Cycle groups for Artin stacks

Virtual classes of Artin stacks

We construct virtual fundamental classes of Artin stacks over a Dedekind domain endowed with a perfect obstruction theory.

The six operations for sheaves on Artin stacks II: Adic Coefficients

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.

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.

Canonical Artin Stacks over Log Smooth Schemes

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.

Twisted stable maps to tame Artin stacks

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.

Artin's axioms, composition and moduli spaces

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.

通过 Artin 叠消解 log-terminal 奇点的 crepant 消解

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。

There is no degree map for 0-cycles on Artin stacks

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.

The six operations for sheaves on Artin stacks I: Finite Coefficients

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.

Artin-Schreier 根叠与群作用的提升

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。

Grothendieck rings of Artin n-stacks

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…

Enhanced adic formalism and perverse t-structures for higher Artin stacks

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…

The Intrinsic Normal Cone For Artin Stacks

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…

On the local quotient structure of Artin stacks

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.

Derived deformations of Artin stacks

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.

论平移接触导出的Artin叠

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。

Stable points on algebraic stacks

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.

Artin algebraization for pairs with applications to the local structure of stacks and Ferrand pushouts

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.

A Cyclic Operad in the Category of Artin Stacks and Gravitational Correlators

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…

A motivic change of variables formula for Artin stacks

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.

Motivic invariants of Artin stacks and 'stack functions'

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…

Tangent Lie algebra of derived Artin stacks

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…

通过Artin栈的 crepant 消解研究弦霍奇数

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。

Virtual fundamental classes of derived stacks I

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.

Flat descent for Artin n-stacks

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.

Companions on Artin stacks

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.

Canonical reduction of stabilizers for Artin stacks with good moduli spaces

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…

Decomposition Theorem for Perverse sheaves on Artin stacks over finite fields

We generalize the decomposition theorem for perverse sheaves to Artin stacks with affine stabilizers over finite fields.

L-series of Artin stacks over finite fields

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.

Non-Archimedean geometry of Artin fans

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 $…

Artin-Schreier Root Stacks

We classify stacky curves in characteristic $p > 0$ with cyclic stabilizers of order $p$ using higher ramification data. This approach replaces the local root stack structure of a tame stacky curve, similar to the local structure of a complex orbifold curve, with a more sensitive structure called an Artin-Schreier root stack, allowing us to incorporate this ramification data directly into the stack. As an application, we compute dimensions of Riemann-Roch spaces for some examples of stacky curves in positive characteristic and suggest a program for computing spaces of modular forms in this set…

Artin-Schreier-Witt 理论在 Stacky 曲线中的应用

Artin-Schreier-Witt Theory for Stacky Curves

本文利用高阶分歧数据和 Artin-Schreier-Witt 理论扩展了正特征下 stacky 曲线的分类，引入了 Artin-Schreier-Witt 根栈作为新技术工具，并应用野 Riemann-Hurwitz 定理计算了部分野 stacky 曲线的典范环。

奇异 Artin 叠的母题积分

Motivic integration for singular Artin stacks

本文将光滑 Artin 叠上的母题积分变量替换公式推广到奇异情形，并借助 warping 叠典范地将任意母题积分表示为 warped 弧上的积分。

A 'Darboux Theorem' for shifted symplectic structures on derived Artin stacks, with applications

This is the fifth in a series arXiv:1304.4508, arXiv:1305,6302, arXiv:1211.3259, arXiv:1305.6428 on the '$k$-shifted symplectic derived algebraic geometry' of Pantev, Toen, Vaquie and Vezzosi, arXiv:1111.3209. This paper extends the previous three from (derived) schemes to (derived) Artin stacks. We prove four main results: (a) If $(X,ω)$ is a $k$-shifted symplectic derived Artin stack for $k<0$ in the sense of arXiv:1111.3209, then near each $x\in X$ we can find a 'minimal' smooth atlas $φ:U\to X$ with $U$ an affine derived scheme, such that $(U,φ^*(ω))$ may be written explicitly in coordinat…

弦性不变量与环状Artin叠

Stringy invariants and toric Artin stacks

本文提出用光滑Artin叠弧上的动机积分计算Gorenstein测度与弦性Hodge数的猜想框架，并在fantastack（环状Artin叠）情形下证明，为环状簇奇点解消提供非分离方法。

An Artin-Rees Theorem and applications to zero cycles

We prove an Artin-Rees type theorem for algebraic cycles and give an application to zero cycles.

