Motivic integration over Deligne-Mumford stacks

Moduli Space of $Λ$-modules on Projective Deligne-Mumford Stacks

In this paper, we define $Λ$-quot-functors on Deligne-Mumford stacks. We prove that the $Λ$-quot-functor is representable by an algebraic space. Then, we construct the moduli space of $Λ$-modules on a projective Deligne-Mumford stack. We prove that this moduli space is a quasi-projective scheme.

Atiyah-Segal theorem for Deligne-Mumford stacks and applications

We prove an Atiyah-Segal isomorphism for the higher $K$-theory of coherent sheaves on quotient Deligne-Mumford stacks over $\C$. As an application, we prove the Grothendieck-Riemann-Roch theorem for such stacks. This theorem establishes an isomorphism between the higher $K$-theory of coherent sheaves on a Deligne-Mumford stack and the higher Chow groups of its inertia stack. Furthermore, this isomorphism is covariant for proper maps between Deligne-Mumford stacks.

Uniformization of Deligne-Mumford curves

We compute the fundamental groups of non-singular analytic Deligne-Mumford curves, classify the simply connected ones, and classify analytic Deligne-Mumford curves by their uniformization type. As a result, we find an explicit presentation of an arbitrary Deligne-Mumford curve as a quotient stack. Along the way, we compute the automorphism 2-groups of weighted projective stacks $\mathcal{P}(n_1,n_2,...,n_r)$. We also discuss connections with the theory of F-groups, 2-groups, and Bass-Serre theory of graphs of groups.

Arithmetic Intersection Theory on Deligne-Mumford Stacks

The arithmetic Chow groups and their product structure are extended from the category of regular arithmetic varieties to regular Deligne-Mumford stacks over the ring of integers in a number field.

On coverings of Deligne-Mumford stacks and surjectivity of the Brauer map

This paper proves a result on the existence of finite flat scheme covers of Deligne-Mumford stacks. This result is used to prove that a large class of smooth Deligne-Mumford stacks with affine moduli space are quotient stacks, and in the case of quasi-projective moduli space, to reduce the question to one concerning Brauer groups of schemes.

Nisnevich Descent for Deligne-Mumford Stacks

We prove the excision theorem for the $K$-theory of perfect complexes on Deligne-Mumford stacks. This is then used to study the Nisnevich site of such stacks. We prove the Nisnevich descent for the $K$-theory of perfect complexes. We also draw some other consequences.

Deriving Deligne-Mumford Stacks with Perfect Obstruction Theories

We give conditions for a n-connective quasicoherent obstruction theory on a Deligne-Mumford stack to come from the structure of a connective spectral Deligne-Mumford stack on the underlying topos.

The Integral (orbifold) Chow Ring of Toric Deligne-Mumford Stacks

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.

On the Bogomolov-Gieseker inequality for tame Deligne-Mumford surfaces

We generalize the Bogomolov-Gieseker inequality for semistable coherent sheaves on smooth projective surfaces to smooth Deligne-Mumford surfaces. We work over positive characteristic $p>0$ and generalize Langer's method to smooth Deligne-Mumford stacks. As applications we obtain the Bogomolov inequality for semistable coherent sheaves on a Deligne-Mumford surface in characteristic zero, and the Bogomolov inequality for semistable sheaves on a root stack over a smooth surface which is equivalent to the Bogomolov inequality for the rational parabolic sheaves on a smooth surface $S$. In a joint a…

The orbifold Chow ring of toric Deligne-Mumford stacks

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.

Embedding Deligne-Mumford stacks into GIT quotient stacks of linear representations

We study how to use a suitably ample locally free sheaf over a proper Deligne-Mumford stack to furnish an embedding of the stack into a geometric invariant theory (GIT) quotient stack constructed from a finite-dimensional linear representation of the general linear group.

Moduli Spaces of Coherent Sheaves on Projective Deligne-Mumford Stacks over Algebraic Spaces

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.

