A derived category approach to generic vanishing

Derived category of filtered objects

For an abelian category C and a filtrant preordered set Lambda, we prove that the derived category of the quasi-abelian category of filtered objects in C indexed by Lambda is equivalent to the derived category of the abelian category of functors from Lambda to C. We apply this result to the study of the category of filtered modules over a filtered ring in a tensor category.

The Derived Category of quasi-coherent sheaves on an Artin stack via model structures

We define the derived category of quasi--coherent modules for certain Artin stacks as the homotopy category of two Quillen monoidal model structures on the corresponding category of unbounded complexes of quasi--coherent modules.

可构造层的阿贝尔范畴的导出范畴

The derived category of the abelian category of constructible sheaves

本文证明了代数簇上有界可构造复形三角范畴与可构造层阿贝尔范畴的有界导出范畴等价，并将 Nori 定理推广至正特征。

A note on the derived category of Enriques surfaces in characteristic 2

We show that the (twisted) derived category "recognizes" the three different kinds of Enriques surfaces in characteristic 2

On the derived category of a weighted projective threefold

We calculate a semi-orthogonal decomposition of the bounded derived category of coherent sheaves on P(1,1,1,3) using a tilting bundle.

Semi-orthogonal decomposability of the derived category of a curve

We show that the bounded derived category of coherent sheaves on a smooth projective curve except the projective line admits no non-trivial semi-orthogonal decompositions.

On the derived category of the Hilbert scheme of points on an Enriques surface

We use semi-orthogonal decompositions to construct autoequivalences of Hilbert schemes of points on Enriques surfaces and of Calabi-Yau varieties which cover them. While doing this, we show that the derived category of a surface whose irregularity and geometric genus vanish embeds into the derived category of its Hilbert scheme of points.

Derived category and Lie algebras

This paper has been withdrawn by the author(s), due the final version in math.QA/0604564

orbifold 射影线的导出范畴的整体维数

Global dimension of the derived category of an orbifold projective line

本文将三角范畴的整体维数定义为其稳定性条件整体维数的下确界，并研究了orbifold射影线导出范畴的整体维数。

Derived equivalence and non-vanishing loci

The paper proposes and motivates a conjecture on the invariance of cohomological support loci under derived equivalence. It contains a proof in the case of surfaces, and explains further developments and consequences.

On equivariant derived categories

We study the equivariant category associated to a finite group action on the derived category of coherent sheaves of a smooth projective variety. We discuss decompositions of the equivariant category and faithful actions, prove the existence of a Serre functor, give a criterion for the equivariant category to be Calabi--Yau, and describe an obstruction for a subgroup of the group of auto-equivalences to act. As application we show that the equivariant category of any symplectic action on the derived category of an elliptic curve is equivalent to the derived category of an elliptic curve.

The Pure derived category of quasi-coherent sheaves

Let X be a quasi-compact and quasi-separated (not necessarily semiseparated) scheme. The category QcoX of all quasi-coherent sheaves of OX-modules has several diferent pure derived categories. Recently, categorical pure derived categories of X have been studied in more details. In this work, we focus on the geometrical purity and find replacements for geometrical pure derived categories of X.

On the derived category of quasi-coherent sheaves on an Adams geometric stack

Let $\mathbf{X}$ be an Adams geometric stack. We show that $D(A_{qc}(\mathbf{X}))$, its derived category of quasi-coherent sheaves, satisfies the axioms of a stable homotopy category defined by Hovey, Palmieri and Strickland. Moreover we show how this structure relates to the derived category of comodules over a Hopf algebroid that determines $\mathbf{X}$.

Derived category of Finite Spaces and Grothendieck Duality

We obtain some fundamental results, as Bokstedt-Neeman Theorem and Grothendieck duality, about the derived category of modules on a finite ringed space. Then we see how these results are transfered to schemes in a simple way and generalized to other ringed spaces.

