This thesis studies the construction of noncompact Einstein manifolds of cohomogeneity one on some vector bundles. Cohomogeneity one vector bundle whose isotropy representation of the principal orbit G/K has two inequivalent irreducible summands has been studied in [Böh99][Win17]. However, the method applied does not cover all cases. This thesis provides an alternative construction with a weaker assumption of G/K admits at least one invariant Einstein metric. Some new Einstein metrics of Taub-NUT type are also constructed. This thesis also provides construction of cohomogeneity one Einstein metrics for cases where G/K is a Wallach space. Specifically, two continuous families of complete smooth Einstein metrics are constructed on vector bundles over CP2, HP2 and OP2 with respective principal orbits the Wallach spaces SU(3)/T2, Sp(3)/(Sp(1)Sp(1)Sp(1)) and F4/Spin(8). The first family is a 1-parameter family of Ricci-flat metrics. All the Ricci- flat metrics constructed have asymptotically conical limits given by the metric cone over a suitable multiple of the normal Einstein metric. All the Ricci-flat metrics constructed have generic holonomy except that the complete metric with G2 holonomy discovered in [BS89][GPP90] lies in the interior of the 1-parameter family on manifold in the first case. The second family is a 2-parameter family of Poincaré–Einstein metrics. / Thesis / Doctor of Philosophy (PhD)

1- Généralisation des fibrés instantons spéciaux sur P^2n+1 qui est appelée les fibrés (b+1)-instantons pondérés sur P^2n+1. On a étudié la stabilité de ces fibrés dans le cas où b=0. On a étudié la déformation de fibrés de Steiner pondérés sur P^2n+1. 2- Généralisation des fibrés de Tango sur P^n qui est appelée les fibrés de Tango pondérés sur P^n. On a étudié la stabilité de ces fibrés vectoriels. On a étudié la déformation de ces fibrés vectoriels. 3- Construction de fibrés vectoriels de rang 3 sur P^4. On a étudié la condition pour avoir des fibrés vectoriels qui ne sont pas isomorphes à une somme directe de trois fibrés en droites. / 1 - Generalization of the special instanton bundles on P^2n+1 which is called the (b+1)-weighted instanton bundles on P^2n+1. The stability of these vector bundles was studied in the case b=0. We studied the deformation of weighted Steiner bundles on P^2n+1. 2 - Generalization of the Tango bundles on P^n which is called the weighted Tango bundles on P^n. The stability of these vector bundles has been studied. The deformation of these vector bundles has been studied. 3 - Construction of vector bundles of rank 3 on P^4. We have studied the condition to have vector bundles that do not isomorphic to a direct sum of three line bundles.

In this thesis we study some questions related to vector bundles over hypersurfaces. More precisely, for hypersurfaces of dimension ≥ 2, we study the extension problem of vector bundles. We find some cohomological conditions under which a vector bundle over an ample divisor of non-singular projective variety, extends as a vector bundle to an open set containing that ample divisor. Our method is to follow the general Groethendieck-Lefschetz theory by showing that a vector bundle extension exists over various thickenings of the ample divisor. For vector bundles of rank > 1, we find two separate cohomological conditions on vector bundles which shows the extension to an open set containing the ample divisor. For the case of line bundles, our method unifies and recovers the generalized Noether-Lefschetz theorems by Joshi and Ravindra-Srinivas. In the last part of the thesis, we make a specific study of vector bundles over elliptic curve.

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 singulares / In 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

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 singulares / In 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

The vector bundles with the basic structure space with affine connection. It is shown that in the present bundles the linear inducts the affine connection, and the curvature of the objects of the present connection is traced. Having defined the concept of the first differential extension of the vector bundles, an indication is made that the linear connection of vector bundles inducts the elongated linear connection of space and linear co-connection, expression form of the linear co-connection components and their interrelation. There are derived commutative formulas of the inducted connection and forms of its components of curvature objects.