On the conjecture of King for smooth toric Deligne-Mumford stacks

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.

Séparation et propriété de Deligne-Mumford des champs de modules d'intersections complètes lisses

We show that the moduli stacks of smooth complete intersections in P^N polarized by O(1) are separated (except for quadrics) and Deligne-Mumford (apart from a few exceptions). ----- On montre que les champs de modules d'intersections completes lisses dans P^N polarisees par O(1) sont separes (sauf dans le cas des quadriques) et de Deligne-Mumford (sauf pour quelques exceptions).

Deformation of a smooth Deligne-Mumford stack via differential graded Lie algebra

For a smooth Deligne-Mumford stack over $\CC$, we define its associated Kodaira-Spencer differential graded Lie algebra and show that the deformation functor of the stack is isomorphic to the deformation functor of the Kodaira-Spencer algebra if the stack is proper over $\CC$.

Cone Theorem via Deligne-Mumford stacks

We prove the cone theorem for varieties with LCIQ singularities using deformation theory of stable maps into Deligne-Mumford stacks. We also obtain a sharper bound on $-(K_X+D)$-degree of $(K_X+D)$-negative extremal rays for projective $\QQ$-factorial log terminal threefold pair $(X,D)$.

Intersection numbers on Deligne-Mumford moduli spaces and quantum Airy curve

We establish the Airy curve case of a conjecture of Gukov and Sułkowski by reducing to Dijkgraaf-Verlinde-Verlinde Virasoro constraints satisfied by the intersection numbers on moduli spaces of algebraic curves.

A note on toric Deligne-Mumford stacks

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.

Le théorème de réduction stable de Deligne et Mumford

The stable reduction theorem of Deligne and Mumford --- The moduli space of smooth projective curves of genus $g$ is a quasi-projective algebraic variety, but is not projective. To understand its geometry, it may be crucial to consider compactifications of this space. By allowing to parameterize as well curves with controlled singularities (the so called stable curves), Deligne and Mumford constructed a projective compactification. The properness of this compactification translates into the stable reduction theorem that they prove, its projectivity is a later theorem of Knudsen and Mumford. Th…

The Orbifold Chow Ring of Hypertoric Deligne-Mumford Stacks

Hypertoric varieties are determined by hyperplane arrangements. In this paper, we use stacky hyperplane arrangements to define the notion of hypertoric Deligne-Mumford stacks. Their orbifold Chow rings are computed. As an application, some examples related to crepant resolutions are discussed.

Gromov-Witten theory of tame Deligne-Mumford stacks in mixed characteristic

We define Gromov-Witten classes and invariants of smooth proper tame Deligne-Mumford stacks of finite presentation over a Dedekind domain. We prove that they are deformation invariants and verify the fundamental axioms. For a smooth proper tame Deligne-Mumford stack over a Dedekind domain, we prove that the invariants of fibers in different characteristics are the same. We show that genus zero Gromov-Witten invariants define a potential which satisfies the WDVV equation and we deduce from this a reconstruction theorem for genus zero Gromov-Witten invariants in arbitrary characteristic.

The realization of the degree zero part of the motivic polylogarithm on abelian schemes in Deligne-Beilinson cohomology

We use Burgos theory of arithmetic Chow groups to exhibit a realization of the degree zero part of the polylogarithm on abelian schemes in Deligne-Beilinson cohomology.

K-Theoretic and Categorical Properties of Toric Deligne--Mumford 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.

On strong exceptional collections of line bundles of maximal length on Fano toric Deligne-Mumford stacks

We study strong exceptional collections of line bundles on Fano toric Deligne-Mumford stacks $\mathbb{P}_{\mathbfΣ}$ with rank of Picard group at most two. We prove that any strong exceptional collection of line bundles generates the derived category of $\mathbb{P}_{\mathbfΣ}$, as long as the number of elements in the collection equals the rank of the (Grothendieck) $K$-theory group of $\mathbb{P}_{\mathbfΣ}$.

