Symmetric groups and the cup product on the cohomology of Hilbert schemes
Manfred Lehn, Christoph Sorger
证明了仿射平面上n元组Hilbert格式的积分上同调环同构于对称群上积分类函数环的滤子所对应的分次环.
The integral cohomology ring of the Hilbert scheme of n-tuples on the affine plane is shown to be isomorphic to the graded ring associated to a filtration of the ring of integral class functions on the symmetric group.
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We study cup products in integral cohomology of the Hilbert scheme of $n$ points on a K3 surface and present a computer program for this purpose. In particular, we deal with the question, which classes can be represented by products of lower degrees.
We study the multiplicative structure of orbifold Hochschild cohomology in an attempt to generalize the results of Kontsevich and Calaque-Van den Bergh relating the Hochschild and polyvector field cohomology rings of a smooth variety. We introduce the concept of linearized derived scheme, and we argue that when $X$ is a smooth algebraic variety and $G$ is a finite abelian group acting on $X$, the derived fixed locus $\widetilde{X^G}$ admits an HKR linearization. This allows us to define a product on the cohomology of polyvector fields of the orbifold $[X/G]$. We analyze the obstructions to ass…
By using vector field techniques, we compute the ordinary and equivariant cohomology rings of Hilbert scheme of points in the projective plane in relation with that of a Grassmann variety.
The cup product of the Hilbert scheme for K3 surfaces
对于任何分次Frobenius代数A，我们将分次Frobenius代数序列A^[n]以这样一种方式联系起来：对于任何具有平凡典范除子的光滑射影曲面X，在（H*X）^[n]和X的n阶Hilbert概型的上同调H*（X^[n]）之间存在一个环的典型同构。
We study the Belkale-Kumar family of cup products on the cohomology of a generalized flag variety. We give an alternative construction of the family using relative Lie algebra cohomology, and in particular, identify the Belkale-Kumar cup product with a relative Lie algebra cohomology ring for every value of the parameter. As a consequence, we extend a fundamental disjointness result of Kostant to a family of Lie algebras. In an appendix, written jointly with Edward Richmond, we extend a Levi movability result of Belkale and Kumar to arbitrary parameters.
By using K-theory, we construct a map from the tangent space to the Hilbert scheme at a point Y to the local cohomology group. And we use this map to answer affirmatively(after slight modification) a question by Mark Green and Phillip Griffiths in [8] on constructing a map from the tangent space to the Hilbert scheme to the tangent space to the cycle group.
Hochschild cohomology of Hilbert schemes of points on surfaces
该研究计算了曲面上点的Hilbert概型的Hochschild上同调，发现其由Hochschild-Serre上同调决定，并通过计算对称商栈的对应上同调推广了前人相关形变理论结果。
Étale cohomology of the complement of a linear subspace arrangement
证明了线性子空间排列的补的l-adic上同调的上积公式。
We revisit Ellingsrud and Strø mme's cellular decomposition of the Hilbert scheme of points in the projective plane. We study the product of cohomology classes defined by the closures of cells, deriving necessary conditions for the non-vanishing of cohomology classes. Though our conditions are formulated in purely combinatorial terms, the machinery for deriving them includes techniques from Biał ynicki-Birula theory: We study closures of Biał ynicki-Birula cells in complete complex varieties equipped with ample line bundles. We prove a necessary condition for two such closures to meet, and app…
Generating series in the cohomology of Hilbert schemes of points on surfaces
在光滑曲面上点的Hilbert格式的有理上同调的研究中，重言丛和切丛的特征类是特别有趣的.在本说明中，我们继续进行这项研究。我们首先收集了文献中单独出现的所有结果，并利用T. Ohmoto关于Hilbert格式上Orbifold Chern类的结果。我们还解释了该主题的算法对应：上同调空间由可用于计算特征类的顶点代数控制。我们提出了一个实现的顶点操作的重写逻辑系统{ \sc Maude}和地址后得到的观察和符号计算。
We give a general formula relating self cup products in cohomology to connecting maps in nonabelian cohomology, and apply it to obtain a formula for the self cup product associated to the Weil pairing.
