Instituto Nacional de Matemática Pura e Aplicada
Complex analytic spaces
Sala 236, Mondays 13:3015:00, Wednesdays 17:0018:30, MarchNovember 2023
Annotation
Course materials
 Part 1: Differential forms and Cauchy theorem
 Part 2: Complex manifolds, sheaves, complex analytic spaces.
 Part 3: Weierstrass preparation theorem.
 Part 4: Ring of germs and their algebraic properties
 Part 5: Galois theory.
 Part 6: Rückert Nullstellensatz.
 Part 7: Smooth and singular points, dimension, finite morphisms.
 Reading:
 Robert C. Gunning, Hugo Rossi, Analytic functions of several complex variables, Sections IIIA, IIIB.
 Normalization and finite dependence:
 M. F. Atiyah, I. G. MacDonald,
"Introduction to Commutative Algebra" ,
Chapter 5.
 B. L. van der Waerden, J. R. Schulenberger, Algebra, Volume II,
Chapter 17.
 J.P. Demailly, Complex analytic and differential geometry,
Chapter II, section 7.
 Slides:
 Handout 8: Smooth points and meromorphic maps
 Handout 9: Divisors and the maximum principle
 Handout 10: Finite morphisms
 Part 8: Remmert and RemmertStein theorem.
 Part 9: Zariski main theorem.
 Part 10: Normalization
 Reading:
 Robin Hartshorne,
Algebraic Geometry, Ch. II, Section 3, Exercise 3.8.
 GertMartin Greuel, Christoph Lossen, Eugenii I. Shustin,
Introduction to singularities and deformations,
Ch. I, Section 1.8
 Grauert H., Remmert R.,
Coherent analytic sheaves, Chapter 8.
 Lecture 22: The integral
closure
(October 23).
 Lecture 23:
Normalization of complex algebraic varieties
(October 25).
 Lecture 24:
Normalization
(October 30).
 Lecture 25:
Locally bounded meromorphic functions
(November 1).
 Part 11: Holomorphic convexity and Stein manifolds
Miscellanea

Class assignment results, for main assignments (updated)
 Exam problems
 Personal problem sets, obtained
randomly with this program.
 Final results (pending)