique correspondante - avait été établi sous diverses hypothèses, que l'introduction de la 1-formalité permet d'élargir. C'est en montrant que pour l'exemple décrit dans cette thèse, ce lien n'existe pas, que l'on prouve que la variété considérée n'est pas formelle.
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Yokura, Shoji y yokura@sci kagoshima-u. ac jp. "Verdier--Riemann--Roch for Chern Class and Milnor Class". ESI preprints, 2000. ftp://ftp.esi.ac.at/pub/Preprints/esi933.ps.
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Couture, Olivier. "Fibres de Milnor dans les familles à un paramètre". Dijon, 1996. http://www.theses.fr/1996DIJOS017.
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Considerons une famille a un paramètre s de fonctions holomorphes a point critique isole a l'origine. Pour une valeur non nulle de s, la théorie de Milnor fournit un plongement de la fibre de Milnor dans la sphère unité. D'autre part, en poussant cette fibre le long des trajectoires du champ de gradient défini au-dessus d'un chemin du plan complexe, on obtient un second plongement dans la sphère. Conjecture p. I. : ces deux plongements sont isotopes. La conjecture est vraie en dimension strictement supérieure a deux. Dans ce travail, on donne quelques éléments de réponse pour des familles en dimension deux. En particulier, on résout la conjecture pour deux types de familles. Pour cela, on construit deux cylindres dont la base commune est le bord de la fibre de Milnor. Ces cylindres sont transverses au niveau du module de la fonction, et l'un d'eux est transverse aux sphères centrées à l’ origine. La conjecture se ramené à montrer que ces cylindres sont isotopes. On vérifie que c'est le cas pour les deux familles particulières
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Williams, Kristopher John. "The Milnor fiber associated to an arrangement of hyperplanes". Diss., University of Iowa, 2011. https://ir.uiowa.edu/etd/1277.
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Let f be a non-constant, homogeneous, complex polynomial in n variables. We may associate to f a fibration with typical fiber F known as the Milnor fiber. For regular and isolated singular points of f at the origin, the topology of the Milnor fiber is well-understood. However, much less is known about the topology in the case of non-isolated singular points. In this thesis we analyze the Milnor fiber associated to a hyperplane arrangement, ie, f is a reduced, homogeneous polynomial with degree one irreducible components in n variables. If n > 2then the origin will be a non-isolated singular point. In particular, we use the fundamental group of the complement of the arrangement in order to construct a regular CW-complex that is homotopy equivalent to the Milnor fiber. Combining this construction with some local combinatorics of the arrangement, we generalize some known results on the upper bounds for the first betti number of the Milnor fiber. For several classes of arrangements we show that the first homology group of the Milnor fiber is torsion free. In the final section, we use methods that depend on the embedding of the arrangement in the complex projective plane (ie not necessarily combinatorial data) in order to analyze arrangements to which the known results on arrangements do not apply.
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Sampaio, Breno Rafael Pinheiro. "Sobre o nÃmero de Milnor de germes de funÃÃes holomorfas". Universidade Federal do CearÃ, 2013. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=13475.
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Nos trabalhos iniciais sobre o nÃmero de Milnor, ele à definido como a dimensÃo do n-Ãsimo grupo de homologia de uma fibra de Milnor. Esse trabalho irà verificar algumas outras equivalÃncias, com o objetivo de mostrar que o nÃmero de Milnor pode ser escrito como a dimenÃÃo de um C-espaÃo vetorial que vem do quociente entre o anel de germes de funÃÃes holomorfas e de seu jacobiano.In early work on the number of Milnor, it is defined as the dimension of the nth homology group of a Milnor fiber. This work will check some other equivalences, with the aim of showing that the number of Milnor can be written as the demension of a vector C-space that comes from the ratio of the germ ring of holomorphic functions and its Jacobian.
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Sampaio, Breno Rafael Pinheiro. "Sobre o número de Milnor de germes de funções holomorfas". reponame:Repositório Institucional da UFC, 2015. http://www.repositorio.ufc.br/handle/riufc/10594.
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SAMPAIO, Breno Rafael Pinheiro. Sobre o número de Milnor de germes de funções holomorfas. 2015. 53 f. Dissertação (Mestrado em Matemática)- Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2015.Submitted by Rocilda Sales (rocilda@ufc.br) on 2015-02-10T15:55:26Z No. of bitstreams: 1 2015_dis_brpsampaio.pdf: 9011044 bytes, checksum: 79ea4e2b63403426b3e68d3c52b699d8 (MD5)
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In early work on the number of Milnor, it is defined as the dimension of the nth homology group of a Milnor fiber. This work will check some other equivalences, with the aim of showing that the number of Milnor can be written as the demension of a vector C-space that comes from the ratio of the germ ring of holomorphic functions and its Jacobian.
Nos trabalhos iniciais sobre o número de Milnor, ele é definido como a dimensão do n-ésimo grupo de homologia de uma fibra de Milnor. Esse trabalho irá verificar algumas outras equivalências, com o objetivo de mostrar que o número de Milnor pode ser escrito como a dimenção de um C-espaço vetorial que vem do quociente entre o anel de germes de funções holomorfas e de seu jacobiano.
22
Geandier, Françoise. "Polynômes de Bernstein et déformations à nombre de Milnor constant". Nice, 1989. http://www.theses.fr/1989NICE4278.
