Canonical Kahler metrics and Arithmetics -- Generalising Faltings heights

TL;DR

This paper generalizes Faltings heights to broader arithmetic varieties, linking them with Kahler-Einstein geometry and the Minimal Model Program, and introduces an arithmetic Yau-Tian-Donaldson conjecture.

Contribution

It extends Faltings heights to general arithmetic varieties and proposes an arithmetic Yau-Tian-Donaldson conjecture, connecting arithmetic and geometric properties.

Findings

Established relations between heights and Kahler-Einstein geometry

Proposed and partially confirmed the arithmetic Yau-Tian-Donaldson conjecture

Extended Faltings heights to broader classes of arithmetic varieties

Abstract

We extend the Faltings modular heights of abelian varieties to general arithmetic varieties and show direct relations with the Kahler-Einstein geometry, the Minimal Model Program, heights of Bost and Zhang, and give some applications. Along the way, we propose arithmetic Yau-Tian-Donaldson conjecture, an equivalence of a purely arithmetic property of variety and its metrical property, and partially confirm it.