Cohomology of compact hyperkähler manifolds

Compact hyperkähler manifolds naturally appear in the classification of complex algebraic varieties with trivial canonical bundle. Their geometry is to a large extent determined by the properties of their cohomology algebra. I will try to present some ideas that are important for the study of hyperkähler manifolds, focusing on the natural Lie algebra action on their cohomology. More specifically, I would like to discuss how this Lie algebra action helps us to understand degenerations of hyperkähler manifolds.