Yves Laszlo, Martin Olsson
In this paper we develop a theory of Grothendieck's six operations of lisse-étale constructible sheaves on Artin stacks which are locally of finite type over suitable regular basis of dimension at most 1.
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In this paper we develop the theory of perverse sheaves on Artin stacks continuing the study in "The six operations for sheaves on Artin stacks I: Finite Coefficients" and "The six operations for sheaves on Artin stacks II: Adic Coefficients" (math.AG/0512097 and math.AG/0603680)
In this paper we develop a theory of Grothendieck's six operations for adic constructible sheaves on Artin stacks continuing the study of the finite coefficients case in math.AG/0512097.
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.
We generalize the decomposition theorem for perverse sheaves to Artin stacks with affine stabilizers over finite fields.
We prove the generic base change theorem for stacks, and give an exposition on the lisse-analytic topos of complex analytic stacks, proving some comparison theorems between various derived categories of complex analytic stacks. This enables us to deduce the decomposition theorem for perverse sheaves on complex Artin stacks with affine stabilizers from the case over finite fields.
There is a well developed intersection theory on smooth Artin stacks with quasi-affine diagonal. However, for Artin stacks whose diagonal is not quasi-finite the notion of the degree of a Chow cycle is not defined. In this paper we propose a definition for the degree of a cycle on Artin toric stacks whose underlying toric varieties are complete. As an application we define the Euler characteristic of an Artin toric stack with complete good moduli space - extending the definition of the orbifold Euler characteristic. An explicit combinatorial formula is given for 3-dimensional Artin toric stack…
We construct an algebraic homology functor for Artin stacks of finite type over a field, and we develop intersection-theoretic properties.
We develop the theory of associating moduli spaces with nice geometric properties to arbitrary Artin stacks generalizing Mumford's geometric invariant theory and tame stacks.
In this note we show that an Artin stack with finite inertia stack is etale locally isomorphic to the quotient of an affine scheme by an action of a general linear group.
We define Reichstein transforms to be certain birational transformations of Artin stacks with good moduli spaces. Our main technical result is that the Reichstein transform of an Artin toric stack is again an Artin toric stack. This leads to our main theorem which states that for Artin toric stacks there is a canonical sequence of Reichstein transforms which produces a Deligne-Mumford toric stack. When the good moduli space of an Artin toric stack is projective our procedure can be interpreted in terms of Kirwan's partial desingularization of geometric invariant theory quotients.
We construct virtual fundamental classes of Artin stacks over a Dedekind domain endowed with a perfect obstruction theory.
We develop a theory of toric Artin stacks extending the theories of toric Deligne-Mumford stacks developed by Borisov-Chen-Smith, Fantechi-Mann-Nironi, and Iwanari. We also generalize the Chevalley-Shephard-Todd theorem to the case of diagonalizable group schemes. These are both applications of our main theorem which shows that a toroidal embedding X is canonically the good moduli space (in the sense of Alper) of a smooth log smooth Artin stack whose stacky structure is supported on the singular locus of X.
This paper is a continuation of our earlier development of a theory of tame Artin stacks. Our main goal here is the construction of an appropriate analogue of Kontsevich's space of stable maps in the case where the target is a tame Artin stack. When the target is a tame Deligne--Mumford stack, the theory was developed by Abramovich and Vistoli, and found a number of applications. The theory for arbitrary tame Artin stacks developed here is very similar, but it is necessary to overcome a number of technical hurdles and to generalize a few questions of foundation.
The derived moduli of perverse sheaves
本文构造了参数化复代数簇和紧实解析簇上可构造层的导出Artin叠，并证明每个反常函数给出反常层的开子叠，推广了通常的特征叠，作为应用构造了与穿孔Riemann曲面相关的上同调Hall代数的新例子。
We prove Artin's axioms satisfy a compatibility for composition of 1-morphisms of stacks in groupoids. Consequently, some natural stacks in groupoids are algebraic, including a common generalization of Vistoli's Hilbert stack and the stack of branchvarieties of Alexeev and Knutson.
