Brauer groups and crepant resolutions

Open Gromov-Witten Theory and the Crepant Resolution Conjecture

We compute open GW invariants for $\mathcal{K}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}$, open orbifold GW invariants for $[\C^3/\Z_2]$, formulate an open crepant resolution conjecture and verify it for this pair. We show that open invariants can be glued together to deduce the Bryan-Graber closed crepant resolution conjecture for the orbifold $[\mathcal{O}_{\mathbb{P}^1}(-1)\oplus\mathcal{O}_{\mathbb{P}^1}(-1)/\Z_2]$ and its crepant resolution $\mathcal{K}_{\mathbb{P}^1\times\mathbb{P}^1}$.

Uniqueness of crepant resolutions and symplectic singularities

We prove the uniqueness of crepant resolutions for some quotient singularities and for some nilpotent orbits. The finiteness of non-isomorphic symplectic resolutions for 4-dimenensional symplectic singularities is proved. We also give an example of symplectic singularity which admits two non-equivalent symplectic resolutions.

Crepant resolutions and Hilb^G(C^4) for certain abelian subgroups for SL(4,C)

Let G be a finite subgroup of SL(n,C), then the quotient C^n/G has a Gorenstein canonical singularity. Bridgeland-King-Reid proved that the G-Hilbert scheme Hilb^G(C^3) gives a crepant resolution of the quotient C^3/G for any finite subgroup G of SL(3,C). However, in dimension 4, very few crepant resolutions are known. In this paper, we will show several examples of crepant resolutions in dimension 4 and show examples in which Hilb^G(C^4) is blow-up of certain crepant resolutions for C^4/G, or Hilb^G(C^4) has singularity.

Crepant Property of Fujiki-Oka Resolutions for Gorenstein Abelian Quotient Singularities

We show a sufficient condition for Fujiki-Oka resolutions of Gorenstein abelian quotient singularities to be crepant in all dimensions by using Ashikaga's continuous fractions. Moreover, we prove that all three dimensional Gorenstein abelian quotient singularities possess a crepant resolution as a corollary. This alternative proof of existence needs only simple computations comparing with the results ever known.

Crepant resolutions of quotient varieties in positive characteristics and their Euler characteristics

In characteristic zero, if a quotient variety has a crepant resolution, the Euler characteristic of the crepant resolution is equal to the number of conjugacy classes of the acting group, by Batyrev's theorem. This is one of the McKay correspondence. It is natural to consider the analogue statement in the positive characteristic. In this paper, we present sequences of crepant resolutions of quotient varieties in the positive characteristic and show that one of the sequences gives a counterexample to the analogue statement of Batyrev's theorem.

Crepant resolutions of a Slodowy slice in a nilpotent orbit closure in $\mathfrak{sl}_N(\mathbb{C})$

One of our results of this article is that every (projective) crepant resolution of a Slodowy slice in a nilpotent orbit closure in $\mathfrak{sl}_N(\mathbb{C})$ can be obtained as the restriction of some crepant resolution of the nilpotent orbit closure. We also show that there is a decomposition of the Slodowy slice into other Slodowy slices with good properties. From this decomposition, one can count the number of crepant resolutions.

Remarks on non-commutative crepant resolutions of complete intersections

We study obstructions to existence of non-commutative crepant resolutions, in the sense of Van den Bergh, over local complete intersections.

Crepant resolution conjecture for $\mathbb{C}^5/\mathbb{Z}_5$

We study the relationship between Gromov-Witten invariants of local $\mathbb{P}^4$ and Gromov-witten invariants of $[\mathbb{C}^5/\mathbb{Z}_5]$ for all genera. We state the crepant resolution conjecture in explicit form and prove this conjecture for $g=2,3.$

Cohomology ring of crepant resolutions of orbifolds

Motivated by physics, we propose two conjectures regarding the cohomology ring of the crepant resolutions of orbifolds and cohomological invariants of K-equivalent manifolds.

