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.

This thesis is concerned with a conjecture of Zilber: that the complex field expanded with the exponential function should be `quasi-minimal'; that is, all its definable subsets should be countable or have countable complement. Our purpose is to study the geometry of this structure and other expansions by holomorphic functions of the complex field without having first to settle any number-theoretic problems, by treating all countable sets on an equal footing. We present axioms, modelled on those for a Zariski geometry, defining a non-first-order class of ``quasi-Zariski'' structures endowed with a dimension theory and a topology in which all countable sets are of dimension zero. We derive a quantifier elimination theorem, implying that members of the class are quasi-minimal. We look for analytic structures in this class. To an expansion of the complex field by entire holomorphic functions $\mathcal{R}$ we associate a sheaf $\mathcal{O}^{\scriptscriptstyle{\mathcal{R}}}$ of analytic germs which is closed under application of the implicit function theorem. We prove that $\mathcal{O}^{\scriptscriptstyle{\mathcal{R}}}$ is also closed under partial differentiation and that it admits Weierstrass preparation. The sheaf defines a subclass of the analytic sets which we call $\mathcal{R}$-analytic. We develop analytic geometry for this class proving a Nullstellensatz and other classical properties. We isolate a condition on the asymptotes of the varieties of certain functions in $\mathcal{R}$. If this condition is satisfied then the $\mathcal{R}$-analytic sets induce a quasi-Zariski structure under countable union. In the motivating case of the complex exponential we prove a low-dimensional case of the condition, towards the original conjecture.

Thesis (Ph. D.)--Brandeis University, 2008. / "UMI:3306459." MICROFILM COPY ALSO AVAILABLE IN THE UNIVERSITY ARCHIVES. Includes bibliographical references.

Univ., FB Mathematik, Habil.-Schr., 2001--Hamburg, 2001.

Thesis (Ph. D. in Mathematics)--Vanderbilt University, Aug. 2006. / Title from title screen. Includes bibliographical references.

We wish to study those domains in Cn,for n ≥ 2, the so-called domains of holomorphy, which are in some sense the maximal domains of existence of the holomorphic functions defined on them. We demonstrate that this study is radically different from that of domains in C by discussing some examples of special types of domains in Cn , n ≥2, such that every function holomorphic on them extends to strictly larger domains. Given a domain in Cn , n ≥ 2, we wish to construct the maximal domain of existence for the holomorphic functions defined on the given domain. This leads to Thullen’s construction of a domain (not necessarily in Cn)spread overCn, the so-called envelope of holomorphy, which fulfills our criteria. Unfortunately this turns out to beavery abstract space, far from giving us sense in general howa domain sitting in Cn can be constructed which is strictly larger than the given domain and such that all the holomorphic functions defined on the given domain extend to it. But with the help of this abstract approach we can give a characterization of the domains of holomorphyin Cn , n ≥ 2. The aforementioned characterization is as follows: adomain in Cn is a domain of holomorphy if and only if it is holomorphically convex. However, holomorphic convexity is a very difficult property to check. This calls for other (equivalent) criteria for a domain in Cn , n ≥ 2, to be a domain of holomorphy. We survey these criteria. The proof of the equivalence of several of these criteria are very technical – requiring methods coming from partial differential equations. We provide those proofs that rely on the first part of our survey: namely, on analytic continuation theorems. If a domain Ω Cn , n ≥ 2, is not a domain of holomorphy, we would still like to explicitly describe a domain strictly larger than Ω to which all functions holomorphic on Ω continue analytically. Aspects of Thullen’s approach are also useful in the quest to construct an explicit strictly larger domain in Cn with the property stated above. The tool used most often in such constructions s called “Kontinuitatssatz”. It has been invoked, without a clear statement, in many works on analytic continuation. The basic (unstated) principle that seems to be in use in these works appears to be a folk theorem. We provide a precise statement of this folk Kontinuitatssatz and give a proof of it.

In this paper we show how to give an intrinsic definition of a constructible circle on the sphere. The classical definition of constructible circle in the plane, using straight edge and compass is there by translated in ters of so called Lenart tools. The process by which we achieve our goal involves concepts from the algebra of Hermitian matrices, complex variables, and Sterographic projection. However, the discussion is entirely elementary throughout and hopefully can serve as a guide for teachers in advanced geometry.

