110.612 COMPLEX GEOMETRY

Fall 2009

Tue Thu 10:30-11:45, Shaffer 202

Instructor: Bernard Shiffman

*This course should be numbered 110.611. The course number was changed by the registrar.

Prerequisites: 601, 605, 607, 615

COURSE INFORMATION: This course is an introduction to analysis on complex manifolds. Here is a brief outline of topics to be covered:

I. Analysis on Cⁿ

· holomorphic functions

· Bergman kernels

· differential forms and currents

· local ring of holomorphic functions, Weierstrass theorems, analytic varieties

· plurisubharmonic functions, Poincaré-Lelong formula

II. Complex Manifolds

· deRham and Dolbeault cohomology

· sheaves and cohomology, deRham and Dolbeault theorems

· meromorphic functions and divisors

· line bundles and Chern classes

· introduction to Kähler manifolds

Primary text:

· Griffiths and Harris, Principles of Algebraic Geometry. (We'll cover mainly Chapter 0, and section 1 of Chapter 1.)

Additional material:

· Bergman kernel (from S. G. Krantz, Function Theory of Several Complex Variables)

· A theorem of Federer

· Complex varieties (Ch. III of Narasimhan, Introduction to the Theory of Analytic Spaces); the entire book is available here (from a JHU IP address)

· Sheaves (from Gunning, Vol. III)

· Long exact homology sequences (from Hatcher, Algebraic Topology) Long exact cohomology sequences are similar; just interchange n-1 with n+1 and raise the indices. The entire book is available here.

· For further reading: Jean-Pierre Demailly, Complex Analytic and Differential Geometry

Assignments:

· #1, due September 15

· #2, due September 29

· #3, due October 6

· #4, due October 20

· #5, due October 27

· #6, due November 3

· #7, due November 12

Last updated 11/5/09