Chapter 6 - Sheaf Theory

Summary

Introduction

In Section 1 of this chapter we present the basic definitions and constructions of sheaf theory with many motivating examples. In section 2 we give an application of sheaf theory to prove the existence and uniqueness of the envelope of holomorphy of a Riemann domain. In section 3 we define the sheaf cohomology groups of a sheaf of groups over a paracompact space using fine resolutions. Amongst the most important results we prove are Leray's theorem and the existence of a canonical, natural isomorphism between Čech cohomology and sheaf cohomology. We conclude with a number of important examples and computations involving the 1st. Chern class.

Sheaves and presheaves

Our aim in this section is to develop the theory of sheaves and show how it provides a unifying topological framework for the study of a diverse range of structures on topological spaces. Our presentation will be geared towards applications in complex analysis and the reader may consult Godement [1] or Tennison [1] for more extensive and general expositions of the theory of sheaves.

Let X be a topological space with topology of open sets U.