Li Chiang, Shi-shyr Roan
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 $n=4,5$.
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Crepant Transformation Conjecture For the Grassmannian Flop
本文通过 abelianization 和“显式重建”方法，证明了 Grassmannian flop 的 Crepant Transformation 猜想的显式形式。
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.
A categorical flop in dimension one
研究一维节点曲线的非交换解析中范畴翻转结构，通过Landau-Ginzburg模型几何解释，并提出区分弱crepant非强crepant解析的中间条件。
On derived equivalence for Abuaf flop: mutation of non-commutative crepant resolutions and spherical twists
本文研究Abuaf 5重翻转的导出等价，利用Toda-Uehara倾斜层和Iyama-Wemyss变异函子补充了Segal的结果，并观察到“翻转-翻转=扭转”和“多重变异=扭转”的结论。
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 …
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。
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 $\ove…
We say that an exact equivalence between the derived categories of two algebraic varieties is tilting-type if it is constructed by using tilting bundles. The aim of this article is to understand the behavior of tilting-type equivalences for crepant resolutions under deformations. As an application of the method that we establish in this article, we study the derived equivalence for stratified Mukai flops and stratified Atiyah flops in terms of tilting bundles.
We compute the motivic Donaldson-Thomas theory of small crepant resolutions of toric Calabi-Yau 3-folds.
I study a sequence of singularities in dimension 4 and above, each given by a cone of rank 1 tensors of a certain signature, which have crepant resolutions whose exceptional loci are isomorphic to cartesian powers of the projective line. In each dimension n, these resolutions naturally correspond to vertices of an (n-2)-simplex, and flops between them correspond to edges of the simplex. I show that each face of the simplex may then be associated to a certain relation between flop functors.
Some faithful algebraic braid twist group actions for 3-fold crepant resolutions
本文研究了3重crepant解析的代数辫子扭群作用，通过构造球形对象配置，展示了当a=9和a=13时，导出的导出范畴分别具有类型D和E的忠实作用。
An E$_7$-Weierstrass model is conjectured to have eight distinct crepant resolutions whose flop diagram is a Dynkin diagram of type E$_8$. In previous work, we explicitly constructed four distinct resolutions, for which the flop diagram formed a D$_4$ sub-diagram. The goal of this paper is to explore those properties of a resolved E$_7$-model which are not invariant under flops. In particular, we examine the fiber degenerations, identify the fibral divisors up to isomorphism, and study violation of flatness appearing over certain codimension-three loci in the base, where a component of the fib…
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.
In this paper, we prove a generalization of Orlov's projectivization formula for the derived category $D^b_{\rm coh} (\mathbb{P}(\mathscr{E}))$, where $\mathscr{E}$ does not need to be a vector bundle; Instead, $\mathscr{E}$ is a coherent sheaf which locally admits two-step resolutions. As a special case, this also gives Orlov's generalized universal hyperplane section formula. As applications, (i) we obtain a blowup formula for blowup along codimension two Cohen-Macaulay subschemes, (ii) we obtain new "flop-flop=twist" results for a large class of flops obtained by crepant resolutions of dege…
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。
Folding of ADE-Dynkin diagrams according to graph automorphisms yields irreducible Dynkin diagrams of ABCDEFG-types. This folding procedure allows to trace back the properties of the corresponding simple Lie algebras or groups to those of ADE-type. In this article, we implement the techniques of folding by graph automorphisms for Hitchin integrable systems. We show that the fixed point loci of these automorphisms are isomorphic as algebraic integrable systems to the Hitchin systems of the folded groups away from singular fibers. The latter Hitchin systems are isomorphic to the intermediate Jac…
We describe a new example of a flop in 5-dimensions, due to Roland Abuaf, with the nice feature that the contracting loci on either side are not isomorphic. We prove that the two sides are derived equivalent.
