Categorical Torelli and Fourier-Mukai partners of hypersurfaces
Abstract
Let X⊂PPM−1 be a smooth hypersurface of degree d<M, and let Ku(X) be its Kuznetsov component, the orthogonal complement of the line bundles OOX,…,OOX(M−d−1) in the derived category. We prove that Ku(X) determines X for every smooth quartic in PP4k+1, k≥1; together with earlier work, this gives categorical Torelli for every smooth hypersurface of degree d<M with dnmid M. The same algebraic reconstruction proves the classical generic Torelli theorem for these quartics, which completes generic Torelli for smooth hypersurfaces whose degree does not divide M, cubic surfaces excepted. When d divides M, the Kuznetsov component is a Calabi--Yau category, as for cubic fourfolds, where categorical Torelli is known to fail. We show that, apart from cubic fourfolds, such a hypersurface has a number of Fourier--Mukai partners bounded by a constant depending only on d and M, and that categorical Torelli holds in sufficiently large degree, in every dimension. The main tool is a deformation theorem: when geometric deformations surject onto the second Hochschild cohomology of the Kuznetsov components, an equivalence deforms together with the hypersurfaces, so statements proved for general hypersurfaces extend to all smooth ones.
Related
- Fourier-Mukai partners and autoequivalences of moduli spaces of vector bundles
- Categorical Torelli theorem for (weighted) hypersurfaces via singularity category
- Fourier-Mukai transforms by a family of stable sheaves and preservation of stability
- Birational maps of moduli spaces of stable sheaves on a K3 surface induced by Fourier-Mukai transforms
- Germwise singular Fourier--Mukai partnerships
BibTeX
@misc{lin2026categorical,
title = {{Categorical Torelli and Fourier-Mukai partners of hypersurfaces}},
author = {Xun Lin and Shizhuo Zhang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05915},
eprint = {2611.05915},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05915},
}
RIS
TY - UNPB
TI - Categorical Torelli and Fourier-Mukai partners of hypersurfaces
AU - Xun Lin
AU - Shizhuo Zhang
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.05915
UR - https://arxiv.org/abs/2611.05915
AB - Let X⊂PP^M−1 be a smooth hypersurface of degree d<M, and let Ku(X) be its Kuznetsov component, the orthogonal complement of the line bundles OO_X,…,OO_X(M−d−1) in the derived category. We prove that Ku(X) determines X for every smooth quartic in PP^4k+1, k≥1; together with earlier work, this gives categorical Torelli for every smooth hypersurface of degree d<M with dnmid M. The same algebraic reconstruction proves the classical generic Torelli theorem for these quartics, which completes generic Torelli for smooth hypersurfaces whose degree does not divide M, cubic surfaces excepted. When d divides M, the Kuznetsov component is a Calabi--Yau category, as for cubic fourfolds, where categorical Torelli is known to fail. We show that, apart from cubic fourfolds, such a hypersurface has a number of Fourier--Mukai partners bounded by a constant depending only on d and M, and that categorical Torelli holds in sufficiently large degree, in every dimension. The main tool is a deformation theorem: when geometric deformations surject onto the second Hochschild cohomology of the Kuznetsov components, an equivalence deforms together with the hypersurfaces, so statements proved for general hypersurfaces extend to all smooth ones.
ER -