Equivalences and stratified flops
Sabin Cautis
我们构造自然等价之间的导出范畴的相干层的局部模型分层Mukai或Atiyah触发器（类型A）。
We construct natural equivalences between derived categories of coherent sheaves on the local models for stratified Mukai or Atiyah flops (of type A).
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Derived equivalence for stratified Mukai flop on G(2,4)
证明了G（2，4）上的分层Mukai触发器存在一个导出等价性.
Extremal contractions, stratified Mukai flops and Springer maps
证明了幂零轨道闭包上的两个相同次数的Springer映射是通过分层Mukai触发器连接的，而后者是通过幂零轨道闭包的自然分解的极值压缩得到的.
On stratified Mukai flops
通过连续爆破光滑子簇，构造了A，D，E_{6, I}型分层Mukai触发器的分解.在E_{6, I}的情况下，我们构造了一个自然函子，它诱导了Chow群之间的同构。
Motivic and quantum invariance under stratified Mukai flops
对于$A_{n,k}, D_{2k+1}$型和$E_{6,I}$型分层向井触发器，证明了纤维积在Chow基模上诱导同构。与（标准）Mukai拖鞋相反，上积产品通常不被保存。对于$A_{n, 2}$、$D_5$和$E_{6, I}$触发器，通过对普通触发器的退化/变形，发现了量子修正。
A simpler kernel for stratified Mukai flops
本文针对分层 Mukai 翻转的导出等价，利用矩阵分解框架，为平面 Grassmann 情形给出了 Fourier-Mukai 核的简单几何构造。
Mukai flops and derived categories II
本文是数学AG/0203287的续篇。E.马克曼在这里我们称之为分层向井翻牌。在本文中，我们观察到，对于一个分层的Mukai触发器：$X \to \bar{X} \leftarrow X^+$，纤维积$X \times_{\bar{X}}X^+$定义了Grothendieck群$K(X) \to K(X^+)$的同构，但是，在导出范畴的水平上，纤维积一般不给出等价。
Flops and Equivalences of derived Categories for Threefolds with only Gorenstein Singularities
本文的主要目的是证明反常点层的Bridgeland模空间对于某些flopping压缩给出了flops，由flops的双有理对应给出的Fourier-Mukai变换是有界导出范畴之间的等价。
We prove that under semismall smooth flops, smooth projective varieties have (non-canonically) isomorphic Chow motives with coefficients in any noetherian local ring $Λ$ by comparing the pushforward of the constant intersection complexes through flopping contractions.
Descendant and Fourier-Mukai equivalences for simple flops
研究简单翻转诱导的后代Gromov-Witten理论之间对应关系及傅里叶-穆卡等价性的兼容性
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 give a new proof of the derived equivalence of a pair of varieties connected by the flop of type $C_2$ in the list of Kanemitsu arXiv:1812.05392, which is originally due to Segal. We also prove the derived equivalence of a pair of varieties connected by the flop of type $A^G_4$ in the same list. The latter proof follows that of the derived equivalence of Calabi--Yau 3-folds in Grassmannians $\text{Gr}(2,5)$ and $\text{Gr}(3,5)$ by Kapustka and Rampazzo arXiv:1711.10231 closely.
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.
Derived Equivalences of Generalized Grassmannian Flops: $D_4$ and $G_2^{\dagger}$ Cases
本文证明 $D_4$ 和 $G_2^{\dagger}$ 型广义格拉斯曼 flop 导出导出等价，基于 Kuznetsov 变异技术对例外对象进行一系列变异，为 DK 猜想提供新证据。
Derived equivalence for the simple flop of type $D_5$
本文证明 $D_5$ 型简单翻折诱导导出等价，为 DK 猜想提供新证据，并扩展至非双有理 Calabi--Yau 五重态及纤维化的导出等价。
Derived equivalence for the simple flop of type $D_4$ via tilting bundles
本文针对 $D_4$ 型简单翻折的局部模型，通过构造两侧的倾斜丛证明其导出等价，并由此推导出次数为 $12$ 的一般 $K3$ 曲面间的导出等价。
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。
Associated to a Mukai flop X ---> X' is on the one hand a sequence of equivalences D(X) -> D(X'), due to Kawamata and Namikawa, and on the other hand a sequence of autoequivalences of D(X), due to Huybrechts and Thomas. We work out a complete picture of the relationship between the two. We do the same for standard flops, relating Bondal and Orlov's derived equivalences to spherical twists, extending a well-known story for the Atiyah flop to higher dimensions.
