The equivariant Euler characteristic of moduli spaces of curves
E. Gorsky
给出了具有n个标记点的亏格g曲线的模空间的S_n -等变欧拉示性数线的一个公式
We give a formula for the S_n - equivariant Euler characteristics of the moduli spaces of genus g curves with n marked points
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We compute the $S_n$-equivariant rational homology of the tropical moduli spaces $Δ_{2,n}$ for $n\leq 8$ using a cellular chain complex for symmetric $Δ$-complexes in Sage.
An equivariant version of the Euler obstruction
对于一个复解析簇的作用的有限群和一个不变的1-形式，我们给出了一个等变版本（与值在伯恩赛德环的组）的局部欧拉阻塞的1-形式，并描述其与等变径向指数的关系定义。这导致了复解析空间的局部欧拉阻塞和全局欧拉阻塞的等变形式。
Equivariant Moduli Theory on $ K3 $ Surfaces
本文研究有限群辛作用下$K3$曲面上层的等变模空间，利用Gieseker与Bridgeland模空间的联系及导出McKay对应，证明其为与点的Hilbert概形形变等价的不可约辛流形。
Let X be a smooth projective variety with the action of the n dimensional torus. The article describes the moduli space of torus equivariant morphisms from stable toric varieties into X as the inverse limit of the GIT quotients of X and their flips when these spaces are enhanced with a naturally associated Deligne-Mumford stack structure. This description is used for constructing a class in the Chow group of the moduli space of dimension dim(X)-n which is invariant to equivariant deformations of X.
On the $S_n$-equivariant Euler characteristic of moduli spaces of hyperelliptic curves
计算了亏格g>1的点超椭圆曲线模空间的$S_n$ -等变欧拉示性数线的母函数。这个答案推广了已知的2和3类的答案，以及J. Bergstrom对任何类和n<8点的答案。
Moduli for Equivariant Vector Bundles of Rank Two on Smooth Toric Surfaces
给出了光滑完备复曲面上秩为2的等变向量丛的完全分类，并构造了此类丛的模空间。这说明是一个直接延续的早期说明，我们制定了一个一般性的描述等变层复曲面品种。在这里，我们给出了这种描述的第一个应用。
A W(E_6)-equivariant projective embedding of the moduli space of cubic surfaces
给出了标记三次曲面模空间的一个显式射影嵌入。这个嵌入在E6型外尔群下是等变的。图像由线性和三次方程的系统定义。为了以最对称的方式表达嵌入，目标将是79维的，然而图像位于9维线性子空间中。
We determine the action of the product of symmetric groups on the cohomology of certain moduli of weighted pointed rational curves. The moduli spaces that we study are of stable rational curves with m+n marked points where the first m marked points are distinct from all the others where as the last n may coincide among themselves. We give a recipe for calculating the equivariant Poincaré polynomials and list them for small m and n.
K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space III: the case r(M)\geq18
在模空间的维数小于或等于2的情况下，证明了利用等变解析挠得到的对合K3曲面的不变量的自守性质。
Given an action of a finite group $G$ on the derived category of a smooth projective variety $X$ we relate the fixed loci of the induced $G$-action on moduli spaces of stable objects in $D^b(\mathrm{Coh}(X))$ with moduli spaces of stable objects in the equivariant category $D^b(\mathrm{Coh}(X))_G$. As an application we obtain a criterion for the equivariant category of a symplectic action on the derived category of a symplectic surface to be equivalent to the derived category of a surface. This generalizes the derived McKay correspondence, and yields a general framework for describing fixed lo…
On the S_n-equivariant Euler characteristic of M_{2,n}
Getzler公式将簇X上不同点的有序元组空间的S_n-等变Hodge-Deligne多项式与X的Hodge-Deligne多项式联系起来。我们得到类似的情况下，当X有一个非平凡的自同构群这个公式。将具有不同自同构群的$\mathcal{M}_2$的所有层集合在一起，导出了$\mathcal{M}_{2,n}$的S_n-等变欧拉示性数的计算公式.
