Rational cohomology of M_3,2

带水平结构的曲线模空间的稳定上同调

The stable cohomology of the moduli space of curves with level structures

本文证明在稳定范围内，带水平结构的曲线模空间与普通曲线模空间的有理上同调相同。

Chen-Ruan cohomology of some moduli spaces, II

Let X be a compact connected Riemann surface of genus at least two. Let r be a prime number and ξa holomorphic line bundle on it such that r is not a divisor of degree(ξ). Let {\mathcal M}_ξ(r) denote the moduli space of stable vector bundles over X of rank r and determinant ξ. By Γwe will denote the group of line bundles L over X such that $L^{\otimes r}$ is trivial. This group Γacts on {\mathcal M}_ξ(r). We compute the Chen-Ruan cohomology of the corresponding orbifold.

On the BV structure on the cohomology of moduli space

The question of vanishing of the BV operator on the cohomology of the moduli space of Riemann surfaces is investigated. The BV structure, which comprises a BV operator and an antibracket, is identified, vanishing theorems are proven, and a counterexample is provided.

The cohomology of the moduli space of abelian varieties

This is a survey on various aspects of the cohomology of the moduli space of abelian varieties

Cohomology of moduli space of cubic fourfolds I

In this paper we compute the cohomology of moduli space of cubic fourfolds with ADE type singularities relying on Kirwan's blowup and Laza's GIT construction. More precisely, we obtain the Betti numbers of Kirwan's resolution of the moduli space. Furthermore, by applying decomposition theorem we obtain the Betti numbers of the intersection cohomology of Baily-Borel compactification of the moduli space.

曲线的分支覆盖模空间的稳定有理上同调分解

Stable rational cohomology splitting of moduli of branched covers of curves

本文研究了曲线分支覆盖模空间的稳定有理上同调，给出了其关于中间覆盖次数的分解，并用因子化同调及带标签的构型空间加以解释，证明了 d=4,5 时稳定奇维上同调类的存在。

Cohomology of the moduli space of smooth plane quartic curves with an odd theta characteristic

We compute the rational cohomology of the moduli space of non-singular non-hyperelliptic complex projective curves of genus 3 with an odd theta characteristic.

On the cohomology of the Losev-Manin moduli space

We determine the cohomology of the Losev-Manin moduli space of pointed genus zero curves as a representation of the product of symmetric groups.

Rational Picard group of moduli of pointed hyperelliptic curves

We determine the rational divisor class group of the moduli spaces of smooth pointed hyperelliptic curves and of their Deligne-Mumford compactification, over the field of complex numbers.

Cohomology of moduli spaces

Some recent progress towards understanding the cohomology of moduli spaces of curves is described. Madsen and Weiss have given a proof of a generalisation of Mumford's conjecture on the stable cohomology of these moduli spaces M_g, and other contributors have made advances related to Faber's conjectures on the tautological ring of M_g.

The high-dimensional cohomology of the moduli space of curves with level structures

We prove that the moduli space of curves with level structures has an enormous amount of rational cohomology in its cohomological dimension. As an application, we prove that the coherent cohomological dimension of the moduli space of curves is at least g-2. Well known conjectures of Looijenga would imply that this is sharp.

Towards the cohomology of moduli spaces of higher genus stable maps

We prove that the orbifold desingularization of the moduli space of stable maps of genus g = 1 recently constructed by Vakil and Zinger has vanishing rational cohomology groups in odd degree k < 10.

Calculating cohomology groups of moduli spaces of curves via algebraic geometry

We compute the first, second, third, and fifth rational cohomology groups of the moduli space of stable n-pointed genus g curves, for all g and n, using (mostly) algebro-geometric techniques.

Rational cohomology of the moduli space of trigonal curves of genus 5

We compute the rational cohomology of the moduli space of trigonal curves of genus 5. We do this by considering their natural embedding in the first Hirzebruch surface and by using Gorinov-Vassiliev's method.

Quantum cohomology of moduli spaces of genus zero stable curves

We investigate the (small) quantum cohomology ring of the moduli spaces of stable n-pointed curves of genus 0. In particular, we determine an explicit presentation in the case n=5 and we outline a computational approach to the case n=6.

黎曼曲面与柄体模空间的最高权上同调

Top weight cohomology of moduli spaces of Riemann surfaces and handlebodies

本研究证明双曲曲面模空间$M_g$中含足够多短测地线的曲面轨迹是柄体映射类群的分类空间，由此推出$M_g$的最高权上同调可单射到柄体映射类群的上同调中。

Moduli of curves on toric varieties and their stable cohomology

We prove that the cohomology of the moduli space of morphisms of a fixed finite degree from a smooth projective curve $C$ of genus $g$ to a complete simplicial toric variety $\mathbb{P}(Σ)$, denoted by the rational polyhedral fan $Σ$, stabilizes. As an arithmetic consequence we obtain a resolution of the Batyrev-Manin conjecture for toric varieties over global function fields in all but finitely many characteristics.