We study the geometric engineering of gauge theories with gauge group Spin(4) and SO(4) using crepant resolutions of Weierstrass models. The corresponding elliptic fibrations realize a collision of singularities corresponding to two fibers with dual graph the affine $A_1$ Dynkin diagram. There are eight different ways to engineer such collisions using decorated Kodaira fibers. The Mordell-Weil group of the elliptic fibration is required to be trivial for Spin(4) and Z/2Z for SO(4). Each of these models have two possible crepant resolutions connected by a flop. We also compute a generating func…
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…
For a finite subgroup G in SL(3,C), Bridgeland, King and Reid proved that the moduli space of G-clusters is a crepant resolution of the quotient C^3/G. This paper considers the moduli spaces M_θ, introduced by Kronheimer and further studied by Sardo Infirri, which coincide with G-Hilb for a particular choice of the GIT parameter θ. For G Abelian, we prove that every projective crepant resolution of C^3/G is isomorphic to M_θfor some parameter θ. The key step is the description of GIT chambers in terms of the K-theory of the moduli space via the appropriate Fourier--Mukai transform. We also unc…
Given any smooth toric surface S, we prove a SYM-HILB correspondence which relates the 3-point, degree zero, extended Gromov-Witten invariants of the n-fold symmetric product stack [Sym^n(S)] of S to the 3-point extremal Gromov-Witten invariants of the Hilbert scheme Hilb^n(S) of n points on S. As we do not specialize the values of the quantum parameters involved, this result proves a strengthening of Ruan's Cohomological Crepant Resolution Conjecture for the Hilbert-Chow morphism from Hilb^n(S) to Sym^n(S) and yields a method of reconstructing the cup product for Hilb^n(S) from the orbifold i…
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…
This is a local version of math.AG/0506534. We shall deal with the deformation of a convex symplectic variety $X$ instead of a projective one. The usual deformation does not work well in the convex case. Instead, we regard $X$ as a Poisson scheme and study its Poisson deformation. One of the application is the following: Let $Y$ be an affine symplectic variety, and assume that $Y$ has two $Q$-factorial crepant terminalizations $X$ and $X'$. If $X$ is non-singular, then $X'$ is non-singular, too. Moreover, when $Y$ has a good $C^*$-action, $X$ and $X'$ have the same kind of singularities.
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…
We study crepant resolutions of Weierstrass models of $\text{SU}(2)\!\times\!\text{SU}(3)$-models, whose gauge group describes the non-abelian sector of the Standard Model. The $\text{SU}(2)\!\times\!\text{SU}(3)$-models are elliptic fibrations characterized by the collision of two Kodaira fibers with dual graphs that are affine Dynkin diagrams of type $\widetilde{\text{A}}_1$ and $\widetilde{\text{A}}_2$. Once we eliminate those collisions that do not have crepant resolutions, we are left with six distinct collisions that are related to each other by deformations. Each of these six collisions…
Derived equivalence for the simple flop of type $G_2^{\dagger}$ via tilting bundles
本文通过倾转包证明了类型$G_2^{\dagger}$简单翻转的局部模型的导出等价性，并给出了非交换crepant解析，该解析与翻转两侧导出等价。
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。
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.
Birational complexity of log Calabi-Yau 3-folds
本文研究log Calabi-Yau 3-折叠的双有理复杂度，证明其取值于{0,2,3}，并构造具有到射影空间收缩的crepant双有理模型，通过标准$\mathbb{P}^1$-链的几何刻画实现。
We consider Calabi--Yau 3-folds of Borcea--Voisin type, i.e. Calabi--Yau 3-folds obtained as crepant resolutions of a quotient $(S\times E)/(α_S\times α_E)$, where $S$ is a K3 surface, $E$ is an elliptic curve, $α_S\in {\rm Aut} (S)$ and $α_E\in {\rm Aut}(E)$ act on the period of $S$ and $E$ respectively with order $n=2,3,4,6$. The case $n=2$ is very classical, the case $n=3$ was recently studied by Rohde, the other cases are less known. First we construct explicitly a crepant resolution, $X$, of $(S\times E)/(α_S\times α_E)$ and we compute its Hodge numbers; some pairs of Hodge numbers we fou…
The index of a 3-fold canonical singularity at a crepant centre is at most 6.
This is an overview article, based on my 2018 Kinosaki lecture, that surveys and announces work on 3-fold flopping contractions, their affine combinatorics, stability conditions, tilting bundles and autoequivalences. Some first applications are given.
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…
A Donaldson-Thomas crepant resolution conjecture on Calabi-Yau 4-folds
本文在Calabi-Yau 4-折叠的翻转解消上定义了$K$理论稳定对不变量和零次Donaldson-Thomas不变量，猜想了闭式公式，并提出了联系这两种理论的翻转解消对应。
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".
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 …
In this paper, we introduce the notion of toric special weak Fano manifolds, which have only special primitive crepant contractions. We study the structure of them, and in particular completely classify smooth toric special weak Fano 4-folds. As a result, we can confirm that almost every smooth toric special weak Fano 4-fold is a weakened Fano manifold, that is, a weak Fano manifold which can be deformed to a Fano manifold.
The zeroth line bundle cohomology on Calabi-Yau three-folds encodes information about the existence of flop transitions and the genus zero Gromov-Witten invariants. We illustrate this claim by studying several Picard number 2 Calabi-Yau three-folds realised as complete intersections in products of projective spaces. Many of these manifolds exhibit certain symmetries on the Picard lattice which preserve the zeroth cohomology.
