Tom Bridgeland
This paper contains some applications of Fourier-Mukai techniques to the birational geometry of threefolds. In particular, we prove that birational Calabi-Yau threefolds have equivalent derived categories. To do this we show how flops arise naturally as moduli spaces of perverse coherent sheaves.
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The main propose of this paper is to show that Bridgeland's moduli space of perverse point sheaves for certain flopping contractions gives the flops, and the Fourier-Mukai transform given by the birational correspondence of the flop is an equivalence between bounded derived categories.
This paper is a sequel to math.AG/0203287. A generalization of the Mukai flop has been studied by E. Markman. Here we call it a stratified Mukai flop. In this paper, we observe that, for a stratified Mukai flop: $X \to \bar{X} \leftarrow X^+$, the fiber product $X \times_{\bar{X}}X^+$ defines an isomorphism of Grothendieck groups $K(X) \to K(X^+)$, but, at the level of derived categories, the fiber product does not generally give an equivalence.
In this note, we shall prove that two smooth projective varieties of dim 2n connected by a Mukai flop have equivalent bounded derived categories. More precisely, let $ϕ: X - - \to X^+$ be a Mukai flop with centers $Y \subset X$ and $Y^+ \subset X^+$. In our case, the natural fuctor $Φ: D(X) \to D(X^+)$ defined by the graph of $ϕ$ is not fully faithful. Instead, let $X \to {\bar X}$ and $X^+ \to {\bar X}$ be the birational contraction maps of the centers, and put ${\hat X} := X \times_{{\bar X} X^+$. Then ${\hat X}$ is a normal crossing variety with two irreducible components. This ${\hat X}$ d…
We construct natural equivalences between derived categories of coherent sheaves on the local models for stratified Mukai or Atiyah flops (of type A).
We prove a comparison formula for the Donaldson-Thomas curve-counting invariants of two smooth and projective Calabi-Yau threefolds related by a flop. By results of Bridgeland any two such varieties are derived equivalent. Furthermore there exist pairs of categories of perverse coherent sheaves on both sides which are swapped by this equivalence. Using the theory developed by Joyce we construct the motivic Hall algebras of these categories. These algebras provide a bridge relating the invariants on both sides of the flop.
We prove that there is a derived equivalence for stratified Mukai flop on G(2,4).
We prove derived McKay correspondence in special cases and the decomposition of toric K-equivalence into flops.
We give a survey for the results in [Yeu20a, Yeu20b, Yeu20c], which attempts to relate the derived categories under general classes of flips and flops. We indicate how the approach fails because of what appears to be a formal problem. We give some ideas, and record some failed attempts, to fix this problem. We also present some new examples.
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 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.
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…
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.
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.
If Y,Z are three-dimensional smooth varieties related by a flop, then Bondal and Orlov conjectured that the derived categories of coherent sheaves on Y and Z are equivalent. This conjecture was recently proved by Bridgeland. Our aim in this paper is to give a partially new proof of Bridgeland's result using non-commutative rings. The new proof also covers some mild singular and higher dimensional situations (including the one in the recent paper by Chen: ``Flops and Equivalences of derived Categories for Threefolds with only Gorenstein Singularities'').
Perverse schobers are conjectural categorical analogs of perverse sheaves. We show that such structures appear naturally in Homological Minimal Model Program which studies the effect of birational transformations such as flops, on the coherent derived categories. More precisely, the flop data are analogous to hyperbolic stalks of a perverse sheaf. In the first part of the paper we study schober-type diagrams of categories corresponding to flops of relative dimension 1, in particular we determine the categorical analogs of the (compactly supported) cohomology with coefficients in such schobers.…
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。
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…
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。
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 autoequivalences of length 2 flops via GIT
本文利用 GIT 和窗口理论研究长度 2 坍缩的理查德曲线的导出自同构，并证明弦性凯勒模空间与长度 2 三维流形的描述一致，恢复了 Donovan-Wemyss 的单束作用。
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 $G_2^{\dagger}$ 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。
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。
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。
Originally a technical tool, the derived category of coherent sheaves over an algebraic variety has become over the last twenty years an important invariant in the birational study of algebraic varieties. Problems of birational invariance and of minimization of the derived category have appeared, inspired by Kontsevich's homological mirror symmetry conjecture and Mori's minimal model program. We present the main conjectures and their proofs in dimension 3 and for particular classes of flops.
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-…
Group Actions from Algebraic Flops
本文通过双模锥为代数翻折收缩构造导出自等价，证明其为局部双边倾斜复形，并由此在导出范畴上实现无限超平面排列基本群的群作用，推广并简化了已有结果。
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。
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.
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.
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…
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 develop a generalization of the $Q$-construction of the first author, Diemer, and the third author for Grassmann flips. This generalization provides a canonical idempotent kernel on the derived category of the associated global quotient stack. The idempotent kernel, after restriction, induces a semi-orthogonal decomposition which compares the flipped varieties. Furthermore its image, after restriction to the geometric invariant theory semistable locus, "opens" a canonical "window" in the derived category of the quotient stack. We check this window coincides with the set of representations u…
We prove the motivic version of the DT/PT-correspondence in \cite{PT} and the motivic flop formula of the curve counting invariants in the derived category of smooth Calabi-Yau threefold DM stacks. The main method we use is Bridgeland's Hall algebra identities and the motivic integration map of Bridgeland and Joyce from the motivic Hall algebra of the abelian category of coherent sheaves on a Calabi-Yau threefold DM stack $Y$ to the motivic quantum torus, which is a Poisson algebra homomorphism.