We compute rational equivariant Chow rings with respect to a torus of quiver moduli spaces. We derive a presentation in terms of generators and relations, use torus localization to identify it as a subring of the Chow ring of the fixed point locus, and we compare the two descriptions.
Equivariant geometry and the cohomology of the moduli space of curves
本文给出了栈的整上同调环的范畴定义。我们表明，商栈的范畴上同调可以确定与等变上同调。通过这种识别，我们表明，Deligne-Mumford商栈的上同调是理性同构的粗模空间的有理上同调。该理论的重点是光滑和稳定的曲线的堆栈。
We compute the weighted Euler characteristic, equivariant with respect to the action of the symplectic group of degree six over the field of two elements, of the moduli space of principally polarized abelian threefolds together with a level two structure.
Equivariant Chow-Witt groups and moduli stacks of elliptic curves
本文引入等变Chow-Witt群定义商栈的Chow-Witt群，计算稳定与光滑椭圆曲线模栈的Chow-Witt环，并给出新生成元的几何解释及$μ_{2n}$分类栈的相应结果。
A geometric parametrization for the virtual Euler characteristic for the moduli spaces of real and complex algebriac curves
我们表明，虚拟欧拉示性数的模空间的$s$ -指向代数曲线的属$g$可以确定从一个多项式在$1/γ$ $γ$允许专业化，通过$γ=1,$的复杂的情况下处理的Harer和Zagier，通过$γ=1/2$，真正的情况下。这个多项式似乎具有几何意义，可能是某些模空间的虚欧拉示性数，但尚未确定。这与一个猜想有关，即不确定的$b=γ^1-1$与通过矩阵模型和Jack对称函数的单元分解的组合不变量有关。发展使用Strebel微分三角形的模空间，和识别的$γ$都作为一个参数，在杰克对称函数和作为一个参数，通过广义Selberg积分矩阵模型。
We compute cohomology of the moduli space of genus three curves with level two structure and some related spaces. In particular, we determine the cohomology groups of the moduli space of plane quartics with level two structure as representations of the symplectic group on a six dimensional vector space over the field of two elements. We also make the analogous computations for some related spaces such as moduli spaces of genus three curves with a marked points and strata of the moduli space of Abelian differentials of genus three.
Equivariant counts of points of the moduli spaces of pointed hyperelliptic curves
研究亏格为$g$的$n$点光滑超椭圆曲线的模空间$\Hh_{g,n}$.为了得到上同调的信息，我们希望使$\s_n$ -等变计数的点的数量定义在模空间的有限域。我们发现递归关系的属，这些号码履行。因此，如果我们可以对低亏格进行$\Hh_{g,n}$的$\s_n$ -等变计数，那么我们可以对每个亏格都这样做。然后发现关于亏格0和1的曲线的信息足以计算所有$g$和$n \leq 7$的$\Hh_{g,n}$的答案。这些结果被应用到亏格为2的稳定曲线的模空间中，得到了$\s_n$ -等变Galois（分别为. Hodge）结构的$\ell$ -adic（resp. Betti）上同调。
A holomorphic torsion invariant of K3 surfaces with involution was introduced by the second-named author. In this paper, we completely determine its structure as an automorphic function on the moduli space of such K3 surfaces. On every component of the moduli space, it is expressed as the product of an explicit Borcherds lift and a classical Siegel modular form. We also introduce its twisted version. We prove its modularity and a certain uniqueness of the modular form corresponding to the twisted holomorphic torsion invariant. This is used to study an equivariant analogue of Borcherds' conject…
The $S_n$-equivariant top weight Euler characteristic of $M_{g,n}$
本文证明了 Zagier 关于 $M_{g,n}$ 顶权上同调的 $S_n$-等变欧拉示性数猜想，给出了其显式公式。
On a formula for the equivariant Euler characteristic of a $G$-sheaf