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On étudie une déformation à un paramètre y, appartenant à un voisinage de l'origine dans C, d'une singularité isolée d'hypersurface de Cn d'équation F(x, y)=0. On prouve d'abord l'existence d'un polynôme de Bernstein générique de F à l'origine, c'est-à-dire correspondant à un opérateur différentiel relatif polynomial en S et en L/y. Si on suppose de plus que la déformation est à nombre de Milnor constant, alors il existe un polynôme de Bernstein en famille de F à l'origine, c'est-à-dire correspondant à un opérateur différentiel relatif polynomial en S. On prouve ensuite que l'existence d'un opérateur différentiel relatif polynomial et unitaire en S, annulant Fs, caractérise les déformations à nombre de Milnor constant. D'autre part, on démontre un théorème de structure des D-modules cohérents à support l'axe des paramètres, ou D désigne le faisceau des opérateurs différentiels relatifs. Ce théorème permet de prouver que lors d'une déformation à nombre de Milnor constant, le polynôme de Bernstein by de F(x, y) à l'origine est égal au polynôme de Bernstein générique de F, pour y non nul voisin de l'origine dans C. Enfin, on montre que le polynôme de Bernstein bo(s) de la fibre spéciale divise by(s)by(s+1). . . By(s+n-1)
23
Lauber, Marianne. "Nombres de Lelong et nombres de Milnor d'une singularité isolée". Grenoble 1, 1993. http://www.theses.fr/1993GRE10028.
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L'objet de cette these est de relier entre elles deux grandeurs importantes, l'une en theorie des courants le nombre de lelong et l'autre en geometrie algebrique locale le nombre de milnor, ou plus generalement la multiplicite. Cette etude nous permet de donner une nouvelle preuve, plus analytique, d'un theoreme du a f. Loeser, qui met en relation les nombres de milnor d'une fonction a singularite isolee et des integrales des formes de chern associees aux fibres non singulieres. Par ailleurs, nous proposons une generalisation des nombres de lelong microlocaux introduits par l. Abrahamsson et nous montrons comment ils peuvent s'interpreter comme des nombres de lelong generalises avec poids
24
Nguyen, Dinh Huu. "On [rho]-generic splitting varieties for Milnor K-symbols mod [rho]". Diss., Restricted to subscribing institutions, 2009. http://proquest.umi.com/pqdweb?did=1905631291&sid=1&Fmt=2&clientId=1564&RQT=309&VName=PQD.
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Zanchetta, Michelle Ferreira. "Números de Lê e classes de Milnor de hipersuperfícies analíticas complexas". Universidade de São Paulo, 2010. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-15042010-161340/.
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Este trabalho está dividido em duas partes distintas. Na primeira parte caracterizamos os números de Lê de polinômios que são rodutos de polinômios de Pham-Brieskorn de mesmo tipo, que denominamos de arranjos de Pham-Brieskorn, obtendo fórmulas para estes números somente utilizando o número de variáveis, os pesos e o grau de homogeneidade destes polinômios. Na segunda parte nos dedicamos a estabelecer relações entre os números de Lê, que é um conceito local, e as classes de Milnor, que são objetos globais que fornecem informações quanto a geometria e topologia de hipersuperfícies analíticas complexas. No contexto geral, usando a hipótese de especialização, relacionamos a classe de Milnor de dimensão máxima de uma hipersuperfície Z numa variedade compacta M com uma soma, sobre os estratos de uma estratificação de Whitney de Z (com estratos conexos) que estão contidos no conjunto singular, em termos do último número de Lê associado a cada estrato. Além disso, obtivemos uma caracterização da classe de Milnor de dimensão mínima via os números de Lê sem usar a hipótese de especialização. Esta classe coincide com o chamado número de Milnor de Parusinski que, assim como os números de Lê, também é uma generalização do número de MilnorThis work is divided into two distinct parts. In the first part we characterize the Lê numbers of polynomials that are products of Pham- Brieskorn polynomials of the same type that we call Pham-Brieskorn arrangements, obtaining formulas to these numbers only using the number of variables, weights and degree of homogeneity of these polynomials. In the second part we are dedicated to establishing relationships between Lê numbers, which is a local concept, and the Milnor classes, which are global objects that provide information about the geometry and topology of complex analytic hypersurfaces. In a general context, using the hypothesis of specialization we relate the top dimensional Milnor class of a hypersurface Z in a compact manifold M with a sum given in terms of the last Lê number associated to each stratum of a Whitney estratification of Z (with connected strata) that are contained in singular set. Moreover, we obtain a characterization of the Milnor class of minimum dimension via the Lê numbers without using the hypothesis of specialization. This class coincides with the Milnor number of Parusinski that, as the Lê numbers, it is also a generalization of the Milnor number
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García, Barroso Evelia. "Courbes polaires et courbure des fibres de milnor des courbes planes". Paris 7, 2000. http://www.theses.fr/2000PA077089.
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Le but de ce travail est d'etudier les proprietes des courbes polaires generiques p(t) d'un germe reduit de courbe analytique complexe plane c a partir des proprietes de c. Nous nous interessons au contact avec les branches de c des branches de p(t). Plus precisement, nous enoncons un theoreme optimal (comme le montre un exemple) de decomposition en paquets des branches de p(t). L'ensemble de ces paquets est indexe par le graphe d'eggers qui ne depend que du type topologique (de sa classe d'equisingularite) de la courbe c donnee. Par construction, toutes les branches d'un meme paquet ont le meme contact avec chacune des branches de c et en consequence un certain nombre des premiers termes du developpement de puiseux de chaque branche de p(t) sont independants de t d'apres l'interpretation du contact comme mesure de coincidence entre les parametrisations. Toutes les proprietes qu'on en deduit ne dependent que du type topologique de c ; en particulier la decomposition en paquets fournit des invariants numeriques du type topologique de c, dont on savait dans le cas ou c est irreductible qu'ils determinent ce type topologique. Cependant ils ne sont pas suffisants pour determiner le type topologique de c quand c est reduite non irreductible. Nous introduisons les invariants polaires partiels et nous montrons qu'ils ne dependent que de la topologie de la courbe c et la determinent completement si nous connaissons de plus la multiplicite a l'origine de chaque branche de c. En utilisant les resultats sur les courbes polaires, nous etudions le comportement asymptotique de la courbure des fibres de milnor associees a la courbe c. Nous montrons que cette courbure se concentre asymptotiquement dans des boules evanescentes a differentes echelles, ces echelles ne dependant que de la topologie de la courbe c.