Riemann-Roch Theorems for Deligne-Mumford Stacks

The goal of this paper is to prove Riemann-Roch type theorems for Deligne-Mumford algebraic stacks. To this end, we introduce a "cohomology with coefficients in representations" and a Chern character, and we prove a Grothendieck-Riemann-Roch theorem for the Riemann-Roch transformation it defines. As a corollary we obtain an Hirzebruch-Riemann-Roch formula for the Euler characteristic of a coherent sheaf, and some formulas for the different topological Euler characteristics of complex algebraic stacks.

Comparison results for derived Deligne-Mumford stacks

In this short note we write down a comparison between the notion of derived Deligne-Mumford stack in the sense of B. Toën and G. Vezzosi [HAG-II] and the one introduced by J. Lurie in [DAG-V]. It is folklore that the two theories yield essentially the same objects, but it is difficult to locate in the literature a precise result, despite being sometimes useful to be able to switch between the two frameworks.

Instantons and framed sheaves on Kähler Deligne-Mumford stacks

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.

光滑环面 Deligne-Mumford 栈上无瑕线丛的渐近性质

Asymptotics of immaculate line bundles on smooth toric Deligne-Mumford stacks

本文研究了光滑环面 Deligne-Mumford 栈上的无瑕线丛，给出了存在无穷多无瑕线丛的判据，肯定回答了 Borisov 和 Wang 的问题，并描述了其渐近行为。

射影 Deligne–Mumford 栈上半稳定对的模空间

Moduli spaces of semistable pairs on projective Deligne-Mumford stacks

本文把半稳定对模空间构造推广至射影 Deligne–Mumford 栈，研究其形变与障碍理论，并在三维光滑情形定义 PT 不变量。

Frobenius morphisms and derived categories on two dimensional toric Deligne-Mumford stacks

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.

Quot Functors for Deligne-Mumford Stacks

Given a separated and locally finitely-presented Deligne-Mumford stack $\cX$ over an algebraic space $S$, and a locally finitely-presented $\OO_{\cX}$-module $\cF$, we prove that the Quot functor $\text{Quot}(\cF/\cX/S)$ is represented by a separated and locally finitely-presented algebraic space over $S$. Under additional hypotheses, we prove that the connected components of $\text{Quot}(\cF/\cX/S)$ are quasi-projective over $S$.

Deligne--Mumford 超栈的粗模超空间的存在性

Existence of coarse moduli superspaces for Deligne--Mumford superstacks

本文证明在 $\mathbb{Z}[1/2]$ 上具有有限惯性的 Deligne--Mumford 超栈存在粗模超空间，推广 Keel--Mori 定理，并给出反例。

Deligne--Mumford 栈的二次加细推前

A Quadratically Enriched Pushforward for Deligne--Mumford Stacks

本文利用 Grothendieck--Serre 对偶，为 Deligne--Mumford 栈定义二次加细推前，进而构造欧拉类、欧拉数及基本类，并据此提出二次加细量子 $K$-不变量的定义。

Lectures on motivic cohomology 2000/2001 (written by Pierre Deligne)

These lecture notes cover four topics. There is a proof of the fact that the functors represented by the motivic Eilenberg-Maclane spaces on the motivic homotopy category coincide with the motivic cohomology defined in terms of the motivic complexes. There is a description of the equivariant motivic homotopy category for a finite flat group scheme (over a noetherian base) together with a new characterization of A^1-equivalences. There is a part where we introduce a class of sheaves called solid sheaves. Finally there is a part where we study functors of the form X -> X/G and X -> X^W and show …

The Vafa-Witten invariants via surface Deligne-Mumford stacks and S-duality

Motivated by the S-duality conjecture of Vafa-Witten, Tanaka-Thomas have developed a theory of Vafa-Witten invariants for projective surfaces using the moduli space of Higgs sheaves. Their definition and calculation prove the S-duality prediction of Vafa-Witten in many cases in the side of gauge group $SU(r)$. In this survey paper for ICCM-2019 we review the S-duality conjecture in physics by Vafa-Witten and the definition of Vafa-Witten invariants for smooth projective surfaces and surface Deligne-Mumford stacks. We make a prediction that the Vafa-Witten invariants for Deligne-Mumford surface…

