A quantization proof of the uniform Yau-Tian-Donaldson conjecture

TL;DR

This paper proves a uniform version of the Yau-Tian-Donaldson conjecture for twisted Kähler-Einstein metrics using quantization and pluripotential theory, providing new criteria for existence of constant scalar curvature Kähler metrics.

Contribution

It introduces a quantization approach to establish the uniform Yau-Tian-Donaldson theorem without relying on non-Archimedean methods, and offers a new computable existence criterion.

Findings

$oldsymbol{ ext{delta-invariant}}$ coincides with the optimal exponent in a Moser-Trudinger inequality

Established a uniform existence theorem for twisted Kähler-Einstein metrics

Provided a new criterion for constant scalar curvature Kähler metrics

Abstract

Using quantization techniques, we show that the $\delta$-invariant of Fujita-Odaka coincides with the optimal exponent in certain Moser-Trudinger type inequality. Consequently we obtain a uniform Yau-Tian-Donaldson theorem for the existence of twisted K\"ahler-Einstein metrics with arbitrary polarizations. Our approach mainly uses pluripotential theory, which does not involve Cheeger-Colding-Tian theory or the non-Archimedean language. A new computable criterion for the existence of constant scalar curvature K\"ahler metrics is also given.