Baohua Fu, Yoshinori Namikawa
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.
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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 $W$.
Let $M_c=M(2,0,c)$ be the moduli space of O(1)-semistable rank 2 torsion-free sheaves with Chern classes $c_1=0$ and $c_2=c$ on a K3 surface $X$ where O(1) is a generic ample line bundle on $X$. When $c=2n\geq4$ is even, $M_c$ is a singular projective variety equipped with a symplectic structure on the smooth locus. In this paper, we show that there is no crepant resolution of $M_{2n}$ for $n\geq 3$. This implies that there is no symplectic desingularization.
Let J be an abelian surface with a generic ample line bundle O(1). For n>0, the moduli space M(2, 0, 2n) of O(1)-semistable sheaves F of rank 2 with Chern classes c_1(F) = 0, c_2(F) = 2n is a singular projective variety, endowed with a holomorphic symplectic structure on the smooth locus. In this paper, we show that there does not exist a crepant resolution of M(2; 0; 2n) for n>1. This certainly implies that there is no symplectic desingularization of M(2, 0, 2n) for n>1.
In this note we prove that the crepant transformation conjecture for a crepant birational transformation of Lawrence toric DM stacks studied in \cite{CIJ} implies the monodromy conjecture for the associated wall crossing of the symplectic resolutions of hypertoric stacks, due to Braverman, Maulik and Okounkov.
Crepant Transformation Correspondence For Toric Stack Bundles
本文通过构造辛变换和 Fourier-Mukai 同构，研究了簇堆叠包在 crepant 墙交叉下的 Gromov-Witten 理论对应关系。
We develop a method of finding a Cox ring of a crepant resolution of a quotient singularity with a torus action and apply it to examples of symplectic quotient singularities in dimension 4. In addition we obtain a bound on the degrees of homogeneous generators of the Cox ring in a more general setup, based on the multigraded Castelnuovo-Mumford regularity and Kawamata-Viehweg vanishing.
We shall prove that the Poisson deformation functor of an affine (singular) symplectic variety is unobstructed. As a corollary, we prove the following result. Theorem: For an affine symplectic variety $X$ with a good $C^*$-action (where its natural Poisson structure is positively weighted), the following are equivalent (1) $X$ has a crepant projective resolution. (2) $X$ has a smoothing by a Poisson deformation.
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。
We consider the quotients $X = V/G$ of a symplectic complex vector space $V$ by a finite subgroup $G \subset Sp(V)$ which admit a smooth crepant resolution $Y \to X$. For such quotients, we prove the homological McKay correspondence conjectured by M. Reid. Namely, we construct a natural basis in the homology space $H_\cdot(Y,\Q)$ whose elements are numbered by the conjugacy classes in the finite group $G$.
We shall prove that any small deformation of a Q-factorial projective symplectic variety with terminal singularities is locally rigid; in other words, it preserves the singularity. In particular, many singular symplectic moduli of semi-stable sheaves on K3 have no smoothings via deformations. As an application of the result, we also prove that the smootheness is preserved under a flop in our symplectic case. We conjecture that a projective symplectic variety has a smoothing by a flat deformation if and only if it has a crepant (symplectic) resolution. This conjecture would be true if the minim…
We give an equivalence of triangulated categories between the derived category of finitely generated representations of symplectic reflection algebras associated with wreath products (with parameter t=0) and the derived category of coherent sheaves on a crepant resolution of the spectrum of the centre of these algebras.
Let $V$ be a finite-dimensional symplectic vector space over a field of characteristic 0, and let $G \subset Sp(V)$ be a finite subgroup. We prove that for any crepant resolution $X \to V/G$, the bounded derived category $D^b(Coh(X))$ of coherent sheaves on $X$ is equivalent to the bounded derived category $D^b_G(Coh(V))$ of $G$-equivariant coherent sheaves on $V$.
