18.117 | Spring 2005 | Graduate

Topics in Several Complex Variables

Lecture Notes

The lecture notes were prepared by Jonathan Campbell, a student in the class. They are available as a single file (PDF - 1.4 MB) or mapped to the lecture topics below. The notes for lectures 16, 17, and 18 are from the Supplementary Notes on Elliptic Operators.

Lec # Topics
Complex Variable Theory on Open Subsets of Cn
1 Functions of one Complex Variable, Cauchy Integral Formula, Taylor Series, Analytic Continuation (PDF)
2 Cauchy Integral Formula (cont.), Inhomogeneous C.R. Equation, Riemann Equation in One Variable, Functions of Several Complex Variables (PDF)
3 The Inhomogeneous Cauchy-Riemann Equation in Several Variables, Hartog’s Theorem (PDF)
4 Applying Hartog’s Theorem, The Dolbeault Complex, Exactness of the Dolbeault Complex on Polydisks (PDF)
5 The Holomorphic Version of the Poincare Lemma (PDF)
6 The Inverse Function Theorem and the Implicit Function Theorem for Holomorphic Mappings (PDF)
Theory of Complex Manifolds, Kaehler Manifolds
7 Complex Manifolds: Affine and Projective Varieties (PDF)
8 Complex Manifolds: Affine and Projective Varieties (cont.) (PDF)
9 Sheaf Theory and Sheaf Cohomology (PDF)
10 The DeRham Theorem for Acyclic Covers (PDF)
11 Identification of Cech Cohomology Groups with the Cohomology Groups of the Dolbeault Complex (PDF)
12 Linear Aspects of Symplectic and Kaehler Geometry (PDF)
13 The Local Geometry of Kaehler Manifolds, Strictly Pluri-subharmonic Functions and Pseudoconvexity (PDF)
14 The Ricci Form and the Kaehler Einstein Equation (PDF)
15 The Fubini Study Metric on CPn (PDF)
Elliptic Operators and Pseudo-differential Operators
16 Differential Operators on Rn and Manifolds (PDF)
17 Smoothing Operators, Fourier Analysis on the n-torus (PDF)
18 Pseudodifferential Operators on Tn and Open Subsets of Tn, Elliptic Operators on Compact Manifolds (PDF)
Hodge Theory on Kaehler Manifolds
19 Systems of Elliptic Operators and Elliptic Operators on Vector Bundles (PDF)
20 Elliptic Complexes and Examples (PDF)
21 Hodge Theory, the *-operator (PDF)
22 Computing the *-operator (PDF)
23 The *-operator in Kaehler Geometry (PDF)
24 The *-operator in Kaehler Geometry (cont.) (PDF)
25 The Symplectic Version of the Hodge Theory (PDF)
26 The Symplectic Version of the Hodge Theory (cont.) (PDF)
27 The Brylinski Conjecture and the Hard Lefchetz Theorem, Hodge Theory on Riemannian Manifolds (PDF)
28 Basic Facts About Representations of SL(2,R), SL(2,R) Modules of Finite H-type (PDF)
29 Hodge Theory on Kaehler Manifolds (PDF)
30 Hodge Theory on Kaehler Manifolds (cont.) (PDF)
Geometric Invariant Theory
31 Actions of Lie Groups on Manifolds, Hamiltonian G Actions on Symplectic Manifolds (PDF)
32 Symplectic Reduction (PDF)
33 Kaehler Reduction and GIT Theory (PDF)
34 Toric Varieties (PDF)
35 The Cohomology Groups of Toric Varieties (PDF)
36 Stanley’s Proof of the McMullen Conjecture (PDF)

Course Info

As Taught In
Spring 2005
Learning Resource Types
Lecture Notes
Problem Sets