A Riemann-Hilbert correspondence for cohomology theories of closed 1-forms
本文基于Kontsevich-Soibelman的工作，建立了闭1形式的Deligne-Malgrange型黎曼-希尔伯特对应，并证明了复曲线上的简单代数1形式同构比较定理的变体。
Let $X$ be a Hilbert modular variety and $\mathbb{V}$ a non-trivial local system over $X$ with infinite monodromy. In this paper we study Saito's mixed Hodge structure (MHS) on the cohomology group $H^k(X,\mathbb{V})$ using the method of Higgs bundles. Among other results we prove the Eichler-Shimura isomorphism, give a dimension formula for the Hodge numbers and show that the mixed Hodge structure is split over $\mathbb{R}$. These results are analogous to Matsushima-Shimura [Annals of Mathematics 78, 1963] in the cocompact case and complement the results in Freitag [Book: Hilbert modular form…
We show that a natural isomorphism between the rational cohomology groups of the two zero-dimensional Hilbert schemes of $n$-points of two surfaces, the affine plane minus the axes and the cotangent bundle of an elliptic curve, exchanges the weight filtration on the first set of cohomology groups with the perverse Leray filtration associated with a natural fibration on the second set of cohomology groups. We discuss some associated hard Lefschetz phenomena.
Let $S$ be the affine plane regarded as a toric variety with an action of the 2-dimensional torus $T$. We study the equivariant Chow ring $A_{K}^*(Hilb^n(S))$ of the punctual Hilbert scheme $Hilb^n(S)$ with equivariant coefficients inverted. We compute base change formulas in $A_{K}^*(Hilb^n(S))$ between the natural bases introduced by Nakajima, Ellingsrud and Strømme, and the classical basis associated with the fixed points. We compute the equivariant commutation relations between creation/annihilation operators. We express the class of the small diagonal in $Hilb^n(S)$ in terms of the equiva…
We show that if X is a smooth complex projective surface with torsion-free cohomology, then the Hilbert scheme X^[n] has torsion-free cohomology for every natural number n. This extends earlier work by Markman on the case of Poisson surfaces. The proof uses Gholampour-Thomas's reduced obstruction theory for nested Hilbert schemes of surfaces.
Integral operators and integral cohomology classes of Hilbert schemes
给出了曲面上点的Hilbert格式的上同调的积分算子方法。它们被用来建立一类曲面上点的Hilbert概型的上同调群的整基（并且确切地说，是所有单连通曲面的积分基）。
Quantum cohomology of the Hilbert scheme of points in the plane
确定了复平面上点的Hilbert概型的等变量子上同调的环结构。量子除子类乘算符是量子Calogero-Sutherland多体系统的非定常变形。证明了Hilbert格式的量子上同调与局部曲线的Gromov-Witten/Donaldson-Thomas对应之间的关系。
Stability of the cohomology rings of Hilbert schemes of points on surfaces
建立了射影曲面X上n点Hilbert格式的上同调环的一些显著性质，由此可以看出这些上同调环在多大程度上依赖于X和n.
The cohomology rings of Hilbert schemes via Jack polynomials
本文推广了Nakajima和Vasserot的早期工作，研究了某些复曲面的Hilbert格式的（等变）上同调环，并建立了它们与Fock空间和Jack多项式的联系.
Vertex algebras and the cohomology ring structure of Hilbert schemes of points on surfaces
利用顶点代数技巧，确定了复数域上任意光滑射影曲面上点的Hilbert格式的上同调环的生成元集.
Hermitian matrices and cohomology of Kähler varieties
给出了上积映射$H^{1}(X,\mathbb{C})\otimes H^{1}(X,\mathbb{C}) \to H^{2}(X,\mathbb{C})$的核维数的上界，其中$X$是不含Albanese纤维化的紧Kähler簇.
