Jian Zhou
We propose a conjecture that relates some local Gromov-Witten invariants of some crepant resolutions of Calabi-Yau 3-folds with isolated singularities with some Donaldson-Thomas type invariants of the moduli spaces of representations of some quivers with potentials.
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We consider the moduli space of the McKay quiver representations associated to the binary polyhedral groups G < SU(2) < SU(3). The derived category of such representations is equivalent to the derived category of coherent sheaves on the corresponding ADE resolution Y=G-Hilb(C^3). Following the ideas of Nagao and Nakajima, by making particular choices of parameters in the space of stability conditions on the equivalent derived categories above, we recover Donaldson-Thomas (DT), Pandharipande-Thomas (PT) and Szendroi (NCDT) moduli spaces. We also compute the Gromov-Witten (GW) partition function…
We study resolutions of singularities of orbit closures in quiver representations. We consider certain resolutions of singularities which have already been constructed by Reineke, and we determine under which conditions they are crepant. More specifically, we deal with quivers of type A3, An, D4. Then, as an application of the results we have found, we construct several Fano and Calabi-Yau varieties as orbital degeneracy loci.
Quiver braid group action for a 3-fold crepant resolution
研究三维环形商奇点(1,2,4)的crepant解析X，证明其导出范畴D(X)存在quiver braiding群作用，该quiver为3环，编码交集数据。
On certain M-theory backgrounds which are a circle fibration over a smooth Calabi-Yau, the quantum theory of M2 branes can be studied in terms of the K-theoretic Donaldson-Thomas theory on the threefold. We extend this relation to noncommutative crepant resolutions. In this case the threefold develops a singularity and classical smooth geometry is replaced by the algebra of paths of a certain quiver. K-theoretic quantities on the quiver representation moduli space can be computed via toric localization and result in certain rational functions of the toric parameters. We discuss in particular t…
Nonprojective crepant resolutions of quiver varieties
本文利用局部射影解消与表示空间开子集取商两种方法，构造并分类了 Nakajima 箭簇簇的真非射影 crepant 解消，给出四维、六维及三顶点箭图等新例子。
Stacky Resolutions of Kleinian Singularities and Nakajima Quiver Varieties
该研究构造克莱因奇点的非交换平展分解，证明其与典范轨形分解Morita等价，确定对应点希尔伯特概型为中岛箭图簇，并刻画其两类锥的结构。
We present an algorithm that finds all toric noncommutative crepant resolutions of a given toric 3-dimensional Gorenstein singularity. The algorithm embeds the quivers of these algebras inside a real 3-dimensional torus such that the relations are homotopy relations. One can project these embedded quivers down to a 2-dimensional torus to obtain the corresponding dimer models. We discuss some examples and use the algorithm to show that all toric noncommutative crepant resolutions of a finite quotient of the conifold singularity can be obtained by mutating one basic dimer model. We also discuss …
Crepant resolutions, mutations, and the space of potentials
本文通过开发网络应用程序，利用带势箭图的变异与 crepant 三角剖分研究 Gorenstein 环面商奇点的 crepant 消解之间的 flop，并发现箭图变异可分类这些消解，同时实现了 McKay 对应的关键构造。
Let $G$ be a polyhedral group $G\subset SO(3)$ of types $\mathbb{Z}/n\mathbb{Z}$, $D_{2n}$ and $\mathbb{T}$. We prove that there exists a one-to-one correspondence between flops of $G$-Hilb$\mathbb{C}^3$ and mutations of the McKay quiver with potential which do not mutate the trivial vertex. This correspondence provides two equivalent methods to construct every projective crepant resolution for the singularities $\mathbb{C}^3/G$, which are constructed as moduli spaces $\mathcal{M}_C$ of quivers with potential for some chamber $C$ in the space $Θ$ of stability conditions. In addition, we study …
We prove that all projective crepant resolutions of Nakajima quiver varieties satisfying natural conditions are also Nakajima quiver varieties. More generally, we classify the small birational models of many Geometric Invariant Theory (GIT) quotients by introducing a sufficient condition for the GIT quotient of an affine variety $V$ by the action of a reductive group $G$ to be a relative Mori Dream Space. Two surprising examples illustrate that our new condition is optimal. When the condition holds, we show that the linearisation map identifies a region of the GIT fan with the Mori chamber dec…
All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces
