Misha Verbitsky

Instituto Nacional de Matemática Pura e Aplicada

Complex analytic spaces

Sala 236, Mondays 13:30-15:00, Wednesdays 17:00-18:30, March-November 2023

Course materials

• Part 1: Differential forms and Cauchy theorem
  • Reading: Henri Cartan, "Elementary theory of analytic functions of one or several complex variables", chapters I-IV.
  • Lecture 1: Cauchy theorem (August 7).
  • Handout 1: Hodge decomposition.
• Part 2: Complex manifolds, sheaves, complex analytic spaces.
  • Reading:
    • S. Ramanan, Global calculus, Chapter 1.
    • H. Grauert, R. Remmert, Coherent Analytic Sheaves, Chapter 1
    • Roger Godement, Topologie algebrique et theorie des faisceaux, Chapitre II.
  • Lecture 2: Sheaves, schemes, complex analytic spaces (August 9).
  • Handout 2: Sheaves and manifolds.
• Part 3: Weierstrass preparation theorem.
  • Reading: Robert C. Gunning, Hugo Rossi, Analytic functions of several complex variables, Sections IIA, IIB, IIC, IID.
  • Lectures:
    • Lecture 3: Germs of functions and Weierstrass preparation theorem (August 14).
    • Lecture 4: The proof of Weierstrass preparation theorem (August 16).
    • Lecture 5: Weierstrass division theorem (August 21).
  • Handout 3: Weierstrass preparation theorem
  • Handout 4: Weierstrass division theorem
• Part 4: Ring of germs and their algebraic properties
  • Reading:
    • M. F. Atiyah, I. G. MacDonald, "Introduction to Commutative Algebra" , Chapters 1-2 and 6-7 (with exercises).
    • B.L. van der Waerden, J.R. Schulenberger, Algebra, Volume II, chapter 16 ("Theory of polynomial ideals")
  • Lecture 6: Noetherianity and germs of complex subvarieties (August 23)
  • Lecture 7: Irreducible decomposition (August 28)
  • Handout 5: Algebraic properties of the ring of germs (Noetherianity, factoriality)
• Part 5: Galois theory.
  • Basic notions of Galois theory
    • Reading: KG Ramanathan, Lectures on the Algebraic Theory of Fields, chapters 1-5.
    • Reading: Jenia Tevelev, Algebra: lecture notes, chapters 1-4.
  • Galois coverings and Galois categories
    • Askold Khovanskii, Galois Theory, Coverings, and Riemann Surfaces
    • Anna Cadoret, Galois Categories
  • Lecture 9: Artin's primitive element theorem (September 4)
  • Handout 6: Crash-course of Galois theory
• Part 6: Rückert Nullstellensatz.
  • Reading: Three lectures on Hilbert Nullstellensatz, Geometria Algébrica I, 2018:
    • 2018, Lecture 1: Hilbert Nullstellensatz
    • 2018, Lecture 2: Hilbert Nullstellensatz and categories
    • 2018, Lecture 3: Strong Nullstellensatz
  • Lecture 8: Regular coordinate systems (August 30).
  • Lecture 10: Rückert Nullstellensatz (September 6).
  • Handout 7: 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:
    • Lecture 11: Discriminant (September 11).
    • Lecture 12: Meromorphic maps and smooth points of complex varieties (September 13).
    • Lecture 13: Finite morphisms (September 18).
    • Lecture 14: Dimension (September 20).
    • Lecture 15: The maximum principle and the maps with finite fibers (September 25).
  • Handout 8: Smooth points and meromorphic maps
  • Handout 9: Divisors and the maximum principle
  • Handout 10: Finite morphisms
• Part 8: Remmert and Remmert-Stein theorem.
  • Slides:
    • Lecture 16: Remmert rank (September 27).
    • Lecture 16½: Finite group quotients (October 18).
    • Lecture 17: Remmert and Remmert-Stein theorem (October 1-3).
    • Lecture 18: Chow theorem and othe applications of Remmert-Stein (October 8).
  • Handout 11: Remmert rank
• Part 9: Zariski main theorem.
  • Lecture 19: Regular local rings and Auslander-Buchsbaum theorem (October 11).
  • Lecture 20: Degree of a map and degree of a field extension (October 16).
  • Lecture 21: Zariski main theorem (October 18).
• Part 10: Normalization
  • Reading:
    • Robin Hartshorne, Algebraic Geometry, Ch. II, Section 3, Exercise 3.8.
    • Gert-Martin 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
  • Reading:
    • J.-P. Demailly, Complex analytic and differential geometry, Chapter I, section 6.
    • H. Grauert, R. Remmert, A. Huckleberry, Theory of Stein spaces
  • Lecture 26: Domains of holomorphy (November 6).
  • Lecture 27: Strong maximum principle (November 13).
  • Lecture 28: Pluri-Laplacian (November 15).
  • Lecture 29: Plurisubharmonic functions (November 27).
  • Lecture 30: Stein manifolds (November 29).

  Miscellanea

  • Written test assignments
    • Test 1: Hodge decomposition (distributed October 23, deadline October 30).
    • Test 2: Weierstrass preparation theorem (distributed October 30, deadline November 6).
    • Test 3: Meromorphic functions (distributed November 3, deadline November 13).
    • Test 4: Germs of complex varieties (distributed November 3, deadline November 13).
    • Test assignment results (updated).
• Class assignment results, for main assignments (updated)
• Exam problems
• Personal problem sets, obtained randomly with this program.
• Final results (pending)

Misha Verbitsky IMPA