Kähler geometry is a natural extension of complex projective geometry where the tools of Hodge theory are still available and allow to study the link between topology and complex geometry.
The easy examples of projective complex manifolds deforming into non-projective ones are the abelian varieties (complex tori). The subject of the lectures will be hyper-Kähler (or quaternionic) manifolds, which also share this property. Although this geometry may seem extremely restricted, there are many (families of) examples, all built via algebraic geometry.