Kähler (& Hyper-kähler) Manifolds

These notes are based on two talks given at the Arithmetic & Algebraic Geometry Seminar of the Korteweg-de Vriesinstituut for mathematics of the Universiteit van Amsterdam. They are intended to give a short introduction to the theory of Kähler manifolds, with a slight focus of applicability to the subject of K3 surfaces. However, they also include other interesting results not related to K3 surfaces, most noteably at the end. Most of these notes (all except the section on Hyper-Kähler manifolds) are based on a course on complex manifolds and an essay written by myself, respectively lectured and supervised by Julius Ross, DPMMS, Cambridge. I will use the summation convention for repeated indices, where convenient.