Prerequisites:
202AB or equivalent + some basic knowledge of differential
forms and smooth manifolds.
Textbooks:Lars Hörmander,
An Introduction to Complex Analysis in Several Variables, 3rd edition,
J.-P. Demailly, Complex Analytic and Differential Geometry, on-line lecture notes.
Syllabus:
Review of the theory of functions of one complex
variables;
Holomorphy, power series in several complex
variables, Hartogs property;
Domains of holomorphy, plurisubharmonic functions,
pseudoconvex domains;
Entire functions, weighted L2 spaces, Bergman
projectors for L2 spaces with quadratic plurisubharmonic weights;
Applications: Berezin-Toeplitz quantization,
Catlin-D'Angelo-Quillen Theorem, analytic proof of the
Nullstallensatz;
Solving the ∂ equation in L2 spaces with global plurisubharmonic weights;
Fefferman/Boutet de Monvel-Sjöstrand asymptotics
for Bergman projectors for "nonlinear" weights;
Quick introduction to complex manifolds, complex
line bundles and powers of line bundles;
Applications of Bergman kernel asymptotics:
Kodaira's embedding theorem and the Catlin-Tian-Yau-Zelditch
asymptotics.
Grading:
You are asked to write a short paper on
a research topic related to the course subject,
for instance, based on a topic from the texts
not covered in class.