本文证明阶为32的例外子群G的商奇点同构于仿射箭图簇，通过GIT商变动将其所有81个射影crepant解构造为超多边形空间，并推广至n角星形箭图，完整描述了相应超多边形空间的双有理几何。
Hilbert schemes for crepant partial resolutions
研究克莱因奇点的crepant部分分解上\(n\)个点的希尔伯特概型，通过构造中岛箭图簇推广和统一相关构造，还计算了该希尔伯特概型的有效锥和可动锥。
All crepant resolutions of hyperpolygon spaces via their Cox rings
本文利用 Cox 环与 Arzhantsev-Derenthal-Hausen-Laface 方法，构造并枚举超多边形空间的全部 crepant 消解，证明其数量为 Hoşten-Morris 数，并刻画相关 GIT 几何商。
In this article, we construct a non-commutative crepant resolution (=NCCR) of a minimal nilpotent orbit closure $\overline{B(1)}$ of type A, and study relations between an NCCR and crepant resolutions $Y$ and $Y^+$ of $\overline{B(1)}$. More precisely, we show that the NCCR is isomorphic to the path algebra of the double Beilinson quiver with certain relations and we reconstruct the crepant resolutions $Y$ and $Y^+$ of $\overline{B(1)}$ as moduli spaces of representations of the quiver. We also study the Kawamata-Namikawa's derived equivalence between crepant resolutions $Y$ and $Y^+$ of $\ove…
We study variations of tautological bundles on moduli spaces of representations of quivers with relations associated with dimer models under a change of stability parameters. We prove that if the tautological bundle induces a derived equivalence for some stability parameter, then the same holds for any generic stability parameter, and any projective crepant resolution can be obtained as the moduli space for some stability parameter. This result is used in 0905.0059 to prove the abelian McKay correspondence without using the result of math/9908027.
Non-commutative crepant resolutions of toric singularities with divisor class group of rank one
本文证明并分类了除子类群秩一的Gorenstein环状奇点的环状非交换凝聚解消（NCCR），通过上集与Iyama-Wemyss突变建立导出等价，并在非金字塔情形下给出二聚体模型实现。
Gale duality and the linearisation map for noncommutative crepant resolutions
本文用几何方法描述了三维非交换奇点解消的箭图模空间上线性化映射的Gale对偶，并利用一对整值矩阵表述Reid配方，为相关猜想提供了新证据。
Moduli of $G$-constellations and crepant resolutions II: the Craw-Ishii conjecture
本文证明了 Craw-Ishii 猜想：对任意有限子群 $G\subset SL_3(\mathbb{C})$，商簇的每个射影 crepant 消解都同构于一般稳定性条件下的 $G$-星座模空间，并利用 McKay 箭图的半不变量构造了 Cox 环的生成元。
The three-dimensional McKay correspondence seeks to relate the geometry of crepant resolutions of Gorenstein $3$-fold quotient singularities $\mathbb{A}^3/G$ with the representation theory of the group $G$. The first crepant resolution studied in depth was the $G$-Hilbert scheme $G\text{-Hilb}\,\mathbb{A}^3$, which is also a moduli space of $θ$-stable representations of the McKay quiver associated to $G$. As the stability parameter $θ$ varies, we obtain many other crepant resolutions. In this paper we focus on the case where $G$ is abelian, and compute explicit inequalities for the chamber of …
5d SCFTs from Isolated Complete Intersection Singularities
本文系统研究了M理论在孤立完全交奇性上构造的5维超共形场论，分类了秩$\leqslant 10$的光滑模型，计算了Coulomb branch与flavor symmetry algebra，并检验了与4d $\mathcal{N}=2$ SCFT相关联的猜想。
We first construct a derived equivalence between a small crepant resolution of an affine toric Calabi-Yau 3-fold and a certain quiver with a superpotential. Under this derived equivalence we establish a wall-crossing formula for the generating function of the counting invariants of perverse coherent systems. As an application we provide certain equations on Donaldson-Thomas, Pandeharipande-Thomas and Szendroi's invariants. Finally, we show that moduli spaces associated with a quiver given by successive mutations are realized as the moduli spaces associated the original quiver by changing the s…
Let $G\subset SL_2(C)\subset SL_3(C)$ be a finite group. We compute motivic Pandharipande-Thomas and Donaldson-Thomas invariants of the crepant resolution $Hilb^G(C^3)$ of $C^3/G$ generalizing results of Gholampour and Jiang who computed numerical DT/PT invariants using localization techniques. Our formulas rely on the computation of motivic Donaldson-Thomas invariants for a special class of quivers with potentials. We show that these motivic Donaldson-Thomas invariants are closely related to the polynomials counting absolutely indecomposable quiver representations over finite fields introduce…
We compute motivic Donaldson-Thomas invariants for crepant resolutions of quotients of affine three-space by even dihedral groups in terms of an affine type D root system, using double dimensional reduction and the representation theory of affine type D quivers.