We study here the crepant resolution correspondence for the torus equivariant descendent Gromov-Witten theories of Hilb(C2) and Sym(C2).The descendent correspondence is obtained from our previous matching of the associated CohFTs by applying Givental's quantization formula to a specific symplectic transformation K. The first result of the paper is an explicit computation of K. Our main result then establishes a fundamental relationship between the Fourier-Mukai equivalence of the associated derived categories (by Bridgeland, King, and Reid) and the symplectic transformation K via Iritani's int…
The aim of this paper is to construct Calabi-Yau 4-folds as crepant resolutions of the quotients of a hyperkähler 4-fold $X$ by a non symplectic involution $α$. We first compute the Hodge numbers of a Calabi-Yau constructed in this way in a general setting and then we apply the results to several specific examples of non symplectic involutions, producing Calabi-Yau 4-folds with different Hodge diamonds. Then we restrict ourselves to the case where $X$ is the Hilbert scheme of two points on a K3 surface $S$ and the involution $α$ is induced by a non symplectic involution on the K3 surface. In t…
This is a continuation of math.AG/0609741. Let Y be an affine symplectic variety with a C^*-action with positive weights, and let π: X -> Y be its crepant resolution. Then πinduces a natural map PDef(X) -> PDef(Y) of Kuranishi spaces for the Poisson deformations of X and Y. In the Part I, we proved that PDef(X) and PDef(Y) are both non-singular, and this map is a finite surjective map. In this paper (Part II), we prove that it is a Galois covering. Markman already obtained a similar result in the compact case, which was a motivation of this paper. As an application, we shall construct explicit…
Let G be a finite group acting on a symplectic complex vector space V. Assume that the quotient V/G has a holomorphic symplectic resolution. We prove that G is generated by "symplectic reflectionsd"', i.e. symplectomorphisms with fixed space of codimension 2 in V. Symplectic resolutions are always semismall. A crepant resolution of V/G is always symplectic. We give a symplectic version of Nakamura conjectures.
The main result of the present paper concerns finiteness properties of Floer theoretic invariants on affine log Calabi-Yau varieties $X$. Namely, we show that: (a) the degree zero symplectic cohomology $SH^0(X)$ is finitely generated and is a filtered deformation of a certain algebra defined combinatorially in terms of a compactifying divisor $\mathbf{D}.$ (b) For any Lagrangian branes $L_0, L_1$, the wrapped Floer groups $WF^*(L_0,L_1)$ are finitely generated modules over $SH^0(X).$ We then describe applications of this result to mirror symmetry, the first of which is an ``automatic generatio…
When the quotient of a symplectic vector space by the action of a finite subgroup of symplectic automorphisms admits as a crepant projective resolution of singularities the Hilbert scheme of regular orbits of Nakamura, then there is a natural isomorphism between the Grothendieck group of this resolution and the representation ring of the group, given by the Bridgeland-King-Reid map. However, this isomorphism is not compatible with the ring structures. For the Hilbert scheme of points on the affine plane, we study the multiplicative behavior of this map.
A Calabi-Yau orbifold is locally modeled on C^n/G where G is a finite subgroup of SL(n, C). In dimension n=3 a crepant resolution is given by Nakamura's G-Hilbert scheme. This crepant resolution has a description as a GIT/symplectic quotient. We use tools from global analysis to give a geometrical generalization of the McKay Correspondence to this case.
Let S be a K3 surface that admits a non-symplectic automorphism $ρ$ of order 3. We divide $S\times \mathbb{P}^1$ by $ρ\timesψ$ where $ψ$ is an automorphism of order 3 of $\mathbb{P}^1$. There exists a threefold ramified cover of a partial crepant resolution of the quotient that is a Calabi-Yau orbifold. We compute the Euler characteristic of our examples and obtain values ranging from 30 to 219.