27
Scully, Stephen. "Quasilinear forms, Milnor K-theory and function fields of hypersurfaces in positive characteristic". Thesis, University of Nottingham, 2013. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.597111.
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The main part of this thesis comprises a study of quasilinear p-forms (i.e., Fermat-type forms of degree p over fields of characteristic p > 0) in the spirit of the theory of non-singular quadratic forms over fields of characteristic different from two. Our approach to the subject centres around the study of a basic discrete invariant of quasilinear p-forms known as the standard splitting pattern, which plays here a similar role to that played by Knebusch's splitting pattern invariant for non-singular quadratic forms in characteristic different from two. This invariant not only captures a certain notion of algebraic complexity for quasilinear p-forms, but also encodes essential information concerning the geometry of their projective zero-loci, which we call quasilinear p+hypersurfaces. We explore here various aspects of this interaction between the alge+ braic and algebra-geometric parts of the theory. Our approach is successful, not only in extending well-known results from the theory of non-singular quadratic forms over fields of characteristic different from 2 into this setting, but in revealing interesting and important new phenomena which are completely absent from the former theory. A second part of this thesis is concerned with studying the behaviour of mod+p Milnor K-groups of fields of characteristic p > 0 under scalar extension to the fW1ction fields of nowhere-smooth hypersurfaces. A unifying theme for all this work is the study of a basic birational invariant of hypersurfaces of the latter type, which we call the norm field.
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Hohlenwerger, Maria Amelia de Pinho Barbosa. "Novos exemplos de NS-pares e de fibrações de Milnor reais não-triviais". Universidade de São Paulo, 2014. http://www.repositorio.ufrb.edu.br/handle/123456789/952.
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Neste trabalho, nos concentramos no estudo da topologia da fibração de Milnor associada
a um germe de aplicação polinomial f : (Rn; 0) ! (Rp; 0) com uma singularidade
isolada na origem. O primeiro resultado é uma extensão da caracterização de germes de
aplicações triviais nos pares de dimensões (n; p) quando n p = 3: Uma caracterização
inicial foi apresentada por Church e Lamotke em 1975. O segundo resultado é a caracterização
de NS-pares (S5;K2); usando a topologia de espaços de configuração. Como
uma consequência desta caracterização, mostramos a existência de germe de aplicação
polinomial real nos pares de dimensões (6; 3) com uma singularidade isolada na origem
tal que sua fibra de Milnor não é difeomorfa a um disco. A existência desses exemplos
coloca um fim ao problema da não-trivialidade proposto por Milnor em 1968 e além disso,
nos permite apresentar um novo resultado sobre a topologia da fibra de Milnor real nos
pares de dimensões (2n; n) e (2n + 1; n); n > 3: Tal resultado garante a existência de
germes de aplicações polinomiais (Rn; 0) ! (Rp; 0); n > p > 2; com uma singularidade
isolada na origem tais que suas fibras de Milnor têm o tipo de homotopia de um buquê
de um número positivo de esferasIn this work, we focus on the study of the topology of the Milnor fibration associated with a polynomial map germ f : (Rn; 0) ! (Rp; 0) with an isolated singularity at the origin. The first result is an extension of the characterization of trivial map germs in the pairs of dimensions (n; p) when n p = 3: An initial characterization was presented by Church and Lamotke in 1975. The second result is a characterization of NS-pairs (S5;K2); using the topology of configuration spaces. As a consequence of this characterization, we show the existence of real polynomial map germs in the pairs of dimensions (6; 3) with an isolated singularity at the origin such that its Milnor fibers are not diffeomorphic to a disc. The existence of such examples ends a non-triviality problem posed by Milnor in 1968 and furthermore, it allows us to show a new result about the topology of the real Milnor fibers in the pairs of dimensions (2n; n) and (2n + 1; n); n > 3: This result ensure the existence of polynomial map germs (Rn; 0) ! (Rp; 0); n > p > 2; with an isolated singularity at the origin such that its Milnor fibers has the homotopy type of a bouquet of a positive number of spheres.
29
Khélif, Anatole. "Generalisation du theoreme de bass-milnor-serre au cas d'anneaux d'entiers non-standards". Paris 7, 1990. http://www.theses.fr/1990PA077052.
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Soit q le corps des fractions d'un modele de l'arithmetique de peano, k une extension finie de q, a l'anneau des entiers sur k, n un entier superieur ou egal a 3. Si k admet un conjugue totalement ordonne (ce qui arrive sinon sera etudie en detail), nous montrerons que tout sous-groupe d'indice fini de sln(a) est un groupe de congruence
30
Monteiro, Amanda. "Estudo de polinômios quase homogêneos via formas de Seifert /". São José do Rio Preto, 2019. http://hdl.handle.net/11449/180974.