Crepant resolutions of log-terminal singularities via Artin stacks
本文证明任意 log-terminal 奇点簇均可由光滑 Artin 叠实现 crepant 消解，并建立相应 McKay 对应，用光滑叠弧空间上的动机积分表达 Q-Gorenstein 簇的弦不变量。
We show that there is no way to define degrees of 0-cycles on Artin stacks with proper good moduli spaces so that (i) the degree of an ordinary point is non-zero, and (ii) degrees are compatible with closed immersions.
On deformations of foliated complex analytic structures II: Functors of Artin rings
本文以Artin环函子为框架，研究叶状非奇异簇的形变，识别一阶形变与障碍。
Artin-Schreier Root Stacks and lifts of group actions
本文证明，当连通仿射代数群G无非平凡特征时，其在光滑射影簇上的作用可提升至相关的Artin-Schreier根叠，并证明了这类根叠上某自伴可逆层的G-线性化存在性。
We introduce a Grothendieck ring of higher Artin stacks generalizing the Grothendieck ring of algebraic varieties. We show that this ring is not trivial by noticing that it factors the invariant "number of rational points over a finite field". We also introduce the notion of "special Artin stacks", which by definition have affine homotopy groups π_{i}, and furthermore unipotent for i>1. Our principal theorem states that the natural inclusion morphism from the Grothendieck ring of varieties to the Grothendieck ring of special Artin stacks is an isomorphism after inverting the class of the affin…
In this sequel of arXiv:1211.5294 and arXiv:1211.5948, we develop an adic formalism for étale cohomology of Artin stacks and prove several desired properties including the base change theorem. In addition, we define perverse t-structures on Artin stacks for general perversity, extending Gabber's work on schemes. Our results generalize results of Laszlo and Olsson on adic formalism and middle perversity. We continue to work in the world of $\infty$-categories in the sense of Lurie, by enhancing all the derived categories, functors, and natural transformations to the level of $\infty$-categories…
We extend Orlov's representability theorem on the equivalence of derived categories of sheaves to the case of smooth stacks associated to normal projective varieties with only quotient singularities.
Artin perverse sheaves
本文研究 Artin ℓ-adic 复形上反常 t-结构的存在性，证明在维数小于2的优概型上成立并给出三维反例，进而在有限域有限型概型上构造反常同伦 t-结构，刻画其单对象并证明无权截断函子的 t-正合性。
Perverse sheaves on the stack of $G$-zips
该研究提出了$G$-zips栈上简单反常层的计算方法，并通过多个示例完成了相关计算，为代数几何领域相关研究提供方法参考。
We extend the construction of the normal cone of a closed embedding of schemes to any locally of finite type morphism of higher Artin stacks and show that in the Deligne-Mumford case our construction recovers the relative intrinsic normal cone of Behrend and Fantechi. We characterize our extension as the unique one satisfying a short list of axioms, and use it to construct the deformation to the normal cone. As an application of our methods, we associate to any morphism of Artin stacks equipped with a choice of a global perfect obstruction theory a relative virtual fundamental class in the Cho…
We view the moduli space of semistable sheaves on a K3 surface as a global quotient stack, and compute its cotangent complex in terms of the universal sheaf on the Quot scheme. Relevant facts on the classical and reduced Atiyah classes are reviewed. We also define the notion of a symplectic stack, and show that it includes all moduli stacks of semistable sheaves on K3 surfaces.
We show that near closed points with linearly reductive stabilizer, Artin stacks are formally locally quotient stacks by the stabilizer. We conjecture that the statement holds etale locally and we provide some evidence for this conjecture. In particular, we prove that if the stabilizer of a point is linearly reductive, the stabilizer acts algebraically on a miniversal deformation space generalizing results of Pinkham and Rim.