本文利用商栈的Riemann-Roch定理，将域上$G$-曲线$X$上$G$-层等变Euler示性数的计算推广到了$dim(X)>1$的高维情形。
Computing the equivariant Euler characteristic of Zariski and etale sheaves on curves
证明了一个等变的Grothendieck-Ogg-Shafarevich公式。这个公式可以被看作是一个著名的公式的经典类似物，推广了经典的Chevalley-Weil公式。我们给出了一个新的方法，这些公式（首先证明了Ellingsrud/Lønsted，Nakajima，Kani和Ksir），也可以应用在étale的情况下。
Epsilon constants and equivariant Arakelov Euler characteristics
研究了支持有限群G的驯服作用的算术簇上厄米层的等变Arakelov-欧拉示性数.群作用的驯服使我们能够在一个特别精细的“射影”算术类群中产生一个等变的阿拉克洛夫-欧拉示性数。然后，我们表明，等变的Arakelov-欧拉示性数的各种复杂的微分确定的L-功能的动机从算术品种使用辛表示的群G的的常数。我们的结果可以被视为第一个高维版本的Cassou-Noguès泰勒辛Artin根数的厄米结构的整数环的特征，其次作为一个有符号的等变版本的布洛赫导体公式。
The $\mathbb{S}_n$-equivariant Chow polynomial of the braid matroid
本文确定了辫形矩阵B_n的S_n等变Chow多项式的生成函数，通过将其与多尺度微分的模空间联系起来，贡献于多尺度微分模空间的拓扑研究。
We make cohomological computations related to the moduli space of genus three curves with symplectic level two structure by means of counting points over finite fields. In particular, we determine the cohomology groups of the quartic locus as representations of the symmetric group on seven elements.
Equivariant Euler characteristics on permutohedral varieties
本文基于Huh的工作，将二项式系数与置换体簇上的相交问题关联，通过等变K-理论将欧拉示性数升级为洛朗多项式，并给出三种计算方法。
Moduli spaces of twisted equivariant G-bundles over a curve
本文构造了紧黎曼曲面上多稳定(θ,c)-扭曲Γ-等变G-丛的粗模空间，推广了Ramanathan的G-丛模空间构造，并给出非连通群Ĝ-丛模空间的GIT构造。
Fixed point loci of moduli spaces of sheaves on toric varieties
该研究推广前人工作给出非奇异环面簇上纯等变层的组合描述，借助GIT构造对应模空间，显式表出Gieseker稳定层模空间的环面作用不动点轨迹，构造出对应Gieseker稳定性的等变线丛，并得到μ-稳定自反层模空间不动点轨迹的组合描述，可用于计算相关欧拉示性数生成函数。
Semi-prorepresentability of formal moduli problems and equivariant structures
本文将半普适性推广至导形变理论，给出形式模问题半亲表示及其在线性约化群下具有等变结构的判据，并推出代数格式与复紧流形的等变形式半普适形变存在性。
K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space II: a structure theorem for r(M)>10
研究了对合K3曲面的不变量的结构，利用等变解析挠量得到了不变量的结构。以前已知不变量表示为模空间上自守形式的Petersson范数。当K3-格的不变子格关于对合的秩严格大于10时，我们证明了这种自守形式表示为显式Borcherds提升与Igusa的Siegel模形式的张量积.
Orbifold Hirzebruch-Riemann-Roch for quotient Deligne-Mumford stacks and equivariant moduli theory on $ K3 $ surfaces
本文通过轨形Mukai配对导出新的轨形HRR公式，用于计算K3曲面上有限群作用下稳定层等变模空间的维数，并证明这些模空间是形变等价于点Hilbert概型的不可约辛流形。
There are (at least) two different approaches to define equivariant analogue of the Euler charateristic for a space with a finite group action. The first one defines it as an element of the Burnside ring of the group. The second approach emerged from physics and includes the orbifold Euler characteristic and its higher order versions. Here we suggest a way to merge the two approaches together defining (in a certain setting) higher order Euler characteristics with values in the Burnside ring of a group. We give Macdonald type equations for these invariants. We also offer generalized ("motivic")…
Cubic structures, equivariant Euler characteristics and lattices of modular forms
本文利用三次结构理论，给出了有限阿贝尔群在Z上具有驯服作用的射影平坦格式上相干层的等变欧拉示性数的不动点Riemann-Roch公式.这个公式支持了一个猜想，关于在多大程度上这种等变欧拉示性数可以从层的限制到不动点轨迹的无穷小邻域来确定。我们的结果被应用于研究代数整数环中具有傅里叶系数的模形式的模结构，以及菱形Hecke算子对模曲线的Jacobian的Mordell-Weil群和Tate-Shafarevich群的作用.
