On the $S_n$-equivariant Euler characteristic of moduli spaces of hyperelliptic curves

On the S_n-equivariant Euler characteristic of M_{2,n}

The Getzler's formula relates the S_n-equivariant Hodge-Deligne polynomial of the space of ordered tuples of distinct points on a given variety X with the Hodge-Deligne polynomial of X. We obtain the analogue of this formula for the case when X has a nontrivial automorphism group. Collecting together all strata of $\mathcal{M}_2$ with different automorphism groups, we derive a formula for the S_n-equivariant Euler characteristic of $\mathcal{M}_{2,n}$.

$M_{g,n}$ 的 $S_n$-等变顶权欧拉示性数

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。

Cohomology of the moduli space of non-hyperelliptic genus four curves

We compute the intersection Betti numbers of the GIT model of the moduli space of Brill-Noether-Petri general curves of genus 4. This space was shown to be the final non-trivial log canonical model for the moduli space of stable genus four curves, under the Hassett-Keel program. The strategy of the cohomological computation relies on a general method developed by Kirwan to calculate the cohomology of GIT quotients of projective varieties, based on the equivariantly perfect stratification of the unstable points studied by Hesselink and others and a partial resolution of singularities.

关于$G$-层的等变Euler示性数的一个公式

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

We prove an equivariant Grothendieck-Ogg-Shafarevich formula. This formula may be viewed as an étale analogue of well-known formulas for Zariski sheaves generalizing the classical Chevalley-Weil formula. We give a new approach to those formulas (first proved by Ellingsrud/Lønsted, Nakajima, Kani and Ksir) which can also be applied in the étale case.

置换体簇上的等变欧拉示性数

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。

Cohomology of moduli spaces of curves of genus three via point counts

In this article we consider the moduli space of smooth $n$-pointed non-hyperelliptic curves of genus 3. In the pursuit of cohomological information about this space, we make $\mathbb{S}_n$-equivariant counts of its numbers of points defined over finite fields for $n \leq 7$. Combining this with results on the moduli spaces of smooth pointed curves of genus 0, 1 and 2, and the moduli space of smooth hyperelliptic curves of genus 3, we can determine the $\mathbb{S}_n$-equivariant Galois and Hodge structure of the ($\ell$-adic respectively Betti) cohomology of the moduli space of stable curves of…

等变环面几何与欧拉-麦克劳林公式——综述

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。

The equivariant Euler characteristic of moduli spaces of curves

We give a formula for the S_n - equivariant Euler characteristics of the moduli spaces of genus g curves with n marked points

The equivariant Euler characteristic of $\mathcal{A}_3[2]$

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.

Epsilon constants and equivariant Arakelov Euler characteristics

We study equivariant Arakelov-Euler characteristics of hermitian sheaves on arithmetic varieties which support a tame action by a finite group G. The tameness of the group action allows us to produce an equivariant Arakelov-Euler characteristic in a particularly fine "projective" arithmetic class group. We then show that the equivariant Arakelov-Euler characteristics of various complexes of differentials determine the epsilon constants of the L-functions of the motives obtained from the arithmetic variety using symplectic representations of the group G. Our results may be viewed firstly as a h…

局部乘积在具有等价增强的非横截贝祖定理中的应用

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。

On equivariant versions of higher order orbifold Euler characteristics

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")…

Equivariant splitting of the Hodge--de Rham exact sequence

Let $X$ be an algebraic curve with an action of a finite group $G$ over a field $k$. We show that if the Hodge-de Rham short exact sequence of $X$ splits $G$-equivariantly then the action of $G$ on $X$ is weakly ramified. In particular, this generalizes the result of Köck and Tait for hyperelliptic curves. We discuss also converse statements and tie this problem to lifting coverings of curves to the ring of Witt vectors of length $2$.

