＜概要＞On the space of Kaehler metrics

On the infinite dimensional space of Kaehler metrics, Mabuchi, Semmes and Donaldson introduce a Weil-Peterson type metric. Under this metric, this space becomes an infinite dimensional symmetric space of non-compact type with semi-negative curvature. Donaldson made several important conjectures concerning the geometric structure of this space; and the resolution of these conjectures of Donaldson has important consequences on Kaehler geometry. For instance, the well known problem of uniqueness of "best metric" in each Kaehler class is settled in recent years through this and related program. In this lecture, I will give an expository account of this program as well as some recent updates on Kaehler geometry.