Complex surfaces.

Complex surfaces are compact complex
manifolds of dimension 2. Their classification
was given by Kodaira in 1960-ies, in a series of
difficult papers. The key results of
Kodaira's classification were improved by Buchdahl
and Lamari in the 1990-ies, who used the advances in
pluripotential theory, obtained by Demailly, Nadel
and Siu. Now the Kodaira's immense oeuvre can
be presented in a more compact (and more
consistent) way.

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

Program:

1. Hopf maximum principle

2. Gauduchon's theorem on existence and uniqueness of Gauduchon metric

3. Some notions of theory of topological vector spaces:
Frechet spaces, Montel spaces, strong and weak duality, reflexive
spaces. Space of currents 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 existence
of special metrics on complex manifolds,
Buchdahl-Lamari theorem.

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

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

The site of the course is
http://verbit.ru/IMPA/Surfaces-2025/