We generalise the techniques of arXiv:0908.1963 to describe derived deformations in simplicial categories. This allows us to consider deformation problems with higher automorphisms, such as chain complexes (which have homotopies) and stacks (which have 2-automorphisms). We also give a general approach for studying deformations of diagrams.
On Shifted Contact Derived Artin Stacks
该研究作为导出接触几何研究的续作，将k<0时k-平移接触导出概形的相关结果推广至导出Artin叠，证明了其接触达布阿特拉斯的存在性，给出了辛化的典范描述，并利用余切叠与平移预量子化结构构造了若干接触导出叠。
We prove formulae for the motives of stacks of coherent sheaves of fixed rank and degree over a smooth projective curve in Voevodsky's triangulated category of mixed motives with rational coefficients.
This paper is largely concerned with constructing coarse moduli spaces for Artin stacks. The main purpose of this paper is to introduce the notion of stability on an arbitrary Artin stack and construct a coarse moduli space for the open substack of stable points. Also, we present an application to coherent cohomology of Artin stacks.
We prove a representability theorem for moduli functors of framed torsion-free sheaves on nonsingular complex projective surfaces, using formal geometry along a curve in the surface. This has as a consequence that a certain restriction morphism for moduli stacks of sheaves has fibers that are schemes.
We define and study infinite root stacks of fine and saturated logarithmic schemes, a limit version of the root stacks introduced by Niels Borne and the second author. We show in particular that the infinite root stack determines the logarithmic structure, and recovers the Kummer-flat topos of the logarithmic scheme. We also extend the correspondence between parabolic sheaves and quasi-coherent sheaves on root stacks to this new setting.
We give a variant of Artin algebraization along closed subschemes and closed substacks. Our main application is the existence of étale, smooth, or syntomic neighborhoods of closed subschemes and closed substacks. In particular, we prove local structure theorems for stacks and their derived counterparts and the existence of henselizations along linearly fundamental closed substacks. These results establish the existence of Ferrand pushouts, which answers positively a question of Temkin-Tyomkin.
We define an Artin stack which may be considered as a substitute for the non-existing (or empty) moduli space of stable two-pointed curves of genus zero. We show that this Artin stack can be viewed as the first term of a cyclic operad in the category of stacks. Applying the homology functor we obtain a linear cyclic operad. We formulate conjectures which assert that cohomology of a smooth projective variety has the structure of an algebra over this homology operad and that gravitational quantum cohomology can naturally expressed in terms of this algebra. As a test for these conjectures we show…
Let $\mathcal{X} \to Y$ be a birational map from a smooth Artin stack to a (possibly singular) variety. We prove a change of variables formula that relates motivic integrals over arcs of $Y$ to motivic integrals over arcs of $\mathcal{X}$. With a view toward the study of stringy Hodge numbers, this change of variables formula leads to a new notion of crepantness for the map $\mathcal{X} \to Y$ that coincides with the usual notion in the special case that $\mathcal{X}$ is a scheme.
In this paper, we study the geometric invariant theory on algebraic spaces, and construct te moduli spaces of $\mathcal{H}$-semistable sheaves on projective Deligne-Mumford stacks over algebraic spaces $S$. We prove that this moduli space is projective over $S$ as an algebraic space.
In this article, we classify the irreducible components of moduli stacks of torsion free sheaves of rank 2 on K3 surfaces of Picard number 1. For ruled surfaces, the components of moduli stacks of torsion free sheaves were classified by C.Walter. Moreover, by virtue of our result, we classify the irreducible components of Brill-Noether loci of Hilbert schemes of points on K3 surfaces.
An invariant I of quasiprojective K-varieties X with values in a commutative ring R is "motivic" if I(X)= I(Y)+I(X\Y) for Y closed in X, and I(X x Y)=I(X)I(Y). Examples include Euler characteristics chi and virtual Poincare and Hodge polynomials. We first define a unique extension I' of I to finite type Artin K-stacks F, which is motivic and satisfies I'([X/G])=I(X)/I(G) when X is a K-variety, G a "special" K-group acting on X, and [X/G] is the quotient stack. This only works if I(G) is invertible in R for all special K-groups G, which excludes I=chi as chi(K*)=0. But we can extend the constru…
In this paper, we propose a definition of the moduli stack of stable relative ideal sheaves, and prove that it is a separated and proper Deligne-Mumford stack. It is the first part of the project of relative Donaldson-Thomas theory of ideal sheaves on projective threefolds, which in the end provides a degeneration formula of Donaldson-Thomas invariants of threefolds.