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Orientador: Michelle Ferreira Zanchetta MorgadoCoorientador: Évelin Meneguesso Barbaresco
Banca: Nivaldo de Goes Grulha Junior
Banca: Maria Gorete Carreira Andrade
Resumo: Dado um polinômio quase homogêneo com singularidade isolada na origem existe associado um polinômio que depende apenas de seus pesos. Motivados por um resultado que garante que dados dois polinômios quase homogêneos com singularidade isolada na origem, eles têm os mesmos pesos se, e somente se, os seus polinômios associados são iguais, fizemos um estudo destes polinômios através das chamadas Formas de Seifert, que são formas sobre o grupo de homologia da fibra de Milnor associadas ao polinômio inicial, definido pelo linking number de dois ciclos. Desenvolvemos a teoria necessária para mostrar que dados dois polinômios quase homogêneos com singularidade isolada na origem, se suas Formas de Seifert forem equivalentes sobre os números reais, então seus polinômios associados são congruentes de uma certa maneira. Ressaltamos que a recíproca deste resultado também é válida e, portanto, existe uma condição necessária e suficiente para que esses polinômios tenham Formas de Seifert reais equivalentes em termos de seus pesos
Abstract: Given a weighted homogeneous polynomials with isolated singularity at the origin there is a polynomial associated that depends only on its weights. Motivated by a result that ensures that given two weighted homogeneous polynomials with isolated singularity at the origin, they have the same weights if, and only if, their associated polynomials are equal, we did a study of these polynomials through the so-called Seifert Forms, which are forms on the homology group of Milnor fiber associated to the initial polynomial, defined by the linking number of two cycles. We develop the necessary theory to show that given two weighted homogeneous polynomials with isolated singularity at the origin, if their Seifert Forms are equivalent on real numbers, then their associated polynomials are congruent in a certain way. We note that the converse of this result is also valid and, therefore, there is a necessary and sufficient condition for these polynomials to have equivalent real Seifert Forms in terms of their weights
Mestre
31
Vial, Charles. "Operations in Milnor K-theory and coniveau filtrations and finite dimensionality for pure motives". Thesis, University of Cambridge, 2009. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.611566.
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Pontigo, Herrera Jessie Diana. "Problème de centre tangentiel et problème de monodromie pour certains Hamiltoniens non-génériques". Thesis, Dijon, 2016. http://www.theses.fr/2016DIJOS001/document.
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Dans le cas générique Yu. S. Ilyashenko a donné une solution pour le problème tangentielle du centre et le probème de la monodromie. Néanmoins, on ne connaît pas la solution pour tous les cas non-génériques. Dans cette thèse on étudie une famille des équations hamiltoniennes non-génériques dont l'hamiltonien est un produit de polynômes réels irréductibles de dégre supérieur ou égal à 1. On étudie cette famille dans le but d'avoir un modèle d'équation hamiltonienne qui nous permette de comprendre d'autres cas non-génériques. Cette famille ne satisfait pas necessairement les conditions de généricité de transversalité à l'infini et n'a pas nécessairement tous les points singuliers aux niveaux distincts. Nous considerons quelques conditions géomètriques sur les hamiltoniens qu'on appelle bon partage du plan proyective réel et bonne multiplicité à l'infini. Ces conditions nous servent pour calculer l'orbite par monodromie des cycles évanescents. On résout le problème de la monodromie pour deux sous-familles dans cette famille d'hamiltoniennes. Une d'elles satisfait que tous les points critiques de type centre sont à des niveux critiques distincts, et l'autre satisfait que l'hamiltonien est invariant par la réflexion par rapport à l'axe des y. En utilisant la solution du problème de la monodromie on résout aussi le problème tangentiel du centre pour ces famillesIn the generic case Yu. S. Ilyashenko gave a solution of the tangential center problem and the monodromy problem. However, a solution for all non-generic cases is not known. In this thesis we study a family of non-generic Hamiltonians, whose Hamiltonian is a product of real polynomials of degree equal or bigger than 1. We study this family with the idea that a good understanding of this Hamiltonian model could help us to understand other non-generic cases later. In this family the genericity assumption of transversality at infinity fails and the coincidence of the critical values for different critical points is allowed. We consider some geometric conditions on the Hamiltonians of this family that we call good divide of the real projective plane and good multiplicity at infinity. These conditions help us to compute the orbit under monodromy of vanishing cycles. We give a solution of the monodromy problem of two sub-families in this family. One of them satisfying that all the center critical points are at different critical levels, and the other satisfying that the Hamiltonian is invariant under the reflection with respect to the y-axis. Using the solution of the monodromy problem we also provide a solution of the tangential center problem for those families
33
Barbosa, Grazielle Feliciani. "Topologia de singularidades e o estudo de seus invariantes". Universidade de São Paulo, 2008. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-08052008-135109/.
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Algumas relações entre A-invariantes de germes de aplicações de coposto 1 equidimensionais f : \'C POT. n\' , 0 \'SETA\' \"C POT.n\', 0 são descritas. O principal resultado estabelece que a soma alternada de números de Milnor dos fechos dos conjuntos Ai na fonte de f é igual a multiplicidade local de f menos n + 1. E existem fórmulas correspondentes para os s-tipos estáveis locais A(\'k IND.1\' ,...\'k IND.s\'). As relações nos garantem condiçõoes para a A-finitude de f e para a A-trivialidade topológica de deformações de f. Também classificamos os germes de aplicações A-simples f : \'C POT.2\', 0 \'SETA\' \'C POT.5\', 0, para multiplicidades 1, 2 e 3Some new relations between A-invariants of equidimensional corank-1 map germs f :\'C POT.n\', 0 \' \'ARROW\' \'C POT.n\', 0 are described. The main local result states that the alternating sum ofthe Milnor numbers of the closures of the Ai sets in the source of f is equal to the local multiplicity of f minus n + 1. And there are corresponding formulas for the s-local stable types A(\'k IND.1\' ,...,\'k IND.s\'). The realations provide simplified (or weaker) conditions for the A-finiteness of f and for the topological A-triviality of deformations of f. We also classify the A-simple germs f : \'C POT.2\', 0 \'ARROW\' \'C POT.5\', 0 for multiplicities 1, 2, and 3
34
Szawlowski, Adrian [Verfasser], Viktor [Akademischer Betreuer] Pidstrygach y Thomas [Akademischer Betreuer] Schick. "The Geometry of the Milnor Number / Adrian Szawlowski. Gutachter: Viktor Pidstrygach ; Thomas Schick. Betreuer: Viktor Pidstrygach". Göttingen : Niedersächsische Staats- und Universitätsbibliothek Göttingen, 2012. http://d-nb.info/1043514619/34.
