The Kähler-Ricci flow

Valentino Tosatti

The Ricci flow is an evolution equation which deforms a Riemannian metric in the direction of its Ricci tensor, with the goal of making it more "round". If the underlying manifold is complex and the initial metric is Kahler, then so are the evolved metrics, and the flow is called the Kähler-Ricci flow. When the manifold is also compact, the flow becomes intimately related to the complex structure of the manifold. If the manifold is algebraic, Song and Tian have recently initiated a program relating the convergence properties of the flow to the minimal model program in birational geometry.

In these lectures I will give an introduction to the Kähler-Ricci flow, and present some results which fit in this framework. Possible topics to be covered are: the characterization of the maximal existence time of the flow, the long time behavior on some minimal Kahler manifolds, the formation of singularities in finite time.