On the rational cohomology of moduli spaces of curves with level structures

We investigate low degree rational cohomology groups of smooth compactifications of moduli spaces of curves with level structures. In particular, we determine $H^k(\sgbar, \Q)$ for $g \ge 2$ and $k \le 3$, where $\sgbar$ denotes the moduli space of spin curves of genus $g$.

结点代数曲线上主$\Bbb C^*$-丛的模栈的上同调

Cohomology of Moduli Stacks of Principal $\Bbb C^*$-Bundles Over Nodal Algebraic Curves

研究结点曲线上主$\Bbb C^*$-丛模栈，用陈类确定其有理上同调代数。

Equivariant geometry and the cohomology of the moduli space of curves

In this expository article we give a categorical definition of the integral cohomology ring of a stack. We show that for quotient stacks the categorical cohomology may be identified with equivariant cohomology. Via this identification we show that for Deligne-Mumford quotient stacks this cohomology is rationally isomorphic to the rational cohomology of the coarse moduli space. The theory is presented with a focus on the stacks of smooth and stable curves.

Generators for the cohomology ring of the moduli of one-dimensional sheaves on $\mathbb{P}^2$

We explore the structure of the cohomology ring of the moduli space of stable 1-dimensional sheaves on $\mathbb{P}^2$ of any degree. We obtain a minimal set of tautological generators, which implies an optimal generation result for both the cohomology and the Chow ring of the moduli space. Our approach is through a geometric study of tautological relations.

余子式与λ连接模的空间

The Determinant of Cohomology and Moduli of $λ$-Connections

本文研究了通过余子式余子式线丛的Atiyah藻酸盐对偶来理解稳定丛的Hodge-Deligne模空间。

Simplicial equations for the moduli space of stable rational curves

In this, largely expository, note, we show how the simplicial structure of the moduli spaces of stable rational curves with marked points allows to produce explicit equations for these spaces. The key argument is an elementary combinatorial statement about the sets of trees with marked leaves.

曲线上 $\PGL_2$-丛模空间的量子上同调

The quantum cohomology of moduli space of $\PGL_2$-bundles on curves

本文针对亏格不小于 2 的光滑曲线，计算其上稳定 $\PGL_2$-丛模空间的量子上同调。

Equivariant cohomology of moduli spaces of genus three curves with level two structure

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.

Hirzebruch 曲面上超椭圆曲线模空间的稳定上同调

Stable cohomology of moduli spaces of hyperelliptic curves on Hirzebruch surfaces

本文计算了 Hirzebruch 曲面上固定亏格超椭圆曲线模空间的稳定上同调，并用带标记点的非嵌入模空间进行描述。

Cohomology Rings of Moduli of Point Configurations on the Projective Line

We describe the Chow rings of moduli spaces of ordered configurations of points on the projective line for arbitrary (sufficiently generic) stabilities. As an application, we exhibit such a moduli space admitting two small desingularizations with non-isomorphic cohomology rings.

$\ell$-adic étale cohomology of the moduli of stable elliptic fibrations

We determine the $\ell$-adic étale cohomology and the eigenvalues of the geometric Frobenius for the moduli stack $\mathcal{L}_{1,12n} := \mathrm{Hom}_{n}(\mathbb{P}^1, \overline{\mathcal{M}}_{1,1})$ of stable elliptic fibrations over $\mathbb{P}^{1}$ with $12n$ nodal singular fibers and a marked Weierstrass section over $\overline{\mathbb{F}}_q$ with $\mathrm{char}(\overline{\mathbb{F}}_q) \neq 2,3$.

Rank one Eisenstein cohomology of local systems on the moduli space of abelian varieties

We give a formula for the Eisenstein cohomology of local systems on the partial compactification of the moduli of principally polarized abelian varieties given by rank 1 degenerations. For genus 2 we give a formula for the full Eisenstein cohomology.

Geometry of the intersection ring and vanishing relations in the cohomology of the moduli space of parabolic bundles on a curve

We study the ring generated by the Chern classes of tautological line bundles on the moduli space of parabolic bundles of arbitrary rank on a Riemann surface. We show the Poincaré duals to these Chern classes have simple geometric representatives. We use this construction to show that the ring generated by these Chern classes vanishes below the dimension of the moduli space, in analogy with the Newstead-Ramanan conjecture for stable bundles.

Equivariant Cohomology of Certain Moduli of Weighted Pointed Rational Curves

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.

