One of the central approaches to the study of complex algebraic varieties is via Hodge theory. It provides, in particular, a connection between algebro-geometric properties of the varieties and their topology. It turns out that Hodge theory can be applied to a wider and more natural class of complex manifolds, namely to Kähler manifolds. These are complex manifolds that carry a special Riemannian metric called Kähler metric. Apart from complex projective manifolds this class contains, for example, all compact complex tori, and a general complex torus is not an algebraic variety. In the course we will give an introduction to the theory of Kähler manifolds.

Prerequisites for the course include: basic theory of differentiable manifolds (vector bundles, connections, differential forms, etc.), sheaf theory (resolutions, cohomology), basic algebraic topology and complex analysis. We will also have to assume without proof some facts about elliptic differential operators that can be found in the literature. The course will be aimed at master students and post-graduates, but motivated bachelor students will also be welcomed. Subjects marked with (*) in the list below are more difficult and will be discussed if time permits.

Tentative syllabus:

1. Hermitian bundles, connections, curvature, Chern classes.
2. Kähler metrics, differential operators on Kähler manifolds.
3. Hodge decomposition, Hodge structures on the cohomology of Kähler manifolds.
4. Lefschetz decomposition and Lefschetz theorems.
5. Positive bundles, Kodaira embedding theorem.
6. (*) Deformations of complex structures and variations of Hodge structures.
7. (*) Calabi's conjecture, Calabi-Yau manifolds, hyperkähler manifolds.

Literature:

1. Voisin C. Hodge theory and complex algebraic geometry. Vol I, II. Cambridge University Press, 2002
2. Wells, R. O. Differential analysis on complex manifolds. Springer-Verlag, 1980
3. Huybrechts D. Complex geometry. And introduction. Springer, 2005
4. Moroianu A. Lectures on Kähler geometry. Cambridge University Press, 2007
5. Demailly J.-P. Complex analytic and differential geometry, online book https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf
6. Besse A. Einstein manifolds, Springer-Verlag, 1987

Please, address Andrey Soldatenkov, aosoldatenkov@gmail.com, for Zoom data.

Financial support. The course is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, Agreement no. 075-15-2022-265).

Lecturer
Soldatenkov Andrey

Institutions
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center