Wei-Ping Li, Zhenbo Qin, Weiqiang Wang
In this note, generalizing earlier work of Nakajima and Vasserot, we study the (equivariant) cohomology rings of Hilbert schemes of certain toric surfaces and establish their connections to Fock space and Jack polynomials.
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The Jack symmetric polynomials $P_λ^{(α)}$ form a class of symmetric polynomials which are indexed by a partition $λ$ and depend rationally on a parameter $α$. They reduced to the Schur polynomials when $α=1$, and to other classical families of symmetric polynomials for several specific parameters. It is well-known that Schur polynomials can be realized as certain elements of homology groups of Grassmann manifolds. The purpose of this paper is to give a similar geometric realization for Jack polynomials. However, spaces which we use are totally different. Our spaces are Hilbert schemes of poin…
The methods of integral operators on the cohomology of Hilbert schemes of points on surfaces are developed. They are used to establish integral bases for the cohomology groups of Hilbert schemes of points on a class of surfaces (and conjecturally, for all simply connected surfaces).
We establish some remarkable properties of the cohomology rings of the Hilbert schemes of n points on a projective surface X, from which one sees to what extent these cohomology rings are (in)dependent of X and n.
Using vertex algebra techniques, we determine a set of generators for the cohomology ring of the Hilbert schemes of points on an arbitrary smooth projective surface over the field of complex numbers.
Using the methods developed in [LQW], math.AG/0009132, we obtain a second set of generators for the cohomology ring of the Hilbert scheme of points on an arbitrary smooth projective surface over the field of complex numbers. These generators have clear and simple geometric as well as algebraic descriptions.
We study the ideals of the rational cohomology ring of the Hilbert scheme X^{[n]} of n points on a smooth projective surface X. As an application, for a large class of smooth quasi-projective surfaces X, we show that every cup product structure constant of H^*(X^{[n]}) is independent of n; moreover, we obtain two sets of ring generators for the cohomology ring H^*(X^{[n]}). Similar results are established for the Chen-Ruan orbifold cohomology ring of the symmetric product. In particular, we prove a ring isomorphism between H^*(X^{[n]}, C) and H^*_{orb}(X^{[n]}/S_n, C) for a large class of smoo…
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 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.
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.
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.
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.
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。
We determine the ring structure of the equivariant quantum cohomology of the Hilbert scheme of points in the complex plane. The operator of quantum multiplication by the divisor class is a nonstationary deformation of the quantum Calogero-Sutherland many-body system. A relationship between the quantum cohomology of the Hilbert scheme and the Gromov-Witten/Donaldson-Thomas correspondence for local curves is proven.
In the study of the rational cohomology of Hilbert schemes of points on a smooth surface, it is particularly interesting to understand the characteristic classes of the tautological bundles and the tangent bundle. In this note we pursue this study. We first collect all results appearing separately in the literature and prove some new formulas using T. Ohmoto's results on orbifold Chern classes on Hilbert schemes. We also explain the algorithmic counterpart of the topic: The cohomology space is governed by a vertex algebra that can be used to compute characteristic classes. We present an implem…
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 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.
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…
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。
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.
We determine the two-point invariants of the equivariant quantum cohomology of the Hilbert scheme of points of surface resolutions associated to type A_n singularities. The operators encoding these invariants are expressed in terms of the action of the affine Lie algebra \hat{gl}(n+1) on its basic representation. Assuming a certain nondegeneracy conjecture, these operators determine the full structure of the quantum cohomology ring. A relationship is proven between the quantum cohomology and Gromov-Witten/Donaldson-Thomas theories of A_n x P^1. We close with a discussion of the monodromy prope…
Let Hilb^p be the Hilbert scheme parametrizing the closed subschemes of P^n with Hilbert polynomial p \in Q[t] over a field K of characteristic zero. By bounding below the cohomological Hilbert functions of the points of Hilb^p we define locally closed subspaces of the Hilbert scheme. The aim of this thesis is to show that some of these subspaces are connected. For this we exploit the ideals constructed by D. Mall. It turns out that these ideals are sequentially Cohen-Macaulay and that their initial ideals and their generic initial ideals coincide for any admissible term order.
Non-commutative Hilbert schemes, introduced by M. V. Nori, parametrize left ideals of finite codimension in free algebras. More generally, parameter spaces of finite codimensional submodules of free modules over free algebras are considered. Cell decompositions of these varieties are constructed, whose cells are parametrized by certain types of forests. Asymptotics for the corresponding Poincare polynomials and properties of their generating functions are discussed.
