Programa de Doutorado: Introdução à Geometria Complexa


Professor: Hossein Movasati
Programa: http://w3.impa.br/~hossein/courses/2017IntroductionToComplexGeometry.html

Pré-requisito: Análise complexa

Teorema de Hartogs no polidisco. Teorema de extensão de Hartogs. Domínios holomorficamente convexos. Pseudoconvexidade. Relações entre convexidade holomorfa e pseudoconvexidade. Lemma e cohomologias de Dolbeault. Problemas de Cousin. Feixes, cohomologia de Cech com coeficientes em feixes. Seqüências exatas curtas e longas associadas. Resoluções finas. Variedades de Stein. Teoremas A e B de Cartan (enunciados) e aplicações. Variedades de Kaehler. Decomposição de Hodge. Relações entre fibrados lineares e divisores. Primeira classe de Chern, Dualidade de Serre.

Referências:
CHABAT, S. - Introduction `a l'Analyse Complexe, Vol.2. MIR, 1990.
GUNNING, R. - Introduction to Holomorphic Functions of Several Variables, Vol. I, II e III Belmont, Ed. Wadsworth, 1990.
FRITZSCHE, K., GRAUERT, H. - From Holomorphic Functions to Complex Manifolds. Graduate Texts in Mathematics, 213. Springer-Verlag, New York, 2002.
VOISIN, C.- Hodge Theory and Complex Algebraic Geometry, I.

Playlist no Youtube:
https://www.youtube.com/playlist?list=PLo4jXE-LdDTSA3nHhZTZzwSQDKmTNd8om

DOWNLOAD

Class 01: (07/03/2017): Holomorphic functions, polydiscs, Osgood's lemma, Cauchy Riemann equations - download
Class 02: (09/03/2017) The C-algebra of holomorphic functions - download
Class 03: (14/03/2017) Germs of analytic varieties, analytic varieties - download
Class 04: (16/03/2017) Embedding dimension, tangent and cotangent space, complex manifolds - download
Class 05: (21/07/2017) Examples of sheaves, sheaves of differential forms, sheaves of ideals, sheaf of invertible functions, constant sheaf - download
Class 06: (23/03/2017) Analytics sheaves and morphism between sheaves, kernel and image of a morphism between sheaves, motivation for Cech cohomology H^1 extension of holomorphic functions from Y to X, Y a subvariety of X - download
Class 07: (28/03/2017) motivation for Cech cohomology H^1 short and long exact sequences, Cech cohomology - download
Class 08: (30/03/2017) Leray Lemma, acyclic sheaves and resolutions - download
Class 09: (04/04/2017) The isomorphism between Cech cohomology with real coefficients and de Rham cohomology - download
Class 10: (06/04/2017) Holomorphically convex domains, Stein varieties - download
Class 11: (11/04/2017) Generalized Cauchy integral formula, Dolbeault's Lemma download
Class 12: (13/04/2017) The isomorphism between Cech cohomology with real coefficients and de Rham cohomology download
Class 13: (25/04/2017) Holomorphically convex domains, Stein varieties. download
Class 14: (07/04/2017) Dolbeault cohomology of a polydisc, Hartog's extension theorem (holomorphic functions outside a compact set extends...) - download
Class 15: (02/05/2017) Positive line bundle in the sense of Kodaira and Grauert, Chern class  - download
Class 16: (04/05/2017) Chern class of a line bundle in Cech and de Rham cohomology - download
Class 17: (09/05/2017)  The equivalence of Kodaira and Grauert positivity of line bundles - download
Class 18: (11/05/2017)  Kodaira vanishing theorem, neighborhoods of analytic varieties - download
Class 19: (16/05/2017)   Kodaira and Grauert vanishing theorem theorems,   Blow-up of  a singular variety - download
Class 20: (18/05/2017)  Kodaira embedding theorem - download
Class 21:(25/05/2017) the equivalence of Kodaira and Grauert positivity of line bundles,download

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