Categorical Torelli theorem for (weighted) hypersurfaces via singularity category
Abstract
We prove categorical Torelli theorems for ordinary and weighted hypersurfaces using the singularity category of the completed affine cone. For Fano hypersurfaces we use the Kuznetsov component, whereas for hypersurfaces of general type we use the full category of graded matrix factorizations. When the degree is coprime to the sum of the weights, the internal grading shift can be expressed in terms of the Serre functor and the cohomological shift. Its dg orbit is therefore intrinsic and, after taking the perfect hull, agrees with the dg singularity category of the completed cone. The formal singularity is then recovered by the categorical Mather--Yau theorem. For ordinary homogeneous equations, the resulting formal equivalence induces a projective linear equivalence; in the weighted setting the dual torus action determines the grading of the Tjurina algebra and hence the weighted homogeneous equation. The argument gives a uniform proof for Fano hypersurfaces under the stated numerical assumptions and proves new categorical Torelli theorems for hypersurfaces of general type.
Related
- Categorical Torelli and Fourier-Mukai partners of hypersurfaces
- Categorical Torelli theorem for smooth Fano varieties
- A Categorical Realization of the (2-)Category of Monoids via Sch{ü}tzenberger Categories and Strict Factorization Systems
- Examples of nontrivial presilting objects in a singularity category
- Finitistic dimension via modules over the singularity category
BibTeX
@misc{lin2026categorical,
title = {{Categorical Torelli theorem for (weighted) hypersurfaces via singularity category}},
author = {Xun Lin and Jieheng Zeng and Shizhuo Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00130},
eprint = {2610.00130},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00130},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Categorical Torelli theorem for (weighted) hypersurfaces via singularity category
AU - Xun Lin
AU - Jieheng Zeng
AU - Shizhuo Zhang
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00130
UR - https://arxiv.org/abs/2610.00130
AB - We prove categorical Torelli theorems for ordinary and weighted hypersurfaces using the singularity category of the completed affine cone. For Fano hypersurfaces we use the Kuznetsov component, whereas for hypersurfaces of general type we use the full category of graded matrix factorizations. When the degree is coprime to the sum of the weights, the internal grading shift can be expressed in terms of the Serre functor and the cohomological shift. Its dg orbit is therefore intrinsic and, after taking the perfect hull, agrees with the dg singularity category of the completed cone. The formal singularity is then recovered by the categorical Mather--Yau theorem. For ordinary homogeneous equations, the resulting formal equivalence induces a projective linear equivalence; in the weighted setting the dual torus action determines the grading of the Tjurina algebra and hence the weighted homogeneous equation. The argument gives a uniform proof for Fano hypersurfaces under the stated numerical assumptions and proves new categorical Torelli theorems for hypersurfaces of general type.
ER -