Nous étudions le déterminant du laplacien sur les fibrés vectoriels sur les graphes et l'utilisons, en lien avec des techniques d'analyse complexe discrète, pour comprendre des modèles de physique statistique. Nous calculons certaines constantes de réseaux, construisons des limites d'échelles d'excursions de la marche aléatoire à boucles effacées sur les surfaces, et étudions certains champs gaussiens et processus déterminantaux. / We study the determinant of the Laplacian on vector bundles on graphs and use it, combined with discrete complex analysis, to study models of statistical physics. We compute exact lattice constants, construct scaling limits for excursions of the loop-erased random walk on surfaces, and study some Gaussian fields and determinantal processes.

Nous allons étudier les propriétés des courbes algébriques sur des surfaces K3 spéciales, du point de vue de la théorie de Brill-Noether.La démonstration de Lazarsfeld du théorème de Gieseker-Petri a mis en lumière l'importance de la théorie de Brill-Noether des courbes admettant un plongement dans une surface K3. Nous allons donner une démonstration détaillée de ce résultat classique, inspirée par les idées de Pareschi. En suite, nous allons décrire le théorème de Green et Lazarsfeld, fondamental pour tout notre travail, qui établit le comportement de l'indice de Clifford des courbes sur les surfaces K3.Watanabe a montré que l'indice de Clifford de courbes sur certaines surfaces K3, admettant un recouvrement double des surfaces de del Pezzo, est calculé en utilisant les involutions non-symplectiques. Nous étudions une situation similaire pour des surfaces K3 avec un réseau de Picard isomorphe à U(m), avec m>0 un entier quelconque. Nous montrons que la gonalité et l'indice de Clifford de toute courbe lisse sur ces surfaces, avec une seule exception déterminée explicitement, sont obtenus par restriction des fibrations elliptiques de la surface. Ce travail est basé sur l'article suivant :M. Ramponi, Gonality and Clifford index of curves on elliptic K3 surfaces with Picard number two, Archiv der Mathematik, 106(4), p. 355–362, 2016.Knutsen et Lopez ont étudié en détail la théorie de Brill-Noether des courbes sur les surfaces d'Enriques. En appliquant leurs résultats, nous allons pouvoir calculer la gonalité et l'indice de Clifford de toute courbe lisse sur les surfaces K3 qui sont des recouvrements universels d'une surface d'Enriques. Ce travail est basé sur l'article suivant :M. Ramponi, Special divisors on curves on K3 surfaces carrying an Enriques involution, Manuscripta Mathematica, 153(1), p. 315–322, 2017. / We study the properties of algebraic curves lying on special K3 surfaces, from the viewpoint of Brill-Noether theory.Lazarsfeld's proof of the Gieseker-Petri theorem has revealed the importance of the Brill-Noether theory of curves which admit an embedding in a K3 surface. We give a proof of this classical result, inspired by the ideas of Pareschi. We then describe the theorem of Green and Lazarsfeld, a key result for our work, which establishes the behaviour of the Clifford index of curves on K3 surfaces.Watanabe showed that the Clifford index of curves lying on certain special K3 surfaces, realizable as a double covering of a smooth del Pezzo surface, can be determined by a direct use of the non-simplectic involution carried by these surfaces. We study a similar situation for some K3 surfaces having a Picard lattice isomorphic to U(m), with m>0 any integer. We show that the gonality and the Clifford index of all smooth curves on these surfaces, with a single, explicitly determined exception, are obtained by restriction of the elliptic fibrations of the surface. This work is based on the following article:M. Ramponi, Gonality and Clifford index of curves on elliptic K3 surfaces with Picard number two, Archiv der Mathematik, 106(4), p. 355-362, 2016.Knutsen and Lopez have studied in detail the Brill-Noether theory of curves lying on Enriques surfaces. Applying their results, we are able to determine and compute the gonality and Clifford index of any smooth curve lying on the general K3 surface which is the universal covering of an Enriques surface. This work is based on the following article:M. Ramponi, Special divisors on curves on K3 surfaces carrying an Enriques involution, Manuscripta Mathematica, 153(1), p. 315-322, 2017.