Semi-orthogonal decomposition of a derived category of a 3-fold with an ordinary double point

We consider semi-orthogonal decompositions of derived categories for 3-dimensional projective varieties in the case when the varieties have ordinary double points.

The derived category of a non generic cubic fourfold containing a plane

We describe an Azumaya algebra on the resolution of singularities of the double cover of a plane ramified along a nodal sextic associated to a non generic cubic fourfold containing a plane. We show that the derived category of such a resolution, twisted by the Azumaya algebra, is equivalent to the Kuznetsov component in the semiorthogonal decomposition of the derived category of the cubic fourfold.

On the derived category of $\bar{M}_{0,n}$

Using Keel's presentation and Orlov's theorem, we give an inductive description of the derived category of moduli spaces of $n$--pointed stable curves of genus zero and some full exceptional collections in it. The detailed calculations are given for $\bar{M}_{0,6}$.

On the derived category of $\mathrm{IGr}(3,8)$

We construct a full exceptional collection of vector bundles in the bounded derived category of coherent sheaves on the Grassmannian $\mathrm{IGr}(3,8)$ of isotropic 3-dimensional subspaces in an 8-dimensional symplectic vector space.

Derived category of coherent sheaves and counting invariants

We survey recent developments on Donaldson-Thomas theory, Bridgeland stability conditions and wall-crossing formula. We emphasize the importance of the counting theory of Bridgeland semistable objects in the derived category of coherent sheaves to find a hidden property of the generating series of Donaldson-Thomas invariants.

The Derived Category of the Intersection of Four Quadrics

The derived category of a general complete intersection of four quadrics in P^{2n-1} has a semi-orthogonal decomposition < O(-2n+9), ..., O(-1), O, D >, where D is the derived category of twisted sheaves on a certain non-algebraic complex 3-fold coming from a moduli problem. In particular, when n=4 we obtain a (twisted) derived equivalence of Calabi-Yau 3-folds predicted by Gross. This differs from Kuznetsov's result in that our construction is geometric and avoids non-commutative varieties.

$\u2119^1$上抛物丛模空间的导出范畴

Derived category of moduli of parabolic bundles on $\mathbb{P}^1$

该研究针对$\u2119^1$上秩2抛物丛模空间的凝聚层有界导出范畴的结构问题，提出了一项相关猜想，为代数几何领域该方向的研究提供了新的探索方向。

On the derived category of the Cayley plane

We describe a maximal exceptional collection on the Cayley plane, the minimal homogeneous projective variety of $E_6$. This collection consists in a sequence of 27 irreducible homogeneous bundles.

关于环带叠片丛的导出范畴

On the derived category of a toric stack bundle

本文研究了分裂环带叠片丛上环面等变凝聚层导出范畴的性质，并给出了其半正交分解。

Equivariant derived category of flat families

We prove the existence of tilting bundles on global quotient stacks that are produced by compatible finite group actions on flat families.

Derived category of V_{12} Fano threefolds

A V_{12} Fano threefold is a smooth Fano threefold X of index 1 with Pic X = Z and (-K_X)^3=12. We show that the bounded derived category of coherent sheaves on any V_{12} threefold X admits a semiorthogonal decomposition consisting of two exceptional bundles and of the derived category of a curve of genus 7. As an application we show that the Fano surface of X (the surface parameterizing conics on X) is canonically isomorphic to the symmetric square of the associated genus 7 curve.

Reconstruction of projective curves from the derived category

We prove that integral projective curves with Cohen-Macaulay singularities over an algebraically closed field of characteristic zero are determined by their triangulated category of perfect complexes.This partially extends a theorem of Bondal and Orlov to the case of singular projective varieties, and in particular shows that all integral complex projective curves are determined by their category of perfect complexes

Derived Category of Fibrations

Let X ->Y be a Zariski locally trivial fibration of smooth complex projective varieties, with fiber F. We give a structure theorem for the derived category of X provided both F and Z have a full strongly exceptional collection of line bundles.

