Yunfeng Jiang
We calculate explicit formulas for the general equivariant Bondal-Orlov functors on the localized K-theory groups for a crepant birational transformation of toric DM stacks. We recall some facts that the Bondal-Orlov functors give equivalences on the bounded derived categories. Applying twice of these functors we get the Seidel-Thomas spherical twists for the derived category.
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Suppose $F\colon \mathcal{D}(X)\to \mathcal{T}$ is an exact functor from the bounded derived category of coherent sheaves on a smooth projective variety $X$ to a triangulated category $\mathcal{T}$. If $F$ possesses left and right adjoints, then the Bondal-Orlov criterion gives a simple way of determining if $F$ is fully faithful. We prove a natural extension to the case when $X$ is a smooth and proper DM stack with projective coarse moduli space.
The Bondal-Orlov Localization Conjecture Holds for Threefolds
该研究证明了满足特定条件的射影态射下导出前推的等价关系，并由此得到拟射影三维代数簇的邦达尔-奥尔洛夫局部化猜想的无特征形式，推进了代数几何中导出范畴的相关猜想研究。
Bondal--Orlov reconstruction for tame stacks with trivial generic stabilizer
该文将 Bondal--Orlov 重构定理推广到具有平凡一般稳定子且典范丛处处非挠的光滑真驯服栈，并利用 Tannaka 对偶和 Grothendieck 对偶，同时推广到非真 Gorenstein 概形。
Cohen-Macaulay modules and the Bondal-Orlov conjecture
本文通过引入局部自由层的 CM-degree 概念构造与 flop 相容的倾斜生成元，并利用 CM-degree 在严格变换下的不变性，在小消解的某些情形下证明了 Bondal-Orlov 猜想。
Bondal-Orlov's reconstruction theorem in noncommutative projective geometry
该研究证明Bondal-Orlov重构定理在非交换射影几何中成立，且非交换射影概形导出范畴间的全忠实正合函子为Fourier-Mukai型，推进了非交换射影几何领域的理论成果。
Homological Bondal-Orlov localization conjecture for rational singularities
本文利用 Hodge 理论证明有理奇点消解下导出范畴的导出前推函子像生成目标三角范畴，从而给出 Bondal-Orlov 局部化猜想的弱版本，并推广到满足自然条件的紧合态射情形。
A generalized Bondal-Orlov full faithfulness criterion for Deligne-Mumford stacks
本文将 Bondal-Orlov 的 Fourier-Mukai 函子全忠实性判据推广到特征零任意域上的光滑真 Deligne-Mumford 叠，并建立代数 stack 真与驯态射导出范畴的基础结果。
Let R be a normal, equi-codimensional Cohen-Macaulay ring of dimension $d\geq 2$ with a canonical module. We give a sufficient criterion that establishes a derived equivalence between the noncommutative crepant resolutions of R. When $d\leq 3$ this criterion is always satisfied and so all noncommutative crepant resolutions of R are derived equivalent. Our method is based on cluster tilting theory for commutative algebras, developed in [IW10].
We show that all toric noncommutative crepant resolutions (NCCRs) of affine GIT quotients of "weakly symmetric" unimodular torus representations are derived equivalent. This yields evidence for a non-commutative extension of a well known conjecture by Bondal and Orlov stating that all crepant resolutions of a Gorenstein singularity are derived equivalent. We prove our result by showing that all toric NCCRs of the affine GIT quotient are derived equivalent to a fixed Deligne-Mumford GIT quotient stack associated to a generic character of the torus. This extends a result by Halpern-Leistner and …
Relative Monoidal Bondal-Orlov
本文将 Bondal-Orlov 重建推广到相对幺半情形，证明在纤维具丰富（反）典范丛时 D^b(X) 上张量三角结构的唯一性。
We combine the Bondal-Uehara method for producing exceptional collections on toric varieties with a result of the first author and Favero to expand the set of varieties satisfying Orlov's Conjecture on derived dimension.
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.
We prove that varieties are uniquely determined by the derived category of $\mathscr{O}[[\hbar]]$-modules with coherent cohomology which is the same as $\mathscr{O}$-modules proved by A. Bondal and D. Orlov. We also generalize a theorem of Orlov.
Bernardi and Tirabassi show the existence of full strong exceptional collections consisting of line bundles on smooth toric Fano $3$-folds under assuming Bondal's conjecture, which states that the Frobenius push-forward of the structure sheaf $\mc O_X$ generates the derived category $D^b(X)$ for smooth projective toric varieties $X$. In this article, we show Bondal's conjecture for smooth toric Fano $3$-folds and also improve their result, using birational geometry.
