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- Type of Document: M.Sc. Thesis
- Language: Farsi
- Document No: 42068 (02)
- University: Sharif University of Technology
- Department: Mathematical Sciences
- Advisor(s): Jafari, Amir
- Abstract:
- Cohomology groups of complex algebraic varieties with coefficients in ℂ, can be considered more than just a vector space and they can equipped with various and rich linear algebra structures which are functorial with respect to morphisms between them. In this thesis we first review classic Hodge theory, Hodge decomposition and Lefschetz decomposition theorems which enable us to introduce the concept of pure Hodge structure on cohomology group H^n (X,C) of a compact Kähler manifold X. Then we define Frölicher spectral sequence for a complex manifold X and show that it will degenerate at E_1 when X is compact and Kähler. For generalization of pure Hodge structures to smooth non projective varieties over ℂ, we need new concept of mixed Hodge structures which was first defined by P.Deligne in 1971. Our main goal is studying Deligne’s proof. We will first define category of mixed Hodge structures and prove it is an abelian category and then by introducing differential forms with logarithmic singularities we will show that cohomology groups of a smooth algebraic variety over ℂ, carry mixed Hodge structures in a functorial way that is generalization of pure Hodge structures on cohomology of smooth projective varieties which defined earlier.
- Keywords:
- Kahler Manifold ; Mixed Hodge Structure ; Differential Forms with Logarithmic Singular ; Frolicher Spectral Sequence ; Leray Spectral Sequence
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