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Huarcaya, Jorge Alberto Coripaco. "Poliedros de Newton e singularidades de polinômios". Universidade de São Paulo, 2011. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-10082011-140945/.
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Neste trabalho, estudamos a relação que existe entre o número de Milnor de um polinômio cômodo ou seja, a soma dos números de Milnor dos pontos singulares isolados deste polinômio, com seu número de Newton. Este número é sempre menor ou igual ao número de Newton e a igualdade entre os números é obtida sempre que o polinômio cômodo possui parte principal Newton não-degenerada no infinitoIn this work, we study the relation between the Milnor number of a polynomial cômodo ie. the sum of Milnor numbers of isolated singular points of polynomial, with the Newton number. This number is always lower than or equal to the Newton number and equality between the numbers is obtained when the polynomial has non-degenerate newtonian principal part at the infinity
36
Bailet, Pauline. "Arrangements d'hyperplans". Phd thesis, Université Nice Sophia Antipolis, 2014. http://tel.archives-ouvertes.fr/tel-01059809.
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Cette thèse étudie la fibre de Milnor d'un arrangement d'hyperplans complexe central, et l'opérateur de monodromie sur ses groupes de cohomologie. On s'intéresse à la problématique suivante : peut-on déterminer l'opérateur de monodromie, ou au moins les nombres de Betti de la fibre de Milnor, à partir de l'information contenue dans le treillis d'intersection de l'arrangement? On donne deux théorèmes d'annulation des sous-espaces propres non triviaux de l'opérateur de monodromie. Le premier résultat s'applique à une large classe d'arrangements, le deuxième à des arrangements de droites projectives tels qu'il existe une droite contenant exactement un point de multiplicité supérieure ou égale à trois. Dans le dernier chapitre, on considère la structure de Hodge mixte des groupes de cohomologie de la fibre de Milnor d'un arrangement central et essentiel dans l'espace complexe de dimension quatre. On donne ensuite l'équivalence entre la trivialité de la monodromie, la nullité des coefficients non entiers du spectre de l'arrangement, et la nullité des nombres de Hodge mixtes des groupes de cohomologie de la fibre de Milnor.
37
Dutertre, Nicolas. "Sur le calcul de la caracteristique d'euler-poincare d'ensembles semi-algebriques et de la fibre de milnor reelle". Rennes 1, 1998. http://www.theses.fr/1998REN10133.
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On etudie dans les deux premiers chapitres la fibre de milnor d'une application analytique de r#n vers r#k. On choisit un germe de fonction g puis on considere l'ensemble des points de la fibre de milnor ou g est positive et l'ensemble des points ou g est negative. On evalue la difference des caracteristiques d'euler des ces ensembles. Le premier chapitre est consacre au cas d'une bifurcation a p parametres d'un germe de fonction analytique n-dimensionel et d'un germe de fonction en les parametres g dont le gradient ne s'annule pas a l'origine. La difference consideree s'exprime comme le degre en 0 d'une application analytique de r#p#+#n. On obtient egalement une formule pour le cas p = 4 et g quelconque. On etudie le cas general dans le deuxieme chapitre. Pour le cas k = 1, on trouve une formule similaire a celle d'une bifurcation et pour le cas k 1, une formule modulo 2 dont on tire une nouvelle preuve de l'invariance topologique du nombre de milnor modulo 2. Le but du troisieme chapitre est d'exprimer la caracteristique d'euler des ensembles semi-algebriques de r#n definis par l'intersection d'une intersection complete algebrique lisse et de plusieurs inegalites polynomiales. Pour ceci, on adapte des techniques de szafraniec et on exprime ces caracteristiques d'euler en fonction des signatures de formes bilineaires. On etudie au passage les fibres d'un polynome propre de r#n a points critiques isoles et on obtient une version globale de la formule d'arnol'd, wall et khimshiasvili. On exprime egalement l'homotopie de la fibre d'un polynome modere de c#n en fonction du rang d'une certaine forme bilineaire symetrique. Le quatrieme chapitre est consacre a l'etude des fibres d'un polynome non propre de r#n.
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Ament, Daiane Alice Henrique. "Invariantes de germes de aplicações". Universidade Federal de São Carlos, 2017. https://repositorio.ufscar.br/handle/ufscar/8976.
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In this work, we show relations between invariants of map germs. First, we consider an analytic function germ f : (X, 0) —(C, 0) on an isolated determinantal singularity and we present a relation between the Euler obstruction of f and the determinantal Milnor number of f. In the particular case where (X, 0) is an isolated complete intersection singularity, we obtain a simple way to calculate the Euler obstruction of f as the difference between the dimension of two algebras. After, we work with map germs f : (X, 0) —— (C2, 0), where (X, 0) is a plane curve with isolated singularity. We introduce the image Milnor number to these map germs and we present a positive answer to the Mond’s conjecture in this context. The Mond’s conjecture proposes an inequality between two other invariants, the A^-codimension and the image Milnor number, in the case of map germs f : (Cn, 0) —(Cn+1, 0) when the dimensions (n,n + 1) is in Mather’s nice dimensions. The conjecture is true for n = 1, 2, and for the cases n > 3 is an open problem.