The Hilbert scheme X^{[a]} of points on a complex manifold X is a compactification of the configuration space of a-element subsets of X. The integral cohomology of X^{[a]} is more subtle than the rational cohomology. In this paper, we compute the mod 2 cohomology of X^{[2]} for any complex manifold X, and the integral cohomology of X^{[2]} when X has torsion-free cohomology. The results of this paper are used in Voisin's work on the universal CH_0 group of cubic hypersurfaces, because the crucial point there is to study the 2-torsion in the Chow group.
We prove that Chow groups of certain non-commutative Hilbert schemes have a basis consisting of monomials in Chern classes of the universal bundle. Furthermore, we realize the cohomology of non-commutative Hilbert schemes as a module over the Cohomological Hall algebra.
We study a natural Hodge module on the Hilbert scheme of four points on affine three-space, which categorifies the Donaldson--Thomas invariant of the Hilbert scheme. We determine the weight filtration on the Hodge module explicitly in terms of intersection cohomology complexes, and compute the E-polynomial of its cohomology. The computations make essential use of a description of the singularity of the Hilbert scheme as the degeneracy locus of the Pfaffian function.
Hochschild Cohomology and Twisted Complexes on Complex Manifolds
本文利用扭分解和扭复形理论证明了Kontsevich关于系数在对角层的积空间的Ext群与系数在切层的外幂的上同调的典范同构中Yoneda积对应于上积的论断.
Generators for the cohomology ring of Hilbert schemes of points on surfaces
利用文献[LQW]，Math.AG/0009132中提出的方法，我们得到了复数域上任意光滑射影曲面上点的Hilbert格式的上同调环的第二组生成元.这些生成元具有清晰而简单的几何和代数描述。
We show that the cohomology ring of Hilbert scheme of $n$-points in the affine plane is isomorphic to the coordinate ring of $\mathbb{G}_{m}$-fixed point scheme of the $n$-th symmetric product of $\mathbb{C}^{2}$ for a natural $\mathbb{G}_{m}$-action on it. This result can be seen as an analogue of a theorem of DeConcini, Procesi and Tanisaki on a description of the cohomology ring of Springer fiber of type A.
Hilbert polynomials of the first and second secant varieties
本文通过描述光滑射影复簇上2或3点的Hilbert方案上充分正线丛的对称幂的上同调群，将其维数用第一和第二割簇的希尔伯特多项式表示，并完全计算了这些多项式。
Let $X=X_1\cup \cdots \cup X_s\subset \mathbb {P}^n$, $n\ge 4$, be a general union of smooth non-special curves with $X_i$ of degree $d_i$ and genus $g_i$ and $d_i\ge \max \{2g_i-1,g_i+n\}$ if $g_i>0$. We prove that $X$ has maximal rank, i.e. for any $t\in \mathbb {N}$ either $h^0(\mathcal{I}_X(t))=0$ or $h^1(\mathcal{I}_X(t))=0$.
The age grading and the Chen-Ruan cup product
我们证明了用于定义Chen-Ruan上同调中杯积的阻塞丛是由所谓的“年龄分级”或“移度数”决定的。实际上，可以使用年龄分级直接计算阻塞束。我们得到一个Kunneth定理陈阮上同调作为一个直接后果的基本性质的年龄分级，并解释如何其他几个结果-包括结合性的杯产品-可以证明以类似的方式。
Filtrations on equivariant quantum cohomology and Hilbert-Poincaré series
本文证明Floer理论在具C*作用的辛流形的等变量子上同调上诱导理想滤过，并由此在普通上同调上生成依赖Floer理论的Hilbert-Poincaré多项式，同时发展了三种等变Floer上同调滤过的结构性质，给出了Calabi-Yau与Fano设定下等变辛上同调的显式表示。
We first show the existence of a weight filtration on the equivariant cohomology of real algebraic varieties equipped with the action of a finite group, by applying group cohomology to the dual geometric filtration. We then prove the compatibility of the equivariant weight filtrations and spectral sequences with K{ü}nneth isomorphism, cup and cap products, from the filtered chain level. We finally induce the usual formulae for the equivariant cup and cap products from their analogs on the non-equivariant weight spectral sequences.
