$\mathbb{G}$-uniform weighted K-stability for models on klt varieties

TL;DR

This paper extends the concept of weighted K-stability to singular varieties, establishing links to the Mabuchi functional and demonstrating stability preservation under polarization perturbations, with implications for weighted extremal metrics.

Contribution

It generalizes weighted K-stability results to singular and weighted settings, connecting stability to functional coercivity and metric existence.

Findings

Weighted K-stability implies Mabuchi functional coercivity on klt varieties.

Stability is preserved under polarization perturbations in the toric case.

Existence of weighted extremal metrics is established under certain conditions.

Abstract

In this paper, we make a generalization of the results in \cite{Li22a} to the singular and weighted setting. In particular, we show that on a polarized projective klt variety, the $\mathbb{G}$-uniform weighted K-stability for models implies the $\mathbb{G}$-coercivity of the weighted Mabuchi functional. In the toric case, we further show that the $(\mathbb{C}^{\times})^n$-uniform $(\mathrm{v},\mathrm{w}\cdot\ell_{\mathrm{ext}})$-weighted K-stability is preserved when perturbing the polarization on the resolution, which implies the existence of the weighted extremal metric(s) on the resolution if the weight function $\mathrm{v}$ is log-concave.