La conjecture de Kobayashi stipule qu'une hypersurface générique X dans CPn+1de degré d>= 2n+1 esthyperbolique complexe, un problème qui a attiré une grande attention récemment, avec l'espoir de mettre au point une théorie de Nevanlinna complète en dimension supérieure.Dans la première partie de cette thèse, notre objectif est de construire des exemples d'hypersurfaces hyperboliques de l'espace projectif dont le degré soit aussi petit que possible. Tout d'abord, en tenant compte du niveau de troncation dans le Second Théorème Principal de Cartan, nous établissons l'hyperbolicité de complémentaires de certaines configurations d'hyperplans avec points de passages, ce qui étend un résultat classique de Bloch-Fujimoto-Green. Ceci nous permet d'amorcer un algorithme récent de Duval, basé sur la méthode de déformation de Zaidenberg, pour créer des sextiques hyperboliques dans CP3, et de construire ainsi des familles d'hypersurfaces hyperboliques X dans CPn+1 de degré =2n+2 pour 2<=n<=5. En adaptant cette technique aux dimensions supérieures, nous obtenons aussi des exemples d'hypersurfaces hyperboliques de degré d>=((n+3)/2)^2 dans CPn+1.Dans la deuxième partie, nous étudions le problème de diminuer le niveau de troncation dans le Second Théorème Principal de Cartan. Noguchi a conjecturé que dans ce théorème, pour une famille de 4 droites en position générale dans CP2, si une courbe holomorphe entière f de C dans CP2 est supposée n'être pas algébriquement dégénérée, alors le niveau de troncation peut être abaissé à 1. En utilisation la théorie de recouvrement d'Ahlfors pour les surfaces, nous proposons une réponse positive dans le cas où la courbe f est proche d'une certaine courbe algébrique c dans CP2, au sens où l'ensemble d'accumulation de f(C) à l'infini, le cluster set de f est contenu dans c. / Kobayashi's conjecture asserts that a generic hypersurface X in CPn+1 having degree d>= 2n+1 is complex hyperbolic, a problem that has attracted much attention recently, also with the hope of setting up a complete higher dimensional Nevanlinna theory.In the first part of this thesis, our goal is to construct examples of hyperbolic hypersurfaces in projective spaces of degree as low as possible. First of all, taking into account the truncation level in Cartan's Second Main Theorem, we establish the hyperbolicity of complements of some configurations of hyperplanes with passage points, extending a classical result of Bloch-Fujimoto-Green. This allows us to launch a recent algorithm of Duval, based on the deformation method of Zaidenberg, on creating hyperbolic sextics in CP3, hence to construct families of hyperbolic hypersurfaces X in CPn+1 having degree d=2n+2 for 2<= n<= 5. Adapting this technique to higher dimensional cases, we also obtain examples of hyperbolic hypersurfaces of degree d>=((n+3)/2)^2 in CPn+1.In the second part, we study the problem of decreasing the truncation level in Cartan's Second Main Theorem. It was conjectured by Noguchi that in this theorem, for a family of 4 lines in general position in CP2, if an entire holomorphic curve from C to CP2 is assumed to be algebraically nondegenerate, then the truncation level can be decreased to 1. Using Ahlfors'theory of covering surfaces, we propose a positive answer in the case where the curve f is close to some algebraic curve c in CP2, in the sense that the set of accumulation points of f(C) at infinity, the cluster set of f is contained in c.

Orientador: Suzete Maria Silva Afonso / Banca: Eliris Cristina Rizziolli / Banca: Luciane Parron Gimenes Arantes / Resumo: O objetivo principal deste trabalho é desenvolver um estudo introdutório, porém detalhado, sobre Análise Complexa e algumas de suas aplicações. Apresentamos o corpo dos números complexos, exploramos as funções complexas de uma variável complexa, exibimos parte da teoria das funções analíticas e parte da teoria de integração complexa. Provamos importantes resultados, tais como o Teorema de Cauchy, o Teorema de Taylor, o Teorema dos Resíduos, entre outros igualmente relevantes. Como aplicação da teoria, destacamos a utilização do Teorema dos Resíduos para determinar a transformada inversa de Laplace de uma função F(s) / Abstract: The main objective of this work is to develop an introductory but detailed study on Complex Analysis and some of its applications. We present the field of the complex numbers, explore the complex functions of a complex variable, exhibit part of the theory of analytic functions, and part of the complex integration theory. We prove important results, such as Cauchy's Theorem, Taylor's Theorem, Residue Theorem, among others equally relevant. As an application of the theory, we highlight the use of the Residue Theorem to determine the inverse Laplace transform of a function F(s) / Mestre

In this thesis, we study integral representations of Herglotz-Nevanlinna functions, that is to say holomorphic functions defined on a product of several copies of the complex upper half-plane having non-negative imaginary part. The manuscript is divided into three parts, beginning with a general introduction followed by two papers. In the general introduction, we familiarize ourselves with the concept of a Herglotz-Nevanlinna function as well as providing a comprehensive introduction into the theory of integral representations for this particular class of functions. Paper I treats exclusively the two-variable case and presents an integral representation of Herglotz-Nevanlinna functions in two complex variables in terms of a real number, two non-negative numbers and a positive Borel measure satisfying two properties. Three properties that hold for the class of measures appearing in such integral representations are also proven. In Paper II, we provide an integral representation for the class of Herglotz-Nevanlinna functions in arbitrarily many complex variables in terms of a real number, a linear term and a positive Borel measure satisfying two properties. Properties of the class of measures appearing in this representation are then discussed in detail as well as alternative descriptions of said class. Finally, a symmetry formula satisfied by Herglotz-Nevanlinna functions is proved at the end.