Frances Kirwan
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.
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This is a survey on various aspects of the cohomology of the moduli space of abelian varieties
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.
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.
We determine the cohomology of the Losev-Manin moduli space of pointed genus zero curves as a representation of the product of symmetric groups.
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.
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.
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.
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.
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 $g$.
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。
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.
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.
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。
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。
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.
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。
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.
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.
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。
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 $k$.
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.
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.
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。
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.
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…
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.
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.
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.
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.
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…
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。
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.
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 $\mathbb{P}^1$.
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…
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…
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.
Cohomology of vector bundles on the moduli space of parabolic connections on $\mathbb{P}^1$ minus $5$ points
本文研究了复射影直线去掉五个点上秩为二的抛物连接的模空间，并计算其上同调，推广了Arinkin关于几何兰格朗日对应的结果，同时给出了该模空间的紧化描述。
Full level-n structures on smooth, complex curves are trivializations of the n-torsion points of their Jacobians. We give an algebraic proof that the etale cohomology of the moduli space of smooth, complex curves of genus at least 2 with "infinite level structure" vanishes in degrees above 4g-5. This yields a new perspective on a result of Harer who showed such vanishing already at finite level via topological methods. We obtain similar results for moduli spaces of stable curves and curves of compact type which are not covered by Harer's methods. The key ingredients in the proof are a vanishin…
Weight 11 compactly supported cohomology of moduli spaces of curves
本文利用图复形研究曲线模空间 $M_{g,n}$ 紧支集上同调的权 11 部分，重点处理 $n=0$，并证明 $M_g$ 上同调在广泛次数中非消失且随 $g$ 指数增长。
In this mostly expository paper we review several known results about the cohomology of moduli spaces of smooth and stable curves, focusing in particular on low degree cohomology. We also give a new proof of Harer's theorem describing the second cohomology group of the moduli space of smooth n-pointed curves of given genus
On the rational cohomology of moduli spaces of Prym curves
研究扭曲普里姆曲线模空间的低次有理上同调群，通过证明其在特定奇数度消失及二度群为代数，涵盖了经典普里姆曲线模空间等，明确了该模空间低次有理上同调群的性质。
Relative étale slices and cohomology of moduli spaces
本文利用Alper-Hall-Rydh技术证明了光滑模栈间光滑映射在具有线性可还原稳定化点附近的局部结构定理，推导出良好模空间的纤维等奇异性和纤维的ℓ-上同调环与交上同调群同构性。
Let M be a moduli space of stable sheaves on a K3 or Abelian surface S. We express the class of the diagonal in the cartesian square of M in terms of the Chern classes of a universal sheaf. Consequently, we obtain generators of the cohomology ring of M. When S is a K3 and M is the Hilbert scheme of length n subschemes, this set of generators is sufficiently small in the sense that there aren't any relations among them in the stable cohomology ring. When S is the cotangent bundle of a Riemann surface, we recover the result of T. Hausel and M. Thaddeus: The cohomology ring of the moduli spaces o…
Weight 11 Compactly Supported Cohomology of Moduli Spaces of Curves in excess four
本文扩展了对过剩四图复形的上同调计算，详细探讨了生成元、计算过程及图复形的定义。
In this paper we study the second integral cohomology of moduli spaces of semistable sheaves on projective K3 surfaces. If $S$ is a projective K3 surface, $v$ a Mukai vector and $H$ a $v-$generic polarization on $S$, we show that $H^{2}(M_{v},\mathbb{Z})$ is a free $\mathbb{Z}-$module of rank 23 carrying a pure weight-two Hodge structure and a lattice structure, with respect to which $H^{2}(M_{v},\mathbb{Z})$ is Hodge isometric to the Hodge sublattice $v^{\perp}$ of the Mukai lattice of $S$. Similar results are proved for Abelian surfaces.