For a finite subgroup $Γ\subset \mathrm{SL}(2,\mathbb{C})$ and for $n\geq 1$, we use variation of GIT quotient for Nakajima quiver varieties to study the birational geometry of the Hilbert scheme of $n$ points on the minimal resolution $S$ of the Kleinian singularity $\mathbb{C}^2/Γ$. It is well known that $X:=\mathrm{Hilb}^{[n]}(S)$ is a projective, crepant resolution of the symplectic singularity $\mathbb{C}^{2n}/Γ_n$, where $Γ_n=Γ\wr\mathfrak{S}_n$ is the wreath product. We prove that every projective, crepant resolution of $\mathbb{C}^{2n}/Γ_n$ can be realised as the fine moduli space of $…
Crepant resolutions of stratified varieties via gluing
本文通过粘合方法研究分层簇的余切消解，证明局部射影余切消解构成可构造层，给出全局射影性判据，并应用于对称幂、Hilbert 方案及箭图簇等例子。
Local forms for the double $A_n$ quiver
本文研究双A_n箭形的非交换奇点理论及其在代数几何和表示论中的应用，通过坐标变换证明了A型势能的单调化结果，并解决了Brown-Wemyss的现实猜想。
Let $Γ$ be a finite group acting linearly on $\C^n$, freely outside the origin. In previous work a generalisation of Kronheimer's construction of moduli of Hermitian-Yang-Mills bundles with certain invariance properties was given. This produced varieties $X_ζ$ (parameterised by $ζ\in\Q^N$) which are partial resolutions of $\C^n/Γ$. In this article, it is shown the same $X_ζ$ can be described as moduli spaces of representations of the McKay quiver associated to the action of $Γ$. It it shown that, for abelian groups, $X_ζ$ are toric varieties defined by convex polyhedra which are the solution s…
The semi-invariant ring as the Cox ring of a GIT quotient
该研究针对满足特定线性化映射同构条件的GIT商，证明其Cox环同构于底空间中θ-稳定轨迹的半不变量环，该结论可应用于多类代数簇，并给出仿射平面n点Hilbert概形Cox环的显式计算。
In this paper, we study and describe the universal Poisson deformation space of hypertoric varieties concretely. In the first application, we show that affine hypertoric varieties as conical symplectic varieties are classified by the associated regular matroids (this is a partial generalization of the result by Arbo and Proudfoot). As a corollary, we obtain a criterion when two quiver varieties whose dimension vector have all coordinates equal to one are isomorphic to each other. Then we describe all 4- and 6-dimensional affine hypertoric varieties as quiver varieties and give some examples of…
Lectures on resolutions à la Kronheimer of orbifold singularities, McKay quivers for Gauge Theories on D3 branes, and the issue of Ricci flat metrics on the resolved three-folds
本讲义回顾了利用推广的Kronheimer构造与McKay对应进行轨形奇点消解，以构建四维N=1规范理论与IIB超引力D3膜解的全息对偶对的研究项目，介绍了其数学基础、待解问题与最终目标。
In this writing we shall address certain beautiful inter-relations between the construction of 4-dimensional supersymmetric gauge theories and resolution of algebraic singularities, from the perspective of String Theory. We review in some detail the requisite background in both the mathematics, such as orbifolds, symplectic quotients and quiver representations, as well as the physics, such as gauged linear sigma models, geometrical engineering, Hanany-Witten setups and D-brane probes. We investigate aspects of world-volume gauge dynamics using D-brane resolutions of various Calabi-Yau singular…
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