Abstract:

Hyper-Kähler manifolds are, in a sense, the quaternionic analogues of Kähler manifolds. As such, many points of view can be taken when studying them. They have a Riemannian metric which is Ricci-flat, and they have three Kähler forms and complex structures (in fact, they have infinitely many), which allow for new constructions, like hyper-Kähler reduction and twistor spaces. We will talk about some of the most important properties of these spaces from the perspectives of Riemannian, symplectic and complex geometry, and we will see some examples, like those arising as moduli spaces of gauge theoretic problems.