An example of crepant resolution conjecture in two steps

We study the relation among the genus 0 Gromov-Witten theories of the three spaces $\mathcal{X}\leftarrow\mathcal{Z}\leftarrow Y$, where $\mathcal{X}=[\c^2/\z_3]$, $\mathcal{Z}$ is obtained by a weighted blowup at the stacky point of $\mathcal{X}$, and $Y$ is the crepant resolution of the $A_2$ singularity. We formulate and verify a statement similar to the Crepant Resolution Conjecture of Bryan and Graber.b

On the Crepant Resolution Conjecture in the Local Case

In this paper we analyze four examples of birational transformations between local Calabi-Yau 3-folds: two crepant resolutions, a crepant partial resolution, and a flop. We study the effect of these transformations on genus-zero Gromov-Witten invariants, proving the Coates-Corti-Iritani-Tseng/Ruan form of the Crepant Resolution Conjecture in each case. Our results suggest that this form of the Crepant Resolution Conjecture may also hold for more general crepant birational transformations. They also suggest that Ruan's original Crepant Resolution Conjecture should be modified, by including appr…

Deformation of tilting-type derived equivalences for crepant resolutions

We say that an exact equivalence between the derived categories of two algebraic varieties is tilting-type if it is constructed by using tilting bundles. The aim of this article is to understand the behavior of tilting-type equivalences for crepant resolutions under deformations. As an application of the method that we establish in this article, we study the derived equivalence for stratified Mukai flops and stratified Atiyah flops in terms of tilting bundles.

乘积簇之商的crepant消解上的稳定条件

Stability conditions on crepant resolutions of quotients of product varieties

本文在乘积簇之商的crepant消解上构造稳定条件，得到任意维数严格Calabi-Yau簇的首批例子，并证明其与商stack导出等价，验证Bondal-Orlov猜想的一个实例。

Crepant resolutions of double covers: On the Cynk-Hulek criterion for crepant resolutions of double cover

A collection $S = \{D_1,\ldots, D_n\}$ of divisors in a smooth variety $X$ is an {\em arrangement} if intersections of all subsets of $S$ are smooth. We show that a double cover of $X$ ramified on an arrangement has a crepant resolution under additional hypotheses. Namely, we assume that all intersection components that change the canonical divisor when blown up satisfy are {\em splayed}, a property of the tangent spaces of the components first studied by Faber. This strengthens a result of Cynk and Hulek, which requires a stronger hypothesis on the intersection components. Further, we study t…

关于希尔伯特-丘创设解析猜想

On the Hilbert-Chow crepant resolution conjecture

本文证明了希尔伯特-丘创设解析猜想在例外曲线类上的成立，并推导了三重积的DT/GW对应关系，提供了墙交叉证明。

Crepant resolutions of 3-dimensional quotient singularities via Cox rings

We study Cox rings of crepant resolutions of quotient singularities $\mathbb{C}^3/G$ where $G$ is a finite subgroup of $SL(3,\mathbb{C})$. We use them to obtain information on the geometric structure of these resolutions, number of different resolutions and relations between them. In particular, we treat explicitly several examples where $G$ contains elements of age 2.

The cohomological crepant resolution conjecture for P(1,3,4,4)

We prove the cohomological crepant resolution conjecture of Ruan for the weighted projective space P(1,3,4,4). To compute the quantum corrected cohomology ring we combine the results of Coates-Corti-Iritani-Tseng on P(1,1,1,3) and our previous results.

Crepant resolutions and brane tilings I: Toric realization

Given a brane tiling, that is, a bipartite graph on a torus, we can associate with it a singular 3-Calabi-Yau variety. In this paper we study its commutative and non-commutative crepant resolutions. We give an explicit toric description of all its commutative crepant resolutions. We also explain how the McKay correspondence in dimension 3 can be interpreted using brane tilings.

Non-commutative crepant resolutions for some toric singularities II

Using the theory of dimer models Broomhead proved that every 3-dimensional Gorenstein affine toric variety Spec R admits a toric non-commutative crepant resolution (NCCR). We give an alternative proof of this result by constructing a tilting bundle on a (stacky) crepant resolution of Spec R using standard toric methods. Our proof does not use dimer models.

Crepant resolutions and brane tilings II: Tilting bundles

Given a brane tiling, that is, a bipartite graph on a torus, we can associate with it a singular 3-Calabi-Yau variety. Using the brane tiling, we can also construct all crepant resolutions of the above variety. We give an explicit toric description of tilting bundles on these crepant resolutions. This result proves the conjecture of Hanany, Herzog and Vegh and a version of the conjecture of Aspinwall.

Crepant Resolutions, Quivers and GW/NCDT Duality

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.

Crepant resolutions of orbit closures in Quiver representations of type A\_n and D\_4

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.