Chow rings, cohomology rings, and point counts of moduli spaces of curves
综述曲线模空间Chow环、上同调环和有限域点计数三个密切相关不变量的最新结果，探讨它们何时由tautological类生成。
Chow and cohomology rings of moduli stacks of plane quartics
本文研究了四次曲线模栈Hacking模栈的Chow环和上同调环，通过堆叠理论加权吹胀构造了一个光滑proper Deligne-Mumford堆栈，解决了KSBA和K-模空间之间的Calabi-Yau墙交叉问题，并计算了该模栈的Poincaré多项式，证明了循环类映射在有理系数下是同构的，并确定了其Chow环的生成元和关系。
Let $(X,\,D)$ be an $m$-pointed compact Riemann surface of genus at least $2$. For each $x \,\in\, D$, fix full flag and concentrated weight system $α$. Let $P \mathcal{M}_ξ$ denote the moduli space of semi-stable parabolic vector bundles of rank $r$ and determinant $ξ$ over $X$ with weight system $α$, where $r$ is a prime number and $ξ$ is a holomorphic line bundle over $X$ of degree $d$ which is not a multiple of $r$. We compute the Chen-Ruan cohomology of the orbifold for the action on $P \mathcal{M}_ξ$ of the group of $r$-torsion points in ${\rm Pic}^0(X)$.
The cohomology ring of the moduli space of stable holomorphic vector bundles of rank n and degree d over a Riemann surface of genus g>1 has a standard set of generators when n and d are coprime. When n=2 the relations between these generators are well understood, and in particular a conjecture of Mumford, that a certain set of relations is a complete set, is known to be true. In this article generalisations are given of Mumford's relations to the cases when n>2 and also when the bundles are parabolic bundles, and these are shown to form complete sets of relations.
Parabolic bundles and the intersection cohomology of moduli spaces of vector bundles on curves
本文提出纯几何框架，利用分解定理分析抛物模空间遗忘映射，几何证明递归公式，将0次模空间交点Betti数计算归约到1次情形。
Stability of representation cohomology sheaves on moduli spaces of bundles
本文证明了向量丛模空间上切层及更一般的表示上同调层的斜率稳定性，适用于亏格至少为2的曲线和连通约化群，并处理了奇异模空间情形。
Cohomology of the moduli space of cubic threefolds and its smooth models
本文计算并比较了三次三重超曲面模空间多种自然几何紧化的上同调，以Kirwan方法为起点，利用分解定理与球商模型边界研究上同调在双有理映射下的行为，并在附录以三次曲面为例说明。
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.
Intersection cohomology of moduli spaces of vector bundles over curves
本文通过将交叉上同调的Hodge-Deligne多项式与Donaldson-Thomas不变量关联，计算了曲线上半稳定向量丛模空间的交叉上同调，并推广至任意遗传范畴中半稳定对象的模空间。
Conical resolutions and cohomology of the moduli spaces of nodal hypersurfaces
本文利用锥形消解及其推广形式，并结合紧 GIT 商的显式描述，计算节点三次、四次平面曲线及三次曲面方程空间与模空间的有理上同调。
Cohomology classes on moduli of curves from Theta Characteristics
本文利用 theta 特征的 Galois–Stiefel–Whitney 类，证明全实域上曲线与主极化阿贝尔簇模栈在指定次数存在非平凡上同调类，并计算亏格 3 情形的 Brauer 群拉回。
Deligne's weight spectral sequence and tautological cohomology of the moduli space of curves
本文通过计算机程序实现Deligne的重量谱序列，计算标号光滑曲线模空间的有理上同调的重量分级，特别针对 genus 五和 genus 三的情况，研究了边界分层中上同调群由塔罗格类生成时的结构。
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.
Let X be a compact connected Riemann surface of genus at least two. We compute the Chen--Ruan cohomology ring of the moduli space of stable PSL(2, C)--bundles of nontrivial second Stiefel--Whitney class over X.
We compute the rational cohomology ring of \bar R_2, the (compactified) moduli space of Prym curves of genus 2. We also recompute the rational cohomology ring of \bar S_2, the moduli space of spin curves of genus 2, thereby correcting some errors made in an article by G. Bini and C. Fontanari, containing a computation of the same ring.
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