The Lawson homology and Deligne-Beilinson cohomology for Fulton-MacPherson configuration spaces

The Chow ring of relative Fulton-MacPherson space

Suppose that X is a nonsingular variety and D is a nonsingular proper subvariety. Configuration spaces of distinct and non-distinct n points in X away from D were constructed by the author and B. Kim in arXiv:0806.3819, by using the method of wonderful compactification. In this paper, we give an explicit presentation of Chow motives and Chow rings of these configuration spaces.

On the biregular geometry of the Fulton-MacPherson compactification

Let $X[n]$ be the Fulton-MacPherson compactification of the configuration space of $n$ ordered points on a smooth projective variety $X$. We prove that if either $n\neq 2$ or $\dim(X)\geq 2$, then the connected component of the identity of $Aut(X[n])$ is isomorphic to the connected component of the identity of $Aut(X)$. When $X = C$ is a curve of genus $g(C)\neq 1$ we classify the dominant morphisms $C[n]\rightarrow C[r]$, and thanks to this we manage to compute the whole automorphism group of $C[n]$, namely $Aut(C[n])\cong S_n\times Aut(C)$ for any $n\neq 2$, while $Aut(C[2])\cong S_2\ltimes …

A Geometric Invariant Theory Compactification of M_{g,n} Via the Fulton-MacPherson Configuration Space

A compactification over $\overline{M}_g$ of $M_{g,n}$ is obtained by considering the relative Fulton-MacPherson configuration space of the universal curve. The resulting compactification differs from the Deligne-Mumford space $\overline{M}_{g,n}$. In case $n=2$, the compactification constructed here and the Deligne-Mumford compactification are essentially the distinct minimal resolutions of the fiber product over $\overline{M}_g$ of the universal curve with itself.

Balanced normal cones and Fulton-MacPherson's intersection theory

Let X be a subscheme of a reduced scheme Y. Then Y has a flat "degeneration to the normal cone" C_X Y of X, and this degeneration plays a key step in Fulton and MacPherson's "basic construction" in intersection theory. The intersection product has a canonical refinement as a sum over the components of C_X Y, for X and Y depending on the given intersection problem. The cone C_X Y is usually not reduced, which leads to the appearance of multiplicities in intersection formulae. We describe a variant of this degeneration, due essentially to Samuel, Rees, and Nagata, in which Y flatly degenerates t…

Polydiagonal compactification of configuration spaces

A smooth compactification X<n> of the configuration space of n distinct labeled points in a smooth algebraic variety X is constructed by a natural sequence of blowups, with the full symmetry of the permutation group S_n manifest at each stage of the construction. The strata of the normal crossing divisor at infinity are labeled by trees with levels and their structure is studied. This is the maximal wonderful compactification in the sense of DeConcini-Procesi, and it has a strata-compatible surjection onto the Fulton-MacPherson compactification. The degenerate configurations added in the compa…

The chow cohomology of affine toric varieties

We study the Fulton-Macpherson Chow cohomology of affine toric varieties. In particular, we prove that the Chow cohomology vanishes in positive degree. We prove an analogous result for the operational $K$-theory defined by Anderson and Payne.

Weighted compactifications of configuration spaces and relative stable degenerations

We study a compactification of the configuration space of n distinct labeled points on an arbitrary nonsingular variety. Our construction provides a generalization of the original Fulton-MacPherson compactification that is parallel to the generalization of the moduli space of n-pointed stable curves carried out by Hassett.

Residual Intersections and Some Applications

We give a new residual intersection decomposition for the refined intersection products of Fulton-MacPherson. Our formula refines the celebrated residual intersection formula of Fulton, Kleiman, Laksov, and MacPherson. The new decomposition is more likely to be compatible with the canonical decomposition of the intersection products and each term in the decomposition thus has simple geometric meaning. Our study is motivated by its applications to some geometric problems. In particular, we use the decomposition to find the distribution of limiting linear subspaces in degenerations of hypersurfa…

Motivic Milnor classes

The Milnor class is a generalization of the Milnor number, defined as the difference (up to sign) of Chern--Schwartz--MacPherson's class and Fulton--Johnson's canonical Chern class of a local complete intersection variety in a smooth variety. In this paper we introduce a "motivic" Grothendieck group $K^{\mathcal Prop}_{\ell.c.i}(\mathcal V/X \to S)$ and natural transformations from this Grothendieck group to the homology theory. We capture the Milnor class, more generally Hirzebruch--Milnor class, as a special value of a distinguished element under these natural transformations. We also show a…

