Complex surfaces are compact complex varieties of dimension 2. Their classification was given by Kodaira in the 1960s, in a series of difficult articles. The main results of Kodaira’s classification were improved by Buchdahl and Lamari in the 1990s, who used the advances in pluripotential theory obtained by Demailly, Nadel and Siu. Now, Kodaira’s immense work can be presented in a more compact (and more consistent) form.

I’ll start with the proof of Gauduchon’s theorem, constructing a special metric on a conformal class of a Hermitian form on any compact complex variety. I will then introduce currents and proceed to the Buchdahl-Lamari theorem, claiming that any complex surface with even $b_1$ is Kahler. If time permits, I will use the Buchdahl-Lamari results to prove the Kahler version of the Nakai-Moishezon theorem and finish with the structure theorem for non-Kahler elliptic surfaces. I would assume basic knowledge of Hodge theory and elliptic equations, but I will state all the relevant definitions and results and explain their context.

Program:
1. Hopf’s maximum principle

2. Gauduchon’s theorem on the existence and uniqueness of Gauduchon’s metric

3. Some notions of the theory of topological vector spaces: Frechet spaces, Montel spaces, strong and weak duality, reflexive spaces. Current space and its reflexivity.

4. Positive currents, Lelong numbers, Demailly’s regularization theorem.

5. Hodge decomposition on non-Kahler surfaces, Harvey-Lawson-Sullivan duality criteria for the existence of special metrics on complex varieties, Buchdahl-Lamari theorem.

6. (*) Kahler’s version of the Nakai-Moishezon theorem: description of Kahler’s cone.

7. (*) Classification of non-Kahler surfaces and the structure theorem for non-Kahler elliptic fibrations.

The course website is http://verbit.ru/IMPA/Surfaces-2025/

 

References:
Buchdahl and Lamari in the 1990s, who used the advances in pluripotential theory made by Demailly, Nadel and Siu. Now Kodaira’s immense body of work can be presented in a more compact (and more consistent) form.