Lectures on Complex Differential Geometry

Course Synopsis

We will start with a revision lecture briskly reviewing the most relevant points of real differential geometry. The rest of the series will focus mainly on complex differential geometry, covering certain supporting materials as we go. Starting from complex manifolds we will cover complex structures on vector spaces, almost complex structures and almost complex manifolds, connections on complex manifolds, the Dolbeault cohomology group and Hodge decomposition, Hermitian structures and Hermitian manifolds, the Kähler form and Kähler manifolds (we may or may not touch briefly on symplectic manifolds), and lastly we will end with a short discussion of Calabi-Yau manifolds.

Course Prerequisites

This course assumes that students are fairly familiar with the following prerequisite material:

• Complex Euclidean Space and Calculus: All of it, with an emphasis on Holomorphic Maps and the Cauchy-Riemann Equations.
• Linear Algebra: Definition of a vector space, Vector Bases, Linear Maps, Change of Bases, Jacobians, Dual Spaces.
• Multivariable Calculus: Partial Derivatives, Chain Rule, Multiple Integral, Smooth Maps.
• Differential Forms: Definition, exterior algebras, exterior derivatives and the wedge product.
• Multilinear Algebra or Tensor Algebra: Definition of a tensor space, index notation and manipulation, Einstein summation convention, Rank, Covariance vs Contravariance.
• Group Theory (Minimal): Definition, Homomorphisms.
• Topology: Definition, Neighbourhoods, Connectedness, Compactness, Continuity, Continuous Maps, Homeomorphisms, Open and Closed Sets.
• Real Differential Geometry (Optional): Topological Manifolds, Charts and Atlases, Differential Manifolds, Tangent and Cotangent Spaces. As we get further along the list I will explain more and more these concepts as we go, the last two especially, so minimal prior contact is necessary.

Reading Material

The main reading material for this course is Lectures on Differential Geometry by Chern, Chen and Lam. For topology review one may consult Topology by James Munkres. Further reading material will be recommended as the course progresses.

Class Forum

I would like to invite everyone who is considering joining to the Discord server for the course. Here you will get announcements, updates, and a forum for discussion of the topics in the course.

Class Timing

Sunday, 9:00 pm