Foundations of Complex Geometry: geometric, topological and analytic aspects.

Author(s)

Lucas Kaufmann Sacchetto

• Document type: Master's Dissertation
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI)
• Defense date: 2012-05-03

Examining board members

Claudio Gorodski; Andrew James Clarke; Paulo Domingos Cordaro

• Advisor: Claudio Gorodski

Abstract

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. (AU)

• FAPESP's process: 10/02525-5 - Complex Differential Geometry
• Grantee: Lucas Kaufmann Sacchetto
• Support Opportunities: Scholarships in Brazil - Master