Neste trabalho, mostramos relações entre invariantes de germes de aplicações. Primeiro, consideramos um germe de funçao analítica f : (X, 0)^(C, 0) sobre uma singularidade determinantal isolada e apresentamos uma relaçao entre a obstrução de Euler de f e o número de Milnor determinantal de f. No caso particular em que (X, 0) e uma interseçao completa com singularidade isolada, obtemos um modo simples de calcular a obstrucao de Euler de f como a diferenca entre dimensães de duas algebras. Depois, trabalhamos com germes de aplicacoes f : (X, 0)^(C2, 0), onde (X, 0) e uma curva plana com singularidade isolada. Introduzimos o número de Milnor da imagem para estes germes de aplicacães e apresentamos uma resposta positiva para a conjectura de Mond neste contexto. A conjectura de Mond propoe uma desigualdade entre outros dois invariantes, a A^-codimensao e o numero de Milnor da imagem, para o caso de germes de aplicacoes f : (Cn, 0)^(Cn+1,0) quando as dimensoes (n,n + 1) estao nas boas dimensoes de Mather. A conjectura e verdadeira para n = 1, 2, e para os casos n > 3 e um problema em aberto.
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Orgogozo, Fabrice. "Groupe fondamental premier à p, nombre de Milnor des singularités isolées, motifs de dimension inférieure ou égale à 1". Phd thesis, Université Pierre et Marie Curie - Paris VI, 2003. http://tel.archives-ouvertes.fr/tel-00004093.
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Dans le premier chapitre, on démontre divers résultats sur le plus grand quotient du groupe fondamental étale premier aux caractéristiques, parmi lesquels la formule de Künneth et l'invariance par changement de corps séparablement clos pour les schémas de type fini sur un corps. Ces énoncés sont déduits de faits généraux sur les images directes de champs, une fois spécialisés au cas des torseurs sous un groupe constant fini d'ordre inversible sur la base. Des résultats analoguespour le groupe fondamental modéré sont également discutés.
Au deuxième chapitre, on déduit de la formule du conducteur, conjecturée par S. Bloch, celle de P. Deligne exprimant, dans le cas d'une singularité isolée, la dimension totale des cycles évanescents en fonction du nombre de Milnor.
En particulier, la formule de Deligne est établie en dimension relative un.
Dans le troisième chapitre, on compare les 1-isomotifs de P. Deligne sur un corps avec la théorie de V. Voevodsky en dimension inférieure à 1.
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Orgogozo, Fabrice. "Groupe fondamental premier à p, nombres de Milnor des singularités isolées, motifs de dimension inférieure ou égale à 1". Paris 6, 2003. https://tel.archives-ouvertes.fr/tel-00004093v2.
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Luchesi, Vanda Maria. "Invariantes de germes de aplicações de C^2 em C^3". Universidade de São Paulo, 2005. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-04072005-122826/.
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Sejam f:(C^2,0) to (C^3,0) um germe de aplicação holomorfa de coposto 1 e f_t uma perturbação estável de f. Os pontos singulares de f_t são cross-caps, pontos duplos ou pontos triplos. O número de cross-caps e pontos triplos de f_t e o número de Milnor da curva de pontos duplos de f_t são invariantes do germe f. Neste trabalho estudamos fórmulas para obter estes invariantes e no caso dos germes quasi-homogêneos relacionamos estes invariantes com a A_e-codimensão de f.Let f:(C^2,0) to (C^3,0) be a holomorphic map-germ with corank 1 and f_t a stable perturbation of f. The singular points of f_t are either cross-caps, double points or triple points. The number of cross-caps and the number of triple points of f_t and the Milnor number of the double points curve of f_t are invariants of the germs f. In this work we study formulas to get these invariants and in the case of quasi-homogeneous germs we relate these invariants with the A_e-codimension of f.
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Curmi, Octave. "Topologie des lissages de singularités non-isolées de surfaces complexes". Thesis, Lille 1, 2019. http://www.theses.fr/2019LIL1I030/document.
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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 singularitéThis 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
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Forey, Arthur. "Invariants motiviques dans les corps valués". Thesis, Paris 6, 2017. http://www.theses.fr/2017PA066557/document.
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Cette thèse est consacrée à définir et étudier des invariants motiviques associés aux ensembles semi-algébriques dans les corps valués. Ceux-ci sont les combinaisons booléennes d'ensembles définis par des inégalités valuatives. L'outil principal que nous utilisons est l'intégration motivique, une forme de théorie de la mesure à valeurs dans le groupe de Grothendieck des variétés définies sur le corps résiduel. Dans une première partie, on définit la notion de densité locale motivique. C'est un analogue valuatif du nombre de Lelong complexe, de la densité réelle de Kurdyka-Raby et de la densité p-adique de Cluckers-Comte-Loeser. C'est un invariant métrique à valeurs dans un localisé du groupe de Grothendieck des variétés. Notre résultat principal est que cet invariant se calcule sur le cône tangent muni de multiplicités motiviques. On établit un analogue de la formule de Cauchy-Crofton locale. On montre enfin que dans le cas d'une courbe plane, la densité motivique est égale à la somme des inverses des multiplicités des branches. L'objet de la seconde partie est de définir un morphisme d'anneau du groupe de Grothendieck des ensembles semi-algébriques sur un corps valué K vers le groupe de Grothendieck de la catégorie d'Ayoub des motifs rigides analytiques sur K. On montre qu'il étend le morphisme qui envoie la classe d'une variété algébrique sur la classe de son motif cohomologique à support compact. Cela fournit donc une notion virtuelle de motif cohomologique à support compact pour les variétés rigides analytiques. On montre également un théorème de dualité permettant de comparer le motif cohomologique de la fibre de Milnor analytique avec la fibre de Milnor motiviqueThis thesis is devoted to define and study some motivic invariants associated to semialgebraic sets in valued fields. They are boolean combinations of sets defined by valuative inequalities. Our main tool is the theory of motivic integration, which is a kind of measure theory with values in the Grothendieck group of varieties defined over the residue field. In the first part, we define the notion of motivic local density. It is a valuative analog of complex Lelong number, Kurdyka-Raby real density and p-adic density of Cluckers- Comte-Loeser. It is a metric invariant with values in a localization of the Grothendieck group of varieties. Our main result is that it can be computed on the tangent cone with motivic multiplicities. We also establish an analog of the local Cauchy-Crofton formula. We finally show that the density of a germ of plane curve defined over the residue field is equal to the sum of the inverses of the multiplicities of the formal branches of the curve. The goal of the second part is to define a ring morphism from the Grothendieck group of semi-algebraic sets defined over a valued field K to the Grothendieck group of Ayoub’s categoryof rigid analytic motives over K. We show that it extends the morphism sending the class of an algebraic variety to the class of its cohomological motive with compact support. This gives a notion of virtual cohomological motive with compact support for rigid analytic varieties. We also show a duality theorem allowing us to compare the cohomological motive of the analytic Milnor fiber with the motivic Milnor fiber
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Joint, Marie-Emmanuelle. "Extensions d'algèbres de Hopf primitivement engendrées". Angers, 2004. http://www.theses.fr/2004ANGE0005.