Let X be a hyperkahler manifold deformation equivalent to a Hilbert scheme of n points on a K3 surface. We compute the graded character formula of the generic Mumford-Tate group representation on the cohomology ring of X, and derive a generating series for deducing the number of canonical Hodge classes on X. The formula indicates the number of Hodge classes on X that remain Hodge under any deformation.
A theory of algebraic cocycles
我们引入的概念，代数上循环的代数模拟的地图的Eilenberg-MacLane空间。利用这些上循环，我们发展了一个复代数簇的“上同调理论”。该理论是双阶的，函的，并承认Gysin映射。它携带天然上积和与$L$同源的配对。在理论中定义了代数丛的陈类。有一个自然变换到（奇异）积分上同调理论，保持上积。对特殊情况进行了计算。在光滑簇上证明了每个代数有理$(p,p)$ -上同调类中存在代数上圈.
Chern Classes of Tautological Sheaves on Hilbert Schemes
我们给出了一个算法描述的行动的陈类重言丛上的Hilbert计划的上同调的点表面内的Nakajima的振子代数的框架。这导致识别的上同调环的希尔伯特计划的仿射平面与一个环的微分算子的Fock空间。最后计算了Hilb^n上直到n=7的线丛的重言式丛的顶Segre类，并给出了生成级数的一个猜想。
We generalize Macdonald's formula for the cohomology of Hilbert schemes of points on a curve from smooth curves to curves with planar singularities: we relate the cohomology of the Hilbert schemes to the cohomology of the compactified Jacobian of the curve. The new formula is a consequence of a stronger identity between certain perverse sheaves defined by a family of curves satisfying mild conditions. The proof makes an essential use of Ngo's support theorem for compactified Jacobians and generalizes this theorem to the relative Hilbert scheme of such families. As a consequence, we generalize …
Quantum cohomology of the Hilbert scheme of points on A_n-resolutions
本文确定了与A_n型奇点有关的表面分解点的Hilbert方案的等变量子上同调的两点不变量。编码这些不变量的算子表示为仿射李代数\hat{gl}（n+1）在其基本表示上的作用。假设一定的非简并性猜想，这些算子决定了量子上同调环的完整结构。证明了量子上同调与A_n x P^1的Gromov-Witten/Donaldson-Thomas理论之间的关系。最后，我们讨论了相关量子微分方程的单值性质，并推广到D型和E型奇点。
Connectedness of Hilbert scheme strata defined by bounding cohomology
设Hilbert ^p是特征为零的域K上的Hilbert多项式p \in Q[t]参数化P^n的闭子概形的Hilbert格式。通过在Hilbert ^p的点的上同调Hilbert函数之下有界，我们定义了Hilbert概型的局部闭子空间。这篇论文的目的是证明这些子空间中的一些是连通的。为此，我们利用D.商场证明了这些理想是序列Cohen-Macaulay理想，并且它们的初始理想和它们的一般初始理想对于任何容许项序都是一致的。
Cohomology of non-commutative Hilbert schemes
非交换Hilbert格式是M. V.Nori，参数化自由代数中有限余维左理想。更一般地，考虑了自由代数上自由模的有限余维子模的参数空间。这些品种的胞腔剖分，其细胞被参数化的某些类型的森林。讨论了相应Poincare多项式的渐近性及其生成函数的性质。
Galois representations modulo $p$ and cohomology of Hilbert modular varieties
