CEU Electronic Theses and Dissertations, 2020
| Author | Menjanahary, Jean Michel |
|---|---|
| Title | Sheaf Cohomology and its Applications in Analytic Geometry |
| Summary | The purpose of this project is to explore the notion of sheaves of abelian groups and sheaf cohomology over a topological space, and apply them to investigate analytic varieties and several problems in analytic geometry. Any sheaf of abelian groups over a topological space X can be considered as an algebraic object as it is a collection of abelian groups, parametrized by the space X. It also can be considered as a topological object by lifting the topology of X under the natural projection given by the parameters of the collection. From this, we apply homological algebra to construct sheaf cohomology, which helps us to investigate the global sections of sheaves. For computation of sheaf cohomology, we describe two methods. The first one is based on fine resolutions, which is useful in case of paracompact Hausdorff spaces. The second one uses Čech cohomology, which is very useful when we have natural open coverings of the space X. As an application, we investigate the notion of Stein varieties in terms of sheaf cohomology. Furthermore, we analyze two problems in the theory of holomorphic functions in several variables, namely, the additive and multiplicative Cousin's problems. |
| Supervisor | Némethi, András |
| Department | Mathematics MSc |
| Full text | https://www.etd.ceu.edu/2020/menjanahary_jean.pdf |
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