Algebraic families of constant scalar curvature K\"ahler metrics

TL;DR

This paper presents a new proof that the existence of Kähler-Einstein metrics on Fano manifolds is Zariski-open, using differential-geometric estimates rather than stability criteria, with implications for constant scalar curvature Kähler metrics.

Contribution

It introduces a novel proof technique for the openness of Kähler-Einstein metrics on Fano manifolds that bypasses stability conditions, extending ideas to constant scalar curvature metrics.

Findings

Zariski-openness of Kähler-Einstein metrics on Fano manifolds

New proof avoiding stability characterizations

Applicability to constant scalar curvature Kähler metrics

Abstract

We give a new proof of the fact that the condition of a Fano manifold admitting a K\"ahler-Einstein metric is Zariski-open (provided that the automorphism group is discrete). This proof does not use the characterisation involving stability. The arguments involve estimates of Futaki invariants obtained from a differential-geometric "volume estimate", and variants of the algebro-geometric arguments of Stoppa. Many of the ideas apply to constant scalar curvature K\"ahler metrics.