Flops are birational transformations which, conjecturally, induce derived equivalences. In many cases an equivalence can be produced as pull-push via a resolution of the birational transformation; when this happens, we have a non-trivial autoequivalence of either sides of the flop known as the \emph{flop-flop autoequivalence}. We prove that such autoequivalence can be realised as the inverse of a spherical twist around a conservative, spherical functor in a natural way. More precisely, we prove that a natural, conservative spherical functor exists in a more general framework and that the flop-…
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。
Differential Complexes and Stratified Pro-Modules
本文引入了层Pro-模范畴和该范畴中诱导对象的概念。我们提出了一个翻译的Morihiko齐藤等价结果使用的对偶语言的亲对象。证明了层Pro-模的导出范畴与Pro-微分复形范畴的等价性。我们还提供了一个比较的概念，晶体的Pro模（P. Deligne在1960年推出）。
Flops and about: a guide
层错出现在辛簇的双有理几何中，如幂零轨道和层的模空间的分解。奇怪的是，在由这种失败相关的变种之间构造派生等价关系，与表示论和纽结同调有关。在本文中，我们将讨论如何构建这样的等价，解释这样做的主要工具（类别李代数行动）和评论各种相关的主题。
Window shifts, flop equivalences and Grassmannian twists
我们引入一类新的自等价，作用于某些向量丛在格拉斯曼上的导出范畴。这些自等价来自格拉斯曼的触发器：它们推广了塞德尔-托马斯球面扭曲，可以被看作是来自标准触发器。我们首先给出了一个简单的代数结构，这是非常适合显式计算。然后，我们给出了一个尺规作图法使用球函子，我们证明是等价的。
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.
Donaldson-Thomas Invariants and Flops
本文证明了两个光滑的和投射的Calabi-Yau三重相关的Donaldson-Thomas曲线计数不变量的一个比较公式。根据Bridgeland的结果，任何两个这样的簇都是等价的.此外，在两边还存在着被这个等价交换的反常凝聚层的范畴对。利用Joyce的理论，我们构造了这些范畴的motivic Hall代数。这些代数提供了一座桥梁，将触发器两侧的不变量联系起来。
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.
The Fano of lines, the Kuznetsov component, and a flop
本文证明了三次超曲面导出范畴的Kuznetsov分量的对称平方等价于直线Fano簇的导出范畴，验证了Galkin猜想，其关键步骤是建立了一个特定12维flop的新导出等价。
We formulate and prove Serre's equivalence for $\mathbb{Z}$-graded rings. When restricted to the usual case of $\mathbb{N}$-graded rings, our version of Serre's equivalence also sharpens the usual one by replacing the condition that $A$ be generated by $A_1$ over $A_0$ by a more natural condition, which we call the Cartier condition. For $\mathbb{Z}$-graded rings coming from flips and flops, this Cartier condition relates more naturally to the geometry of the flip/flop in question. We also interpret Grothendieck duality as an instance of Greenlees-May duality for graded rings. These form the b…
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。
Flops of G-Hilb and equivalences of derived categories by variation of GIT quotient
对于SL（3，C）中的有限子群G，Bridgeland，King和Reid证明了G-簇的模空间是商C^3/G的一个可分解.本文考虑由Kronheimer引入并由Sardo Infirri进一步研究的模空间M_θ，它与G-Hilb在GIT参数θ的特定选择下一致.对于G的Abel算子，证明了C^3/G的每一个投射收缩分解对某个参数θ同构于M_θ.关键的一步是通过适当的傅立叶-向井变换的模空间的K-理论的GIT室的描述。我们还揭示了位于相邻GIT室中的参数的模量M_θ的导出范畴之间的显式等价性。
We study derived categories of Gorenstein varieties X and X^+ connected by a flop. We assume that the flopping contractions f: X \to Y, f^+: X^+ \to Y have fibers of dimension bounded by 1 and Y has canonical hypersurface singularities of multiplicity 2. We consider the fiber product W=X \times_Y X^+ with projections p: W \to X, q: W \to X^+ and prove that the flop functors F = Rq_* Lp^*: D^b(X) \to D^b(X^+), F^+= Rp_*Lq^*: D^b(X^+) \to D^b(X) are equivalences, inverse to those constructed by M. Van den Bergh. The composite F^+ \circ F: D^b(X) \to D^b(X) is a non-trivial auto-equivalence. When…
