{
  "abstract": "Let $F=G/B$ be the complete flag variety of a connected semisimple complex linear algebraic group. We prove that if a smooth Fano variety $X$ has the same dimension and Picard number as $F$, then the existence of a continuous map $X\\to F$ of odd topological degree implies $X$ is isomorphic to $F$. In particular, a smooth Fano variety homotopy equivalent to $F$ is isomorphic to $F$. The algebraic input is a numerical characterization of complete flags: a positive-dimensional smooth Fano variety $X$ is a complete flag variety if and only if its pseudoindex $ι_X$ is at least two and ${\\rm ker}({\\rm Sym}^2N^1(X)_{\\mathbb R}\\to N^2(X)_{\\mathbb R})$ contains a positive-definite quadratic form. The pseudoindex condition can equivalently be replaced by the divisibility of $-K_X$ by two in ${\\rm Pic}(X)$. The key observation is that a single definite quadratic relation excludes surfaces in every fiber of any elementary contraction; standard contraction theorems then yield smooth ${\\mathbb P}^1$-fibrations and the characterization of complete flags applies. For the odd-degree theorem, Poincaré duality and a mod-two Wu-class argument show that the anticanonical class is divisible by two; comparison of the integral coroot lattices then identifies the flag. Most proofs were developed through extensive discussions with OpenAI's ChatGPT (GPT-5.6 Sol and GPT-6), based on mathematical directions supplied by the author and arguments suggested by ChatGPT; see the AI Disclosure.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41237",
  "author_total": 1,
  "authors": [
    {
      "name": "Kiwamu Watanabe"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/36xb5m49q9jkx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41237",
  "fulltext_url": null,
  "id": "36xb5m49q9jkx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41237",
  "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.AT"
    ],
    [
      "arxiv",
      "math.CV"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "arxiv",
      "math.RT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41237",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/36xb5m49q9jkx.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41237",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41237"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Odd-Degree Rigidity and a Numerical Characterization of Complete Flag Varieties",
  "title_text": "Odd-Degree Rigidity and a Numerical Characterization of Complete Flag Varieties",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/36xb5m49q9jkx",
  "venue": "arXiv",
  "versions": []
}