We consider geometrical problems on Gorenstein hypersurface orbifolds of dimension $n \geq 4$ through the theory of Hilbert scheme of group orbits. For a linear special group $G$ acting on $\CZ^n$, we study the $G$-Hilbert scheme, $\hl^G(\CZ^n)$, and crepant resolutions of $\CZ^n/G$ for $G$=the $A$-type abelian group $ A_r(n)$. For $n=4$, we obtain the explicit structure of $\hl^{A_r(4)}(\CZ^4)$. The crepant resolutions of $\CZ^4/A_r(4)$ are constructed through their relation with $\hl^{A_r(4)}(\CZ^4)$, and the connections between these crepant resolutions are found by the "flop" procedure of …
We prove the transformation formula of Donaldson-Thomas (DT) invariants counting two dimensional torsion sheaves on Calabi-Yau 3-folds under flops. The error term is described by the Dedekind eta function and the Jacobi theta function, and our result gives evidence of a 3-fold version of Vafa-Witten's S-duality conjecture. As an application, we prove a blow-up formula of DT type invariants on the total spaces of canonical line bundles on smooth projective surfaces. It gives an analogue of the similar blow-up formula in the original S-duality conjecture by Yoshioka, Li-Qin and Göttsche.
NCCRs of cones over del Pezzo surfaces
本文研究了非交换交换解在非终端的反canonical锥体上的应用，证明了所有几何螺旋通过突变相互连接，利用例外集合的多边形进行证明。
Let $X$ and $X'$ be nonsingular projective $3$-folds related by a flop of a disjoint union of $(-2)$-curves. We prove a flop formula relating the Donaldson-Thomas invariants of $X$ to those of $X'$, which implies some simple relations among BPS state counts. As an application, we show that if $X$ satisfies the GW/DT correspondence for primary insertions and descendants of the point class, then so does $X'$. We also propose a conjectural flop formula for general flops.
Given a quasi-projective 3-fold X with only Gorenstein terminal singularities, we prove that the flop functors beginning at X satisfy higher degree braid relations, with the combinatorics controlled by a real hyperplane arrangement H. This leads to a general theory, incorporating known special cases with degree 3 braid relations, in which we show that higher degree relations can occur even for two smooth rational curves meeting at a point. This theory yields an action of the fundamental group of the complexified complement of H on the derived category of X, for any such 3-fold that admits indi…
For a cyclic group $G$ acting on a smooth variety $X$ with only one character occurring in the $G$-equivariant decomposition of the normal bundle of the fixed point locus, we study the derived categories of the orbifold $[X/G]$ and the blow-up resolution $\widetilde Y \to X/G$. Some results generalise known facts about $X = A^n$ with diagonal $G$-action, while other results are new also in this basic case. In particular, if the codimension of the fixed point locus equals $|G|$, we study the induced tensor products under the equivalence $D^b(\widetilde Y) \cong D^b([X/G])$ and give a 'flop-flop…
Derived deformation theory of crepant curves
本文确定了Calabi-Yau三维流形中光滑有理曲线的完整导出形变理论，给出非交换形变的超势代数描述，验证弦理论预测，并提出推广Artin曲面结果的三维流形有理曲线可收缩性猜想。
We prove that the functor of noncommutative deformations of every flipping or flopping irreducible rational curve in a 3-fold is representable, and hence associate to every such curve a noncommutative deformation algebra. This new invariant extends and unifies known invariants for flopping curves in 3-folds, such as the width of Reid, and the bidegree of the normal bundle. It also applies in the settings of flips and singular schemes. We show that the noncommutative deformation algebra is finite dimensional, and give a new way of obtaining the commutative deformations of the curve, allowing us…
Window categories for a simple $9$-fold flop of Grassmannian type
本文针对 Grassmann 型简单 9 折翻折，构造四个与推前函子交换的导出等价，通过识别 Artin 栈上的窗口范畴实现，并应用于非双有理 Calabi-Yau 三维簇的导出等价新证明。
We study elliptic fibrations that geometrically engineer an SU(2)$\times$ G$_2$ gauge theory realized by Weierstrass model for the collision III+$\text{I}_0^{*\text{ns}}$. We construct the four possible crepant resolutions of such a Weierstrass model and show that they form a chain of four minimal models connected by flops. We compute the generating function for the Euler characteristic of these crepant resolutions. In the case of a Calabi-Yau threefold, we consider the compactification of M-theory and F-theory on an SU(2)$\times$ G$_2$-model to a five and six-dimensional supergravity with eig…
Two 3-fold flops are exhibited, both of which have precisely one flopping curve. One of the two flops is new, and is distinct from all known algebraic D4-flops. It is shown that the two flops are neither algebraically nor analytically isomorphic, yet their curve-counting Gopakumar-Vafa invariants are the same. We further show that the contraction algebras associated to both are not isomorphic, so the flops are distinguished at this level. This shows that the contraction algebra is a finer invariant than various curve-counting theories, and it also provides more evidence for the proposed analyt…
This is an expository article on the recent studies of Ruan's crepant resolution/flop conjecture and its possible relations to the K-theory integral structure in quantum cohomology.