高维Artin栈的增强六运算与基变换定理

Enhanced six operations and base change theorem for higher Artin stacks

本文利用Lurie的稳定$\infty$-范畴理论，为高维Artin栈建立平展上同调中Grothendieck六运算及基变换定理，并推广了Gabber的perverse t-结构，通过增强到$\infty$-范畴层面解决了同伦相干性问题。

Constructible functions on Artin stacks

Let K be an algebraically closed field, X a K-scheme, and X(K) the set of closed points in X. A constructible set C in X(K) is a finite union of subsets Y(K) for finite type subschemes Y in X. A constructible function f : X(K) --> Q has f(X(K)) finite and f^{-1}(c) constructible for all nonzero c. Write CF(X) for the Q-vector space of constructible functions on X. Let phi : X --> Y and psi : Y --> Z be morphisms of C-varieties. MacPherson defined a Q-linear "pushforward" CF(phi) : CF(X) --> CF(Y) by "integration" w.r.t. the topological Euler characteristic. It is functorial, that is, CF(psi o …

p-adic and motivic measure on Artin n-stacks

We construct a notion of p-adic measure on Artin n-stacks which are strongly of finite type over the ring of p-adic integers. We also prove the rationality of of the Poincare series and the Serre series associated with such stacks. Finally, we use motivic integration to show that the rationality of the Serre series is uniform with respect to p.

Artin algebraization and quotient stacks

This article contains a slightly expanded version of the lectures given by the author at the summer school "Algebraic stacks and related topics" in Mainz, Germany from August 31 to September 4, 2015. The content of these lectures is purely expository and consists of two main goals. First, we provide a treatment of Artin's approximation and algebraization theorems following the ideas of Conrad and de Jong which rely on a deep desingularization result due to Neron and Popescu. Second, we prove that under suitable hypotheses, algebraic stacks are etale locally quotients stacks in a neighborhood o…

del Pezzo 类型的标记 Artin--Schelter 曲面

Marked Artin--Schelter surfaces of del Pezzo types

本文引入 del Pezzo 类型的 Artin--Schelter 曲面（非交换曲面，含 del Pezzo 曲面为特例），并证明其标记模栈与亏格 1 曲线及若干线丛元组的模栈双有理。

Deformation quantisation for unshifted symplectic structures on derived Artin stacks

We prove that every $0$-shifted symplectic structure on a derived Artin $n$-stack admits a curved $A_{\infty}$ deformation quantisation. The classical method of quantising smooth varieties via quantisations of affine space does not apply in this setting, so we develop a new approach. We construct a map from DQ algebroid quantisations of unshifted symplectic structures on a derived Artin $n$-stack to power series in de Rham cohomology, depending only on a choice of Drinfeld associator. This gives an equivalence between even power series and certain involutive quantisations, which yield anti-inv…

Artin v-叠的雅可比准则

A Jacobian Criterion for Artin $v$-stacks

本文提出了一种用于Artin v-叠的雅可比准则，并展示了其在几何朗兰兹计划中的应用。

Dimensional reduction in cohomological Donaldson-Thomas theory

For oriented $-1$-shifted symplectic derived Artin stacks, Ben-Bassat-Brav-Bussi-Joyce introduced certain perverse sheaves on them which can be regarded as sheaf theoretic categorifications of the Donaldson-Thomas invariants. In this paper, we prove that the hypercohomology of the above perverse sheaf on the $-1$-shifted cotangent stack over a quasi-smooth derived Artin stack is isomorphic to the Borel-Moore homology of the base stack up to a certain shift of degree. This is a global version of the dimensional reduction theorem due to Davison. We give two applications of our main theorem. Firs…