本文的目的是将椭圆模形式的一些算术结果推广到Hilbert模形式的情形。在这些结果中，让我们提到：（1）控制的形象伽罗瓦表示模$p$，（2）Hida的同余准则以外的一组明确的素数$p$，和（3）自由的积分上同调的希尔伯特模簇的某些地方组成部分的Hecke代数和Gorenstein性质，这些地方的代数。本文通过研究Hilbert模形式的模$p$伽罗瓦表示来研究Hilbert模形式的算术，主要工具是惯性群在$p$上的素数作用.为了确定这个动作，我们计算Hodge-Tate（resp. Fontaine-Laffaille）权重的$p$ -adic（分别Hilbert模簇的模$p$）标上同调。本文的上同调部分受到Mokrane、Polo和Tilouine关于Siegel模簇上同调的工作的启发，并建立在数学NT/0212071和数学NT/0212072的尺规作图法的基础上。
Let $F$ be a totally real field in which $p$ is unramified. We prove that, if a cuspidal overconvergent Hilbert cuspidal form has small slopes under $U_p$-operators, then it is classical. Our method follows the original cohomological approach of Coleman. The key ingredient of the proof is giving an explicit description of the Goren-Oort stratification of the special fiber of the Hilbert modular variety. A byproduct of the proof is to show that, at least when $p$ is inert, of the rigid cohomology of the ordinary locus has the same image as the classical forms in the Grothendieck group of Hecke …
Hilbert scheme strata defined by bounding cohomology
设Hilbert ^p是特征为零的域K上的Hilbert多项式p \in Q[t]参数化P^n的闭子概形的Hilbert格式。通过在Hilbert ^p的点的上同调Hilbert函数之下有界，我们定义了Hilbert概型的局部闭子空间。本文的目的是表明，这些子空间的一些连接。为此，我们利用D.商场证明了这些理想是序列Cohen-Macaulay理想，并且它们关于逆字典序项的初始理想是一般初始理想。
Ideals of the cohomology rings of Hilbert schemes and their applications
研究光滑射影曲面X上n点Hilbert概型X^{[n]}的有理上同调环的理想.作为应用，对一大类光滑拟投射曲面X，证明了H^*（X^{[n]}）的每个上积结构常数与n无关，并得到了上同调环H^*（X^{[n]}）的两组环生成元.对对称积的Chen-Ruan轨道重上同调环也得到了类似的结果。特别地，对于一类具有数值平凡标准类的光滑拟投射曲面，证明了H^*（X^{[n]}，C）与H^* orb}（X^{[n]}/S_n，C）之间的环同构.
Equivariant cohomology of incidence Hilbert schemes and loop algebras
令$S$为仿射平面$\C^2$以及适当的$\mathbb T = \C^*$动作。设$\hil{m,m+1}$为关联Hilbert格式。平行于\cite{LQ}，我们构造了一个作用于$\hil{m,m+1}$的中次等变上同调群的直和$$\Wft = \bigoplus_{m=0}^{+\infty}H^{2(m+1)}_{\mathbb T}(S^{[m,m+1]})$$的无限维李代数.该代数与无限维海森堡代数的圈代数有关。此外，我们还研究了$\Wft$的三种不同线性基之间的变换。我们的结果适用于环结构的普通上同调的$\hil{m,m+1}$和环的对称函数在无穷多个变量。
Twisted cohomology of the Hilbert schemes of points on surfaces
我们计算了局部常数系统中曲面上点的Hilbert格式的上同调空间。为此，我们概括了I。Grojnoswki和H. Nakajima的描述的普通上同调在福克空间表示扭曲的情况下。我们进一步推广M。Lehn关于Virasoro代数对扭曲情形的作用的工作。基于M的工作。Lehn和Ch. Sorger的方法，我们给出了扭曲情况下的杯积的一个明确的描述，只要曲面有一个数值平凡的典范除子。我们制定我们的结果的方式，他们适用于在同等程度上的投射和非投射的情况下。作为我们方法的一个应用，我们给出了广义Kummer簇的上同调环和一系列偶数维Calabi-Yau流形的显式模型.