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La notion d'algèbre de Hopf joue un grand rôle en mathématique et en physique. En topologie algébrique, la notion fondamentale d'espace de lacets, nous conduit à nous intéresser particulièrement aux algèbres de Hopf graduées connexes. Les travaux initialisés en 1965, par J. Milnor et J. Mooore, étendus par ceux de D. Anick et S. Halperin, ont mis en évidence l'intérêt de la notion d'algèbre de Hopf primitivement engendrée. Le théorème de structure démontré par Y. Félix, S. Halperin et J-C Thomas caractérise ces algèbres à l'aide d'une extension centrale d'algèbres de Hopf. Le sujet de cette thèse est la classification des extensions d'algèbres de Hopf primitivement engendrées. En particulier nous développons une technique de calculs des classes d'extension à l'aide d'une formule de Campbell-Haussdorf, et nous illustrons par quelques exemples de nature purement topologique les résultats algébriques que nous avons obtenusThe notion of Hopf algebra pla,ys an important part in mathematics and physics. In algebraic topology, the fundamental notion of loop space, leads us to graded connected Hopf agebras. The work begun in 1965 by J. ~Zilnor and J. Moore and extended bv those of D. Anick and S. Halperin, showed the interest of the notion of primitivelv ~Hopf algebra. The structure theorem proved by Y. Felix, S. Halperin and J-C. Thomas characterizes those algebras by mean central extension of Hopf algebras. The subject of this the5is is the classification of extensions of primitively generated Hopf algebras. In particular. We develop an explicit computation for the extension classes using the Campbell-Haussdorf formula. We illustrate with some purely naturally topological exemples the algebraic results that we have proved
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Souza, Taciana Oliveira. "Teoremas de (H,G)-coincidências para variedades e classificação global de singularidades isoladas em dimensões (6,3)". Universidade de São Paulo, 2013. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-10062013-161959/.
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Este trabalho é constituido por duas partes. Na primeira parte, obtivemos algumas generalizações do clássico Teorema de Borsuk-Ulam em termos de (H,G)-coincidências. Na segunda parte, estendemos a caracterização dos germes de aplicações triviais, em codimensão 3, pelas fibrações de Milnor iniciada por Church e Lamotke em [11]. Usamos essa caracterização na classificação global de singularidades isoladas em dimensões (6, 3)This work consists of two parts. In the first part, we obtain some generalizations of the classical Borsuk-Ulam Theorem in terms of (H,G)-coincidences. In the second part, we extend the characterization of trivial map germs, in codimension 3, by the Milnor fibrations started by Church and Lamotke in [11]. We use this characterization in the global classification of isolated singularities in dimensions (6, 3)
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Santos, Duilio Ferreira. "Elementos da teoria algébrica das formas quadráticas e de seus anéis graduados". Universidade de São Paulo, 2015. http://www.teses.usp.br/teses/disponiveis/45/45131/tde-22062016-104927/.
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Neste trabalho procuramos realizar uma apresentação autocontida sobre os conceitos da teoria algébrica de formas quadráticas e sobre os anéis graduados que surgiram no desenvolvimento desta teoria. Iniciamos procurando esclarecer o sentido da equivalência entre as várias acepções do conceito de forma quadrática. Após a apresentação de ingredientes e resultados geométricos, fazemos um extrato da teoria dos anéis de Witt, conceito que originou a moderna teoria algébrica de formas quadráticas. Disponibilizamos os elementos fundamentais para a formulação das teorias de cohomologia, nos concentrado no desenvolvimento da teoria de cohomologia profinita e, sobretudo, galoisiana. Descrevemos os funtores K0, K1 e K2 da K-teoria clássica e também a K-teoria de Milnor, que é mais adequada para formular questões sobre formas quadráticas. Finalizamos o trabalho com a apresentação de alguns conceitos da Teoria dos Grupos Especiais, uma codificação em primeira-ordem da teoria algébrica das formas quadráticas e exemplificamos sua importância, fornecendo um extrato da prova realizada por Dickmann-Miraglia da conjectura de Marshall sobre assinaturas, que se baseia fortemente nesta teoria.In this work I try to provide a self-contained presentation on the concepts of algebraic theory of quadratic forms and on the graded rings that have emerged in the development of this theory. I started trying to clarify the meaning of \"equivalence\"between the various meanings of the concept of quadratic form. After the presentation of geometrical ingredients and results, we make an extract of the theory of Witt rings, a concept that originated the modern algebraic theory of quadratic forms. It is provided the key elements for the formulation of cohomology theories, focusing on the development of profinite cohomology theory and, especially, on galoisian cohomology. Are described the functors K0, K1 and K2 of classical K-theory and also the Milnor K-theory, which is more appropriate to formulate questions about quadratic forms. The dissertation is finished with the presentation of some concepts of the Theory of Special Groups, a first-order encoding of algebraic theory of quadratic forms, and with an example its importance by providing an extract of proof by Dickmann-Miraglia of the Marshalls conjecture on signatures, which relies heavily on this theory.