Hodge structure on the cohomology of the moduli space of Higgs bundles

We study the moduli space of Higgs bundles on a compact Riemann surface. It was shown by Thaddeus and Hausel (in rank 2) and Markman (in general rank) that the rational cohomology ring of this space is generated by universal classes. In this paper, we use their results to compute the Hodge structure on the cohomology of this moduli space. We show that the natural Hodge structure on H^k is pure of weight k.

非正则抛物Higgs丛模空间上同调的生成元

Generators for the cohomology of the moduli space of irregular parabolic Higgs bundles

该研究证明非正则抛物Higgs丛模空间上同调环纯部分的生成元构成，并将类似结果推广到正则完全旗情形下的抛物与强抛物Higgs丛模空间。

The motivic structures $\mathsf{LS}_{12}$ and $\mathsf{S}_{16}$ in the cohomology of moduli spaces of curves

We study the appearances of $\mathsf{LS}_{12}$ and $\mathsf{S}_{16}$ in the weight-graded compactly supported cohomology of moduli spaces of curves. As applications, we prove new nonvanishing results for the middle cohomology groups of $\mathcal{M}_9$ and $\mathcal{M}_{11}$ and give evidence to support the conjecture that the dimension fo $H^{2g + k}_c(\mathcal{M}_g)$ grows at least exponentially with $g$ for almost all $k$.

Cohomology of the moduli stack of algebraic vector bundles

Let $\mathscr{V}\mathrm{ect}_n$ be the moduli stack of vector bundles of rank $n$ on schemes. We prove that, if $E$ is a Zariski sheaf of ring spectra which is equipped with finite quasi-smooth transfers and satisfies the projective bundle formula, then $E^*(\mathscr{V}\mathrm{ect}_{n,S})$ is freely generated by Chern classes $c_1,\dotsc,c_n$ over $E^*(S)$ for any scheme $S$. Examples include all multiplicative localizing invariants.

Cohomology of the moduli space of degree two Enriques surfaces

We compute the intersection Betti numbers of the GIT model of the moduli space of numerically polarized Enriques surfaces of degree 2. The strategy of the cohomological calculation relies on a general method developed by Kirwan to compute 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, called Kirwan blow-up.

On the Cohomology of Moduli of Vector Bundles

We compute some Hodge and Betti numbers of the moduli space of stable rank $r$ degree $d$ vector bundles on a smooth projective curve. We do not assume $r$ and $d$ are coprime. In the process we equip the cohomology of an arbitrary algebraic stack with a functorial mixed Hodge structure. This Hodge structure is computed in the case of the moduli stack of rank $r$, degree $d$ vector bundles on a curve. Our methods also yield a formula for the Poincare polynomial of the moduli stack that is valid over any ground field.

Fano箭图模空间的量子上同调

Quantum Cohomology of a Fano Quiver Moduli Space

本文计算了作为精细箭图模空间的指标为3的本原Fano 6-维簇$Y$的量子上同调环，推导出Schubert型子簇的量子Chevalley公式，并验证了Dubrovin猜想。

Stable cohomology of the moduli space of trigonal curves

We prove that the rational cohomology $H^i(\mathcal{T}_g;\mathbf{Q})$ of the moduli space of trigonal curves of genus $g$ is independent of $g$ in degree $i<\lfloor g/4\rfloor.$ This makes possible to define the stable cohomology ring as $H^\bullet(\mathcal{T}_g;\mathbf{Q})$ for a sufficiently large $g.$ We also compute the stable cohomology ring, which turns out to be isomorphic to the tautological ring. This is done by studying the embedding of trigonal curves in Hirzebruch surfaces and using Gorinov-Vassiliev's method.

Notions of purity and the cohomology of quiver moduli

We explore several variations of the notion of purity for the action of Frobenius on schemes defined over finite fields. In particular, we study how these notions are preserved under certain natural operations like quotients for principal bundles and also geometric quotients for reductive group actions. We then apply these results to study the cohomology of quiver moduli. We prove that a natural stratification of the space of representations of a quiver with a fixed dimension vector is equivariantly perfect and from it deduce that each of the $l$-adic cohomology groups of the quiver moduli spa…

On the cohomology of moduli spaces of (weighted) stable rational curves

We give a recursive algorithm for computing the character of the cohomology of the moduli space $\Mmb_{0,n}$ of stable $n$-pointed genus zero curves as a representation of the symmetric group $\Sg_n$ on $n$ letters. Using the algorithm we can show a formula for the maximum length of this character. Our main tool is connected to the moduli spaces of weighted stable curves introduced by Hassett.

Rational cohomology of the moduli spaces of pointed genus 1 curves

We give a combinatorial description of the rational cohomology of the moduli spaces of pointed genus 1 curves with $n$ marked points and level $N$ structures. More precisely, we explicitly describe the $E_2$ term of the Leray spectral sequence of the forgetful mapping $\EuScript{M}_{1,n}(N)\to\EuScript{M}_{1,1}(N)$ and show that the result is isomorphic to the rational cohomology of $\EuScript{M}_{1,n}(N)$ as a rational mixed Hodge structure equipped with an action of the symmetric group $\mathfrak{S}_n$. The classical moduli space $\EuScript{M}_{1,n}$ is the particular case N=1.