Total Cohomology of Hilbert schemes of Orbisurfaces I: Reduction to Stacky Loci
本文将Nakajima的曲面希尔伯特概型理论推广到轨曲面，通过稳定子分层因式分解，研究栈位点局部的Quot概型，获全上同调生成级数及Borel-Moore同调作用，为相关研究提供基础。
Let C be a proper, integral, locally planar curve, and consider its Hilbert schemes of points C^[n]. We define 4 creation/annihilation operators acting on the rational homology groups of these Hilbert schemes and show that the operators satisfy the relations of a Weyl algebra. The action of this algebra is similar to that defined by Grojnowski and Nakajima for a smooth surface. As a corollary, we compute the cohomology of C^[n] in terms of the cohomology of the compactified Jacobian of C together with an auxiliary grading on the latter. This recovers and slightly strenghtens a formula recently…
We study the cup product on the Hochschild cohomology of the stack quotient [X/G] of a smooth quasi-projective variety X by a finite group G. More specifically, we construct a G-equivariant sheaf of graded algebras on X whose G-invariant global sections recover the associated graded algebra of the Hochschild cohomology of [X/G], under a natural filtration. This sheaf is an algebra over the polyvector fields T^{poly}_X on X, and is generated as a T^{poly}_X-algebra by the sum of the determinants det(N_{X^g}) of the normal bundles of the fixed loci in X. We employ our understanding of Hochschild…
Hilbert schemes on blowing ups and the free Boson
本文通过中岛-吉冈构造，将曲面上爆破的秩1反常凝聚层模空间及希尔伯特概型的上同调分别等同于原曲面希尔伯特概型上同调与费米子/玻色子福克空间的张量积，并给出无限维代数作用的几何对应。
We study a topological method to calculate the first Milnor fiber cohomology of a defining polynomial of a reduced projective hyperplane arrangement $X$ of degree $d$. We can show the vanishing of a monodromy eigenspace of the first Milnor fiber cohomology with eigenvalue of order $m\ge 2$ if $X\setminus(X^{[(m)]}\cup X^{\langle 3\rangle})$ or more generally $X\setminus(X^{[(m)]}\cup X^{\langle 3\rangle}\cup X_d)$ is connected. Here $X^{[(m)]}$ is the set of points of $X$ with multiplicity divisible by $m$, and $X^{\langle 3\rangle}:=\bigcup_{i,j,k}X_i\cap X_j\cap X_k$ with $X_i$ the irreducib…
Ideals with maximal local cohomology modules
刻画了多项式环的理想类，使得其分次局部上同调模的Hilbert级数是极大的。
Equivariant cohomology, symmetric functions and the Hilbert schemes of points on the total space of the invertible sheaf O(-2) over the projective line
设X是由射影直线上的可逆层O（-2）的全空间给出的拟射影辛曲面。设Hilbert X是X上点的Hilbert概型族。本文给出并证明了一个闭公式，它表示Hilb X上的任意乘性示性类，并给出了Hilb X上同调的标准Fock空间描述.作为一个附带结果，我们还导出了Hilb X的切丛的Chern特征标的一个公式.这里发现的结果是另一个步骤走向一个完整的描述的切丛的希尔伯特计划的一般拟射影曲面的公式在这里产生某些系数的描述。最后，我们还给出了一个表示在Hilb X上重言丛上赋值的任何乘性示性类的闭公式。
Conjectures on the cohomology of the Grassmannian
本文从某个自然滤子的Hilbert级数出发，给出了Grassmannian的上同调环上的一系列逐次较弱的结构.
