Complex surfaces are compact complex manifolds of dimension 2. Their classification was given by Kodaira in the 1960s, in a series of difficult papers. The main results of Kodaira's classification were improved by Buchdahl and Lamari in the 1990s, who used 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) way.

I will begin with the proof of Gauduchon's theorem, constructing a special metric on a conformal class of a Hermitian form on any compact complex manifold. Then, I will introduce the currents and proceed to the Buchdahl-Lamari theorem, claiming that any complex surface with $b_1$ pair 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 relevant definitions and results and explain their context.

Program:
1. Hopf's maximum principle

2. Gauduchon's Theorem on the Existence and Uniqueness of the Gauduchon Metric

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

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

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

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