Cubic structures, equivariant Euler characteristics and lattices of modular forms

We use the theory of cubic structures to give a fixed point Riemann-Roch formula for the equivariant Euler characteristics of coherent sheaves on projective flat schemes over Z with a tame action of a finite abelian group. This formula supports a conjecture concerning the extent to which such equivariant Euler characteristics may be determined from the restriction of the sheaf to an infinitesimal neighborhood of the fixed point locus. Our results are applied to study the module structure of modular forms having Fourier coefficients in a ring of algebraic integers, as well as the action of diam…

The Euler Characteristic Formula for Logarithmic Minimal Degenerations of Surfaces with Kodaira Dimension Zero and its application to Calabi-Yau Threefolds with a pencil

In this paper, the Euler characteristic formula for projective logarithmic minimal degenerations of surfaces with Kodaira dimension zero over a 1-dimensional complex disk is proved under a reasonable assumption and as its application, the singularity of logarithmic minimal degenerations are determined in the abelian or hyperelliptic case. By globalizing this local analysis of singular fibres via generalized canonical bundle formulae due to Fujino-Mori, we bound the number of singular fibres of abelian fibred Calabi-Yau threefolds from above,which was previously done by Oguiso in the potentiall…

亏格为三的聚焦规范修正KdV方程超椭圆解的闭合实平面曲线

Closed real plane curves of hyperelliptic solutions of focusing gauged modified KdV equation of genus three

本研究将欧拉弹性线推广，针对亏格为三的超椭圆曲线，通过调节模参数与初始条件，构造出满足聚焦规范修正KdV方程、超越欧拉8字形弹性线的闭合实平面曲线。

On Euler Characteristic of equivariant sheaves

Let $k$ be an algebraically closed field of characteristic $p>0$ and let $\ell$ be another prime number. O. Gabber and F. Loeser proved that for any algebraic torus $T$ over $k$ and any perverse $\ell$-adic sheaf $\calF$ on $T$ the Euler characteristic $χ(\calF)$ is non-negative. We conjecture that the same result holds for any perverse sheaf $\calF$ on a reductive group $G$ over $k$ which is equivariant with respect to the adjoint action. We prove the conjecture when $\calF$ is obtained by Goresky-MacPherson extension from the set of regular semi-simple elements in $G$. From this we deduce th…

Equivariant Euler characteristics of $\overline{\mathscr{M}}_{g, n}$

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 $0\le g' \le g,$ …

关于托尔矢量束欧拉特征的注记

Note on Euler characteristic of a toric vector bundle

本文基于凸链与托尔等变向量束的关系，证明凸链在格点上的求和等于向量束的欧拉特征。

等变环面几何与Euler-Maclaurin公式

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。

Local Euler obstructions for determinantal varieties

The goal of this note is to explain a derivation of the formulas for the local Euler obstructions of determinantal varieties of general, symmetric and skew-symmetric matrices, by studying the invariant de Rham complex and using character formulas for simple equivariant $D$-modules. These calculations are then combined with standard arguments involving Kashiwara's local index formula and the description of characteristic cycles of simple equivariant $D$-modules. The formulas are implicit in the work of Boe and Fu, and in the case of general matrices they have also been obtained recently by Gaff…

Equivariant approach to weighted projective curves

We investigate group actions on the category of coherent sheaves over weighted projective lines. We show that the equivariant category with respect to certain finite group action is equivalent to the category of coherent sheaves over a weighted projective curve, where the genus of the underlying smooth projective curve and the weight data are given by explicit formulas. Moreover, we obtain a trichotomy result for all such equivariant relations. This equivariant approach provides an effective way to investigate smooth projective curves via weighted projective lines. As an application, we give a…

Equivariant virtual Betti numbers

We define a generalised Euler characteristic for arc-symmetric sets endowed with a group action. It coincides with equivariant homology for compact nonsingular sets, but is different in general. We lay emphasis on the particular case of $Z/2\Z$, and give an application to the study of the singularities of Nash function germs via an analog of the motivic zeta function of Denef & Loeser.