We calculate the Euler characteristic of associated vector bundles over the moduli spaces of stable parabolic bundles on smooth curves. Our method is based on a wall-crossing technique from Geometric Invariant Theory, certain iterated residue calculus and the tautological Hecke correspondence. Our work was motivated by the results of Teleman and Woodward on the index of K-theory classes on moduli stacks.
On the moduli space of semi-stable plane sheaves with Euler characteristic one and supported on sextic curves
研究复射影平面上支撑于六次曲线且具有欧拉示性数1的Gieseker半稳定层的模空间.我们确定所有这样的层的长度为1的局部自由分解。我们将模空间分解成若干层，这些层自然地作为某些代数群的模作用的子项出现。在某些情况下，我们给出具体的几何描述的地层。
The Euler characteristic of local systems on the moduli of genus 3 hyperelliptic curves
对于非负整数的划分$lambda=\{lambda_1 \geq λ_2 \geq λ_3 \}$，通过适当的分层，计算了亏格为3的超椭圆曲线的模空间上局部系统$V_λ$的欧拉示性数.对于某些低次$λ$，利用有限域上曲线的计数，对$V_λ$的运动欧拉示性数作了一个猜想.
Equivariant toric geometry and Euler-Maclaurin formulae -- an overview
本文综述了射影环面簇上等变 motivic Chern 类与 Hirzebruch 示性类的最新进展，利用局部化和等变 Hirzebruch-Riemann-Roch 形式体系建立加权及一般凝聚层系数的欧拉-麦克劳林型公式，并推广到面的内部和 Minkowski 加项。
On Euler Characteristic of equivariant sheaves
设$k$是特征$p>0$的代数闭域，$\ell$是另一个素数. O. Gabber和F. Loeser证明了对于$k$上的任何代数环面$T$和$T$上的任何反常$\ell$ -adic层$\calF$，欧拉示性数$χ(\calF)$是非负的.我们猜想，对于$k$上的约化群$G$上的任意反常层$\calF$，对于伴随作用是等变的，同样的结果也成立。证明了当$G$的正则半单元素集通过Goresky-MacPherson扩张得到$\calF$时的猜想.由此我们推出，猜想成立的$G$的半单秩1。
On the Euler characteristic of Kronecker moduli spaces
结合稳定模表示模空间Poincaré多项式的MPS退化公式和局部化理论，将确定这些模空间的欧拉示性数归结为计算某些树的组合问题.我们利用这一事实，以获得一个上界的欧拉示性数的情况下，克罗内克图。我们还导出了MPS退化公式中出现的某些模空间的欧拉示性数的公式。
Let $\overline{\mathscr{M}}_{g, n}$ be the moduli space of $n$-pointed stable genus $g$ curves, and let $\mathscr{M}_{g, n}$ be the moduli space of $n$-pointed smooth curves of genus $g.$ In this paper, we obtain an asymptotic expansion for the characteristic of the free modular operad $\mathbb{M}\mathcal{V}$ generated by a stable $\mathbb{S}$-module $\mathcal{V},$ allowing to effectively compute $\mathbb{S}_{n}$-equivariant Euler characteristics of $\overline{\mathscr{M}}_{g, n}$ in terms of $\mathbb{S}_{n'}$-equivariant Euler characteristics of $\mathscr{M}_{g'\!, n'}$ with $0\le g' \le g,$ …
Equivariant Hodge polynomials of heavy/light moduli spaces
本文研究Hassett重/轻权模空间及其光滑轨迹的等变Hodge-Deligne多项式，给出变量替换公式，用已知模空间的S_n-等变生成函数计算(S_m×S_n)-等变生成函数。
The Euler characteristic of local systems on the moduli of curves and abelian varieties of genus three
我们展示了如何计算的欧拉示性数的局部系统相关联的不可约表示的辛群的亏格3上的模空间的曲线的亏格3和模空间的主要极化阿贝尔簇的3维。
Equivariant toric geometry and Euler-Maclaurin formulae
本文研究了拟射影环面簇的等变motivic Chern类和Hirzebruch特征类，通过等变K-理论和局部化方法推广了多个经典公式，并基于motivic特征类方法给出了全维格多面体的Euler-Maclaurin型公式的代数几何证明及其加权与局部推广。
On the Euler numbers of certain moduli spaces of curves and points
本文在一个光滑射影簇中确定了一维闭子概形模空间的拓扑欧拉数，该簇允许一个具有1维纤维的Zapriki-局部平凡纤维化.主要的方法是使用虚拟霍奇多项式和环面行动。这些结果可能会对相应的Donaldson-Thomas不变量有所启发。
We give a combinatorial proof for a multivariable formula of the generating series of type D Young walls. Based on this we give a motivic refinement of a formula for the generating series of Euler characteristics of Hilbert schemes of points on the orbifold surface of type D.
