The main goal of this short course is to present a proof of Calabi_Yau theorem. It was first conjectured by Calabi that any volume form on a compact Kahler manifold can be realized as the volume form associated to a Kahler metric. In a seminal work, Yau proved Calabi's conjecture. A very important consequence of Calabi-Yau theorem is the existence of Kahler-Ricci flat metrics on compact Kahler manifolds with trivial canonical bundle.

Outline of the course:
1) background material on complex manifolds, Hermitian and Kahler metrics, Ricci form
2) Some background on elliptic PDE such as Schauder estimates
3) Yau's original proof on apriori $C^0$ estimate
4) Some advance approach to Calabi-Yau theorem
5) an overview on complex Monge-Ampere via Pluripotential theory (if time permitted)