Since the work of Mikhail Kapranov in [Kap], it is known that the shifted tangent complex $\mathbb{T}_X[-1]$ of a smooth algebraic variety $X$ is endowed with a weak Lie structure. Moreover any complex of quasi-coherent sheaves on $X$ is endowed with a weak Lie action of this tangent Lie algebra. This action is given by the Atiyah class of $E$. We will generalize this result to (finite enough) derived Artin stacks, without any smoothness assumption. This in particular applies to (finite enough) singular schemes. This work uses tools of both derived algebraic geometry and $\infty$-category theo…
Stringy Hodge numbers via crepant resolutions by Artin stacks
本研究证明簇可通过光滑Artin栈实现crepant消解等价于其具有对数典范奇点，给出弦霍奇数的扭弧积分公式，还定义了光滑Artin栈的弦霍奇数并复现轨形霍奇数，为弦霍奇数的上同调解释提供了可行方向。
We provide stacky generalizations of classical gauge-theoretic results inspired by Donaldson, the Uhlenbeck-Yau theorem and variants due to Bando and his collaborators. Moreover, we show an application of this machinery in the study of ALE spaces.
Equivalences of derived categories of sheaves on tame stacks
本文将Orlov表示定理推广至光滑、proper且tame的代数stack，证明了完全忠实函子与Fourier--Mukai变换的关联，扩展了Kawamata关于Deligne--Mumford stack的先前成果。
Perverse sheaves with nilpotent singular support on the stack of coherent sheaves on an elliptic curve
本文通过定义椭圆曲线上凝聚层模栈的分层，显式描述了全局幂零锥的不可约分支，建立了球面 Eisenstein 层与这些分支之间的双射，并参数化了具有幂零奇异支集的反常层。
Microlocal perverse sheaves form a stack on the cotangent bundle of a complex manifold that is the analogue of the stack of perverse sheaves on the manifold itself. We give an embedding of the stack of microlocal perverse sheaves into a simpler stack on the cotangent bundle which we call melded Morse systems. The category of melded Morse systems is elementary, in the sense that an object is a collection of vector spaces and a collection of linear maps between them satisfying certain composition relations.
A geometric stack is a quasi-compact and semi-separated algebraic stack. We prove that the quasi-coherent sheaves on the small flat topology, Cartesian presheaves on the underlying category, and comodules over a Hopf algebroid associated to a presentation of a geometric stack are equivalent categories. As a consequence, we show that the category of quasi-coherent sheaves on a geometric stack is a Grothendieck category. We associate, in a 2-functorial way, to a 1-morphism of geometric stacks $f$, an adjunction $(f^*, f_*)$ for the corresponding categories of quasi-coherent sheaves that agrees w…
Perverse sheaves on smooth toric varieties and stacks
本文将光滑环面簇及其轨道分层上的反常层范畴显式刻画为某代数上的有限维模范畴，并推广到等变情形以描述环面轨形上的反常层，同时比较了相关导出范畴。
We construct the étale motivic Borel-Moore homology of derived Artin stacks. Using a derived version of the intrinsic normal cone, we construct fundamental classes of quasi-smooth derived Artin stacks and demonstrate functoriality, base change, excess intersection, and Grothendieck-Riemann-Roch formulas. These classes also satisfy a general cohomological Bézout theorem which holds without any transversity hypotheses. The construction is new even for classical stacks and as one application we extend Gabber's proof of the absolute purity conjecture to Artin stacks.