Vector bundles near negative curves: moduli and local Euler characteristic
研究了不可约曲线$\ell = CP^1$的二维邻域$\ell^2 = -k$上向量丛的模空间，给出了这些模作为分层空间的显式构造。给出了$Z_k$上丛的局部全纯欧拉示性数的精确界，并证明了具有给定数值不变量的丛族的存在性.我们的数值计算使用Macaulay 2算法进行，该算法可在http://www.maths.ed.ac.uk/~s0571100/Instanton/下载。
On the Virtual Euler Characteristic of the Moduli Space of Stable Pairs on Surfaces
本文研究了环面上稳定对理论，确定了固定点轨迹上的虚拟切空间，提出计算虚拟欧拉示性数的程序并以射影平面为例说明，通过特例验证支持了有理性与对称性猜想。
Euler Characteristics of Moduli Spaces of Curves
设${mathcal M}_g^n$是亏格为g的n点Riemann曲面的模空间。用${\bar {\mathcal M}}_g^n$表示${mathcal M}_g^n$的Deligne-Mumford紧化。本文计算了${\bar {\mathcal M}}_g^n$对任意g和n（n>2-2g）的轨道折叠和普通欧拉示性数.
For the moduli spaces of Abelian differentials, the Euler characteristic is one of the most basic intrinsic topological invariants. We give a formula for the Euler characteristic that relies on intersection theory on the smooth compactification by multi-scale differentials. It is a consequence of a formula for the full Chern polynomial of the cotangent bundle of the compactification. The main new technical tools are an Euler sequence for the cotangent bundle of the moduli space of Abelian differentials and computational tools in the Chow ring, such as normal bundles to boundary divisors.
Let $G$ be an algebraic group and $Γ$ a finite subgroup of automorphisms of $G$. Fix also a possibly ramified $Γ$-covering $\widetilde{X} \to X$. In this setting one may define the notion of $(Γ,G)$-bundles over $\widetilde{X}$ and, in this paper, we give a description of these objects in terms of $\mathcal{H}$-bundles on $X$, for an appropriate group $\mathcal{H}$ over $X$ which depends on the local type of the $(Γ,G)$-bundles we intend to parametrize. This extends, and along the way clarifies, an earlier work of Balaji and Seshadri.
Local multiplicities for an equivariantly enriched non-transverse Bézout's theorem
本文提出等价动机同调理论中的度数和局部度数，用于研究一般域上的等价计数问题。通过定义等价动机度数和局部度数，证明了局部到全局的公式，并应用这些构造定义了向量丛的欧拉数，展示了其在有限群作用下的计算方法。
We prove an analog of the Verlinde formula on the moduli space of semistable meromorphic G-Higgs bundles over a smooth curve for a reductive group G whose fundamental group is free. The formula expresses the graded dimension of the space of sections of a positive line bundle as a finite sum whose terms are indexed by formal solutions of a generalized Bethe ansatz equation on the maximal torus of G. In an appendix, Constantin Teleman proves a vanishing theorem for the higher cohomology of positive line bundles on the stack of Higgs bundles.
