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.

**Assessment**

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.

**Certification**

Edukite courses are free to study. To successfully complete a course you must submit all the assignment of the course as part of the assessment. Upon successful completion of a course, you can choose to make your achievement formal by obtaining your Certificate at a cost of £49.

Having an Official Edukite Certification is a great way to celebrate and share your success. You can:

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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 | ||

Assessment | |||

Submit Your Assignment | 00:00:00 | ||

Certification | 00:00:00 |

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