We show that the moduli stacks of semistable sheaves on smooth projective varieties are analytic locally on their coarse moduli spaces described in terms of representations of the associated Ext-quivers with convergent relations. When the underlying variety is a Calabi-Yau 3-fold, our result describes the above moduli stacks as critical locus analytic locally on the coarse moduli spaces. The results in this paper will be applied to the wall-crossing formula of Gopakumar-Vafa invariants defined by Maulik and the author.
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 $\mathbf{X}$.
We prove two flat descent statements for Artin n-stacks. We first show that an n-stack for the etale topology which is an Artin n-stack in the sense of HAGII, is also an n-stack for the fppf topology. Moreover, an n-stack for the fppf topology which possess a fppf n-atlas is an Artin n-stack (i.e. possesses a smooth n-atlas). We deduce from these results some comparison statements between fppf and etale (non-ablelian) cohomolgies. This paper is written in the setting of derived algebraic geometry and its results are also valid for derived Artin n-stacks.
We present a complete generalization of Kirwan's partial desingularization theorem on quotients of smooth varieties. Precisely, we prove that if $\mathcal{X}$ is an irreducible Artin stack with stable good moduli space $\mathcal{X} \to X$, then there is a canonical sequence of birational morphisms of Artin stacks $\mathcal{X}_n \to \mathcal{X}_{n-1} \to \ldots \to \mathcal{X}_0 = \mathcal{X}$ with the following properties: (1) the maximum dimension of a stabilizer of a point of $\mathcal{X}_{k+1}$ is strictly smaller than the maximum dimension of a stabilizer of $\mathcal{X}_k$ and the final s…
Raynaud--Gruson characterized flat and pure morphisms between affine schemes in terms of projective modules. We give a similar characterization for non-affine morphisms. As an application, we show that every quasi-coherent sheaf is the union of its finitely generated quasi-coherent subsheaves on any quasi-compact and quasi-separated algebraic stack.
Motivated by the enhanced gauge symmetry phenomenon of the physics literature and mirror symmetry, this paper constructs an action of an Artin group on the derived category of coherent sheaves of a smooth quasiprojective threefold containing a configuration of ruled surfaces described by a finite type Dynkin diagram. The action extends over deformations of the threefold via a compatible action of the corresponding reflection group on the base of its deformation space. All finite type Dynkin diagrams are realized.
Deligne's conjecture that $\ell$-adic sheaves on normal schemes over a finite field admit $\ell'$-companions was proved by L. Lafforgue in the case of curves and by Drinfeld in the case of smooth schemes. In this paper, we extend Drinfeld's theorem to smooth Artin stacks and deduce Deligne's conjecture for coarse moduli spaces of smooth Artin stacks. We also extend related theorems on Frobenius eigenvalues and traces to Artin stacks.
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,…
Moduli of sheaves on ribbons
本文研究了带状结构上无挠层模栈的几何性质，通过完全类型引入分层并分析其性质与闭包关系，揭示了Fano、Calabi-Yau和典范极化情形间的分叉以描述不可约分支，并计算了模栈在给定层处的切空间。
The purpose of this article is to study the role of Artin fans in tropical and non-Archimedean geometry. Artin fans are logarithmic algebraic stacks that can be described completely in terms of combinatorial objects, so called Kato stacks, a stack-theoretic generalization of K. Kato's notion of a fan. Every logarithmic algebraic stack admits a tautological strict morphism $ϕ_\mathcal{X}:\mathcal{X}\rightarrow\mathcal{A}_\mathcal{X}$ to an associated Artin fan. The main result of this article is that, on the level of underlying topological spaces, the natural functorial tropicalization map of $…
We reprove the Lefschetz trace formula for stacks (in the context of derived categories and the six operations for stacks developed by Laszlo and Olsson), and give the meromorphic continuation of L-series (in particular, zeta functions) of Artin stacks over a finite field. We also give an upper bound for the weights of the cohomology groups of stacks, and an "independence of l" result for a certain class of quotient stacks.
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