通过 Artin 叠消解 log-terminal 奇点的 crepant 消解

Crepant resolutions of log-terminal singularities via Artin stacks

本文证明任意 log-terminal 奇点簇均可由光滑 Artin 叠实现 crepant 消解，并建立相应 McKay 对应，用光滑叠弧空间上的动机积分表达 Q-Gorenstein 簇的弦不变量。

Dynkin diagrams and crepant resolutions of quotient singularities

Let $V$ be a complex vector space on which a finite group $G$ acts by linear transformations. Let $W = V \oplus V^*$ be the sum of $V$ with its dual $V^*$. We prove that if the quotient $W/G$ admits a smooth crepant resolution, then the subgroup $G \subset Aut V$ is generated by complex reflections. We also obtain some results on the structure of smooth crepant resolutions of the quotients $W/G$, where $W$ is a symplectic vector space, and $G \subset Aut W$ is a finite group of symplectic linear transformations of the vector space $W$.

Automorphisms of crepant resolutions for quotient spaces

A formula for calculating the Lefschetz number of an automorphism acting on a crepant resolution for a quotient of a Kahler manifold derived from an equivariant version of McKay correspondence. The latter is proven in some cases. As an application the Lefschetz numbers of of involutions acting on Calabi-Yau threefolds and their mirrors are compared.

Crepant resolutions of weighted projective spaces and quantum deformations

We compare the Chen-Ruan cohomology ring of the weighted projective spaces $\IP(1,3,4,4)$ and $\IP(1,...,1,n)$ with the cohomology ring of their crepant resolutions. In both cases, we prove that the Chen-Ruan cohomology ring is isomorphic to the quantum corrected cohomology ring of the crepant resolution after suitable evaluation of the quantum parameters. For this, we prove a formula for the Gromov-Witten invariants of the resolution of a transversal ${\rm A}_3$ singularity.

The Crepant Resolution Conjecture for 3-dimensional flags modulo an involution

After fixing a non-degenerate bilinear form on a vector space V we define an involution of the manifold of flags F in V by taking a flag to its orthogonal complement. When V is of dimension 3 we check that the Crepant Resolution Conjecture of J. Bryan and T. Graber holds: the genus zero (orbifold) Gromov-Witten potential function of [F / Z_2] agrees (up to unstable terms) with the genus zero Gromov-Witten potential function of a crepant resolution Y of the quotient scheme F / Z_2, after setting a quantum parameter to -1, making a linear change of variables, and analytically continuing coeffici…

Framed motivic Donaldson-Thomas invariants of small crepant resolutions

For an arbitrary integer $r\geq 1$, we compute $r$-framed motivic PT and DT invariants of small crepant resolutions of toric Calabi-Yau $3$-folds, establishing a "higher rank" version of the motivic DT/PT wall-crossing formula. This generalises the work of Morrison and Nagao. Our formulae, in particular their relationship with the $r=1$ theory, fit nicely in the current development of higher rank refined DT invariants.

On the Abuaf-Ueda Flop via Non-Commutative Crepant Resolutions

The Abuaf-Ueda flop is a 7-dimensional flop related to $G_2$ homogeneous spaces. The derived equivalence for this flop was first proved by Ueda using mutations of semi-orthogonal decompositions. In this article, we give an alternative proof for the derived equivalence using tilting bundles. Our proof also shows the existence of a non-commutative crepant resolution of the singularity appearing in the flopping contraction. We also give some results on moduli spaces of finite-length modules over this non-commutative crepant resolution.

奇异对数结构与对数crepant对数消解 I

Singular Log Structures and Log Crepant Log Resolutions I

本文定义一类三维奇异对数概形，提出其存在对数crepant对数消解的猜想，并通过示例及与已有猜想和Gross–Siebert纲领的联系提供支撑。

Non-commutative crepant resolutions for some toric singularities I

We give a criterion for the existence of non-commutative crepant resolutions (NCCR's) for certain toric singularities. In particular we recover Broomhead's result that a 3-dimensional toric Gorenstein singularity has a NCCR. Our result also yields the existence of a NCCR for a 4-dimensional toric Gorenstein singularity which is known to have no toric NCCR.