闭合实平面曲线的超椭圆解：聚焦加权修正KdV方程的亏格g解

Closed real plane curves of hyperelliptic solutions of focusing gauged modified KdV equation of genus $g$

本文研究了聚焦加权修正KdV方程的闭合实平面超椭圆曲线解，通过超椭圆曲线数据构造解，并推广了欧拉弹性曲线。

超椭圆轨迹上的万有 Picard 层的 Chow 环

The Chow ring of the universal Picard stack over the hyperelliptic locus

本文确定了超椭圆轨迹上万有 Picard 层的有理 Chow 环及其生成元与关系，并计算了整 Picard 群，证明了其 Brauer 类阶由 $d-g$ 奇偶性决定。

通过联络与留数的曲面等变黎曼-罗赫定理

Equivariant Riemann-Roch for Surfaces via Connections and Residues

本文通过联络与留数方法，结合除子剥离和双有理技巧，为有限群作用下的光滑射影曲面上的等变欧拉示性数提供了统一计算程序，并给出ADE奇点修正类及实例验证。

典范曲线的等变Koszul上同调

Equivariant Koszul Cohomology of Canonical Curves

本文研究了典范曲线的等变Koszul上同调，结合等变欧拉特征和黎曼-罗奇定理，推导出显式公式。

Positivity and Kleiman transversality in equivariant K-theory of homogeneous spaces

We prove the conjectures of Graham-Kumar and Griffeth-Ram concerning the alternation of signs in the structure constants for torus-equivariant K-theory of generalized flag varieties G/P. These results are immediate consequences of an equivariant homological Kleiman transversality principle for the Borel mixing spaces of homogeneous spaces, and their subvarieties, under a natural group action with finitely many orbits. The computation of the coefficients in the expansion of the equivariant K-class of a subvariety in terms of Schubert classes is reduced to an Euler characteristic using the homol…

Indices of collections of equivariant 1-forms and characteristic numbers

If two G-manifolds are G-cobordant then characteristic numbers corresponding to the fixed point sets (submanifolds) of subgroups of G and to normal bundles to these sets coincide. We construct two analogues of these characteristic numbers for singular complex G-varieties where G is a finite group. They are defined as sums of certain indices of collections of 1-forms (with values in the spaces of the irreducible representations of subgroups). These indices are generalizations of the GSV-index (for isolated complete intersection singularities) and the Euler obstruction respectively.

The Euler number of a $C^*$-invariant subvariety in $P^n$

In this note we show that the Euler number of a projective variety $C^*$-equivariantly embedded into a projective space $P^n$ is bounded by $n+1$, as conjectured by Carrell and Sommese.

Young walls and equivariant Hilbert schemes of points in type D

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.

K-理论中的可分解性

Factorisability in K-theory

本文通过详细证明Okounkov引理，展示了环面作用下可分解序列的生成函数与某些sheaves的欧拉特征生成函数之间的关系。

一个代数 $C_2$-等变 Bézout 定理

An algebraic $C_2$-equivariant Bézout's theorem

该论文将 Bézout 定理推广到 $C_2$-等变情形，利用 Burnside 环系数的普通 $C_2$-上同调和扩展分次定义欧拉类，并将其表示为等变秩和度数，同时与常系数和 Borel 上同调结果比较。

$\overline{\mathcal{M}}_{1, n}(\mathbb{P}^r, d)$的$S_n$等变欧拉示性数

The $S_n$-equivariant Euler characteristic of $\overline{\mathcal{M}}_{1, n}(\mathbb{P}^r, d)$

本文通过分解模空间、利用环面作用推导闭式公式，计算了Kontsevich模空间$\overline{\mathcal{M}}_{1, n}(\mathbb{P}^r, d)$的$S_n$等变欧拉示性数，建立了其几何与对称函数、图染色计数的联系。

Euler characteristics of cominuscule quantum K-theory

We prove an identity relating the product of two opposite Schubert varieties in the (equivariant) quantum K-theory ring of a cominuscule flag variety to the minimal degree of a rational curve connecting the Schubert varieties. We deduce that the sum of the structure constants associated to any product of Schubert classes is equal to one. Equivalently, the sheaf Euler characteristic map extends to a ring homomorphism defined on the quantum K-theory ring.

