通过奇点范畴的（加权）超曲面的范畴Torelli定理
Categorical Torelli theorem for (weighted) hypersurfaces via singularity category
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中文总结 AI 辅助
本文利用完备仿射锥的奇点范畴，在数值条件下统一证明了Fano及一般型（加权）超曲面的范畴Torelli定理，并通过形式等价恢复方程结构。
中文摘要 AI 辅助
我们利用完备仿射锥的奇点范畴证明了普通超曲面和加权超曲面的范畴Torelli定理。对于Fano超曲面，我们使用Kuznetsov分量；而对于一般型超曲面，我们使用分次矩阵分解的完整范畴。当次数与权重之和互素时，内部分次平移可以用Serre函子和上同调平移来表达。因此其dg轨道是内在的，并且在取完美包络后，与完备锥的dg奇点范畴一致。然后通过范畴Mather-Yau定理恢复形式奇点。对于普通齐次方程，所得形式等价诱导出射影线性等价；在加权情形中，对偶环面作用决定了Tjurina代数的分次，从而决定了加权齐次方程。该论证在所述数值假设下为Fano超曲面给出了统一证明，并为一般型超曲面证明了新的范畴Torelli定理。
英文摘要
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.
发表机构
- The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
- Hunan Normal University（湖南师范大学）
- Sun Yat-sen University（中山大学）
机构由 AI 辅助整理，请以论文原文为准。
