Hyper-Kählerian varieties are natural generalizations of K3 surfaces. As with complex tori, these varieties exist naturally in the compact Kählerian framework, but those that are projective and thus belong to algebraic geometry are dense in moduli space. If we restrict ourselves to projective hyper-Kählerian varieties, their study is linked (through the study of moduli spaces via the application of periods) to Shimura varieties and automorphic forms. The lecture introduces the results of Hodge theory and deformation theory needed to show that the deformations of compact Kählerian varieties with trivial canonical fibers (in particular hyper-Kählerian varieties) are unobstructed and that the application of periods is a local isomorphism on the period domain, which is defined by the Beauville-Bogomolov form whose existence is a remarkable topological property of these varieties and has considerable consequences for their topology. Various versions of Torelli's theorems are stated. In addition to Hodge theory, the following two aspects of the subject are covered:
- complex differential geometry: Kähler-Einstein metrics, quaternionic structure and twistor lines;
- algebraic geometry: construction of hyper-Kählerian varieties and study of their deformation class.