Euler number of Instanton Moduli space and Seiberg-Witten invariants
本文证明了在黎曼四维流形上，当Euler数与特征和为零时，由Seiberg-Witten不变量给出了拓扑扭曲N=4的Yang-Mills理论的一个配分函数.当可以应用消失定理时，配分函数是瞬子模空间的欧拉数之和。并得到了用瞬子数$k$标记的欧拉数与Seiberg-Witten不变量之间的关系。本文的所有计算都是在不假设对偶的情况下进行的。
Cherednik algebras, W algebras and the equivariant cohomology of the moduli space of instantons on A^2
在A^2上U_r-瞬子的模空间的等变同调空间上构造了gl_r的仿射W-代数的表示，并识别了相应的模.作为推论，我们给出了SU（r）群纯N=2规范理论的AGT猜想的一个版本的证明。Maulik和Okounkov已经宣布了另一个证明。我们的方法使用了Witt代数W_{1+\infty}的泛包络代数的适当变形，它被证明作用于上述同调空间（对于任何r），并专门用于所有W（gl_r）。这种变形又是由GL_n的球面退化双仿射Hecke代数的极限构造的，当n趋于无穷大时。
Let $X$ be a complex scheme acted on by an affine algebraic group $G$. We prove that the Atiyah class of a $G$-equivariant perfect complex on $X$, as constructed by Huybrechts and Thomas, is $G$-equivariant in a precise sense. As an application, we show that, if $G$ is reductive, the obstruction theory on the fine relative moduli space $M\to B$ of simple perfect complexes on a $G$-invariant smooth projective family $Y\to B$ is $G$-equivariant. The results contained here are meant to suggest how to check the equivariance of the natural obstruction theories on a wide variety of moduli spaces equ…
Euler characteristics of moduli spaces of torsion free sheaves on toric surfaces
给定一个光滑环面簇$X$，环面$T$的作用提升到$X$上稳定层的模空间$\mathcal{M}$。利用Klyacho的开创性工作，可以给出不动点轨迹$\mathcal{M}^T$的相当明确的组合描述（如作者早期工作所示）。在本文中，我们应用这种描述的情况下，无挠的层光滑复曲面$S$。得到了这种模空间的欧拉示性数的母函数的一般表达式。生成函数表示为经典GIT中出现的线性子空间的稳定构形的某些模空间的欧拉示性数。该表达式适用于$S$、极化、秩和第一陈类的任何选择。专门针对各种例子，使我们能够计算一些新的以及已知的生成函数。
The Automorphism Equivariant Hitchin Index
本文针对一维复环面下黎曼曲面自同构对秩2奇次固定行列式Higgs丛模空间上T等变行列式线丛上同调的作用，定义并研究自同构等变Hitchin指标，证明了其基于商曲面抛物Higgs丛模空间上同调配对的计算公式。
The weight two compactly supported Euler characteristic of moduli spaces of curves
本文通过图复形与操作子方法推导出曲线模空间权二紧支撑等变欧拉示性数的生成函数公式。
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.
Stable spherical varieties and their moduli
本文引入了稳定球面簇的概念，它包括约化群$G$下的球面簇及其平坦等变退化。给定任意射影空间$\bP$，其中$G$线性作用，我们构造了$\bP$上稳定球面簇的模空间，即$(X,f)$对，其中$X$是稳定球面簇，$f : X \to \bP$是有限等变态射.这个空间是射影的，它的不可约分量是有理的。它推广了偶对$(X,D)$的模空间，其中$X$是稳定的复曲面簇，$D$是不含轨道的$X$上的有效的充分Cartier因子. $\bP$的等变自同构群作用在我们的模空间上; $\bP$上的球面簇及其稳定极限只形成了2个轨道。这个模空间的一个变种给出了另一种观点，以紧化的subjectents薄舒伯特细胞所建造的Kapranov和Lafforgue。
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