The Kähler-Ricci flow on compact Kähler manifolds

Abstract

These lecture notes are based on five hours of lectures given at the Park City Math Institute in the summer of 2013. The notes are intended to be a leisurely introduction to the Kähler-Ricci flow on compact Kähler manifolds. They are aimed at graduate students who have some background in differential geometry, but do not necessarily have any knowledge of Kähler geometry or the Ricci flow. There are exercises throughout the text. The goal is that by the end, the reader will learn the basic techniques in the Kähler-Ricci flow and know enough to be able to explore the current literature.The material covered by these notes is as follows. In the first lecture, we give a quick introduction to some of the main definitions and tools of Kähler geometry. In Lecture 2, we introduce the Kähler-Ricci flow and give some simple examples, before stating, and in Lecture 3 proving, the maximal existence time theorem for the flow. In Lecture 4 we prove long time convergence results in the cases when the manifold has negative or zero first Chern class. Finally in Lecture 5 we discuss more recent work on the behavior of the flow on Kähler surfaces. We also "go beyond" the Kähler-Ricci flow by discussing a new flow on complex manifolds called the Chern-Ricci flow.The Kähler-Ricci flow started as a small branch of the study of Hamilton's Ricci flow, but by now is itself a vast area of research. As a consequence, we have had to omit many topics. For the interested reader seeking more complete expository sources: the chapter [66] by Jian Song and the author contains many of the results of these notes and much more; the works [4,8,31] (in the same volume as [66]) and the more general survey [53] are excellent sources of information.The author thanks Matt Gill who was the teaching assistant for this course, for his help in writing the exercises. In addition, thanks go to the organizers, Hubert Bray, Greg Galloway, Rafe Mazzeo and Natasa Sesum, of the research program of the 2013 PCMI Summer Session for giving the author the opportunity to participate in this exciting event. Discussions with researchers and graduate students at the Park City Math Institute were invaluable in shaping the form of these notes. The author also thanks Valentino Tosatti for some helpful comments on a previous version of these notes.

KAWA lecture notes on the Kähler–Ricci flow

“…There are already two excellent set of lecture notes on the Kähler-Ricci flow, by Song-Weinkove [68] and Weinkove [88]. While preparing these notes, I have benefitted greatly from these references, and in fact the exposition in Section 3 follows [68,88] rather closely (I decided to keep this material here because many similar arguments are used in later sections). On the other hand, in Sections 4 and 5, which form the bulk of these notes, I have decided to focus on rather recent results which are not contained in [68,88].…”

Positivity in Foliated Manifolds and Geometric Applications

“…[56,53]) which is an application of Yau's Schwarz Lemma [59] and Royden's trick [34]. We refer to [32,52] for the simplified calculation.…”

A Parabolic Monge–Ampère Type Equation of Gauduchon Metrics

“…We can find more details about this standard discussion in the proof of [38,Theorem 3.1] (cf. [34,48]).…”