利用环面 Artin 叠对二项式簇的去奇异化

Desingularization of binomial varieties using toric Artin stacks

本文利用 fantastacks 概念对代数闭域上的二项式簇进行有效去奇异化，大幅减少步骤和图表数，并扩展到多个二项式超曲面的部分同时正规交叉去奇异化。

关于 Artin 栈与 Lie 群胚的 étale 图册、Poisson 结构与量子化的注记

A note on étale atlases for Artin stacks and Lie groupoids, Poisson structures and quantisation

本文解释了 Artin 栈扩张为非负分次交换上链代数函子的方法，利用 stacky 仿射概形的 étale 图册构建小 étale 景，从而高效表述偏移 Poisson 结构与量子化，并将类似构造推广至 Lie 群胚与可微及解析栈。

Hom stacks

We study Hom 2-functors parameterizing 1-morphisms of algebraic stacks, and prove that it is representable by an algebraic stack under certain conditions, using Artin's criterion. As an application we study Picard 2-functors which parameterizes line bundles on algebraic stacks.

Géométricité artinienne de l'$\infty$-champs des éléments de Maurer-Cartan

In this paper, based on the same techniques as in \cite{BENZ08} for the construction of a formally smooth map $V \to Perf$ from the variety of complexes to the Artin $\infty$-stack $Perf$, we explicitly construct a new Artin $ \infty $-stack of Maurer-Cartan elements of a dg-category $ \mathcal {P} $ denoted by $\mathcal{MC_P} $, with a map $ V \to \mathcal{MC_P} $ which we show to be formally smooth.

$p$-adic Hodge theory for Artin stacks

This work is devoted to the study of integral $p$-adic Hodge theory in the context of Artin stacks. For a Hodge-proper stack, using the formalism of prismatic cohomology, we establish a version of $p$-adic Hodge theory with the étale cohomology of the Raynaud generic fiber as an input. In particular, we show that the corresponding Galois representation is crystalline and that the associated Breuil-Kisin module is given by the prismatic cohomology. An interesting new feature of the stacky setting is that the natural map between étale cohomology of the algebraic and the Raynaud generic fibers is…

光滑 $p$-adic 分析 Artin 堆栈的局部结构

Local structure of smooth \texorpdfstring{$p$}{p}-adic analytic Artin stacks

本文证明了光滑 $p$-adic 分析 Artin 堆栈的局部轻质-紧致结构，并展示了其 $!$-good 性。

Cohomological invariants of algebraic stacks

The purpose of this paper is to lay the foundations of a theory of invariants in étale cohomology for smooth Artin stacks. We compute the invariants in the case of the stack of elliptic curves, and we use the theory we developed to get some results regarding Brauer groups of algebraic spaces.

Local properties of good moduli spaces

We study the local properties of Artin stacks and their good moduli spaces, if they exist. We show that near closed points with linearly reductive stabilizer, Artin stacks formally locally admit good moduli spaces. We also give conditions for when the existence of good moduli spaces can be deduced from the existence of etale charts admitting good moduli spaces.

超越扭曲弧：为可约群的 McKay 对应关系

Beyond twisted arcs: a McKay correspondence for reductive groups

本文提出扭曲映射的概念，用于研究Artin堆栈，并通过动机变量替换公式建立线性可约群的McKay对应关系。

Compactified Picard stacks over the moduli stack of stable curves with marked points

In this paper we give a construction of algebraic (Artin) stacks endowed with a modular map onto the moduli stack of n-pointed stable curves of genus g, for g greater than 2. These stacks are smooth, irreducible and have dimension 4g-3+n, yielding a geometrically meaningful compactification of the degree d universal Picard stack over the moduli stack of smooth curves with marked points.