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。
Let π: Y -> X be a crepant projective resolution of an affine symplectic variety X with a good C^*-action. We interpret the second cohomology H^2(Y, C) in two ways. First, H^2(Y, C) is the Picard group of Y tensorised with C. By the ample cones of different crepant resolutions of X, there is a natural chamber structure in H^2(Y, C). The second interpretation of H^2(Y, C) is the base space of the universal Poisson deformation $\mathcal Y$ of Y. Let D \subset H^2(Y, C) be the locus where the corresponding Poisson varieties are not affine. Then D is the union of finite number of hyperplanes, whic…
Wall Crossing and the Fourier-Mukai Transform for Grassmann Flops
本文证明光滑基上相对 Grassmann 翻折的 crepant 变换猜想，通过解析延拓与辛变换联系相应 GIT 商空间的 $I$-函数，并验证该辛变换与 Iritani 积分结构相容、由 $K$-理论中的 Fourier-Mukai 变换诱导。
A new addition to the zoo of isolated symplectic singularities
本文在 $\mathbb{C}^4/G_5$ 的crepant部分解中发现了一个新的4维孤立辛奇点，其射影切锥为非约化，并求出了其全部12个 $\mathbb{Q}$-factorial terminalisations 及商奇点的24个。
We prove that a quotient singularity $\mathbb{C}^n/G $ by a finite subgroup $G\subset SL_n(\mathbb{C})$ has a crepant resolution only if $G $ is generated by junior elements. This is a generalization of the result of Verbitsky [V]. We also give a procedure to compute the Cox ring of a minimal model of a given $\mathbb{C}^n/G$ explicitly from information of $G$. As an application, we investigate the smoothness of minimal models of some quotient singularities. Together with work of Bellamy and Schedler, this completes the classification of symplectically imprimitive quotient singularities which …
In this paper, we shall prove that any two (projective) symplectic resolutions of a nilpotent orbit closure in a classical simple Lie algebra are connected by a finite sequence of diagrams which are locally trivial families of Mukai flops of type A or of type D. A Mukai flop of type A is the same as a stratified Mukai flop in our previous paper "Mukai flops and derived categories II". On the other hand, a Mukai flop of type D comes from a nilpotent orbit in so(2k). Let G^+ and G^- be two connected components of the orthogonal Grassmannian G_{iso}(k,2k). Then the closure of this orbit has two S…
Let M be a closed oriented smooth 4-manifold admitting symplectic structures. If M is minimal and has b^+=1, we prove that there is a unique symplectic canonical class up to sign, and any real second cohomology class of positive square is represented by symplectic forms. Similar results hold when M is not minimal.
Towards the classification of symplectic linear quotient singularities admitting a symplectic resolution
本文研究了辛线性商奇点的分类问题，证明除了可能的39种情况外，不存在辛解析，将问题缩减为有限多个开放案例。
Equivariant Cohomological Crepant Resolution Conjecture for ADE-orbifolds
本文证明了ADE轨形[C^2/G]的C^*-等变上同调创消解猜想，通过McKay对应和特征标理论，将Bryan-Gholampour变换化为统一根系恒等式，并给出等变Chen-Ruan乘积的显式表示。
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 $…
We formulate a Crepant Resolution Correspondence for open Gromov-Witten invariants (OCRC) of toric Lagrangian branes inside Calabi-Yau 3-orbifolds by encoding the open theories into sections of Givental's symplectic vector space. The correspondence can be phrased as the identification of these sections via a linear morphism of Givental spaces. We deduce from this a Bryan-Graber-type statement for disk invariants, which we extend to arbitrary topologies in the Hard Lefschetz case. Motivated by ideas of Iritani, Coates-Corti-Iritani-Tseng and Ruan, we furthermore propose 1) a general form of the…
We prove the generalised McKay correspondence for isolated singularities using Floer theory. Given an isolated singularity \C^n/G for a finite subgroup G in SL(n,\C) and any crepant resolution Y, we prove that the rank of positive symplectic cohomology SH_+(Y) is the number of conjugacy classes of G, and that twice the age grading on conjugacy classes is the \Z-grading on SH_+(Y) by the Conley-Zehnder index. The generalised McKay correspondence follows as SH_+(Y) is naturally isomorphic to ordinary cohomology H(Y), due to a vanishing result for full symplectic cohomology. In the Appendix we co…
Plumbings of lens spaces and crepant resolutions of compound $A_n$ singularities
本文通过证明两种版本的同调镜像对称性，建立了复合$A_n$奇点的回缩消解与仿射$A_n$透镜空间管道之间的等价关系。
This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety $X$. The motivation for this is Kontsevich's homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results is that when $\dim X \geq 2$, our braid group actions are always faithful. We describe conjectural mirror symmetries between smoothings and resolutions of singularities that lead us to find examples of braid group actions arising from crepant resolutions of various singularities. Relations with the McKa…
We formulate a Crepant Resolution Correspondence for open Gromov-Witten invariants (OCRC) of toric Calabi-Yau orbifolds by viewing the open theories as sections of Givental's symplectic vector space and the correspondence as a linear map of Givental spaces which identifies them. We deduce a Bryan-Graber-type statement for disk invariants and extend it to arbitrary genus zero topologies in the Hard Lefschetz case. Upon leveraging Iritani's theory of integral structures to equivariant quantum cohomology, we conjecture a general form of the symplectomorphism entering the OCRC which arises from a …
Small symplectic $4$-manifolds via contact gluing and some applications
本文介绍了一种通过接触粘合构造小辛4流形的简化方法，并探讨了其在奇异拉格朗日RP²嵌入、接触类型棱镜面嵌入以及Q-同调球辛填充分类中的应用。
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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