• JoomlaWorks Simple Image Rotator
  • JoomlaWorks Simple Image Rotator
  • JoomlaWorks Simple Image Rotator
  • JoomlaWorks Simple Image Rotator
  • JoomlaWorks Simple Image Rotator
  • JoomlaWorks Simple Image Rotator
  • JoomlaWorks Simple Image Rotator
  • JoomlaWorks Simple Image Rotator
  • JoomlaWorks Simple Image Rotator
  • JoomlaWorks Simple Image Rotator
 
  Bookmark and Share
 
 
Master's Dissertation
DOI
10.11606/D.45.2012.tde-18062012-194224
Document
Author
Full name
Lucas Kaufmann Sacchetto
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2012
Supervisor
Committee
Gorodski, Claudio (President)
Clarke, Andrew James
Cordaro, Paulo Domingos
Title in Portuguese
Fundamentos da geometria complexa: aspectos geométricos, topológicos e analiticos.
Keywords in Portuguese
Classes de Chern
Cohomologia de Feixes
Geometria Complexa
Teorema da Decomposição de Hodge
Teorema das (1 1) -classes de Lefschetz
Teorema dos Hiperplanos de Lefschetz
Teorema ``Difícil'' de Lefschetz
Variedades de Kähler
Abstract in Portuguese
Este trabalho tem como objetivo apresentar um estudo detalhado dos fundamentos da Geometria Complexa, ressaltando seus aspectos geométricos, topológicos e analíticos. Começando com materiais preliminares, como resultados básicos sobre funções holomorfas de uma ou mais variáveis e a definição e primeiros exemplos de variedades complexas, passamos a uma introdução à teoria de feixes e sua cohomologia, ferramenta indispensável para o restante do trabalho. Após um estudo sobre fibrados de linha e divisores damos atenção à Geometria de Kähler e alguns de seus resultados centrais, como por exemplo o Teorema da Decomposição de Hodge, o Teorema ``Difícil'' e o Teorema das $(1,1)$-classes de Lefschetz. Em seguida, nos dedicamos ao estudo dos fibrados vetoriais complexos e sua geometria, abordando os conceitos de conexões, curvatura e Classes de Chern. Terminamos o trabalho descrevendo alguns aspectos da topologia de variedades complexas, como o Teorema dos Hiperplanos de Lefschetz e algumas de suas consequências.
Title in English
Foundations of Complex Geometry: geometric, topological and analytic aspects.
Keywords in English
Chern Classes
Complex Geometry
Hard Lefschetz Theorem
Hodge Decomposition Theorem
Kähler Manifolds
Lefschetz Theorem on (1 1)-classes
Leschetz Hyperplane Theorem.
Sheaf Cohomology
Abstract in English
The main goal of this work is to present a detailed study of the foundations of Complex Geometry, highlighting its geometric, topological and analytical aspects. Beginning with a preliminary material, such as the basic results on holomorphic functions in one or more variables and the definition and first examples of a complex manifold, we move on to an introduction to sheaf theory and its cohomology, an essential tool to the rest of the work. After a discussion on divisors and line bundles we turn attention to Kähler Geometry and its central results, such as the Hodge Decomposition Theorem, the Hard Lefschetz Theorem and the Lefschetz Theorem on $(1,1)$-classes. After that, we study complex vector bundles and its geometry, focusing on the concepts of connections, curvature and Chern classes. Finally, we finish by describing some aspects of the topology of complex manifolds, such as the Lefschetz Hyperplane Theorem and some of its consequences.
 
WARNING - Viewing this document is conditioned on your acceptance of the following terms of use:
This document is only for private use for research and teaching activities. Reproduction for commercial use is forbidden. This rights cover the whole data about this document as well as its contents. Any uses or copies of this document in whole or in part must include the author's name.
Publishing Date
2012-07-03
 
WARNING: Learn what derived works are clicking here.
All rights of the thesis/dissertation are from the authors
Centro de Informática de São Carlos
Digital Library of Theses and Dissertations of USP. Copyright © 2001-2019. All rights reserved.