Wall-Crossings in Toric Gromov-Witten Theory II: Local Examples

In this paper we analyze six examples of birational transformations between toric orbifolds: three crepant resolutions, two crepant partial resolutions, and a flop. We study the effect of these transformations on genus-zero Gromov-Witten invariants, proving the Coates-Corti-Iritani-Tseng/Ruan form of the Crepant Resolution Conjecture in each case. Our results suggest that this form of the Crepant Resolution Conjecture may also hold for more general crepant birational transformations. They also suggest that Ruan's original Crepant Resolution Conjecture should be modified, by including appropria…

Crepant resolution conjecture in all genera for type A singularities

We prove an all genera version of the Crepant Resolution Conjecture of Ruan and Bryan-Graber for type A surface singularities. We are based on a method that explicitly computes Hurwitz-Hodge integrals described in an earlier paper and some recent results by Liu-Xu for some intersection numbers on the Deligne-Mumford moduli spaces. We also generalize our results to some three-dimensional orbifolds.

Crepant Resolutions of C^n/A_1(n) and Flops of n-Folds for n = 4,5

In this article, we determine the explicit toric variety structure of $\hl^{A_1(n)}(\CZ^n)$ for $n=4,5$, where $A_1(n)$ is the special diagonal group of all order 2 elements. Through the toric data of $\hl^{A_1(n)}(\CZ^n)$, we obtain certain toric crepant resolutions of $\CZ^n/A_1(n)$, and the different crepant resolutions are connected by flops of $n$-folds for $n=4,5$.

通过 Cox 环研究超多边形空间的所有 crepant 消解

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。

Ricci-flat Kähler metrics on crepant resolutions of Kähler cones

We prove that a crepant resolution of a Ricci-flat Kähler cone X admits a complete Ricci-flat Kähler metric asymptotic to the cone metric in every Kähler class in H^2_c(Y,R). This result contains as a subcase the existence of ALE Ricci-flat Kähler metrics on crepant resolutions of X=C^n /G, where G is a finite subgroup of SL(n,C). We consider the case in which X is toric. A result of A. Futaki, H. Ono, and G. Wang guarantees the existence of a Ricci-flat Kähler cone metric if X is Gorenstein. We use toric geometry to construct crepant resolutions.

On the Crepant Resolution Conjecture for Gromov-Witten Gravitational Ancestors in All Genera for Surface Singularities

We state a version of the crepant resolution conjecture for total ancestor potentials for surface singularities, and reduce the conjecture to the quantum McKay correspondence conjecture of J.Bryan and A.Gholampour and a vanishing conjecture for Hurwitz-Hodge integrals. In particular, for singularities of type A, we prove the conjecture. We also suggest an approach towards a proof for the general cases by Teleman's reconstruction theorem for semisimple cohomological field theories.

Dimer models and crepant resolutions

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.

Quantum Cohomology and Crepant Resolutions: A Conjecture

We give an expository account of a conjecture, developed by Coates--Corti--Iritani--Tseng and Ruan, which relates the quantum cohomology of a Gorenstein orbifold X to the quantum cohomology of a crepant resolution Y of X. We explore some consequences of this conjecture, showing that it implies versions of both the Cohomological Crepant Resolution Conjecture and of the Crepant Resolution Conjectures of Ruan and Bryan--Graber. We also give a "quantized" version of the conjecture, which determines higher-genus Gromov--Witten invariants of X from those of Y.

Crepant resolution and the holomorphic anomaly equation for C^3/Z_3

We study the orbifold Gromov-Witten theory of the quotient C^3/Z_3 in all genera. Our first result is a proof of the holomorphic anomaly equations in the precise form predicted by B-model physics. Our second result is an exact crepant resolution correspondence relating the Gromov-Witten theories of C^3/Z_3 and local CP2. The proof of the correspondence requires an identity proven in the Appendix by T. Coates and H. Iritani.

Generating toric noncommutative crepant resolutions

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 …

Non-commutative crepant resolutions

We introduce the notion of a ``non-commutative crepant'' resolution of a singularity and show that it exists in certain cases. We also give some evidence for an extension of a conjecture by Bondal and Orlov, stating that different crepant resolutions of a Gorenstein singularity have the same derived category.

四维有限商奇点的亏格消解

Crepant resolutions of four-dimensional finite quotients

该研究针对Dais-Henk-Ziegler的猜想给出反例，确定有限群G⊂SL₄(ℤ)对应的ℂ⁴/G是否存在射影亏格消解，提出十条准则并讨论相关问题。

The Crepant Resolution Conjecture

For orbifolds admitting a crepant resolution and satisfying a hard Lefschetz condition, we formulate a conjectural equivalence between the Gromov-Witten theories of the orbifold and the resolution. We prove the conjecture for the equivariant Gromov-Witten theories of the nth symmetric product of the complex plane and the Hilbert scheme of n points in the plane.

