Introduction to Kähler Manifolds

This chapter provides an introduction to the basic results on the topology of compact Kähler manifolds that underlie and motivate Hodge theory. This chapter consists of five sections which correspond, roughly, to the five lectures in the course given during the Summer School at the International Centre for Theoretical Physics (ICTP). The five topics under discussion are: complex manifolds; differential forms on complex manifolds; symplectic, Hermitian, and Kähler structures; harmonic forms; and the cohomology of compact Kähler manifolds. There are also two appendices. The first collects some results on the linear algebra of complex vector spaces, Hodge structures, nilpotent linear transformations, and representations of sl(2,ℂ) and serves as an introduction to many other chapters in this volume. The second contains a new proof of the Kähler identities by reduction to the symplectic case.