Relative Chow stability and optimal weights

TL;DR

This paper establishes the equivalence between two notions of balanced embeddings for polarized Kähler manifolds, providing a GIT characterization and linking optimal weights to automorphism actions.

Contribution

It proves the equivalence of relative balanced and σ-balanced embeddings, answering a key open question and connecting stability notions with automorphism group actions.

Findings

Equivalence between relative balanced and σ-balanced embeddings.

GIT characterization of σ-balanced embedding existence.

Relation of optimal weights to automorphism group actions.

Abstract

For a polarized K\"ahler manifold $(X, L)$, we show the equivalence between relative balanced embeddings introduced by Mabuchi and $\sigma$-balanced embeddings introduced by Sano, answering a question of Hashimoto. We give a GIT characterization of the existence of a $\sigma$-balanced embedding, and relate the optimal weight $\sigma$ to the action of $\mathrm{Aut}_0(X,L)$ on the Chow line of $(X, L)$.