Introduction to Complex Geometry,
March-June 2017.
- Class 1: (07/03/2017): Holomorphic functions, polydiscs,
Osgood's lemma, Cauchy Riemann equations.
- Class 2: (09/03/2017) The C-algebra of holomorphic functions,
- Class 3: (14/03/2017) Germs of analytic varieties, analytic
varieties
- Class 4: (16/03/2017) Embedding dimension, tangent and cotangent
space, complex manifolds,
- Extra class 5: (17/03/2017) Weierstrass uniformization theorem, the
ring of germs of holomorphic functions is Noetherian
- Class 6: (21/07/2017) Examples of sheaves, sheaves of
differential forms, sheaves of ideals, sheaf of invertible functions,
constant sheaf.
- Class 7: (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.
- Extra class 8: (24/03/2017) Extensions theorems in codimension one
with the boundedness condition, extension theorem in codimension two, Hartogs extension theorem.
- Class 9: (28/03/2017) motivation for Cech cohomology H^1
short and long exact sequences, Cech cohomology,
- Class 10: (30/03/2017) Leray Lemma, acyclic sheaves and
resolutions.
- Extra class 11: (31/03/2017) , Generalized Cauchy integral formula, Dolbeault's Lemma
- Class 12: (04/04/2017) The isomorphism between Cech cohomology
with real coefficients and de Rham cohomology
- Class 13: (06/04/2017)Holomorphically convex domains, Stein varieties.
- Extra
Class 14: (07/04/2017) Dolbeault cohomology of a polydisc, Hartog's
extension theorem (holomorphic functions outside a compact set
extends.....)
- Class 16 (11/04/2017) Cartan's A and B theorem, application to extension of holomorphic functions, Stein covering.
- Class 17 (13/04/2017) Line bundles, Examples of analytic varieties, complex tori, projective varieties, ....
- Extra Class 18: Domain of holomorphy is the same same as holomorphically convex.
- Class 19 (18/04/2017) Strongly convex and plurisubharmonic functions,
- Class 20 (20/04/2017) Pseudoconvex domains
- Class 21 (02/05/2017), Positive line bundle in the sense of Kodaira and Grauert, Chern class
- Class 21 (04/05/2017), Chern class of a line bundle in Cech and de
Rham cohomology,
- Class 22 (09/05/2017) the equivalence of Kodaira and Grauert positivity of line bundles,
- Class 23 (11/05/2017) Kodaira vanishing theorem, neighborhoods of analytic varieties
- Class 24 (16/05/2017) Kodaira and Grauert vanishing theorem theorems, Blow-up of a singular variety.
- Calss 25 (18/05/2017) Kodaira embedding theorem.
- Class 26 (23/05/2017) The case of curves inside surfaces I
- Class 27 (25/05/2017) The case of curves inside surfaces II
- The rest of the course is the proof of Hodge Decomposition and Serre duality based on the first book of Voisin.
Exams: First
exam 24/04/2017, Second exam 29/05/2017, third exam 23/06/2017