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 stacks.
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.
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)
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.
We prove that every balanced 1-dimensional polyhedral complex arises as the tropicalization of a smooth curve over a non-Archimedean field mapping to a toric Artin fan, namely the quotient of a toric variety by its dense torus.
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 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.
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.
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.
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.
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.
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.
The purpose of this paper and its prequel (Toric Stacks I) is to introduce and develop a theory of toric stacks which encompasses and extends the notions of toric stacks defined in [Laf02, BCS05, FMN10, Iwa09, Sat12, Tyo12], as well as classical toric varieties. While the focus of the prequel is on how to work with toric stacks, the focus of this paper is how to show a stack is toric. For toric varieties, a classical result says that any normal variety with an action of a dense open torus arises from a fan. In [FMN09, Theorem 7.24], it is shown that a smooth separated DM stack with an action o…
The purpose of this paper and its sequel (Toric Stacks II) is to introduce and develop a theory of toric stacks which encompasses and extends the notions of toric stacks defined in [Laf02, BCS05, FMN10, Iwa09, Sat12, Tyo12], as well as classical toric varieties. In this paper, we define a \emph{toric stack} as a quotient of a toric variety by a subgroup of its torus (we also define a generically stacky version). Any toric stack arises from a combinatorial gadget called a \emph{stacky fan}. We develop a dictionary between the combinatorics of stacky fans and the geometry of toric stacks, stress…
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…
Simplicial toric stack bundles are smooth Deligne-Mumford stacks over smooth varieties with fibre a toric Deligne-Mumford stack. We compute the Grothendieck $K$-theory of simplicial toric stack bundles and study the Chern character homomorphism.
Any toric Deligne-Mumford stack is a $μ$-gerbe over the underlying toric orbifold for a finite abelian group $μ$. In this paper we give a sufficient condition so that certain kinds of gerbes over a toric Deligne-Mumford stack are again toric Deligne-Mumford 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…
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 introduce extended toric Deligne-Mumford stacks. We use an extended toric Deligne-Mumford stack to get the toric stack bundle and compute its orbifold Chow ring. Finally we generalize one result of Borisov, Chen and Smith so that the orbifold Chow ring of the toric stack bundle and the Chow ring of its crepant resolution are fibres of a flat family.
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.
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 formula for the orbifold Chow ring of semi-projective toric DM stacks, generalizing the orbifold Chow ring formula of projective toric DM stacks by Borisov-Chen-Smith. We also consider a special kind of semi-projective toric DM stacks, the Lawrence toric DM stacks. We prove that the orbifold Chow ring of a Lawrence toric DM stack is isomorphic to the orbifold Chow ring of its associated hypertoric DM stack studied in \cite{JT}.
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.
We explicitly calculate the Grothendieck $K$-theory ring of a smooth toric Deligne-Mumford stack and define an analog of the Chern character. In addition, we calculate $K$-theory pushforwards and pullbacks for weighted blowups of reduced smooth toric DM stacks.
We give a precise relation between the mirror transformation and the Seidel elements for weak Fano toric Deligne-Mumford stacks. Our result generalizes the corresponding result for toric varieties proved by González and Iritani in \cite{GI}.
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…
Generalizing toric varieties, we introduce toric Deligne-Mumford stacks which correspond to combinatorial data. The main result in this paper is an explicit calculation of the orbifold Chow ring of a toric Deligne-Mumford stack. As an application, we prove that the orbifold Chow ring of the toric Deligne-Mumford stack associated to a simplicial toric variety is a flat deformation of (but is not necessarily isomorphic to) the Chow ring of a crepant resolution.
For a toric Deligne-Mumford (DM) stack, we can consider a certain generalization of the Frobenius endomorphism. For such an endomorphism on a two-dimensional toric DM stack, we show that the push-forward of the structure sheaf generates the bounded derived category of coherent sheaves on the toric DM stack. We also choose a full strong exceptional collection from the set of direct summands of the push-forward in several examples of two dimensional toric DM orbifolds.
