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The aim of this course is to cover harmonic theory on complex manifolds, the Hodge decomposition theorem, the Hard Lefschetz theorem, and Vanishing theorems. Some results and tools on deformation and uniformization of complex manifolds are also discussed in this course. If the students are expecting to know about some information about several complex variables then this course is the right course for them.


This course does not involve any written exams. Students need to answer 5 assignment questions to complete the course, the answers will be in the form of written work in pdf or word. Students can write the answers in their own time. Each answer needs to be 200 words (1 Page). Once the answers are submitted, the tutor will check and assess the work.


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Course Credit: MIT

Course Curriculum

Module: 01
Functions of one Complex Variable CV 00:05:00
Functions of Several Complex Variables CV 00:05:00
The Inhomogeneous Cauchy-Riemann Equation in Several Variables, Hartog’s Theorem 00:05:00
Exactness of the Dolbeault Complex on Polydisks 00:05:00
The Holomorphic Version of the Poincare Lemma 00:10:00
The Inverse Function 00:05:00
Module: 02
Complex Manifolds: Affine and Projective Varieties 00:10:00
Complex Manifolds: Affine and Projective Varieties (cont.) 00:05:00
Sheaf Theory and Sheaf Cohomology 00:05:00
The DeRham Theorem for Acyclic Covers 00:05:00
Identification of Cech Cohomology 00:05:00
Linear Aspects of Symplectic and Kaehler Geometry 00:10:00
Module: 03
The Local Geometry of Kaehler Manifolds, Strictly Pluri-subharmonic Functions and Pseudoconvexity 00:10:00
The Ricci Form and the Kaehler Einstein Equation 00:10:00
The Fubini Study Metric on CPn 00:10:00
Differential Operators on Rn and Manifolds 00:10:00
Smoothing Operators, Fourier Analysis on the n-torus 00:10:00
Pseudodifferential Operators on Tn 00:10:00
Module: 04
Systems of Elliptic Operators and Elliptic Operators on Vector Bundles 00:10:00
Elliptic Complexes and Examples 00:10:00
Hodge Theory, the *-operator 00:10:00
Computing the *-operator 00:10:00
The *-operator in Kaehler Geometry 00:10:00
The *-operator in Kaehler Geometry (cont.) 00:10:00
The Symplectic Version of the Hodge Theory 00:10:00
Module: 05
The Symplectic Version of the Hodge Theory (cont.) 00:10:00
Hodge Theory on Riemannian Manifolds 00:10:00
Basic Facts About Representations of SL(2,R), SL(2,R) Modules of Finite H-type 00:10:00
Hodge Theory on Kaehler Manifolds 00:10:00
Hodge Theory on Kaehler Manifolds (cont.) 00:10:00
Actions of Lie Groups on Manifolds, Hamiltonian G Actions on Symplectic Manifolds 00:10:00
Module: 06
Symplectic Reduction 00:10:00
Kaehler Reduction and GIT Theory 00:10:00
Toric Varieties 00:10:00
The Cohomology Groups of Toric Varieties 00:10:00
Stanley’s Proof of the McMullen Conjecture 00:10:00
Submit Your Assignment 00:00:00
Certification 00:00:00

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