The aim of this paper is to extend some arithmetic results on elliptic modular forms to the case of Hilbert modular forms. Among these results let's mention : (1) the control of the image of the Galois representation modulo $p$, (2) Hida's congruence criterion outside an explicit set of primes $p$, and (3) the freeness of the integral cohomology of the Hilbert modular variety over certain local components of the Hecke algebra and the Gorenstein property of these local algebras. We study the arithmetic of the Hilbert modular forms by studying their modulo $p$ Galois representations and our main…
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 …
Let Hilb^p be the Hilbert scheme parametrizing the closed subschemes of P^n with Hilbert polynomial p\in Q[t] over a field K of characteristic zero. By bounding below the cohomological Hilbert functions of the points of Hilb^p we define locally closed subspaces of the Hilbert scheme. The aim of this paper is to show that some of these subspaces are connected. For this we exploit the edge ideals constructed by D. Mall. It turns out that these ideals are sequentially Cohen-Macaulay and that their initial ideals with respect to the reverse lexicographic term order are generic initial ideals.
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…
Let $S$ be the affine plane $\C^2$ together with an appropriate $\mathbb T = \C^*$ action. Let $\hil{m,m+1}$ be the incidence Hilbert scheme. Parallel to \cite{LQ}, we construct an infinite dimensional Lie algebra that acts on the direct sum $$\Wft = \bigoplus_{m=0}^{+\infty}H^{2(m+1)}_{\mathbb T}(S^{[m,m+1]})$$ of the middle-degree equivariant cohomology group of $\hil{m,m+1}$. The algebra is related to the loop algebra of an infinite dimensional Heisenberg algebra. In addition, we study the transformations among three different linear bases of $\Wft$. Our results are applied to the ring stru…
We calculate the cohomology spaces of the Hilbert schemes of points on surfaces with values in locally constant systems. For that purpose, we generalise I. Grojnoswki's and H. Nakajima's description of the ordinary cohomology in terms of a Fock space representation to the twisted case. We further generalise M. Lehn's work on the action of the Virasoro algebra to the twisted case. Building on work by M. Lehn and Ch. Sorger, we then give an explicit description of the cup-product in the twisted case whenever the surface has a numerically trivial canonical divisor. We formulate our results in a w…
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 the birational geometry of deformations of Hilbert schemes of points on the projective plane P2. We complete the picture in the paper of Daniele Arcara, Aaron Bertram, Izzet Coskun and Jack Huizenga by giving an explicit correspondence between the stable base locus walls on the Neron-Severi space and the actual walls on the Bridgeland stability space. We show that the birational geometry of a deformed Hilb P2 is different from that of Hilb P2.
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。
Given a closed complex manifold $X$ of even dimension, we develop a systematic (vertex) algebraic approach to study the rational orbifold cohomology rings $\orbsym$ of the symmetric products. We present constructions and establish results on the rings $\orbsym$ including two sets of ring generators, universality and stability, as well as connections with vertex operators and W algebras. These are independent of but parallel to the main results on the cohomology rings of the Hilbert schemes of points on surfaces as developed in our earlier works joint with W.-P. Li. We introduce a deformation o…
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.
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。
We describe the ring structure of the cohomology of the Hilbert scheme of points for a smooth surface X. When the canonical class K_X = 0, this was done by Lehn and Sorger, extending earlier work when X = C^2. Their approach does not generalize. Instead we recast this problem as a question of finding integrals of motion for a Hamiltonian which describes ``intersection with the boundary''. To do so, we use the identification of the cohomology of the Hilbert schemes with a Fock space modelled on the lattice H(X). With this identification, Lehn computed the operator of intersection with the bound…
We give an algorithmic description of the action of the Chern classes of tautological bundles on the cohomology of Hilbert schemes of points on surfaces within the framework of Nakajima's oscillator algebra. This leads to an identification of the cohomology ring of Hilbert schemes of the affine plane with a ring of differential operators on a Fock space. We end with the computation of the top Segre classes of tautological bundles associated to line bundles on Hilb^n up to n=7, and give a conjecture for the generating series.
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 …
Let X be the quasi-projective symplectic surface that is given by the total space of the invertible sheaf O(-2) over the projective line. Let Hilb X be the family of Hilbert schemes of points on X. We give and prove a closed formula expressing any multiplicative characteristic class evaluated on the Hilb X in terms of the standard Fock space description of the cohomology of the Hilb X. As a side result, we also deduce a formula for the Chern character of the tangent bundles of the Hilb X. The results found here are another step towards a complete description of the tangent bundle of the Hilber…
We characterize the class of ideals of a polynomial ring such that the hilbert series of their graded local cohomology modules is maximal.
The equivariant and ordinary cohomology rings of Hilbert schemes of points on the minimal resolution C^2//G for cyclic G are studied using vertex operator technique, and connections between these rings and the class algebras of wreath products are explicitly established. We further show that certain generating functions of equivariant intersection numbers on the Hilbert schemes and related moduli spaces of sheaves on C^2//G are tau functions of 2-Toda hierarchies.
