212: Several Complex Variables, Fall 2019
202AB or equivalent; some knowledge of manifolds and differential forms.
- 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)
- Lars Hörmander, The Analysis of Linear Partial Differential Operators I (Section 9.4)
- Lars Hörmander, The Analysis of Linear Partial Differential Operators II (Section 15.1)
- 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)
- Robert Berman, Bo Berndtsson, Johannes Sjöstrand, Asymptotics of Bergman kernels
- Review of the theory of functions of one complex
- Holomorphy, power series in several complex
variables, the Hartogs principle
- Domains of holomorphy, plurisubharmonic functions,
- Solving non-homogeneous
Cauchy--Riemann equations in spaces defined using
global strictly plurisubharmonic weights (Hörmander's L2 estimates)
- Solving partial differential equations with holomorphic coefficients: theorems of
Cauchy--Kovalevskaya, Zerner and Bony--Shapira
- Local Bergman kernel asymptotics: a semiclassical version of the theorems of Fefferman and Boutet de Monvel--Sjöstrand
- Bergman kernel asymptotics for powers of positive line bundles over compact
complex manifolds; the Catlin--Tian--Yau--Zelditch asymptotics, the Kodaira embedding theorem.
There will be biweekly homework assignments covering basic aspects of the course.