Incidence Hilbert schemes and infinite dimensional Lie algebras
给定一个投影曲面$S$，利用对应关系，构造了一个作用于所有$n$上的关联Hilbert格式$S^{[n,n+1]}$的上同调群的直和$\Wfock$的无限维李代数.该代数与无限维海森堡代数的扩展有关。空间$\Wfock$是这个代数的最高权表示。我们的结果提供了一个表示理论解释的Cheah的生成函数的贝蒂数的关联希尔伯特计划。由此得到了关联Hilbert格式的上同调群的一个可加基.
We show that the periods of the holomorphic Eisenstein series of level 1, parallel weight 2, on a Hilbert modular surface are not rational, even up to scaling. This is deduced from a study of the mixed Hodge structure on the cohomology of the Hilbert modular surface, where we find extension classes related to the units of the real quadratic field. We prove similar results for Hilbert modular varieties of all dimensions, and produce extensions of Galois representations in étale cohomology. The key point is to study the restriction of the canonical extension of the Hodge bundle to the boundary o…
Given a family of based CW-pairs $(\underline{X},\underline{A})=\{(X;A)\}^m_{i=1}$ together with an abstract simplicial complex $K$ with $m$ vertices, there is an associated based CW-complex $Z(K;(\underline{X},\underline{A}))$ known as a generalized moment-angle complex. The decomposition theorem of \cite{bbcg}, \cite{bbcg2} splits the suspension of $Z(K; (\underline{X}, \underline{A}))$ into a bouquet of spaces determined by the full sub-complexes of $K$. Thatdecomposition theorem is used here to describe the ring structure for the cohomology of Z(K; (\underline{X}, \underline{A})). Explicit…
Quantum cohomology of the Hilbert scheme of points on an elliptic surface
本文确定了椭圆曲面上点的 Hilbert 方案中收缩曲线类的除数量子乘法，用 Fock 空间算子表达，并推广至等变情形，证明基于墙交叉与 GW/PT 对应。
Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles
设$X$为光滑拟射影代数曲面，$L$为$X$上的线丛.设$X^{[n]}$为$X$上$n$点的Hilbert概型，$L^{[n]}$为自然地与$X$上的线丛$L$相关联的$X^{[n]}$上的重言式丛。我们根据$\B{D}^b_{\perm_n}(X^n)$中$\perm_n$ -等变层的复$\comp{\mc{C}}_L$，显式地计算了Bridgeland-King-Reid等价$\bkrh : \B{D}^b(X^{[n]}) \ra \B{D}^b_{\perm_n}(X^n)$的重言丛$L^{[n]}$的象$\bkrh(L^{[n]})$。此外，我们给出了一个表征的图像$\bkrh(L^{[n]} \tens ... \tens L^{[n]})$的超衍生光谱序列$E^{p,q}_1$相关的衍生$k$倍张量功率的复杂$\comp{\mc{C}}_L$。研究这个谱序列的$\perm_n$ -不变量，可以得到希尔伯特-周态射重言丛的双张量幂和广义$k$ -折叠外幂的直接图像，在这两种情况下提供Danila-Brion型公式。这就很容易计算出$X^{[n]}$的上同调，其值在$L^{[n]} \tens L^{[n]}$和$Λ^k L^{[n]}$中。
Hilbert schemes and symmetric products: a dictionary
给定偶数维闭复流形$X$，我们发展了一种系统的（顶点）代数方法来研究对称积的有理orbifold上同调环$\orbsym$。我们目前的建设和建立环$\orbsym$，包括两套环生成元，普遍性和稳定性，以及与顶点运营商和W代数的连接结果。这些结果独立于但平行于我们与W. P. Li.我们介绍了变形的orbifold的上积，并解释它是如何反映在修改的顶点算子的对称产品的情况下。作为推论，我们得到了一个新的证明之间的同构的有理上同调环的Hilbert计划和环$\orbsym$（经过一些修改的符号），当X是一个射影曲面的数值平凡的标准类;我们表明，没有符号修改是必要的，如果两个上同调环使用复系数。
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