published_or_final_version / Mathematics / Master / Master of Philosophy

Thesis (M. Phil.)--University of Hong Kong, 2010. / Includes bibliographical references (p. 86-88). Also available in print.

This thesis is devoted to the study of holomorphic Poisson structures and Lie algebroids, and their relationship with differential equations, singularity theory and noncommutative algebra. After reviewing and developing the basic theory of Lie algebroids in the framework of complex analytic and algebraic geometry, we focus on Lie algebroids over complex curves and their application to the study of meromorphic connections. We give concrete constructions of the corresponding Lie groupoids, using blowups and the uniformization theorem. These groupoids are complex surfaces that serve as the natural domains of definition for the fundamental solutions of ordinary differential equations with singularities. We explore the relationship between the convergent Taylor expansions of these fundamental solutions and the divergent asymptotic series that arise when one attempts to solve an ordinary differential equation at an irregular singular point. We then turn our attention to Poisson geometry. After discussing the basic structure of Poisson brackets and Poisson modules on analytic spaces, we study the geometry of the degeneracy loci---where the dimension of the symplectic leaves drops. We explain that Poisson structures have natural residues along their degeneracy loci, analogous to the Poincar\'e residue of a meromorphic volume form. We discuss the local structure of degeneracy loci that have small codimensions, and place strong constraints on the singularities of the degeneracy hypersurfaces of log symplectic manifolds. We use these results to give new evidence for a conjecture of Bondal. Finally, we discuss the problem of quantization in noncommutative projective geometry. Using Cerveau and Lins Neto's classification of degree-two foliations of projective space, we give normal forms for unimodular quadratic Poisson structures in four dimensions, and describe the quantizations of these Poisson structures to noncommutative graded algebras. As a result, we obtain a (conjecturally complete) list of families of quantum deformations of projective three-space. Among these algebras is an ``exceptional'' one, associated with a twisted cubic curve. This algebra has a number of remarkable properties: for example, it supports a family of bimodules that serve as quantum analogues of the classical Schwarzenberger bundles.

This thesis is devoted to the study of holomorphic Poisson structures and Lie algebroids, and their relationship with differential equations, singularity theory and noncommutative algebra. After reviewing and developing the basic theory of Lie algebroids in the framework of complex analytic and algebraic geometry, we focus on Lie algebroids over complex curves and their application to the study of meromorphic connections. We give concrete constructions of the corresponding Lie groupoids, using blowups and the uniformization theorem. These groupoids are complex surfaces that serve as the natural domains of definition for the fundamental solutions of ordinary differential equations with singularities. We explore the relationship between the convergent Taylor expansions of these fundamental solutions and the divergent asymptotic series that arise when one attempts to solve an ordinary differential equation at an irregular singular point. We then turn our attention to Poisson geometry. After discussing the basic structure of Poisson brackets and Poisson modules on analytic spaces, we study the geometry of the degeneracy loci---where the dimension of the symplectic leaves drops. We explain that Poisson structures have natural residues along their degeneracy loci, analogous to the Poincar\'e residue of a meromorphic volume form. We discuss the local structure of degeneracy loci that have small codimensions, and place strong constraints on the singularities of the degeneracy hypersurfaces of log symplectic manifolds. We use these results to give new evidence for a conjecture of Bondal. Finally, we discuss the problem of quantization in noncommutative projective geometry. Using Cerveau and Lins Neto's classification of degree-two foliations of projective space, we give normal forms for unimodular quadratic Poisson structures in four dimensions, and describe the quantizations of these Poisson structures to noncommutative graded algebras. As a result, we obtain a (conjecturally complete) list of families of quantum deformations of projective three-space. Among these algebras is an ``exceptional'' one, associated with a twisted cubic curve. This algebra has a number of remarkable properties: for example, it supports a family of bimodules that serve as quantum analogues of the classical Schwarzenberger bundles.

Connus des mécaniciens de la géométrie de Poisson, les algébroïdes de Lie transitifs sont ici étudiés du point de vue de leurs sections afin de développer un formalisme algébrique plus proche de celui développé par les théories de jauge. Ici, les algébroïdes de Lie transitifs s'apparentent à une généralisation des champs de vecteurs sur la variété de base. Ce mémoire de thèse a pour objet l'étude des connexions généralisées sur les algébroïdes de Lie transitifs et la construction de théories de jauge. Les connexions ordinaires sur les algébroïdes de Lie transitifs sont définies par des 1-formes de connexion de l'algébroïde de Lie à valeurs dans son noyau et vérifiant une contrainte de normalisation sur ce noyau. En relâchant cette contrainte, on construit l'espace des 1-formes de connexions généralisées qui se décomposent, à l'aide d'une connexion ordinaire de fond, comme la somme d'une connexion ordinaire et d'un paramètre purement algébrique définit sur le noyau. Dans l'esprit des théories Yang-Mills, une action invariante de jauge est définie comme la “norme” de la courbure associée à une connexion généralisée. De cette action, il découle un lagrangien composé des termes des théories de jauge de type Yang-Mills-Higgs : le terme cinétique associé aux champs de jauge et le terme de couplage minimal pour un champ tensoriel scalaire plongé dans un potentiel quartique. La réduction du groupe de symétrie de la théorie s'effectue par une redistribution des degrés de liberté dans l'espace fonctionnel des champs de la théorie. Il résulte de ces manipulations la définition d'une théorie de type Yang-Mills dont les bosons vecteurs sont des champs massifs. / Transitive Lie algebroids are usually studied from the point of view of the geometry of Poisson. Here, they are preferentially defined in terms of sections of fiber bundle in order to get close to the formalism of the gauge field theory. Then, transitive Lie algebroids can be seen as a generalization of vector fields on the base manifold. This PhD thesis is concerned with the study of generalized connections on transitive Lie algebroids and the construction of gauge theories. Ordinary connections on transitive Lie algebroids are defined as the subset of 1-forms on Lie algebroids with values in its kernel which fulfill a normalization constraint on this kernel. By relaxing this constraint, we build the space of generalized connection 1- forms. Using a background connection, we show that any generalized connections can be decomposed as the sum of an ordinary connection and a purely algebraic parameter defined on the kernel. As in Yang-Mills theories, we define a gauge invariant functional action as the “norm” of the curvature associated to a generalized connection. Then, the Lagrangian associated to this action forms a Yang-Mills-Higgs type model composed with the field strength associated to gauge fields and a minimal coupling with a tensorial scalar field embedded into a quartic potential. In the case of Atiyah Lie algebroids, the symmetry group of the theory can be reduced by using an appropriate rearrangement of the degrees of freedom in the functional space of fields. We thus obtain a Yang-Mills type theory describing massive vector bosons.