Mukai flops and derived categories
本文证明了由Mukai触发器连接的dim 2n的两个光滑投射簇具有等价的有界导出范畴。更准确地说，假设$ϕ: X - - \to X^+$是一个中心为$Y \subset X$和$Y^+ \subset X^+$的向井翻牌。在我们的例子中，由$ϕ$的图定义的自然因子$Φ: D(X) \to D(X^+)$不是完全忠实的。相反，令$X \to {\bar X}$和$X^+ \to {\bar X}$为中心的双有理收缩映射，并将${\hat X} := X \times_{{\bar X} X^+$。则${\hat X}$是具有两个不可约分支的正规交叉簇。这个${\hat X}$定义了一个函子$Ψ: D(X) \to D(X^+)$。我们将证明这个$Ψ$是等价的。最近，Wierzba和Wisniewski宣布，两个双有理等价的复射影辛4-折叠是由一个有限序列的Mukai触发器连接的。我们的结果表明，$D(X)$是一个双有理不变量的复射影辛4倍。
Derived categories of toric varieties III
我们证明了派生麦凯对应在特殊情况下和分解的复曲面K-等价到触发器。
Flops, flips and perverse point sheaves on threefold stacks
我们考虑两种情况下的双有理变换的Q-Gorenstein三倍：终端触发器的情况下，一个弗朗西亚翻转。在第一种情况下，我们表明，如果一个取代的三重由他们的规范覆盖堆栈，触发器符合模堆栈的反常点层。这增加了Kawamata的一个结果，显示了导出的等价性。我们的构造有一个缺点，即模空间是使用堆栈的表示来定义的。在第二种情况下，我们表明，如果采取一个Francia翻转$X \to Y$，并取代$X$由其规范覆盖堆栈，那么翻转符合一个版本的Bridgelan的模空间的pwrverse点层涉及一个新的反常。这是Kawamata的另一个结果。同样有一个缺点--结果的证明依赖于翻转的存在，这与布里奇兰和陈的结果形成鲜明对比，其中相关的翻转出现为构造的结果。
The motivic Satake equivalence using perverse Nori motives
本文提出了一种基于 perverse Nori 动机的 Satake 等价性理论，扩展了 Mirković-Vilonen 的几何 Satake 等价性。
Three-dimensional flops and non-commutative rings
如果Y，Z是由一个翻转相关的三维光滑簇，那么Bondal和Orlov证明Y和Z上的相干层的导出范畴是等价的。这个猜想最近被Bridgeland证明了。本文的目的是利用非交换环给出Bridgeland结果的部分新证明。新的证明也涵盖了一些温和的奇异和高维的情况（包括陈的最近论文中的一个："只有Gorenstein奇异性的三重的导出范畴的触发和等价"）。
Moduli stack of stable curves from a stratified homotopy viewpoint
本文加强了Charney和Lee的结果，证明了稳定曲线范畴能捕获模栈的分层同伦类型，并通过外行进等价对可构造层进行分类。
We prove that the stable endomorphism rings of rigid objects in a suitable Frobenius category have only finitely many basic algebras in their derived equivalence class and that these are precisely the stable endomorphism rings of objects obtained by iterated mutation. The main application is to the Homological Minimal Model Programme. For a 3-fold flopping contraction $f \colon X \to \mathrm{Spec}\ R$, where $X$ has only Gorenstein terminal singularities, there is an associated finite dimensional algebra $A_{\mathrm{con}}$ known as the contraction algebra. As a corollary of our main result, th…
We show that for many moduli spaces M of torsion sheaves on K3 surfaces S, the functor D(S) -> D(M) induced by the universal sheaf is a P-functor, hence can be used to construct an autoequivalence of D(M), and that this autoequivalence can be factored into geometrically meaningful equivalences associated to abelian fibrations and Mukai flops. Along the way we produce a derived equivalence between two compact hyperkaehler 2g-folds that are not birational, for every g >= 2. We also speculate about an approach to showing that birational moduli spaces of sheaves on K3 surfaces are derived-equivale…
$K_1(Var)$ is presented by stratified birational equivalences
本文完整呈现了变种的$K_1$群，通过改进Gillet-Grayson的$G$-构造定义了变种的$K$-理论谱。研究在技术上扩展了精确范畴的$K$-理论结果，概念上引入了生成$K_1$的广义自同构（双精确方块）.
