Hyperkahler manifolds
Hyperkahler geometry is geometry of compact
holomorphically symplectic manifolds of Kahler type.
I would give an introduction to the main subjects of
hyperkahler geometry. The course should be accessible to
anybody with basic knowledge of complex algebraic
geometry, differential geometry and Hodge theory,
though I am going to repeat the definitions and
state clearly all the results I am going to use.
Here is the list of possible topics which could
be covered.
1. Levi-Civita connection and its holonomy. Berger's
classification of Riemannian holonomy.
2. Kahler manifolds and holonomy. Calabi-Yau
theorem. Hyperkahler manifolds and special holonomy.
Twistor spaces. Spinors and Clifford algebras. Bochner
vanishing and Bogomolov decomposition theorem.
3. K3 surfaces and their deformation theory.
4. Hyperkahler reduction and quiver spaces.
5. Deformations of hyperkahler manifolds. Global
Torelli theorem.
6. Instantons, stable bundles, Kobayashi-Hitchin correspondence.
Hyperholomorphic bundles and their deformation theory.
7. Trianalytic subvarieties and their desingularization.
Existence and non-existence of trianalytic subvarieties.
8. Supersymmetry and the structure theorem for the
cohomology ring.
9. Matsushita theorem about Lagrangian fibrations
on hyperkahler manifolds. Elliptic fibrations on K3
surfaces. Existence of Lagrangian fibrations and
their deformation theory (Voisin).
10. Hodge monodromy group, mapping class group, automorphism
group of a hyperkahler manifold. Automorphisms of a K3
surface. Classification of automorphisms. Construction
of hyperkahler manifolds with prescribed automorphism groups.
11. Ergodic properties of the mapping class group action
and the applications of ergodicity.
12. MBM classes and the structure of the Kahler cone.
Proof of the Kawamata-Morrison conjecture about the
polyhedral structure of the Kahler cone.
The course page: http://verbit.ru/IMPA/HK-2023/
Literature
A. Besse, ``Einstein manifolds''.
Lectures on Kahler geometru, Andrei Moroianu
http://moroianu.perso.math.cnrs.fr/tex/kg.pdf
Complex analytic and differential geometry, J.-P. Demailly
http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf
Lectures on Kahler manifolds, W. Ballmann
http://people.mpim-bonn.mpg.de/hwbllmnn/notes.html
C. Voisin, ``Hodge theory''.
D. Huybrechts, ``Complex Geometry - An Introduction''