K-stability and Fujita approximation

TL;DR

This paper develops a formula for non-Archimedean Monge-Ampère measures of big models, linking algebraic geometry and non-Archimedean analysis, with implications for stability conjectures in complex geometry.

Contribution

It introduces a new formula for non-Archimedean Monge-Ampère measures and connects it to the uniform Yau-Tian-Donaldson conjecture via Zariski decompositions.

Findings

Derived a positive intersection formula for non-Archimedean Mabuchi functional

Reduced the Yau-Tian-Donaldson conjecture to a conjecture on Zariski decompositions

Established a link between algebraic models and non-Archimedean measures

Abstract

We derive a formula for non-Archimedean Monge-Amp\`{e}re measures of big models. As applications, we derive a positive intersection formula for non-Archimedean Mabuchi functional, and further reduce the uniform Yau-Tian-Donaldson conjecture for polarized manifolds to a conjecture about the existence of approximate Zariski decompositions that satisfy some asymptotic vanishing condition.