三维环形商奇点的crepant解析的quiver braiding群作用

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。

Rigid HYM Connections on Tautological Bundles over ALE Crepant Resolutions in Dimension Three

For $G$ a finite subgroup of ${\rm SL}(3,{\mathbb C})$ acting freely on ${\mathbb C}^3{\setminus} \{0\}$ a crepant resolution of the Calabi-Yau orbifold ${\mathbb C}^3\!/G$ always exists and has the geometry of an ALE non-compact manifold. We show that the tautological bundles on these crepant resolutions admit rigid Hermitian-Yang-Mills connections. For this we use analytical information extracted from the derived category McKay correspondence of Bridgeland, King, and Reid [J. Amer. Math. Soc. 14 (2001), 535-554]. As a consequence we rederive multiplicative cohomological identities on the cre…

通过哈密顿约化构造幂零锥的收缩部分解

Crepant partial resolutions of the nilpotent cone via Hamiltonian reduction

本文通过哈密顿约化方法，构造了单连通半单代数群幂零锥的收缩射影部分解，并实现了其通用泊松形变的基变换。

Moduli of $G$-constellations and crepant resolutions I: the abelian case

For a finite abelian subgroup $G\subset SL_n(\mathbb{C})$, we study whether a given crepant resolution $X$ of the quotient variety $\mathbb{C}^n/G$ is obtained as a moduli space of $G$-constellations. In particular we show that, if $X$ admits a natural $G$-constellation family in the sense of Logvinenko over it with all fibers being indecomposable as $\mathbb{C}[\mathbb{C}^n]$-modules, then $X$ is isomorphic to the normalization of a fine moduli space of $G$-constellations.

Crepant Resolutions of Gorenstein Toric Singularities and Upper Bound Theorem

A necessary condition for the existence of torus-equivariant crepant resolutions of Gorenstein toric singularities can be derived by making use of a variant of the classical Upper Bound Theorem which is valid for simplicial balls.

Refined open non-commutative Donaldson-Thomas invariants for small crepant resolutions

The aim of this paper is to study an analog of non-commutative Donaldson-Thomas theory corresponding to the refined topological vertex for small crepant resolutions of toric Calabi-Yau 3-folds. We define the invariants using dimer models and provide "wall-crossing" formulas. In particular, we get normalized generating functions which are unchanged under "wall-crossing".

On steady non-commutative crepant resolutions

We introduce special classes of non-commutative crepant resolutions (= NCCR) which we call steady and splitting. We show that a singularity has a steady splitting NCCR if and only if it is a quotient singularity by a finite abelian group. We apply our results to toric singularities and dimer models.

特征 2 下 $\mathbb{A}^4/A_4$ 的 crepant 消解

Crepant resolution of $\mathbb{A}^4/A_4$ in characteristic 2

本文在特征 2 下构造了 $\mathbb{A}^4/A_4$ 的 crepant 消解，并通过计算其 Euler 数给出正特征 McKay 对应类似命题的新反例。

Hurwitz-Hodge Integrals, the E6 and D4 root systems, and the Crepant Resolution Conjecture

Let G be the group A_4 or Z_2xZ_2. We compute the integral of λ_g on the Hurwitz locus H_G\subset M_g of curves admitting a degree 4 cover of P^1 having monodromy group G. We compute the generating functions for these integrals and write them as a trigonometric expression summed over the positive roots of the E_6 and D_4 root systems respectively. As an application, we prove the Crepant Resolution Conjecture for the orbifolds [C^3/A_4] and [C^3/(Z_2xZ_2)].

Crepant resolution of trihedral singularities

The purpose of this paper is to construct a crepant resolution of quotient singularities by trihedral groups ( finite subgroups of SL(3,C) of certain type ), and prove that each Euler number of the minimal model is equal to the number of conjugacy classes. Trihedral singularities are 3-dimensional version of D_n-singularities, and they are non-isolated and many of them are not complete intersections. The resolution is similar to one of D_n-singularities. There is a nice combination of the toric resolution and Calabi-Yau resolution.