Bivariant Chern classes and Grothendieck transformations

The existence of bivariant Chern classes was conjectured by W.Fulton and R.MacPherson and proved by J.P.Brasselet for cellular morphisms of analytic varieties. In this paper we show that restricted to morphisms whose target varieties are possible singular but (rational) homology manifolds, the bivariant Chern classes (with rational coefficients) are uniquely determined. In the final sections we construct a unique bivariant Chern class satisfying a suitable normalization condition. In fact, it will be a special case of a general construction of unique Grothendieck transformations, which in a se…

曲面希尔伯特概形的$\boldsymbol{\text{CohFT}}$的普适性

Universality of $\mathsf{CohFT}$ of $\mathsf{Hilb}_n(S)$

该研究确定曲面希尔伯特概形的上同调场论在极端曲线类中由曲面特征数与仿射平面希尔伯特概形理论普适决定，并据此算出所有曲面希尔伯特概形的亏格1极端Gromov–Witten不变量。

Motivic bivariant characteristic classes

The relative Grothendieck group $K_0(\m V/X)$ is the free abelian group generated by the isomorphism classes of complex algebraic varieties over $X$ modulo the "scissor relation". The motivic Hirzebruch class ${T_y}_*: K_0(\m V /X) \to H_*^{BM}(X) \otimes \bQ[y]$ is a unique natural transformation satisfying that for a nonsingular variety $X$ the value ${T_y}_*([X \xrightarrow {\op {id}_X} X])$ of the isomorphism class of the identity $X \xrightarrow {id_X} X$ is the Poincaré dual of the Hirzebruch cohomology class of the tangent bundle $TX$. It "unifies" the well-known three characteristic cl…

Equivariant characteristic classes of singular complex algebraic varieties

Homology Hirzebruch characteristic classes for singular varieties have been recently defined by Brasselet-Schuermann-Yokura as an attempt to unify previously known characteristic class theories for singular spaces (e.g., MacPherson-Chern classes, Baum-Fulton-MacPherson Todd classes, and Goresky-MacPherson L-classes, respectively). In this note we define equivariant analogues of these classes for singular quasi-projective varieties acted upon by a finite group of algebraic automorphisms, and show how these can be used to calculate the homology Hirzebruch classes of global quotient varieties. We…

Towards an Intersection Chow Cohomology Theory for GIT Quotients

We study the Fulton-Macpherson operational Chow rings of good moduli spaces of properly stable, smooth, Artin stacks. Such spaces are étale locally isomorphic to geometric invariant theory quotients of affine schemes, and are therefore natural extensions of GIT quotients. Our main result is that, with rational coefficients, every operational class can be represented by a so-called topologically strong cycle on the corresponding stack. Moreover, this cycle is unique modulo rational equivalence on the stack. Using out methods, we prove that if $X$ is the good moduli space of a properly stable, s…

Ordinarity of configuration spaces and of wonderful compactifications

We prove the following: (1) if $X$ is ordinary, the Fulton-MacPherson configuration space $X[n]$ is ordinary for all $n$; (2) the moduli of stable $n$-pointed curves of genus zero is ordinary. (3) More generally we show that a wonderful compactification $X_\sg$ is ordinary if and only if $(X,\sg)$ is an ordinary building set. This implies the ordinarity of many other well-known configuration spaces (under suitable assumptions).

Integral points and effective cones of moduli spaces of stable maps

Consider the Fulton-MacPherson configuration space of $n$ points on $¶^1$, which is isomorphic to a certain moduli space of stable maps to $¶^1$. We compute the cone of effective ${\mathfrak S}_n$-invariant divisors on this space. This yields a geometric interpretation of known asymptotic formulas for the number of integral points of bounded height on compactifications of $\SL_2$ in the space of binary forms of degree $n\ge 3$.

Mixed Hodge structures of configuration spaces

The symmetric group S_n acts freely on the configuration space of n distinct points in a quasi-projective variety. In this paper, we study the induced action of the symmetric group S_n on the de Rham cohomology of this space, using mixed Hodge theory, combined with methods from the theory of symmetric functions. (We prove a motivic version of this as well.) As an application of our results, we calculate the S_n-equivariant Hodge polynomial of the Fulton-MacPherson compactification X[n] of the configuration space.

