About J-flow, J-balanced metrics, uniform J-stability and K-stability

TL;DR

This paper explores the relationship between J-balanced metrics, stability notions, and their algebraic characterizations, extending the understanding of the J-flow in complex geometry.

Contribution

It establishes a purely algebro-geometric criterion for the existence of J-balanced metrics, linking them to Chow stability, and discusses stability criteria beyond integral classes.

Findings

J-balanced metrics characterized by Chow stability

Criteria for uniform J-stability and K-stability

Extension to non-integral Kähler classes

Abstract

From the work of Dervan-Keller, there exists a quantization of the critical equation for the J-flow. This leads to the notion of J-balanced metrics. We prove that the existence of J-balanced metrics has a purely algebro-geometric characterization in terms of Chow stability, complementing the result of Dervan-Keller. We also obtain various criteria that imply uniform J-stability and uniform K-stability. Eventually, we discuss the case of K\"ahler classes that may not be integral over a compact manifold.