Equivariant Chern classes of singular algebraic varieties with group actions

We define the equivariant Chern-Schwartz-MacPherson class of a possibly singular algebraic variety with a group action over the complex number field (or a field of characteristic 0). In fact, we construct a natural transformation from the equivariant constructible function functor to the equivariant homology functor (in the sense of Totaro-Edidin-Graham), which may be regarded as MacPherson's transformation for (certain) quotient stacks. We discuss on other type Chern/Segre classes and give some applications generalizing orbifold Euler characteristics and Thom polynomials of singularities. The…

Grothendieck ring of varieties with finite groups actions

We define a Grothendieck ring of varieties with finite groups actions and show that the orbifold Euler characteristic and the Euler characteristics of higher orders can be defined as homomorphisms from this ring to the ring of integers. We describe two natural $λ$-structures on the ring and the corresponding power structures over it and show that one of these power structures is effective. We define a Grothendieck ring of varieties with equivariant vector bundles and show that the generalized ("motivic") Euler characteristics of higher orders can be defined as homomorphisms from this ring to t…

曲线模空间权二紧支撑欧拉示性数

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。

On the orbifold Euler characteristics of dual invertible polynomials with non-abelian symmetry groups

In the framework of constructing mirror symmetric pairs of Calabi-Yau manifolds, P. Berglund, T. Hübsch and M. Henningson considered a pair $(f,G)$ consisting of an invertible polynomial $f$ and a finite abelian group $G$ of its diagonal symmetries and associated to this pair a dual pair $(\widetilde{f}, \widetilde{G})$. A. Takahashi suggested a generalization of this construction to pairs $(f, G)$ where $G$ is a non-abelian group generated by some diagonal symmetries and some permutations of variables. In a previous paper, the authors showed that some mirror symmetry phenomena appear only und…

$\mathbb{A}^1$-欧拉特征数的低对称幂与分裂的托里纤维品种

$\mathbb{A}^1$-Euler Characteristic of Low Symmetric Powers and Split Toric Varieties

本文计算了低对称幂和分裂托里纤维品种的$\mathbb{A}^1$-欧拉特征数，并展示了其与Pajwani-Pál构造的幂结构预测的一致性。

环面簇上的层的模空间的不动点轨迹

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。

Genus-Zero Mirror Principle For Two Marked Points

We study a generalization of Lian-Liu-Yau's notion of Euler data in genus zero and show that certain sequences of multiplicative equivariant characteristic classes on Kontsevich's stable map moduli with markings induce data satisfying the generalization. In the case of one or two markings, this data is explicitly identified in terms of hypergeometric type classes, constituting a complete extension of Lian-Liu-Yau's mirror principle in genus zero to the case of two marked points and establishing a program for the general case. We give several applications involving the Euler class of obstructio…

Euler characteristic of stable envelopes

In this paper we prove a formula relating the equivariant Euler characteristic of $K$-theoretic stable envelopes to an object known as the index vertex for the cotangent bundle of the full flag variety. Our formula demonstrates that the index vertex is the power series expansion of a rational function. This result is a consequence of the 3d mirror self-symmetry of the variety considered here. In general, one expects an analogous result to hold for any two varieties related by 3d mirror symmetry.

K-Theoretic Descendent Series for Hilbert Schemes of Points on Surfaces

We study the holomorphic Euler characteristics of tautological sheaves on Hilbert schemes of points on surfaces. In particular, we establish the rationality of K-theoretic descendent series. Our approach is to control equivariant holomorphic Euler characteristics over the Hilbert scheme of points on the affine plane. To do so, we slightly modify a Macdonald polynomial identity of Mellit.

Ansatz for $(-1)^{n-1}\nabla p_n$

We construct a special family of equivariant coherent sheaves on the Hilbert scheme on $n$-points in the affine plane. The equivariant Euler characteristic of these sheaves are closely related to the symmetic functions $(-1)^{n-1} \nabla p_n$. We prove a higher cohomology vanishing result of these sheaves. It follows from the Bridgeland-King-Reid correspondence that there is an effective $S_n$ module underlying the aforementioned family of symmetric functions.