Hall algebras in the derived category and higher rank DT invariants

We remark that the combination of the works of Ben-Bassat-Brav-Bussi-Joyce and Alper-Hall-Rydh imply the conjectured local description of the moduli stacks of semi-Schur objects in the derived category of coherent sheaves on projective Calabi-Yau 3-folds. This result was assumed in the author's previous papers to apply wall-crossing formulas of DT type invariants in the derived category, e.g. DT/PT correspondence, rationality, etc. We also show that the above result is applied to prove the higher rank version of DT/PT correspondence and rationality.

On the realization functor of the derived category of mixed motives

We give an alternative construction of the Betti realization functor on the derived category of motives of complex algebraic varieties via the category of CW complexes instead of the category of complex analytic spaces. In particular we show that the functor we define via the category of CW complexes coincide with Ayoub's one. We deduce from this construction that Ayoub's realization functor on geometric motives factors trough Nori motives and that the image of this functor on the morphisms between the motive of a point and a shift of a Tate twist of the motive with compact support of a comple…

The linear dual of the derived category of a scheme

This paper has been withdrawn because Proposition 2.2 (c) is false. This invalids the main results of section 2 and 3. We thank A. Canonaco for pointing us the error.

Cohomological descent theory for a morphism of stacks and for equivariant derived categories

In the paper we answer the following question: for a morphism of varieties (or, more generally, stacks), when the derived category of the base can be recovered from the derived category of the covering variety by means of descent theory? As a corollary, we show that for an action of a reductive group on a scheme, the derived category of equivariant sheaves is equivalent to the category of objects, equipped with an action of the group, in the ordinary derived category.

Derived equivalence and non-vanishing loci II

We prove a few cases of a conjecture on the invariance of cohomological support loci under derived equivalence by establishing a concrete connection with the related problem of the invariance of Hodge numbers. We use the main case in order to study the derived behavior of fibrations over curves.

曲线上凝聚层的导出范畴与稳定性条件

Derived category of coherent systems on curves and stability conditions

针对亏格$g>0$的光滑射影曲线$C$，刻画其凝聚层有界导出范畴的Bridgeland稳定性条件的开轨迹，证明该稳定性流形可检测曲线的Brill--Noether理论。

On the Derived Category of the Cayley Grassmannian

We construct a full exceptional collection consisting of vector bundles in the derived category of coherent sheaves on the so-called Cayley Grassmannian, the subvariety of the Grassmannian $\mathrm{Gr}(3, 7)$ parameterizing 3-subspaces that are annihilated by a general 4-form. The main step in the proof of fullness is a construction of two self-dual vector bundles which is obtained from two operations with quadric bundles that might be interesting in themselves.

关于$\u2e62IGr(3, 7)$的有界导出范畴

On the bounded derived category of $\mathsf{IGr}(3, 7)$

本研究构造了迷向格拉斯曼簇$\u2e62IGr(3,7)$凝聚层有界导出范畴的极小莱夫谢茨分解，并证明其存在由等变向量丛构成的完全例外系，推进了相关代数几何研究。

On the derived category of 1-motives, I

We consider the category of Deligne 1-motives over a perfect field k of exponential characteristic p and its derived category for a suitable exact structure after inverting p. As a first result, we provide a fully faithful embedding into an etale version of Voevodsky's triangulated category of geometric motives. Our second main result is that this full embedding "almost" has a left adjoint, that we call \LAlb. Applied to the motive of a variety we thus get a bounded complex of 1-motives, that we compute fully for smooth varieties and partly for singular varieties. As an application we give mot…

非交换沙漏 II 1/2：具有良构奇点的双有理联络的导出范畴

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。

On the derived category of 1-motives

This is the final version of the 2007 preprint titled "On the derived category of 1-motives, I". It has been substantially expanded to contain a motivic proof of (two thirds of) Deligne's conjecture on 1-motives with rational coefficients, hence the new title. Compared to the 2007 preprint, the additions mainly concern an abstract theory of realisations with weight filtrations; Deligne's conjecture is tackled though them by an adjunction game.

