18.117 | Spring 2005 | Graduate

Topics in Several Complex Variables


Lec # topics
Complex Variable Theory on Open Subsets of Cn
1 Functions of one Complex Variable, Cauchy Integral Formula, Taylor Series, Analytic Continuation
2 Cauchy Integral Formula (cont.), Inhomogeneous C.R. Equation, Riemann Equation in One Variable, Functions of Several Complex Variables
3 The Inhomogeneous Cauchy-Riemann Equation in Several Variables, Hartog’s Theorem
4 Applying Hartog’s Theorem, The Dolbeault Complex, Exactness of the Dolbeault Complex on Polydisks
5 The Holomorphic Version of the Poincare Lemma
6 The Inverse Function Theorem and the Implicit Function Theorem for Holomorphic Mappings
Theory of Complex Manifolds, Kaehler Manifolds
7 Complex Manifolds: Affine and Projective Varieties
8 Complex Manifolds: Affine and Projective Varieties (cont.)
9 Sheaf Theory and Sheaf Cohomology
10 The DeRham Theorem for Acyclic Covers

Identification of Cech Cohomology Groups with the Cohomology Groups of the Dolbeault Complex

12 Linear Aspects of Symplectic and Kaehler Geometry
13 The Local Geometry of Kaehler Manifolds, Strictly Pluri-subharmonic Functions and Pseudoconvexity
14 The Ricci Form and the Kaehler Einstein Equation
15 The Fubini Study Metric on CPn
Elliptic Operators and Pseudo-differential Operators
16 Differential Operators on Rn and Manifolds
17 Smoothing Operators, Fourier Analysis on the n-torus
18 Pseudodifferential Operators on Tn and Open Subsets of Tn, Elliptic Operators on Compact Manifolds
Hodge Theory on Kaehler Manifolds
19 Systems of Elliptic Operators and Elliptic Operators on Vector Bundles
20 Elliptic Complexes and Examples
21 Hodge Theory, the *-operator
22 Computing the *-operator
23 The *-operator in Kaehler Geometry
24 The *-operator in Kaehler Geometry (cont.)
25 The Symplectic Version of the Hodge Theory
26 The Symplectic Version of the Hodge Theory (cont.)
27 The Brylinski Conjecture and the Hard Lefchetz Theorem, Hodge Theory on Riemannian Manifolds
28 Basic Facts About Representations of SL(2,R), SL(2,R) Modules of Finite H-type
29 Hodge Theory on Kaehler Manifolds 
30 Hodge Theory on Kaehler Manifolds (cont.)
Geometric Invariant Theory
31 Actions of Lie Groups on Manifolds, Hamiltonian G Actions on Symplectic Manifolds
32 Symplectic Reduction
33 Kaehler Reduction and GIT Theory
34 Toric Varieties
35 The Cohomology Groups of Toric Varieties
36 Stanley’s Proof of the McMullen Conjecture

Course Info

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