Recent progress in K\"ahler geometry

TL;DR

This paper reviews recent advances in Kähler geometry, focusing on extremal Kähler metrics, obstructions to their existence, the metric structure of Kähler potentials, and developments in Kähler Ricci flow.

Contribution

It summarizes recent progress in understanding the metric structure of Kähler potentials and their applications to existence problems and Ricci flow in Kähler geometry.

Findings

Advances in understanding extremal Kähler metrics

New insights into obstructions to metric existence

Progress in Kähler Ricci flow analysis

Abstract

In recent years, there are many progress made in K\"ahler geometry. In particular, the topics related to the problems of the existence and uniqueness of extremal K\"ahler metrics, as well as obstructions to the existence of such metrics in general K\"ahler manifold. In this talk, we will report some recent developments in this direction. In particular, we will discuss the progress recently obtained in understanding the metric structure of the infinite dimensional space of Kaehler potentials, and their applications to the problems mentioned above. We also will discuss some recent on Kaehler Ricci flow.