Six operations and Lefschetz-Verdier formula for Deligne-Mumford stacks

Laszlo and Olsson constructed Grothendieck's six operations for constructible complexes on Artin stacks in étale cohomology under an assumption of finite cohomological dimension, with base change established on the level of sheaves. In this article we give a more direct construction of the six operations for complexes on Deligne-Mumford stacks without the finiteness assumption and establish base change theorems in derived categories. One key tool in our construction is the theory of gluing finitely many pseudofunctors developed in arXiv:1211.1877. As an application, we prove a Lefschetz-Verdie…

移除Deligne-Mumford堆叠正性定理中的无挠假设

Removing the Torsion-free Hypothesis in a Positivity Theorem on Deligne-Mumford Stacks

本文在Casalaina-Martin与Zhjeqi的正性定理中移除无挠假设，解答其提出的问题，主要结果借助ChatGPT 5.5 Pro、Danus系统等生成式AI获得。

The Tanaka-Thomas's Vafa-Witten invariants via surface Deligne-Mumford stacks

We provide a definition of Vafa-Witten invariants for projective surface Deligne-Mumford stacks, generalizing the construction of Tanaka-Thomas on the Vafa-Witten invariants for projective surfaces inspired by the S-duality conjecture. We give calculations for a root stack over a general type quintic surface, and quintic surfaces with ADE singularities. The relationship between the Vafa-Witten invariants of quintic surfaces with ADE singularities and the Vafa-Witten invariants of their crepant resolutions is also discussed.

One-dimensional substacks of the moduli stack of Deligne-Mumford stable curves

There is a well-known stratification of the moduli space $M_g$ of Deligne-Mumford stable curves of genus $g$ by the loci of stable curves with a fixed number $i$ of nodes, where $i \le 3g-3$. The associated moduli stack ${\cal M}_g$ admits an analogous stratification. Our main objects of study are those one-dimensional substacks of the moduli stack, which are the irreducible components of the stratum corresponding to stable curves with exactly $3g-4$ nodes. We describe these substacks explicitely as quotient stacks, and relate them to other, and simpler, moduli stacks of (permutation classes o…

Bondal-Orlov Fully Faithfulness Criterion for Deligne-Mumford Stacks

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.

Chern classes of Deligne-Mumford stacks and their coarse moduli spaces

Let $X$ be a complex projective algebraic variety with Gorenstein quotient singularities and $\X$ a smooth Deligne-Mumford stack having $X$ as its coarse moduli space. We show that the CSM class $c^{SM}(X)$ coincides with the pushforward to $X$ of the total Chern class $c(T_{I\X})$ of the inertia stack $I\X$. We also show that the stringy Chern class $c_{str}(X)$ of $X$, whenever is defined, coincides with the pushforward to $X$ of the total Chern class $c(T_{II\X})$ of the double inertia stack $II\X$. Some consequences concerning stringy/orbifold Hodge numbers are deduced.

在Deligne-Mumford栈上的Hodge模和Higgs丛背景下的正性

Positivity in the context of Hodge modules and Higgs bundles on Deligne-Mumford stacks

将Popa-Wu和Popa-Schnell关于光滑射影簇上Hodge模和Higgs丛的正性结果推广到具有射影粗模空间的光滑真DM栈上。

叶状结构、斜率稳定性及Deligne-Mumford栈上对数典范丛的正性

Foliations, slope stability, and positivity of log canonical bundles on Deligne-Mumford stacks

将Campana-Păun关于光滑射影簇上叶状结构、斜率稳定性及对数典范丛正性的结果推广到具有射影粗模空间的光滑真DM栈上，作为将Viehweg双曲性结果推广到DM栈及特定KSBA模空间系列论文的第三部分。

