{
  "abstract": "Let $X$ be a complex klt Fano variety. We study the rational total index $τ_X$ and the nef complexity $c_X=\\dim X+ρ(X)-τ_X$. Previous joint work of the author with Enwright, Filipazzi, Gongyo, Moraga, Svaldi, and Wang established the corresponding nonnegativity and zero-case characterization in the broader framework of a special class of generalized log Calabi--Yau pairs. Here we give a direct proof in the Fano setting, prove that there is no value strictly between zero and one, and classify all klt Fano varieties with $c_X<2$, without assuming $\\mathbb Q$-factoriality. At complexity one, after splitting off projective-space factors, the remaining factor is a normal quadric of Picard number one, a linear blow-up of projective space, or an integral $(1,1)$ hypersurface in a product of projective spaces. If $1<c_X<2$, only two weighted-projective families occur; in particular, no smooth Fano variety has nef complexity strictly between one and two. We also determine the spectrum when the dimension is allowed to vary: \\[ \\{0\\}\\cup\\left\\{2-\\frac2q \\ \\middle| \\ q\\in\\mathbb Z_{\\geq2}\\right\\} \\cup(\\mathbb Q\\cap[2,\\infty)). \\] Most proofs in this paper were developed from arguments generated by OpenAI's ChatGPT through discussions initiated and guided by the author; see the statement on the use of generative AI.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27676",
  "author_total": 1,
  "authors": [
    {
      "name": "Kiwamu Watanabe"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ckabhgh6t4jv1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.27676",
  "fulltext_url": null,
  "id": "ckabhgh6t4jv1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27676",
  "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"
    ],
    [
      "arxiv",
      "math.CV"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7223123190",
  "pdf_url": "https://arxiv.org/pdf/2610.27676",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ckabhgh6t4jv1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.27676",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27676"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Nef complexity of klt Fano varieties: the full spectrum and classification below two",
  "title_text": "Nef complexity of klt Fano varieties: the full spectrum and classification below two",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/ckabhgh6t4jv1",
  "venue": "arXiv",
  "versions": []
}