{
  "abstract": "We formulate a dg-categorical analogue of the Cartan determinant conjecture using numerical Grothendieck groups. Let $D$ be a smooth proper dg category over an algebraically closed field $k$. The Euler pairing on $K_0(D)$ descends to a nondegenerate bilinear form on the numerical Grothendieck group $K_{\\mathrm{num}}(D)$, and hence defines a numerical Cartan determinant $\\det M_χ(D)$. While the classical Cartan determinant conjecture predicts that $\\det M_χ(A)=1$ for finite-dimensional smooth algebras $A$, this equality is no longer expected for general smooth proper dg categories. However, we propose the following positivity question: $\\det M_χ(D)>0.$ Our main result proves this positivity when $D=\\operatorname{Perf}(X)$ for a large class of smooth projective varieties $X$. More precisely, we show that $\\det M_χ(X)>0$ if either $\\dim X$ is odd, or $X$ has even dimension and satisfies the numerical Hodge standard conjecture.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29598",
  "author_total": 3,
  "authors": [
    {
      "name": "Yeqin Liu"
    },
    {
      "name": "Yu Shen"
    },
    {
      "name": "Ziyu Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e0edt48jza49w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29598",
  "fulltext_url": null,
  "id": "e0edt48jza49w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29598",
  "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|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214300849",
  "pdf_url": "https://arxiv.org/pdf/2609.29598",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e0edt48jza49w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29598",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29598"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Cartan Determinants in algebraic geometry",
  "title_text": "Cartan Determinants in algebraic geometry",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/e0edt48jza49w",
  "venue": "arXiv",
  "versions": []
}