110.611 COMPLEX GEOMETRY

Fall 2007

MTW 11, Hodson 305

Instructor: Bernard Shiffman

Text: Griffiths and Harris, Principles of Algebraic Geometry

Prerequisites: 601, 605, 607, 615

COURSE INFORMATION: This course is an introduction to analysis on complex manifolds. We'll cover mainly Chapter 0, and section 1 of Chapter 1, as well as additional material not in the text. Here is a brief outline of topics to be covered:

I. Analysis on Cⁿ

· holomorphic functions

· differential forms and currents

· Bergman and Szegö kernels

· local ring of holomorphic functions, Weierstrass theorems

· analytic varieties, 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

· Kähler manifolds, introduction to Hodge theory

Assignments:

· #1, due September 17

· #2, due September 24

· #3, due October 1

· #4, due October 16

· #5, due October 29

· #6, due November 5

· #7, due November 12

· #8: Read pp. 20-27 in Griffiths and Harris

· #9, due December 3

· #10, due December 10

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) ;
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.

· Bicomplex & Leray's Theorem

· Divisors on complex manifolds

· Poincaré-Lelong II