Multiplicative models for configuration spaces of algebraic varieties

W. Fulton and R. MacPherson described a Sullivan dg-algebra model for the space of n-configurations of labeled points in a smooth compact complex algebraic variety X. I. Kriz then gave a simpler model that depends only on the cohomology ring of X. We construct an even simpler and substantially smaller model for this configuration space. Using this smaller model, we also define another, new dg-algebra model, for the space of n-configurations in the variety X with a point removed. Following an idea of V.G. Drinfel'd, we endow the collection of these new "punctured" models with a simplicial bigra…

Modular interpretation of a non-reductive Chow quotient

The space of n distinct points and a disjoint parameterized hyperplane in projective d-space up to projectivity---equivalently, configurations of n distinct points in affine d-space up to translation and homothety---has a beautiful compactification introduced by Chen-Gibney-Krashen. This variety, constructed inductively using the apparatus of Fulton-MacPherson configuration spaces, is a parameter space of certain pointed rational varieties whose dual intersection complex is a rooted tree. This generalizes $\overline{M}_{0,n}$ and shares many properties with it. In this paper, we prove that the…

带点曲线模空间的上同调

Cohomology of moduli spaces of pointed curves

本文延续相关文献研究带点曲线模空间的上同调，证明了任意光滑射影曲线$C$上$n$个有序不同点空间的Fulton-MacPherson紧化$C[n]$的贝蒂数分布随$n$趋于无穷呈渐近高斯分布。

Cech cover of the complement of the discriminant variety. Part II: Deformations of Gauss-skizze

The configuration space of $n$ marked points on the complex plane is considered. We investigate a decomposition of this space by so-called Gauss-skizze i.e. a class of graphs being forests, introduced by Gauss. It is proved that this decomposition is a semi-algebraic topological stratification. It also forms a cell decomposition of the configuration space of $n$ marked points. Moreover, we prove that classical tools from deformation theory, ruled by a Maurer--Cartan equation, can be used only locally for Gauss-skizze. We prove that the deformation of the Gauss-skizze is governed by a Hamilton-…

Hirzebruch classes and motivic Chern classes for singular spaces

In this paper we study some new theories of characteristic homology classes for singular complex algebraic (or compactifiable analytic) spaces. We introduce a motivic Chern class transformation mC_{*}: K_{0}(var/X)-> G_{0}(X)[y], which generalizes the total λ-class of the cotangent bundle to singular spaces. Here K_{0}(var/X) is the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration, and G_{0}(X) is the Grothendieck group of coherent sheaves of O_{X}-modules. We define a natural transformation …

Deformations of algebras over operads and Deligne's conjecture

In present paper we develop the deformation theory of operads and algebras over operads. Free resolutions (constructed via Boardman-Vogt approach) are used in order to describe formal moduli spaces of deformations. We apply the general theory to the proof of Deligne's conjecture. The latter says that the Hochschild complex of an associative algebra carries a canonical structure of a dg-algebra over the chain operad of the little discs operad. In the course of the proof we construct an operad of geometric nature which acts on the Hochschild complex. It seems to be different from the brace opera…

重新审视 $\mathbb{P}^1$ 上 8 点模空间

Revisiting the moduli space of 8 points on $\mathbb{P}^1$

本文证明 $\mathbb{P}^1$ 上 8 点模空间的 Deligne-Mostow 同构不能提升为 Kirwan 爆开与环面紧化间的态射，且两空间非 $K$-等价，并借助 Fulton-MacPherson 紧化揭示其与极小模型纲领的联系。

图配置空间、合同腺与形式性

Graphical configuration spaces, Contractads and Formality

本文研究图配置空间的合同腺结构，证明其在特定条件下形式性，并探讨了循环图的共形式性障碍及二维合同腺的细胞结构。

$\overline{\mathcal{M}}_{0,n}$ 的上同调上的表示

Representations on the cohomology of $\overline{\mathcal{M}}_{0,n}$

本文利用拟映射模空间的壁交叉给出对称群作用于 $\overline{\mathcal{M}}_{0,n}$ 上同调特征标的闭公式，并证明特定上同调群为置换表示，方法推广至 Fulton-MacPherson 紧化及稳定映射模空间。