Fundamentos da Geometria Complexa: aspectos geom\'etricos, topol\'ogicos e anal\'iticos

TL;DR

This work provides a comprehensive overview of complex geometry, covering foundational concepts, sheaf theory, Kähler geometry, vector bundles, and topological results, emphasizing the interconnectedness of these areas.

Contribution

It offers a detailed synthesis of complex geometry's core topics, integrating geometric, topological, and analytical perspectives in a unified presentation.

Findings

Key results in Kähler geometry, including Hodge and Lefschetz theorems.

Insights into the topology of complex manifolds, such as the Lefschetz Hyperplane Theorem.

Development of the theory of complex vector bundles and their characteristic classes.

Abstract

The main goal of this work is to present a detailed study of the foundations of Complex Geometry, highlighting its geometrical, 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 K\"ahler 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.