Hermitian structures on the derived category of coherent sheaves

The main objective of the present paper is to set up the theoretical basis and the language needed to deal with the problem of direct images of hermitian vector bundles for projective non-necessarily smooth morphisms. To this end, we first define hermitian structures on the objects of the bounded derived category of coherent sheaves on a smooth complex variety. Secondly we extend the theory of Bott-Chern classes to these hermitian structures. Finally we introduce the category $\oSm_{\ast/\CC}$ whose morphisms are projective morphisms with a hermitian structure on the relative tangent complex.

关于 IGr(3,9) 的导出范畴

On the derived category of IGr(3, 9)

本文为奇异性迷向格拉斯曼流形 IGr(3,9) 构造了极小 Lefschetz 分解，其例外对象为 Sp_9-等变向量丛，支持了 Kuznetsov-Smirnov 版本的 Dubrovin 猜想。

Derived category of toric varieties with Picard number three

We construct a full, strongly exceptional collection of line bundles on the variety X that is the blow up of the projectivization of the vector bundle O_{P^{n-1}}\oplus O_{P^{n-1}}(b) along a linear space of dimension n-2, where b is a non-negative integer.

The De Rham, complex Hodge and $p$-adic Hodge realization functors on the derived category of relative motives over a field of characteristic zero

We introduce the categories of geometric mixed Hodge modules on algebraic varieties over a subfield $k\subset\mathbb C$, and for a prime number $p$, the categories of geometric $p$-adic mixed Hodge modules on algebraic varieties over a subfield $k\subset K\subset\mathbb C_p$ of a p-adic field $K$. We then give an Hodge realization functor on the de derived category of relative motives over $k\subset\mathbb C$, and for a prime number $p$ a $p$-adic Hodge realization functor on the derived category of relative motives over $k\subset K\subset\mathbb C_p$.

Derived category of moduli of pointed curves -- I

This is the first paper in the sequence devoted to derived category of moduli spaces of curves of genus $0$ with marked points. We develop several approaches to describe it equivariantly with respect to the action of the symmetric group permuting marked points. We construct an equivariant full exceptional collection on the Losev-Manin space which categorifies derangements.

No phantoms in the derived category of curves over arbitrary fields, and derived characterizations of Brauer-Severi varieties

In this paper we show that the derived category of Brauer-Severi curves satisfies the Jordan-Hölder property and cannot have quasi-phantoms, phantoms or universal phantoms. In this way we obtain that quasi-phantoms, phantoms or universal phantoms cannot exist in the derived category of smooth projective curves over a field $k$. Moreover, we show that a $n$-dimensional Brauer-Severi variety is completely characterized by the existence of a full weak exceptional collection consisting of pure vector bundles of length $n+1$, at least in characteristic zero. We conjecture that Brauer-Severi varieti…

Derived category of toric fibrations

The derived category of bounded complexes of coherent sheaves is one of the most important algebraic invariants of a smooth projective variety. An important approach to understand derived categories is to construct full strongly exceptional sequences. The problem of characterizing smooth projective varieties which have a full strongly exceptional collection and investigate whether there is one consisting of line bundles is a classical and important question in Algebraic Geometry. Not all smooth projective varieties have a full strongly exceptional collection of coherent sheaves. In this paper …

Around the nearby cycle functor for arithmetic $\mathscr{D}$-modules

We will establish a nearby and vanishing cycle formalism for the arithmetic $\mathscr{D}$-module theory following Beilinson's philosophy. As an application, we define smooth objects in the framework of arithmetic $\mathscr{D}$-modules whose category is equivalent to the category of overconvergent isocrystals.

