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Browse materialsMIT 18.117 · Spring 2005

Topics in Several Complex Variables

Subjects: Algebra · Geometry · Mathematics

Art: Anonymous painter,15th century, Westerner and Arab practicing geometry 15th century manuscript, 15th… · Public domain

Instructor: Prof. Victor Guillemin
Term: Spring 2005
Level: Graduate
What's inside: 1 lecture note · 42 other files · 3 web pages
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Source: MIT OpenCourseWare 18.117 Topics in Several Complex Variables, Spring 2005, Prof. Victor Guillemin — original course page on ocw.mit.edu ↗. License: CC BY-NC-SA 4.0. Study order: MIT's own listing order, section by section. Taken from MIT's published course files, downloaded from ocw.mit.edu on September 23, 2026. How RedLectern is built.

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This resource includes information on differential operators, smoothing operators, fourier analysis… (18117_lec16_18.pdf)PDF Download

MIT OpenCourseWare, Prof. Victor Guillemin, 18.117 Topics in Several Complex Variables, Spring 2005.

License: CC BY-NC-SA 4.0 · Edition: Spring 2005 · Open the original course on OCW ↗

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Topics in Several Complex Variables

Before you start

What's here, and what isn't

Materials by section

Lecture Notes

Assignments

Other materials from this course

Lecture 1: Functions of one Complex Variable, Cauchy Integral Formula, Taylor Series, Analytic Continuation

Lecture 2: Cauchy Integral Formula (cont.), Inhomogeneous C.R. Equation, Riemann Equation in One Variable, Functions of Several Complex Variables

Lecture 3: The Inhomogeneous Cauchy-Riemann Equation in Several Variables, Hartog’s Theorem

Lecture 4: Applying Hartog’s Theorem, The Dolbeault Complex, Exactness of the Dolbeault Complex on Polydisks

Lecture 5: The Holomorphic Version of the Poincare Lemma

Lecture 6: The Inverse Function Theorem and the Implicit Function Theorem for Holomorphic Mappings

Lecture 7: Complex Manifolds: Affine and Projective Varieties

Lecture 8: Complex Manifolds: Affine and Projective Varieties (cont.)

Lecture 9: Sheaf Theory and Sheaf Cohomology

Lecture 10: The DeRham Theorem for Acyclic Covers

Lecture 11: Identification of Cech Cohomology Groups with the Cohomology Groups of the Dolbeault Complex

Lecture 12: Linear Aspects of Symplectic and Kaehler Geometry

Lecture 13: The Local Geometry of Kaehler Manifolds, Strictly Pluri-subharmonic Functions and Pseudoconvexity

Lecture 14: The Ricci Form and the Kaehler Einstein Equation

Lecture 15: The Fubini Study Metric on CPⁿ

Lecture 16: Differential Operators on Rⁿ and Manifolds

Lecture 17: Smoothing Operators, Fourier Analysis on the n-torus

Lecture 18: Pseudodifferential Operators on Tⁿ and Open Subsets of Tⁿ, Elliptic Operators on Compact Manifolds

Lecture 19: Systems of Elliptic Operators and Elliptic Operators on Vector Bundles

Lecture 20: Elliptic Complexes and Examples

Lecture 21: Hodge Theory, the *-operator

Lecture 22: Computing the *-operator

Lecture 23: The *-operator in Kaehler Geometry

Lecture 24: The *-operator in Kaehler Geometry (cont.)

Lecture 25: The Symplectic Version of the Hodge Theory

Lecture 26: The Symplectic Version of the Hodge Theory (cont.)

Lecture 27: The Brylinski Conjecture and the Hard Lefchetz Theorem, Hodge Theory on Riemannian Manifolds

Lecture 28: Basic Facts About Representations of SL(2,R), SL(2,R) Modules of Finite H-type

Lecture 29: Hodge Theory on Kaehler Manifolds

Lecture 30: Hodge Theory on Kaehler Manifolds (cont.)

Lecture 31: Actions of Lie Groups on Manifolds, Hamiltonian G Actions on Symplectic Manifolds

Lecture 32: Symplectic Reduction

Lecture 33: Kaehler Reduction and GIT Theory

Lecture 34: Toric Varieties

Lecture 35: The Cohomology Groups of Toric Varieties

Lecture 36: Stanley’s Proof of the McMullen Conjecture