In this paper, we develop several techniques for computing the higher G-theory and K-theory of quotient stacks. Our main results for computing these groups are in terms of spectral sequences. We show that these spectral sequences degenerate in the case of many toric stacks, thereby providing an efficient computation of their higher K-theory. We apply our main results to give explicit description for the higher K-theory for many smooth toric stacks. As another application, we describe the higher K-theory of toric stack bundles over smooth base schemes.
In this paper we study the integral Chow ring of toric Deligne-Mumford stacks. We prove that the integral Chow ring of a semi-projective toric Deligne-Mumford stack is isomorphic to the Stanley-Reisner ring of the associated stacky fan. The integral orbifold Chow ring is also computed. Our results are illustrated with several examples.
Let X be an irreducible affine T-variety. We consider families of affine stable toric T-varieties over X and give a description of the corresponding moduli space as the quotient stack of an open subscheme in a certain toric Hilbert scheme under the action of a torus.
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…
We give a new description of the data needed to specify a morphism from a scheme to a toric Deligne-Mumford stack. The description is given in terms of a collection of line bundles and sections which satisfy certain conditions. As applications, we characterize any toric Deligne-Mumford stack as a product of roots of line bundles over the rigidified stack, describe the torus action, describe morphisms between toric Deligne-Mumford stacks with complete coarse moduli spaces in terms of homogeneous polynomials, and compare two different definitions of toric stacks.
We study Givental's Lagrangian cone for the quantum orbifold cohomology of toric stack bundles and prove that the I-function gives points in the Lagrangian cone, namely we construct an explicit slice of the Lagrangian cone defined by the genus $0$ Gromov-Witten theory of a toric stack bundle.
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 study the bounded derived categories of torus-equivariant coherent sheaves on smooth toric varieties and Deligne-Mumford stacks. We construct and describe full exceptional collections in these categories. We also observe that these categories depend only on the PL homeomorphism type of the corresponding simplicial complex.
We construct full strong exceptional collections of line bundles on smooth toric Fano Deligne-Mumford stacks of Picard number at most two and of any Picard number in dimension two. It is hoped that the approach of this paper will eventually lead to the proof of the existence of such collections on all smooth toric nef-Fano Deligne-Mumford stacks.
We study line bundles on toric DM stacks $\mathbb{P}_{\mathbfΣ}$ of dimension two. We give a combinatorial criterion of when infinitely many line bundles on $\mathbb{P}_{\mathbfΣ}$ have trivial cohomology. We further discuss the structure of the set of such line bundles.
In this paper we will introduce a certain type of morphisms of log schemes (in the sense of Fontaine, Illusie, and Kato) and investigate their moduli. Then by applying this we define a notion of toric algebraic stacks over arbitrary schemes, which may be regarded as torus embeddings within the framework of algebraic stacks, and study some basic properties. Furthermore, we study the stack-theoretic analogue of toroidal embeddings.
We use the mirror theorem for toric Deligne-Mumford stacks, proved recently by the authors and by Cheong-Ciocan-Fontanine-Kim, to compute genus-zero Gromov-Witten invariants of a number of toric orbifolds and gerbes. We prove a mirror theorem for a class of complete intersections in toric Deligne-Mumford stacks, and use this to compute genus-zero Gromov-Witten invariants of an orbifold hypersurface.
We compute the Grothendieck and Picard groups of a complete smooth toric Deligne-Mumford stack by using a suitable category of graded modules over a polynomial ring.
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 prove the following results for toric Deligne-Mumford stacks, under minimal compactness hypotheses: the Localization Theorem in equivariant K-theory; the equivariant Hirzebruch-Riemann-Roch theorem; the Fourier--Mukai transformation associated to a crepant toric wall-crossing gives an equivariant derived equivalence.
