Complex Manifolds

NGA course

19 September - 31 October 2023

This is the homepage of the postgraduate course I am giving on Complex Manifolds, as part of the National Graduate Academy Coursework Initiative. Please register to attend the course by clicking there.

Course Description

Definition and examples of smooth manifolds and smooth maps, and of complex manifolds and holomorphic maps. The tangent bundle and its complexification. Differential forms. The decomposition of the complexified tangent bundle. The “del” operator. Complex line bundles and connections. The curvature of a connection. The Chern connection. Cohomology, and the classification of line bundles. Kahler manifolds.

Prerequisites

Undergraduate degree in mathematics. Basic knowledge of topology (eg topological spaces, continuous maps, homeomorphisms, compactness), multivariable calculus (vector fields, integration) and linear algebra.

Lecture Notes

Complex Manifolds NGA Course Lecture Notes. Updated on an ongoing basis!

Format

There will be 6 lectures, of 2 hours each, and a written test at the end. The test is only necessary for those doing the course for official credits, but others might want to write it too in order to cement their learning. Tutorial problems will also be given in class.

Lectures take place on Tuesdays from 2pm-4pm, starting 19 September. Online via Teams, or locally in 1006A in the Stellenbosch Mathematics Division. Recordings will be available afterwards in Teams. Note that there will be no lecture on Tuesday 26 September.

Lectures

Textbook

Our reference texts will be:

  • For preparatory material on smooth manifolds: Eduard Looijenga, Lecture Notes on Smooth Manifolds
  • For complex manifolds: Yohann Le Floch, A Brief Introduction to Berezin-Toeplitz Operators on Compact Manifolds, Springer, CRM Short Courses, 2018.

Additional fun reading