Categorical Torelli theorem for smooth Fano varieties
Abstract
Let X be a smooth Fano variety and H an ample line bundle with −KX=rH and r≤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 𝒪X⊥ of its structure sheaf in Db(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 −qKX is very ample and the corresponding embedding is cut out by equations of degree at most (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.
Related
- Categorical Torelli theorem for (weighted) hypersurfaces via singularity category
- A Strong Global Torelli theorem for OG6 type varieties
- Torelli theorem for the moduli spaces of twisted vector bundles on smooth projective curves
- Slope stability of tangent bundles of smooth toric Fano varieties
- Categorical Torelli and Fourier-Mukai partners of hypersurfaces
BibTeX
@misc{lin2026categorical,
title = {{Categorical Torelli theorem for smooth Fano varieties}},
author = {Xun Lin and Shizhuo Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25428},
eprint = {2610.25428},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25428},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Categorical Torelli theorem for smooth Fano varieties
AU - Xun Lin
AU - Shizhuo Zhang
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25428
UR - https://arxiv.org/abs/2610.25428
AB - Let X be a smooth Fano variety and H an ample line bundle with −K_X=rH and r≤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 𝒪_X^⊥ of its structure sheaf in 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 (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.
ER -