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/
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”