Window equivalences via categorical sl(2) actions
本文通过 Knörrer 周期性，统一了格拉斯曼余切丛分层 Mukai 翻转的两种导出等价方法，即几何范畴 sl(2) 作用与魔窗范畴。
Categories of Constructible Sheaves
本文研究了可构造sheaf范畴的等价性问题，并探讨了局部常数sheaf和可构造sheaf的基本性质。
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…
Topological exodromy with coefficients
本文证明了分层空间上可构造层的导出∞-范畴与出口路径∞-范畴的表示∞-范畴之间的exodromy等价的最强版本，并推广到更一般的系数和分层空间。
Simple complexes on a flopped curve
本文研究了 (compound) du Val 奇点的 crepant blow-ups，分类了无负自扩张的凝聚层复形，并据此证明了 Bridgeland stability manifold 的连通性及基本倾斜复形间的平移与迭代 mutation 关联性。
Grassmannian twists, derived equivalences and brane transport
本文基于在波恩举行的String-Math 2012上的演讲，与Ed Segal的联合论文。我们表现出相应的格拉斯曼触发器导出的等价。这些等价性的构造受到Herbst-Hori-Page关于规范线性sigma模型的膜运输的工作的启发：特别是，我们定义了对应于其等级限制规则的“窗口”。然后，我们将展示如何组成我们的等价产生有趣的自等价，我们描述为扭曲和cotiwits某些球形函子。我们的证明使用自然长的确切序列的丛格拉斯曼称为扭曲的Lascoux复合体，或楼梯复合体。我们给这些紧凑的描述。我们还涉及一些相关的发展，并通过一些扩展的例子。
Invariance of orientation data for ind-constructible Calabi-Yau $A_{\infty}$ categories under derived equivalence
我们研究的方向数据，介绍了Kontsevich和Soibelman，以定义良好的整合映射从motivic霍尔代数的3维卡-丘范畴环的动机。我们首先通过一个示例来演示方向数据在本文中的作用，然后再讨论技术细节。我们给出了一个帐户的情况下，紧支撑层的非紧卡-丘三重的定向数据。最后，我们研究这种结构的行为下拉回沿准等价的类别，证明Kontsevich和Soibelman的猜想，关于这种行为，也有一些更强的定理，关于触发器和更一般的倾斜。
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 show the existence of semiorthogonal decompositions of Donaldson-Thomas categories for $(-1)$-shifted cotangent derived stacks associated with $Θ$-stratifications on them. Our main result gives an analogue of window theorem for categorical DT theory, which has applications to d-critical analogue of D/K equivalence conjecture, i.e. existence of fully-faithful functors (equivalences) under d-critical flips (flops). As an example of applications, we show the existence of fully-faithful functors of DT categories for Pandharipande-Thomas stable pair moduli spaces under wall-crossing at super-rig…
Perverse schobers are categorical analogs of perverse sheaves. Examples arise from varieties admitting flops, determined by diagrams of derived categories of coherent sheaves associated to the flop: in this paper we construct mirror partners to such schobers, determined by diagrams of Fukaya categories with stops, for examples in dimensions 2 and 3. Interpreting these schobers as supported on loci in mirror moduli spaces, we prove homological mirror symmetry equivalences between them. Our construction uses the coherent-constructible correspondence and a recent result of Ganatra-Pardon-Shende t…
Exodromy beyond conicality
本文证明紧子解析分层和实代数簇的分层具有有限出口路径∞-范畴，通过引入exodromic分层类，建立其粗化和态射性质，推广了exodromy定理。
A version of the Bondal-Orlov conjecture, proved by Bridgeland, states that if $X$ and $Y$ are smooth complex projective threefolds linked by a flop, then they are derived equivalent. Van den Bergh gave a new proof of Bridgeland's theorem using the notion of a NCCR, which is in particular a ring $A$ together with a derived equivalence between $X$ and $A$. This ring $A$ is constructed as an endomorphism ring of a decomposable module, and hence admits an idempotent $e$. Donovan and Wemyss define the contraction algebra $A_\mathrm{con}$ to be the quotient of $A$ by $e$; it is a finite-dimensional…
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