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Nakajima, Evandro Alves. "Campos de vetores em variedades singulares". Universidade de São Paulo, 2013. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-11112013-151324/.
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Neste trabalho estudamos índices de campos de vetores em variedades regulares e em variedades com singularidades isoladas. O principal resultado e o Teorema de Poincaré-Hopf que relaciona a característica de Euler de uma variedade com o índice de Poincaré-Hopf do campo. Para intersecções completas com singularidades isoladas, vemos também algumas variações deste teorema que relacionam a característica de Euler com o índice de Schwartz, o índice GSV e o número de Milnor da fibra genéricaIn this work we study some indices of vector fields on regular manifolds, and on manifolds with isolated singularity. The main result is the Poincare-Hopf Theorem, which connects the Euler characteristic with the Poincare-Hopf index of the field. For complete intersections with isolated singularities, we also study some variations of this theorem, which connects the Euler characteristic with the Schwartz index, the GVS index and the Milnor number of the generic fiber
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Abdallah, Nancy. "Cohomologie des courbes planes algébriques". Phd thesis, Université Nice Sophia Antipolis, 2014. http://tel.archives-ouvertes.fr/tel-01064511.
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On décrit dans cette thèse les dimensions des groupes quotients gradués associés à la cohomologie du complémentaire d'une courbe plane par rapport à la filtration de Hodge en fonction de certains invariants géométriques. Le cas des courbes à singularités ordinaires est détaillé. En particulier, on trouve le polynôme de Hodge-Deligne d'une courbe C quelconque à singularités isolées et celui de son complémentaire duquel on déduit les nombres de Hodge mixtes ainsi que les nombres de Betti correspondants. Dans le cas des courbes dont les singularités sont des nœuds et des points triples ordinaires, on donne des relations importantes avec l'algèbre de Milnor du polynôme homogène f qui définit C, les syzygies de l'idéal Jacobien de f et la filtration par l'ordre de pôle du groupe cohomologique d'ordre 2 du complémentaire de la courbe.
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Clutha, Mahana. "Bounding Betti numbers of sets definable in o-minimal structures over the reals". Thesis, University of Bath, 2011. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.547627.
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A bound for Betti numbers of sets definable in o-minimal structures is presented. An axiomatic complexity measure is defined, allowing various concrete complexity measures for definable functions to be covered. This includes common concrete measures such as the degree of polynomials, and complexity of Pfaffian functions. A generalisation of the Thom-Milnor Bound [17, 19] for sets defined by the conjunction of equations and non-strict inequalities is presented, in the new context of sets definable in o-minimal structures using the axiomatic complexity measure. Next bounds are produced for sets defined by Boolean combinations of equations and inequalities, through firstly considering sets defined by sign conditions, then using this to produce results for closed sets, and then making use of a construction to approximate any set defined by a Boolean combination of equations and inequalities by a closed set. Lastly, existing results [12] for sets defined using quantifiers on an open or closed set are generalised, using a construction from Gabrielov and Vorobjov [11] to approximate any set by a compact set. This results in a method to find a general bound for any set definable in an o-minimal structure in terms of the axiomatic complexity measure. As a consequence for the first time an upper bound for sub-Pfaffian sets defined by arbitrary formulae with quantifiers is given. This bound is singly exponential if the number of quantifier alternations is fixed.
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Siesquén, Nancy Carolina Chachapoyas. "Invariantes de variedades determinantais". Universidade de São Paulo, 2014. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-13022015-100258/.
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Neste trabalho estudamos variedades determinantais essencialmente isoladas (EIDS), definidas por W. Èbeling e S. M. Gusen-Zade em [23]. Este tipo de singularidades é uma generalização das singularidades isoladas. A variedade determinantal genérica Mtm, n é o subconjunto das matrizes m X n, tais que o posto seja menor que t, onde t ≤ min{n;m}. Uma variedade X ⊂ CN é determinantal se é definida como a pré-imagem de uma função holomorfa F : CN → Mm;n, sobre a variedade determinantal genérica M tIn this work, we study the essentially isolated determinantal singularities (EIDS), which have been defined by W. Èbeling and S. M. Gusen-Zade in the article [23]. This type of singularities is a natural generalization of isolated ones. A generic determinantal variety Mtm;n is a subset of the space of m X n matrices, given by matrices of rank less than t, where t ≤ min. A variety X ⊂ CN is determinantal if X is defined as the pre-image of Mtm;n by a holomorphic function F : CN → Mm;n with the condition codim X = codim Mtm;n. Determinantal varieties have isolated singularity if N ≤ (n - t + 2)(m - t + 2) and they admit smoothing if N < (n - t +2)(m - t +2). Several recent works investigate determinantal variety with isolated singularities. The Milnor number of a surface was defined in [35, 31] and the vanishing Euler characteristic was studied in [31]. In this work we study the set of limits of tangent hyperplanes to determinantal varieties X2 ⊂ C4 and X3 ⊂ C5 to give a characterization of this set by the fact that the Milnor number of its section with the surface in the first case or the 3-dimensional determinantal variety in the second case is not minimum. The first case is studied by Jawad Snoussi in [38]. We also prove that if X is a d- dimensional EIDS and H and H\' are strongly general hyperplans, if P ⊂ H and P\' are linear plans of codimension d - 2 contained in H and H\', the Milnor number of the surfaces X ∩ P and X ∩ P\' are equal. In the case that the generic section is a curve the result has been proved in [26]. We study the Nash transformation of an EIDS and give sufficient conditions for this transformation to be smooth. Another aim of our study is the Euler obstruction of essentially isolated determinantal singularities. We obtain inductive formulas associating the Euler obstruction with the vanishing Euler characteristic of the essencial smoothing of their generic sections. We study the determinantal variety with singular set of dimension 1 to illustrate the results.