Lectures on Stability and Constant Scalar Curvature

TL;DR

This paper introduces current research trends in stability conditions for Kähler metrics of constant scalar curvature, discussing classical and new stability notions, analytic methods, and their interrelations in complex differential geometry.

Contribution

It provides an overview of recent developments in stability criteria like K-stability and Donaldson's GIT, connecting them with analytic techniques for constant scalar curvature Kähler metrics.

Findings

Overview of classical and new stability notions

Discussion of analytic methods like energy estimates and Monge-Ampère equations

Connections between stability conditions and geometric analysis

Abstract

An introduction is provided to some current research trends in stability in geometric invariant theory and the problem of Kaehler metrics of constant scalar curvature. Besides classical notions such as Chow-Mumford stability, the emphasis is on several new stability conditions, such as K-stability, Donaldson's infinite-dimensional GIT, and conditions on the closure of orbits of almost-complex structures under the diffeomorphism group. Related analytic methods are also discussed, including estimates for energy functionals, Tian-Yau-Zelditch approximations, estimates for moment maps, complex Monge-Ampere equations and pluripotential theory, and the Kaehler-Ricci flow