Misha Verbitsky

Instituto Nacional de Matemática Pura e Aplicada

Complex variables

Sala 344, Mondays, Fridays, 13:30-14:00, March-June 2019

Course materials

• Part 1: Differential forms and Cauchy theorem
  • Reading assignment: Henri Cartan, "Elementary theory of analytic functions of one or several complex variables", chapters I-IV.
  • Lecture 1: Cauchy theorem (March 25).
  • Handout 1: Hodge decomposition.
• Part 2: Complex manifolds, almost complex manifolds and sheaves.
  • Reading assignment: S. Ramanan, Global calculus, Chapter 1.
  • Handout 2: Sheaves and manifolds.
• Part 3: Weierstrass preparation theorem.
  • Reading assignment: Robert C. Gunning, Hugo Rossi, Analytic functions of several complex variables, Sections IIA, IIB.
  • Handout 3: Weierstrass preparation theorem.
• Part 4: Weierstrass division theorem.
  • Reading assignment: Robert C. Gunning, Hugo Rossi, Analytic functions of several complex variables, Sections IIC, IID.
  • Handout 4: Weierstrass division theorem.
• Part 5: Ring of germs: algebraic properties
  • Reading assignment: M. F. Atiyah, I. G. MacDonald, "Introduction to Commutative Algebra" , Chapters 1-2 and 6-7 (with exercises).
  • Reading assignment: B.L. van der Waerden, J.R. Schulenberger, Algebra, Volume II, chapter 16 ("Theory of polynomial ideals")
  • Handout 5: Ring of germs (Noetherianity, factoriality).
• Part 6-7: Galois theory.
  • Coursera: Ekaterina Amerik, Introduction to Galois Theory, lecture 4.
  • Reading assignment: KG Ramanathan, Lectures on the Algebraic Theory of Fields, chapters 1-5.
  • Reading assignment: 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
  • Handout 6: Galois theory I.
  • Handout 7: Artin's primitive element theorem.
• Part 8: Hilbert's Nullstellensatz.
  Reading assignment: three lectures on Hilbert's Nullstellensatz, Geometria Algébrica I, 2018:
  • Hilbert Nullstellensatz
  • Hilbert Nullstellensatz and categories
  • Strong Nullstellensatz
  • Handout 8: Hilbert's Nullstellensatz.
• Part 9-12: Meromorphic functions, divisors, dimension, finite morphisms.
  • Reading assignment: 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.
  • Handout 9: Rückert's Nullstellensatz.
  • Handout 10: Smooth points and meromorphic maps.
  • Handout 11: Divisors, dimension, maximum principle.
  • Handout 12: Finite morphisms.

• Test assignment 1: Grassmann algebra (April 15). Results.
• Test assignment 2: Holomorphic functions and complex structures (May 13). Results.
• Test assignment 3: Almost complex structures on manifolds (May 20). Results.
• Test assignment 4: Weierstrass preparation theorem (May 27). Results.
• Exam (July 6).

Misha Verbitsky IMPA