{
  "abstract": "Let $X$ be a smooth Fano variety and $H$ an ample line bundle with $-K_X=rH$ and $r\\le\\dim X-2$. We prove that the appropriately defined Kuznetsov component of $X$ with respect to $H$, together with its rotation functor, determines $(X,H)$ when $H$ is very ample, and more generally when $H$ is globally generated and the image of the morphism defined by $|H|$ has a hypersurface as projective dual. As a consequence we prove Perry's conjecture, which predicts that a smooth Fano variety $X$ is determined up to isomorphism by the orthogonal complement $\\mathcal{O}_X^{\\perp}$ of its structure sheaf in $\\mathrm{D}^b(X)$, for all smooth Fano varieties of dimension at least three with very ample anticanonical bundle, and more generally with globally generated anticanonical bundle whose image has a hypersurface as dual. We also prove it when a multiple $-qK_X$ is very ample and the corresponding embedding is cut out by equations of degree at most $(\\mathrm{dim} X-1)/q$. The idea is to read off $X$ from its hyperplane sections. We show that the Kuznetsov component with its rotation functor contains the residual categories of all hyperplane sections of $X$, singular ones included. The sections whose categories are not smooth form the projective dual of $X$, and classical projective duality recovers $X$. When $H$ is not very ample, the nodes of a single tangent hyperplane section, defined over a function field and detected by its singularity category, recover $X$ as a finite cover of its image. Applications include Fano hypersurfaces, all prime Fano threefolds of index one and genus at least three, cyclic covers of quadrics, Fano complete intersections and Gushel--Mukai varieties.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25428",
  "author_total": 2,
  "authors": [
    {
      "name": "Xun Lin"
    },
    {
      "name": "Shizhuo Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/99z7w9zt4aj8j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25428",
  "fulltext_url": null,
  "id": "99z7w9zt4aj8j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25428",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7223064334",
  "pdf_url": "https://arxiv.org/pdf/2610.25428",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/99z7w9zt4aj8j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25428",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25428"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Categorical Torelli theorem for smooth Fano varieties",
  "title_text": "Categorical Torelli theorem for smooth Fano varieties",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/99z7w9zt4aj8j",
  "venue": "arXiv",
  "versions": []
}