Topics in Complex Geometry

Course Overview

This PhD-level course offers an overview to the fundamental concepts and techniques of complex differential geometry.

The aim of the course is to provide an overview of the main ideas in classical complex geomert and some of the most important results of the 20^th century in the aera:

• Kodaira embedding theorem
• Kodaira-Spencer deformation theory and the Tian-Todorov lemma
• Yau's proof of the Calabi-Yau conjecture

Prerequisites: Complex analysis and differential geometry. Familiarity with Riemannian geometry and vector bundles is desirable.

Notes and references

I will mainly draw from

• D. Huybrechts, Complex Geometry: An Introduction,
• J.-P. Demailly, Complex Analytic and Differential Geometry,
• A. Moroianu, Kähler geometry,
• S. Kobayashi, Differential Geometry of Complex Vector Bundles,
• K. Kodaira, Complex Manifolds and Deformation of Complex Structures
• R. O. Wells, Differential Analysis on Complex Manifolds

The lecture notes of the course are available here. (Last update: 29^th of November, 2025)

If you find any mistakes or typos, please let me know by dropping me an email.

Slides

The slides of the course are available here:

• Lecture 1 (3^rd of November)

Exercises

This a compilation of the exercises related to the course. There will not be an official solutions, but I am be happy to discuss any questions you may have via email.

The exercises are available here (Last update: 29^th of October,2026)