Introduction to Complex Geometry, March-June 2017.

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