模态映射的分解与到德利涅-穆迪堆栈的通用映射

Factorizations of Moduli Morphisms and Universal Maps to Deligne-Mumford Stacks

本文研究了模态空间映射的分解，构造了简化堆栈结构但保留更多信息的中间堆栈，并证明了在一定条件下存在到满足良好模性质的堆栈的通用映射，展示了该映射本身是有效的模空间映射。

德利涅-芒福德栈上非正则霍奇滤子的$E_1$退化

$E_1$-Degeneration for Irregular Hodge Filtrations on Deligne--Mumford Stacks

针对粗空间拟射影的光滑分离德利涅-芒福德栈，利用Kontsevich复形的有限平展下降，将射影栈情形归约为Esnault–Sabbah–Yu的定理，证明其非正则霍奇滤子在所有有理指标处$E_1$退化，结果适用于各类栈紧化。

Riemann-Roch for Deligne-Mumford stacks

We give a simple proof of the Riemann-Roch theorem for Deligne-Mumford stacks using the equivariant Riemann-Roch theorem and the localization theorem in equivariant K-theory together with some basic commutative algebra of Artin rings.

Note on orbifold Chow ring of semi-projective toric Deligne-Mumford 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}.

Deligne-Mumford 栈上的可动曲线类与斜率稳定性

Movable curve classes and slope stability on Deligne-Mumford stacks

将光滑射影簇上关于可动曲线类和相干层的斜率稳定性的结果推广到具有射影粗模空间的光滑真DM栈上，并建立了Bogomolov-Gieseker不等式。

德利涅-芒福德堆叠上的下降障碍与布饶尔-马林障碍

Descent and Brauer-Manin Obstructions on Deligne-Mumford Stacks

该研究在具有拟射粗模空间的有限型光滑分离德利涅-芒福德堆叠上，证明下降障碍包含于布饶尔-马林障碍，且下降障碍与 étale 布饶尔-马林障碍、迭代下降障碍一致，布饶尔-马林障碍也与迭代布饶尔-马林障碍一致。

The de Rham comparison theorem for Deligne-Mumford stacks

The de Rham comparison theorem for varieties, first proved by Faltings, gives the de Rham cohomology of a variety in terms of its p-adic etale cohomology. We extend this theorem to proper, smooth Deligne-Mumford stacks. Two approaches are given, which both in the end reduce the problem to the already known comparison theorem for varieties. The first approach employs the formalism of Weil cohomologies. Unfortunately, this does not result in a complete proof of the comparison theorem, as the author was unable to prove the required compatibility between intersections and cup products. Nevertheles…

A construction of the Deligne--Mumford orbifold

We define the Deligne Mumford orbifold axiomatically by a universal mapping property, show that this universal mapping property is equivalent to an infinitessimal universal mapping property, and use the latter to give an existence proof.

Deligne-Mumford 叠的广义 Bondal-Orlov 全忠实性判据

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。

An analytic construction of the Deligne-Mumford compactification of the moduli space of curves

In 1969, P. Deligne and D. Mumford compactified the moduli space of curves. Their compactification is a projective algebraic variety, and as such, it has an underlying analytic structure. Alternatively, the quotient of the augmented Teichmueller space by the action of the mapping class group gives a compactification of the moduli space. We put an analytic structure on this compact quotient and prove that with respect to this structure, it is canonically isomorphic (as an analytic space) to the Deligne-Mumford compactification.

The Balmer spectrum of certain Deligne-Mumford stacks

We consider a Deligne-Mumford stack $X$ which is the quotient of an affine scheme $\operatorname{Spec}A$ by the action of a finite group $G$ and show that the Balmer spectrum of the tensor triangulated category of perfect complexes on $X$ is homeomorphic to the space of homogeneous prime ideals in the group cohomology ring $H^*(G,A)$.