几何Batyrev-Manin-Peyre理论对于加法群等价紧缩的渐进行为

Geometric Batyrev-Manin-Peyre for equivariant compactifications of additive groups

本文研究了给定族在光滑射影曲线上的截面模空间的渐进行为，当通用纤维为有限维向量空间的等价紧缩时，证明了在合适的Grothendieck概形环中，这些模空间的类在某种归一化下收敛到非零有效元素，为Batyrev-Manin-Peyre猜想的动机版本提供了肯定答案。

Chern classes of reductive groups and an adjunction formula

In this paper, I construct noncompact analogs of the Chern classes of equivariant vector bundles over complex reductive groups. For the tangent bundle, these Chern classes yield an adjunction formula for the Euler characteristic of complete intersections in reductive groups. In the case where the complete intersection is a curve, this formula gives an explicit answer for the Euler characteristic and the genus of the curve.

带超多重态的N=2理论的等变局域化

Equivariant Localization for N=2 Theories with Hypermultiplets

Ω背景下的等变局域化技术被应用于CP²上的N=2超对称场论，计算等变关联函数并与低能场论结果对比，确定了不同规范群配分函数的差异因子，还结合S对偶讨论了相关结果。

Generalised Elliptic Functions

We consider multiply periodic functions, sometimes called Abelian functions, defined with respect to the period matrices associated with classes of algebraic curves. We realise them as generalisations of the Weierstras P-function using two different approaches. These functions arise naturally as solutions to some of the important equations of mathematical physics and their differential equations, addition formulae, and applications have all been recent topics of study. The first approach discussed sees the functions defined as logarithmic derivatives of the sigma-function, a modified Riemann t…

Exact results for ${\cal N}=2$ supersymmetric gauge theories on compact toric manifolds and equivariant Donaldson invariants

We provide a contour integral formula for the exact partition function of ${\cal N}=2$ supersymmetric $U(N)$ gauge theories on compact toric four-manifolds by means of supersymmetric localisation. We perform the explicit evaluation of the contour integral for $U(2)$ ${\cal N}=2$ theory on $\mathbb{P}^2$ for all instanton numbers. In the zero mass case, corresponding to the ${\cal N}=4$ supersymmetric gauge theory, we obtain the generating function of the Euler characteristics of instanton moduli spaces in terms of mock-modular forms. In the decoupling limit of infinite mass we find that the ge…

Mirror Principle I

We propose and study the following Mirror Principle: certain sequences of multiplicative equivariant characteristic classes on Kontsevich's stable map moduli spaces can be computed in terms of certain hypergeometric type classes. As applications, we compute the equivariant Euler classes of obstruction bundles induced by any concavex bundles -- including any direct sum of line bundles -- on $¶^n$. This includes proving the formula of Candelas-de la Ossa-Green-Parkes hence completing the program of Candelas et al, Kontesevich, Manin, and Givental, to compute rigorously the instanton prepotential…

万有皮卡堆栈的欧拉特征

Euler characteristics of the universal Picard stack

本文研究了万有皮卡堆栈的欧拉特征，通过组合变换推导出闭式公式，基于Chan-Faber-Galatius-Payne和Gorsky的公式。

Hall-Littlewood polynomials and vector bundles on the Hilbert scheme

Let $E$ be the bundle defined by applying a polynomial representation of $GL_n$ to the tautological bundle on the Hilbert scheme of $n$ points in the complex plane. By a result of Haiman, the Cech cohomology groups $H^i(E)$ vanish for all $i>0$. It follows that the equivariant Euler characteristic with respect to the standard two-dimensional torus action has nonnegative coefficients in the torus variables $z_1,z_2$, because they count the dimensions of the weight spaces of $H^0(E)$. We derive a very explicit asymmetric formula for this Euler characteristic which has this property, by expanding…