Cohomology jump loci in the moduli spaces of vector bundles

Two decades ago, as part of their work of generic vanishing theorems, Green-Lazarsfeld showed that over a compact Kahler manifold $X$, the cohomology jump loci in the $Pic^τ(X)$ are all translates of subtori. In this paper, we generalize this result to higher dimensional vector bundles. It is showed by Nadel that locally the moduli space of vector bundles with vanishing chern classes is canonically isomorphic to a quadratic cone in the Zariski tangent space of a point. We prove that under the isomorphism, the cohomology jump loci are defined by linear equations.

行列式平凡曲线上秩2 Higgs丛模空间交集上同调的爆破公式

A blowing-up formula for the intersection cohomology of the moduli of rank 2 Higgs bundles over a curve with trivial determinant

本文证明了行列式平凡曲线上秩2 Higgs丛模空间交集上同调的爆破公式，并据此在技术假设下推导出其庞加莱多项式。

Framed quiver moduli, cohomology, and quantum groups

Framed quiver moduli parametrize stable pairs consisting of a quiver representation and a map to a fixed graded vector space. Geometric properties and explicit realizations of framed quiver moduli for quivers without oriented cycles are derived, with emphasis on their cohomology. Their use for quantum group constructions is discussed.

Tautological and non-tautological cohomology of the moduli space of curves

After a short exposition of the basic properties of the tautological ring of the moduli space of genus g Deligne-Mumford stable curves with n markings, we explain three methods of detecting non-tautological classes in cohomology. The first is via curve counting over finite fields. The second is by obtaining length bounds on the action of the symmetric group S_n on tautological classes. The third is via classical boundary geometry. Several new non-tautological classes are found.

Jacobi cohomology, local geometry of moduli spaces, and Hitchin connections

The main aim of this paper is to develop general algebraic and cohomological tools for the study of the local geometry of moduli and parameter spaces in Algebraic Geometry, culminating in the so-called Hitchin (or KZ) (projective) connection over the Moduli of curves and the proof of its flatness. Among other things, we notably give a 'modular' (i.e. in terms of the modular problem) construction of the differential operators- of any order- acting on (general)modular vector bundles, and in particular for the Lie algebra of vector fields on a (general) Moduli space, with its natural action on fu…

Intersection cohomology of moduli spaces of sheaves on surfaces

We study intersection cohomology of moduli spaces of semistable vector bundles on a complex algebraic surface. Our main result relates intersection Poincare polynomials of the moduli spaces to Donaldson-Thomas invariants of the surface. In support of this result, we compute explicitly intersection Poincare polynomials for sheaves with rank two and three on ruled surfaces.

亏格0中多重尺度微分模空间的上同调

Cohomology of Moduli Space of Multiscale Differentials in Genus 0

本文证明了亏格0多重尺度微分模空间的有理上同调环由边界除子生成，利用Chow-Künneth性质与边界除子交集的不可约性，并在光滑情形下将结论推广至整上同调环。

The Atiyah-Bott formula for the cohomology of the moduli space of bundles on a curve

This paper is a companion of the paper "Weil's conjecture for function fields" by J. Lurie and the author. We present a different exposition of essentially the same algebro-geometric proof of the Atiyah-Bott for the cohomology of Bun(G), which subsequently leads to the proof of the Tamagawa number formula for the volume of the automorphic space for function fields.

On the Cohomology of Moduli Space of Parabolic Connections

We study the moduli space of logarithmic connections of rank $2$ on $\mathbb{P}^1 \setminus \{ t_1, \dots, t_5 \}$ with fixed spectral data. The aim of this paper is to compute the cohomology of such space, and this computation will be used to extend the results of Geometric Langlands Correspondence due to D. Arinkin to the case where this type of connections have five simple poles on $\mathbb{P}^1$.

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…

Cohomology of the moduli space of Hecke cycles

Let $X$ be a smooth projective curve of genus $g \ge 3$ and let $M_0$ be the moduli space of semistable bundles over $X$ of rank 2 with trivial determinant. Three different desingularizations of $M_0$ have been constructed by Seshadri \cite{Se1}, Narasimhan-Ramanan \cite{NR}, and Kirwan \cite{k5}. In this paper, we construct a birational morphism from Kirwan's desingularization to Narasimhan-Ramanan's, and prove that the Narasimhan-Ramanan's desingularization (called the moduli space of Hecke cycles) is the intermediate variety between Kirwan's and Seshadri's as was conjectured recently in \ci…

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.