arXiv subjects

Jieheng Zeng

Publications and source records attributed to Jieheng Zeng.

8 recordsLinked to original sources

arXiv2026

Categorical Torelli theorem for (weighted) hypersurfaces via singularity category

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.

math.AG↗

arXiv2026

Shifted double Poisson structures and noncommutative Poisson extensions

We develop a theory of noncommutative Poisson extensions. For an augmented dg algebra $A$, we show that any shifted double Poisson bracket on $A$ induces a graded Lie algebra structure on the reduced cyclic homology. Under the Kontsevich--Rosenberg principle, we further prove that the noncommutative Poisson extension is compatible with noncommutative Hamiltonian reduction. Moreover, we show that shifted double Poisson structures are independent of the choice of cofibrant resolutions and that they induce shifted Poisson structures on the derived moduli stack of representations.

math.RT↗

arXiv2025

Singularity categories and singular loci of certain quotient singularities

Let $V$ be a finite dimensional $k$-vector space, where $k$ is an algebraic closed field of characteristic zero. Let $G \subseteq \mathrm{SL}(V)$ be a finite abelian group, and denote by $S$ the $G$-invariant subring of the polynomial ring $k[V]$. It is shown that the singularity category $D_{sg}(S)$ recovers the reduced singular locus of $\mathrm{Spec}(S)$.

math.AG↗

arXiv2025

On the graded singularity category of Abelian quotient singularities, I. Smooth categorical compactification

Given a singularity which is the quotient of an affine space $V$ by a finite Abelian group $G \subseteq \mathrm{SL}(V)$, we study the DG enhancement $\mathcal{D}^{b}(\mathrm{tails}(k[V]^G))$ of the bounded derived category of the non-commutative projective space $\mathrm{tails}(k[V]^G)$ and the DG enhancement $\mathcal{D}_{sg}^{\mathbb{Z}}(k[V]^G)$ of its graded singularity category. In this paper, we construct smooth categorical compactifications, in the sense of Efimov, of $\mathcal{D}^b(\mathrm{tails}(k[V]^G))$ and $\mathcal{D}_{sg}^{\mathbb{Z}}(k[V]^G)$ respectively, via the non-commutative crepant resolution of $k[V]^G$. We give explicit constructions of canonical classical generators in the kernels of such compactifications.

math.AG↗

arXiv2025

DG singular equivalence and singular locus

For a commutative Gorenstein Noetherian ring $R$, we construct an affine scheme $X$ solely from DG singularity category $S_{dg}(R)$ of $R$ such that there is a finite surjective morphism $X \rightarrow \mathrm{Spec}(R /I)$, where $\mathrm{Spec}(R /I)$ is the singular locus in $\mathrm{Spec}(R)$. As an application, for two such rings with equivalent DG singularity categories, we prove that the singular loci in their affine schemes have the same dimension.

math.AC↗

arXiv2024

Tilting objects in singularity categories of toric Gorenstein varieties

We study certain toric Gorenstein varieties with isolated singularities which are the quotient spaces of generic unimodular representations by the one-dimensional torus, or by the product of the one-dimensional torus with a finite abelian group. Based on the works of Špenko and Van den Bergh [Invent. Math. 210 (2017), no. 1, 3-67] and Mori and Ueyama [Adv. Math. 297 (2016), 54-92], we show that the singularity categories of these varieties admit tilting objects, and hence are triangle equivalent to the perfect categories of some finite dimensional algebras.

math.AG↗

arXiv2023

Batalin-Vilkovisky algebra structure on Poisson manifolds with diagonalizable modular symmetry

We study the ``twisted" Poincaré duality of smooth Poisson manifolds, and show that, if the modular vector field is diagonalizable, then there is a mixed complex associated to the Poisson complex, which, combining with the twisted Poincaré duality, gives a Batalin-Vilkovisky algebra structure on the Poisson cohomology. This generalizes the previous results obtained by Xu for unimodular Poisson manifolds. We also show that the Batalin-Vilkovisky algebra structure is preserved under Kontsevich's deformation quantization, and in the case of polynomial algebras it is also preserved by Koszul duality.

math.DG↗

arXiv2022

Derived categories and Calabi-Yau algebras

We study the derived equivalence of Calabi-Yau algebras and show that, for two derived Morita equivalent algebras, if one is Calabi-Yau, then so is the other. Keywords: Derived equivalence, Calabi-Yau algebra

math.RA↗