It is known that every three dimensional Gorenstein toric singularity has a crepant resolution. Although it is not unique, all crepant resolutions are connected by repeating the operation "flop". On the other hand, this singularity also has a non-commutative crepant resolution (= NCCR) which is constructed from a consistent dimer model. Such an NCCR is given as the endomorphism ring of a certain module which we call splitting maximal modifying module. In this paper, we show that all splitting maximal modifying modules are connected by repeating the operation "mutation" of splitting maximal mod…
Let $f\colon X\to\mathrm{Spec}\, R$ be a 3-fold flopping contraction, where $X$ has at worst Gorenstein terminal singularities and $R$ is complete local. We describe the space of Bridgeland stability conditions on the null subcategory $\mathscr{C}$ of the bounded derived category of $X$, which consists of those complexes that derive pushforward to zero, and also on the affine subcategory $\mathscr{D}$, which consists of complexes supported on the exceptional locus. We show that a connected component of stability conditions on $\mathscr{C}$ is the universal cover of the complexified complement …
As a standard example in toric geometry, the Atiyah flop of a $(-1,-1)$-curve in a smooth 3-fold can be described combinatorially in terms of the two possible triangulations of a square cone. The flop of $(-2,0)$-curve cannot be realised in terms of toric geometry. Nevertheless, we explain how to construct a cone $σ$ in an integral affine manifold with singularities associated to the singularity $(P\in Y)$ at the base of a $(-2,0)$-flop. The two sides of the flop can then be described combinatorially in terms of two different subdivisions of $σ$. As an interesting byproduct of our construction…
A 3-fold terminal quotient singularity X=C^3/G admits the economic resolution Y-> X, which is "close to being crepant". This paper proves that the economic resolution Y is isomorphic to a distinguished component of a moduli space of certain G-equivariant objects using the the King stability condition $θ$ introduced by Kędzierski.
We study how Gromov-Witten invariants of projective 3-folds transform under a standard flop and a small extremal transition in the algebro-geometric setting from the recent development of algebraic relative Gromov-Witten theory and its applications. This gives an algebro-geometric account of Witten's wall-crossing formula for correlation functions of the descendant nonlinear sigma model in adjacent geometric phases of a gauge linear sigma model and of the symplectic approach in an earlier work of An-Min Li and Yongbin Ruan on the same problem. A terse account from the stringy and the symplecti…
Box graphs, or equivalently Coulomb phases of three-dimensional N=2 supersymmetric gauge theories with matter, give a succinct, comprehensive and elegant characterization of crepant resolutions of singular elliptically fibered varieties. Furthermore, the box graphs predict that the phases are organized in terms of a network of flop transitions. The geometric construction of the resolutions associated to the phases is, however, a difficult problem. Here, we identify a correspondence between box graphs for the gauge algebras su(2k+1) with resolutions obtained using toric tops and generalizations…
Torsion pairs and 3-fold flops
本文研究了 3 折 flops 合约的局部导出范畴中 t-结构的分类，并探讨了 torsion pairs 在模块范畴中的应用。
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…
We define relative Gromov-Witten invariants and establish a general gluing theory of pseudo-holomorphic curves for symplectic cutting and contact surgery. Then, we use our general gluing theory to study the change of GW-invariants of Calabi-Yau 3-folds tranform under flops and extremal transitions. We prove a complete formula for the change of GW-invariants of any genus transform under flop and a general type I extremal transition. Other extremal transition will be handled in a subsequent paper.
In this article, we examine the arithmetic aspect of the Kummer-surface-type CY 3-folds $\hat{T/G}$, characterized by the crepant resolution of 3-torus-orbifold $T/G$ with only isolated singularities. Up to isomorphisms, there are only two such space $\hat{T/G}$ with $|G|=3, 7$, and both $T$ carrying the structure of triple-product structure of a CM elliptic curve. The (Griffiths) intermediate Jacobians of these $\hat{T/G}$ are identified explicitly as the corresponding elliptic curve appeared in the structure of $T$. We further provide the $\QZ$-structure of $\hat{T/G}$ and verify their modul…
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