Maths 212: Several Complex Variables, Fall 2019

TuTh 11:00am--12:30pm 740 Evans



Professor  Maciej Zworski
email: zworski@math.berkeley.edu
Office: 801 Evans Hall
Office hours: by appointment

Prerequisites: 202AB or equivalent; some knowledge of manifolds and differential forms.

Textbooks:
  1. Lars Hörmander, An Introduction to Complex Analysis in Several Variables, 3rd edition, (Chapters I and II) (A helpful commentary by John Erik Fornaess can be found here)
  2. Lars Hörmander, The Analysis of Linear Partial Differential Operators I (Section 9.4)
  3. Lars Hörmander, The Analysis of Linear Partial Differential Operators II (Section 15.1)
  4. Dmitry Khavinson, Holomorphic Partial Differential Equations and Classical Potential Theory (Chapters 1-8; these cover Section 9.4 of 2 but with more examples and background)
  5. Robert Berman, Bo Berndtsson, Johannes Sjöstrand, Asymptotics of Bergman kernels

    Syllabus:

    1. Review of the theory of functions of one complex variables
    2. Holomorphy, power series in several complex variables, the Hartogs principle
    3. Domains of holomorphy, plurisubharmonic functions, pseudoconvex domains
    4. Solving non-homogeneous Cauchy--Riemann equations in spaces defined using global strictly plurisubharmonic weights (Hörmander's L2 estimates)
    5. Solving partial differential equations with holomorphic coefficients: theorems of Cauchy--Kovalevskaya, Zerner and Bony--Shapira
    6. Local Bergman kernel asymptotics: a semiclassical version of the theorems of Fefferman and Boutet de Monvel--Sjöstrand
    7. Bergman kernel asymptotics for powers of positive line bundles over compact complex manifolds; the Catlin--Tian--Yau--Zelditch asymptotics, the Kodaira embedding theorem.

    Grading:

    There will be biweekly homework assignments covering basic aspects of the course.