{
  "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 $\\widehat{\\mathrm K}$-polystability is equivalent to uniform $\\mathrm K^β$-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 $\\mathrm{K}^β$-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 $\\mathrm K^β$-semistability and prove the quantitative convergence of the corresponding stability thresholds.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15333",
  "author_total": 1,
  "authors": [
    {
      "name": "Theodoros Stylianos Papazachariou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9bvaa9ner8mhp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15333",
  "fulltext_url": null,
  "id": "9bvaa9ner8mhp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15333",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2608|Geometry and complex manifolds"
    ]
  ],
  "openalex": "W7213287025",
  "pdf_url": "https://arxiv.org/pdf/2609.15333",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9bvaa9ner8mhp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15333",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15333"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "An algebraic comparison of $\\widehat{\\mathrm K}$-stability and $\\mathrm K^β$-stability",
  "title_text": "An algebraic comparison of K̂-stability and K^β-stability",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/9bvaa9ner8mhp",
  "venue": "arXiv",
  "versions": []
}