Derived categories of Gushel-Mukai varieties

We study the derived categories of coherent sheaves on Gushel-Mukai varieties. In the derived category of such a variety, we isolate a special semiorthogonal component, which is a K3 or Enriques category according to whether the dimension of the variety is even or odd. We analyze the basic properties of this category using Hochschild homology, Hochschild cohomology, and the Grothendieck group. We study the K3 category of a Gushel-Mukai fourfold in more detail. Namely, we show that this category is equivalent to the derived category of a K3 surface for a certain codimension 1 family of rational…

整全阿廷动机 II：反常动机与阿廷消失定理

Integral Artin motives II: Perverse motives and Artin Vanishing Theorem

本文研究整系数阿廷平展动机范畴上的反常动机t-结构，构造了近似的反常同伦t-结构，在基概型维数≤2时得到了反常动机t-结构，证明其在维数4时不存在，且依赖于阿廷消失定理的类似结论。

Grassmannian twists on the derived category via spherical functors

We construct new examples of derived autoequivalences for a family of higher-dimensional Calabi-Yau varieties. Specifically, we take the total spaces of certain natural vector bundles over Grassmannians G(r,d) of r-planes in a d-dimensional vector space, and define endofunctors of the bounded derived categories of coherent sheaves associated to these varieties. In the case r=2 we show that these are autoequivalences using the theory of spherical functors. Our autoequivalences naturally generalize the Seidel-Thomas spherical twist for analogous bundles over projective spaces.

Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves

We study homological mirror symmetry for Del Pezzo surfaces and their mirror Landau-Ginzburg models. In particular, we show that the derived category of coherent sheaves on a Del Pezzo surface X_k obtained by blowing up CP^2 at k points is equivalent to the derived category of vanishing cycles of a certain elliptic fibration W_k:M_k\to\C with k+3 singular fibers, equipped with a suitable symplectic form. Moreover, we also show that this mirror correspondence between derived categories can be extended to noncommutative deformations of X_k, and give an explicit correspondence between the deforma…

A counterexample to vanishing conjectures for negative $K$-theory

In a 2006 article Schlichting conjectured that the negative {\it K--}theory of any abelian category must vanish. This conjecture was generalized in a 2019 article by Antieau, Gepner and Heller, who hypothesized that the negative {\it K--}theory of any category with a bounded {\it t--}structure must vanish. Both conjectures will be shown to be false.

Introduction to the Category of Derived Motivic Spectra

We formalize an abstraction of Grothendieck's philosophy of motives and construct a category of derived motivic spectra in the Segal category $\mathbb{R} \underline{\text{Hom}} ((\text{dSt}_k)^{\text{op}}_{/F}, \text{Top})$ ($\text{dSt}_k$ the Segal category of derived stacks on s$k$-Alg, Top = $L \text{Set}_Δ$ the Segal category of simplicial sets), thereby providing a starting point for a construction of a stable motivic homotopy category in the Segal setting.

The moduli stack and motivic Hall algebra for the bounded derived category

We give an alternate formulation of pseudo-coherence over an arbitrary derived stack X. The full subcategory of pseudo-coherent objects forms a stable sub-infinity-category of the derived category associated to X. Using relative Tor-amplitude we define a derived stack classifying pseudo-coherent objects. For reasonable base schemes, this classifies the bounded derived category. In the case that X is a projective derived scheme flat over the base, we show the moduli is locally geometric and locally of almost finite type. Using this result, we prove the existence of a derived motivic Hall algebr…

Uniqueness of dg enhancements for the derived category of a Grothendieck category

We prove that the derived category of a Grothendieck abelian category has a unique dg enhancement. Under some additional assumptions, we show that the same result holds true for its subcategory of compact objects. As a consequence, we deduce that the unbounded derived category of quasi-coherent sheaves on an algebraic stack and the category of perfect complexes on a noetherian concentrated algebraic stack with quasi-finite affine diagonal and enough perfect coherent sheaves have a unique dg enhancement. In particular, the category of perfect complexes on a noetherian scheme with enough locally…