An algebraic comparison of K̂-stability and Kβ-stability
Abstract
We establish a uniform quantitative comparison between the non-Archimedean Mabuchi functional and the Darvas--Zhang quantised Mabuchi functional. For a smooth polarised variety with finite automorphism group, we provide an algebraic proof that uniform K̂-polystability is equivalent to uniform Kβ-stability for every sufficiently large rational β>1, without passing through the existence of a cscK metric. The main input is a square-root estimate for directional derivatives of non-Archimedean energies, proved by intersection theory. We also study a reduced Kβ-stability condition when the identity component of the polarised automorphism group is reductive and the Futaki character vanishes. We conclude by characterising K-semistability in terms of asymptotic Kβ-semistability and prove the quantitative convergence of the corresponding stability thresholds.
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BibTeX
@misc{papazachariou2026algebraic,
title = {{An algebraic comparison of $\widehat\mathrm K$-stability and $\mathrm K^β$-stability}},
author = {Theodoros Stylianos Papazachariou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15333},
eprint = {2609.15333},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15333},
}
RIS
TY - UNPB
TI - An algebraic comparison of K̂-stability and K^β-stability
AU - Theodoros Stylianos Papazachariou
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15333
UR - https://arxiv.org/abs/2609.15333
AB - We establish a uniform quantitative comparison between the non-Archimedean Mabuchi functional and the Darvas--Zhang quantised Mabuchi functional. For a smooth polarised variety with finite automorphism group, we provide an algebraic proof that uniform K̂-polystability is equivalent to uniform K^β-stability for every sufficiently large rational β>1, without passing through the existence of a cscK metric. The main input is a square-root estimate for directional derivatives of non-Archimedean energies, proved by intersection theory. We also study a reduced K^β-stability condition when the identity component of the polarised automorphism group is reductive and the Futaki character vanishes. We conclude by characterising K-semistability in terms of asymptotic K^β-semistability and prove the quantitative convergence of the corresponding stability thresholds.
ER -