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…
We give a series of successively weaker conjectures on the cohomology ring of the Grassmannian, starting with the Hilbert series of a certain natural filtration.
Given a projetive surface $S$, using correspondences, we construct an infinite dimensional Lie algebra that acts on the direct sum $\Wfock$ of the cohomology groups of the incidence Hilbert schemes $S^{[n,n+1]}$ over all $n$. The algebra is related to an extension of an infinite dimensional Heisenberg algebra. The space $\Wfock$ is a highest weight representation of this algebra. Our result provides a representation-theoretic interpretation of Cheah's generating function of Betti numbers of the incidence Hilbert schemes. As a consequence, an additive basis of the cohomology group of the incide…
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 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…
We compute some cohomology spaces for the symmetric power of the tautological bundle tensor the determinant bundle on the punctual Hilbert scheme of a complex smooth projective surface.
We express the product of the cohomology ring of the Hilbert scheme in terms of the center of the algebra of the symmetric group. We give a conjecture for the case of crepant resolutions of symplectic quotient singularities.
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。
Let $X$ a smooth quasi-projective algebraic surface, $L$ a line bundle on $X$. Let $X^{[n]}$ the Hilbert scheme of $n$ points on $X$ and $L^{[n]}$ the tautological bundle on $X^{[n]}$ naturally associated to the line bundle $L$ on $X$. We explicitely compute the image $\bkrh(L^{[n]})$ of the tautological bundle $L^{[n]}$ for the Bridgeland-King-Reid equivalence $\bkrh : \B{D}^b(X^{[n]}) \ra \B{D}^b_{\perm_n}(X^n)$ in terms of a complex $\comp{\mc{C}}_L$ of $\perm_n$-equivariant sheaves in $\B{D}^b_{\perm_n}(X^n)$. We give, moreover, a characterization of the image $\bkrh(L^{[n]} \tens ... \ten…
We give a new interpretation of Kozsul cohomology, which is equivalent under the Bridgeland-King-Reid equivalence to Voisin's Hilbert scheme interpretation in dimensions 1 and 2, but is different in higher dimensions. As an application, we prove that the dimension $K_{p,q}(B,L)$ is a polynomial in $d$ for $L=dA+P$ with $A$ ample and $d$ large enough.
In this paper we review various strikingly parallel algebraic structures behind Hilbert schemes of points on surfaces and certain finite groups called the wreath products. We explain connections among Hilbert schemes, wreath products, infinite-dimensional Lie algebras, and vertex algebras. As an application we further describe the cohomology ring structure of the Hilbert schemes. We organize this exposition around several general principles which have been supported by various works on these subjects.
We show that when $d \geq 3$, the Hilbert scheme $Hilb_{dT+1-\binom{d-1}{2}}(G(k,n))$ has 2 components, even though elements in both components have the same cohomology class. Moreover, we show that the Hilbert scheme associated to the Hilbert polynomial $\binom{T+n}{n}-\binom{T+n-d}{n}$ in Grassmannian has at most 2 connected components.
Lecture notes of an algebraic geometry graduate course. The topics covered are as follows. Cohomology: ext sheaves and groups, cohomology with support, local cohomology, local duality. Duality: relative duality, Cohen-Macaulay schemes. Morphisms of schemes: flatness and base change and related topics. Hilbert and Quotient schemes.
Consider a family of integral complex locally planar curves whose relative Hilbert scheme of points is smooth. The decomposition theorem of Beilinson, Bernstein, and Deligne asserts that the pushforward of the constant sheaf on the relative Hilbert scheme splits as a direct sum of shifted semisimple perverse sheaves. We will show that no summand is supported in positive codimension. It follows that the perverse filtration on the cohomology of the compactified Jacobian of an integral plane curve encodes the cohomology of all Hilbert schemes of points on the curve. Globally, it follows that a fa…
We prove the standard conjectures for complex projective varieties that are deformations of the Hilbert scheme of points on a K3 surface. The proof involves Verbitsky's theory of hyperholomorphic sheaves and a study of the cohomology algebra of Hilbert schemes of K3 surfaces.
To any graded Frobenius algebra A we associate a sequence of graded Frobenius algebras A^[n] in such a way that for any smooth projective surface X with trivial canonical divisor there is a canonical isomorphism of rings between (H*X)^[n] and the cohomology H*(X^[n]) of the n-th Hilbert scheme of X.
Cohomology rings of character varieties
本报告介绍了黎曼曲面特征簇上同调环的最新进展，包括 $P=W$ 猜想的证明、零维 COHA 的计算，以及上同调环与 $\mathbb{C}^2$ 的 Hilbert scheme 的联系与辛几何关系。
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