Height on GIT quotients and Kempf-Ness theory

TL;DR

This paper advances the understanding of heights on GIT quotient varieties by generalizing existing bounds and developing a Kempf-Ness theory in Berkovich spaces, bridging algebraic and analytic perspectives.

Contribution

It extends Burnol's height construction to GIT quotients and introduces a Kempf-Ness theory in Berkovich spaces, unifying algebraic and analytic approaches.

Findings

Generalized height bounds for semi-stable points

Proved Burnol's formula in the Berkovich setting

Developed a Kempf-Ness theory for Berkovich spaces

Abstract

In this paper we study heights on quotient varieties in the sense of Geometric Invariant Theory (GIT). We generalise a construction of Burnol and we generalise diverse lower bounds of the height of semi-stable points due to Bost, Zhang, Gasbarri and Chen. In order to prove Burnol's formula for the height on the quotient we develop a Kempf-Ness theory in the setting of Berkovich analytic spaces, completing the former work of Burnol.