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…
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…
Monodromy of Calabi-Yau threefold flops via grade restriction rule and their quantum Kahler moduli
本文基于等级限制规则和窗口类别，给出了Calabi-Yau三重域翻转相关Monodromy的精确表达式，并通过量子Kähler模空间的判别式进一步细化Monodromy形式，利用其与嵌套环链基本群的关系进行验证。
We show that the derived category of a general Enriques surface can be realized as a semiorthogonal component in the derived category of a smooth Fano variety with a diagonal Hodge diamond.
We consider two cases of birational transformations of Q-Gorenstein threefolds: the case of a terminal flop and the case of a Francia flip. In the first case we show that, if one replaces the threefolds by their canonical covering stacks, the the flop coincides with the moduli stack of perverse point sheaves. This adds to a result of Kawamata showing the derived equivalence. Our construction has the drawback that the moduli space is defined using a presentation of the stack. In the second case we show that, if one takes a Francia flip $X \to Y$ and replaces $X$ by its canonical covering stack,…
Stratified flops show up in the birational geometry of symplectic varieties such as resolutions of nilpotent orbits and moduli spaces of sheaves. Constructing derived equivalences between varieties related by such flops is, strangely enough, related to areas in representation theory and knot homology. In this paper we discuss how to construct such equivalences, explain the main tool for doing this (categorical Lie algebra actions) and comment on various related topics.
We study the mirror operation of the Atiyah flop in symplectic geometry. We formulate the operation for a symplectic manifold with a Lagrangian fibration. Furthermore we construct geometric stability conditions on the derived Fukaya category of the deformed conifold and study the action of the mirror Atiyah flop on these stability conditions.
We prove that the derived categories for toric varieties have complete exceptional collections.
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…
We consider the structure of the derived categories of coherent sheaves on Fano threefolds with Picard number 1 and describe a strange relation between derived categories of different threefolds. In the Appendix we discuss how the ring of algebraic cycles of a smooth projective variety is related to the Grothendieck group of its derived category.
Equivariant birational types and derived categories
本文从导出范畴的视角研究了有理曲面和三维簇的等变双有理几何。
We extend some of the results of Bondal-Orlov on the equivalence of derived categories to the case of orbifolds by using the category of coherent orbifold sheaves.
On the structure of equivariant derived categories
本文讨论等变凝聚层导出范畴与GIT商导出范畴的结构，强调受限局部上同调理论，并展示双有理变形影响、范畴补全及广义辫群作用等应用。
We develop a general deformation theory of objects in homotopy and derived categories of DG categories. The main result is a general pro-representability theorem for the corresponding deformation functor.
Let $S$ be a scheme contained in $\mathbf{Sch}_{fppf}$. Let $\mathfrak{X}$ be a Noetherian separated algebraic space over $S$. In this paper, we interpret localizing subcategories of the derived category of $\mathfrak{X}$ by using subsets of $|\mathfrak{X}|$.
We give a generalization of the theorem of Bondal and Orlov about the derived categories of coherent sheaves on intersections of quadrics revealing its relation to projective duality. As an application we describe the derived categories of coherent sheaves on Fano 3-folds of index 1 and degrees 12, 16 and 18.
We prove two theorems on the derived categories of toric varieties, the existence of an exceptional collection consisting of sheaves for a divisorial extraction and the finiteness of Fourier-Mukai partners.
A survey article for AMS Summer Institute at Seattle in 2005.
We prove that if two abelian varieties have equivalent derived categories then the derived categories of the smooth stacks associated to the corresponding Kummer varieties are equivalent as well. The second main result establishes necessary and sufficient conditions for the existence of equivalences between the twisted derived categories of two Kummer surfaces in terms of Hodge isometries between the generalized transcendental lattices of the corresponding abelian surfaces.
We investigate conditions for a Fourier-Mukai transform between derived categories of coherent sheaves on smooth projective stacks endowed with actions by finite groups to lift to the associated equivariant derived categories. As an application we give a condition under which a global quotient stack cannot be derived equivalent to a variety. We also apply our techniques to generalised Kummer stacks and symmetric quotients.
This paper is based on a talk at a conference "JDG 2017: Conference on Geometry and Topology". We survey recent progress on the DK hypothesis connecting the birational geometry and the derived categories stating that the K-equivalence of smooth projective varieties should correspond to the equivalence of their derived categories, and the K-inequality to the fully faithful embedding.
We extend the conjecture on the derived equivalence and K-equivalence to the logarithmic case and prove it in the toric case.
We investigate equivalences between the categories of perfects complexes of the quotients of two smooth projective schemes by the action of a finite group. As a result we give a necessary and sufficient condition for an equivalence between the equivariant derived categories to descend to the categories of perfect complexes.
We discuss derived categories of coherent sheaves on algebraic varieties. We focus on the case of non-singular Calabi-Yau varieties and consider two unsolved problems: proving that birational varieties have equivalent derived categories, and computing the group of derived autoequivalences. We also introduce the space of stability conditions on a triangulated category and explain its relevance to these two problems.
We discuss relations between the motives of two varieties with equivalent derived categories of coherent sheaves.
In this paper we prove first a general theorem on semiorthogonal decompositions in derived categories of coherent sheaves for flat families over a smooth base. Based on the results of math.AG/0510670, we then show that the derived categories of coherent sheaves on flag varieties of classical type are generated by complete exceptional collections. Finally, we find complete exceptional collections in the derived categories of some homogeneous spaces of the symplectic groups of small rank.
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