Strata of toric hyperplane arrangements, zonotope lattice points, and the Bondal-Thomsen collection
本文证明了有向环面超平面排列的层与带形多胞形格点间的双射，并关联了层维数与最小面维数，进而探讨了与环面簇及Bondal-Thomsen生成元的联系。
A new proof of the Bondal-Orlov reconstruction using Matsui spectra
本文利用 Matsui 谱证明拟射影簇的完美导出范畴的谱嵌入关系，并据此给出 Bondal-Orlov 与 Ballard 重构定理的新证明，进而重构阿贝尔簇的 Fourier-Mukai 伙伴。
These lecture notes were prepared for the workshop ``Algebraic Geometry: Presentations by Young Researchers'' in Snowbird, Utah, July 2004, and for the autumn school in Lukecin, Poland, September 2004. In six lectures I attempted to present a non-technical introduction to derived categories of sheaves on varieties, including some important results like: -- the description of the derived category of P^n; -- Serre duality; -- the Bondal-Orlov result about derived categories of varieties with ample or antiample canonical class; -- the Mukai-Bondal-Orlov-Bridgeland criterion for equivalence; -- eq…
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.
In this short note we show how results of Orlov and Toën imply that any equivalence between the derived categories of coherent sheaves on two varieties lifts to an equivalence at the level of dg-categories. This establishes the link between the noncommutative geometry practised by the school of Bondal-Orlov, and the variant developed by Kontsevich and Tabuada. As an application we recover Orlov's result that the derived category determines the Chow motive with rational coefficients up to Tate twists.
The aim of this mainly expository note is to point out that, given an Fourier-Mukai functor, the condition making it fully faithful is an instance of \emph{generic vanishing}. We test this point of view on some fairly classical examples, including the strong simplicity criterion of Bondal and Orlov, the standard flip and the Mukai flop.
Noncommutative Hourglasses II 1/2: Derived Category of Birational Linkage with Well-Formed Singularities
本文针对奇点为环面上奇点内完全交的良构簇，将有界凝聚层导出范畴的经典公式推广到奇异情形，并讨论终复三维簇及其栈间态射的Bondal-Orlov局部化问题。
In this paper, we study derived categories of certain toric varieties with Picard number three that are blowing-up another toric varieties along their torus invariant loci of codimension at most three. We construct strong full exceptional collections by using Orlov's blow-up formula and mutations.
Generating systems, generalized Thomsen collections and derived categories of toric varieties
本文研究了 toric 变形的生成系统和广义汤姆森集合，验证了 Bondal 关于导出范畴生成的猜想。
We prove the following result of Bondal's: that there is a fully faithful embedding $κ$ of the perfect derived category of a proper toric variety into the derived category of constructible sheaves on a compact torus. We compare this result to a torus-equivariant version considered in joint work with Fang, Liu, and Zaslow. There we showed that in the torus-equivariant version the image of the embedding is cut out by microlocal conditions. To establish a similar characterization of the image of $κ$ is an open problem.
DK Conjecture for Some $K$-inequivalences from Grassmannians
本文从Grassmann流形构造非环面K-等价例子，并证明其满足Bondal-Orlov与Kawamata的DK猜想，从而将DK猜想从环面情形推广到新的非环面实例。
We prove that the bounded derived category of coherent sheaves with proper support is equivalent to the category of locally-finite, cohomological functors on the perfect derived category of a quasi-projective scheme over a field. We introduce the notions of pseudo-adjoints and Rouquier functors and study them. As an application of these ideas and results, we extend the reconstruction result of Bondal and Orlov to Gorenstein projective varieties.
The coherent-constructible correspondence is a relationship between coherent sheaves on a toric variety X, and constructible sheaves on a real torus T. This was discovered by Bondal, and explored in the equivariant setting by Fang, Liu, Treumann and Zaslow. In this paper we collect partial results towards a proof of the non equivariant coherent-constructible correspondence. Also, we give applications to the construction of tilting complexes in the derived category of toric DM stacks.
We introduce a global Landau-Ginzburg model which is mirror to several toric Deligne-Mumford stacks and describe the change of the Gromov-Witten theories under discrepant transformations. We prove a formal decomposition of the quantum cohomology D-modules (and of the all-genus Gromov-Witten potentials) under a discrepant toric wall-crossing. In the case of weighted blowups of weak-Fano compact toric stacks along toric centres, we show that an analytic lift of the formal decomposition corresponds, via the $\widehatΓ$-integral structure, to an Orlov-type semiorthogonal decomposition of topologic…
Derived categories of quadric bundles and moduli stacks of spinor sheaves
本文通过旋量sheaf的模堆，证明了二次束族的Kuznetsov组件与扭曲导出范畴的等价性，并提供了Brauer类的几何解释。
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