Nous traitons ici de questions d’effectivité dans les problèmes de Mordell-Lang et de Schanuel où la notion de hauteur algébrique joue un rôle central.Dans un premier temps nous revisitions la méthode de Vojta-Faltings dans un cadre général, en y incluant notamment un procédé de descente uniforme qui permet d’optimiser le nombre de recours au pesant mécanisme d’approximation diophantienne. Nous proposons ensuite une application de ce résultat au problème de Mordell-Lang plus Bogomolov dans le tore, qui consiste à décrire un sousensemble algébrique X comme réunion de translatés de sous-tores inclus dans X moyennant de se restreindre à un sous-groupe de rang fini épaissi. Nous nous appuyons en particulier sur un énoncé d’Amoroso et Viada concernant le problème de Bogomolov dans ce contexte et améliorons les bornes antérieures obtenues par Rémond.Dans un second temps, nous établissons une version du théorème de Schanuel dans le cadre d’un espace adélique hermitien sur un corps de nombres. Nous donnons une estimation asymptotique du nombre de points projectifs de hauteur bornée pour une hauteur définie par une famille de normes sur les complétés en chaque place, vérifiant certaines conditions mais sans hypothèse de pureté dans le cas ultramétrique. Le terme reste obtenu est totalement explicite et linéaire en le régulateur du corps de nombres grâce au recours à une méthode introduite par Schmidt. Nous traitons également plusieurs applications de ce résultat, notamment aux problèmes de Dedekind-Weber et de Loher-Masser. / We are dealing here with effectiveness matters about the Mordell-Lang and Schanuel problems where algebraic heights play a central role.At the first time, we modify the Vojta-Faltings method in a general context by including some uniform descending process which has the advantage to optimize the number of iterations of the heavy Diophantine approximation mechanism. We then propose an application to the toric Mordell-Lang plus Bogomolov problem whose aim is to describe an algebraic subset X as the union of translates of closed, irreducible subgroups included in X when restricted to some enlarged, finite rank subgroup. In particular we use a theorem of Amoroso and Viada about the Bogomolov problem in this context and we improve the previous bound given by Rémond.At the second time, we prove a version of the theorem of Schanuel in the setting of a Hermitian adelic vector bundle over a number field. We give an asymptotic estimate for the number of projective points of bounded height for heights given by a family of norms over the completions at each place, satisfying several conditions but no purity hypothesis in the ultrametric case. The error term is totally explicit and linear with respect to the regulator of the number field through the use of Schmidt’s method. We finally give some applications of our result in particular to the Dedekind-Weber and Loher-Masser problems.

The curvature of a contraction T in the Cowen-Douglas class is bounded above by the curvature of the backward shift operator. However, in general, an operator satisfying the curvature inequality need not be contractive. In this thesis, we characterize a slightly smaller class of contractions using a stronger form of the curvature inequality. Along the way, we find conditions on the metric of the holomorphic Hermitian vector bundle E corresponding to the operator T in the Cowen-Douglas class which ensures negative definiteness of the curvature function. We obtain a generalization for commuting tuples of operators in the Cowen-Douglas class. Secondly, we obtain an explicit formula for the curvature of the jet bundle of the Hermitian holomorphic bundle E f on a planar domain Ω. Here Ef is assumed to be a pull-back of the tautological bundle on gr(n, H ) by a nondegenerate holomorphic map f :Ω →Gr (n, H ). Clearly, finding relationships amongs the complex geometric invariants inherent in the short exact sequence 0 → Jk(Ef ) → Jk+1(Ef ) →J k+1(Ef )/ Jk(Ef ) → 0 is an important problem, whereJk(Ef ) represents the k-th order jet bundle. It is known that the Chern classes of these bundles must satisfy c(Jk+1(Ef )) = c(Jk(Ef )) c(Jk+1(Ef )/ Jk(Ef )). We obtain a refinement of this formula: trace Idnxn ( KJk(Ef )) - trace Idnxn ( KJk-1(Ef ))= KJk(Ef )/ Jk-1(Ef )(z).