Dan Edidin, Ryan Richey
We show that a cone theorem for ${\mathbbA}^1-homotopy invariant contravariant functors implies the vanishing of the positive degree part of the operational Chow cohomology rings of a large class of affine varieties. We also discuss how this vanishing relates to a number of questions about representing Chow cohomology classes of GIT quotients in terms of equivariant cycles.
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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.
We show that the equivariant Chow cohomology ring of a toric variety is naturally isomorphic to the ring of integral piecewise polynomial functions on the associated fan. This gives a large class of singular spaces for which localization holds in equivariant Chow cohomology with integer coefficients. We also compute the equivariant Chow cohomology of toric prevarieties and general complex hypertoric varieties in terms of piecewise polynomial functions.
We compare the cohomology ring of the flag variety $FL_n$ and the Chow cohomology ring of the Gelfand-Zetlin toric variety $X_{GZ}$. We show that $H^*(FL_n, \mathbb{Q})$ is the Gorenstein quotient of the subalgebra $L$ of $A^*(X_{GZ}, \mathbb{Q})$ generated by degree $1$ elements. We compute these algebras for $n=3$ to see that, in general, the subalgebra $L$ does not have Poincare duality.
Let $f: X \rightarrow A$ be an abelian cover from a complex algebraic variety with quotient singularities to an abelian variety. We show that $f^*$ induces an isomorphism between the rational cohomology rings $H^\bullet(A, \mathbb{Q})$ and $H^\bullet(X, \mathbb{Q})$ if and only if $f^*$ induces an isomorphism between the Chow rings with rational coefficients $\mathrm{CH}^\bullet(A)_{\mathbb{Q}}$ and $\mathrm{CH}^\bullet(X)_{\mathbb{Q}}$.
We show that the pseudoeffective cone of $k$-cycles on a complete complexity one $T$-variety is rational polyhedral for any $k$, generated by classes of $T$-invariant subvarieties. When $X$ is also rational, we give a presentation of the Chow groups of $X$ in terms of generators and relations, coming from the combinatorial data defining $X$ as a $T$-variety.
Auel-Bigazzi-Böhning-Graf von Bothmer proved that if a proper smooth variety $X$ over a field $k$ of characteristic $p>0$ has universally trivial Chow group of $0$-cycles, the cohomological Brauer group of $X$ is universally trivial as well. In this paper, we generalize their argument to arbitrary unramified mod $p$ étale motivic cohomology groups. We also see that the properness assumption on the variety $X$ can be dropped off by using the Suslin homology together with a certain tame subgroup of the unramified cohomology group.
Higher Chow groups with finite coefficients and refined unramified cohomology
本文证明域上拟射影等维概形的Bloch有限系数高阶圈类映射自然嵌入涉及Schreieder精细未分歧上同调的长正合序列，证明后者满足局部化序列，并猜想其为动机上同调理论。
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。
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…
Chow theory of toric variety bundles
本文研究了托尔奇簇包的Chow理论，通过扇形位移规则和等价乘法性，建立了普通和等价上同调与Minkowski权重及分段多项式函数的对应关系，并应用于对数枚举几何中的平衡条件。
For a toric variety X_P determined by a rational polyhedral fan P in a lattice N, Payne shows that the equivariant Chow cohomology of X_P is the Sym(N)--algebra C^0(P) of integral piecewise polynomial functions on P. We use the Cartan-Eilenberg spectral sequence to analyze the associated reflexive sheaf on Proj(N), showing that the Chern classes depend on subtle geometry of P and giving criteria for the splitting of the sheaf as a sum of line bundles. For certain fans associated to the reflection arrangement A_n, we describe a connection between C^0(P) and logarithmic vector fields tangent to …
The Bloch-Beilinson-Murre conjectures predict the existence of a descending filtration on Chow groups of smooth projective varieties which is functorial with respect to the action of correspondences and whose graded parts depend solely on the topology -- i.e. the cohomology -- of $X$. In this paper, we wish to explore, at the cost of having to assume general conjectures about algebraic cycles, how the coniveau filtration on the cohomology of $X$ has an incidence on the Chow groups of $X$. However, by keeping such assumptions minimal, we are able to prove some of these conjectures either in low…
On the Chow and cohomology rings of moduli spaces of stable curves
本文研究稳定曲线模空间 $\overline{\mathcal{M}}_{g,n}$ 的有理 Chow 环与上同调环何时由赘言类生成，并证明若干低亏格、低标记点情形下二者同构且为赘言环。
We study $T$-linear schemes, a class of objects that includes spherical and Schubert varieties. We provide a localization theorem for the equivariant Chow cohomology of these schemes that does not depend on resolution of singularities. Furthermore, we give an explicit presentation of the equivariant Chow cohomology of possibly singular complete spherical varieties admitting a smooth equivariant envelope (e.g. group embeddings).
We introduce extended toric Deligne-Mumford stacks. We use an extended toric Deligne-Mumford stack to get the toric stack bundle and compute its orbifold Chow ring. Finally we generalize one result of Borisov, Chen and Smith so that the orbifold Chow ring of the toric stack bundle and the Chow ring of its crepant resolution are fibres of a flat family.
We introduce the category of finite Chow-Witt correspondences over a perfect field k of characteristic not 2. We then use them to define bigraded generalized motivic cohomology groups of a smooth scheme over k and begin the study of their relationship with ordinary motivic cohomology groups.
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。
We explicitly describe cycle-class maps c_H from motivic cohomology to absolute Hodge cohomology, for smooth quasi-projective and (some) proper singular varieties, and compute special cases of the latter. For smooth projective varieties, we also study Hodge-theoretically defined "higher Abel-Jacobi maps" on the kernel of c_H; this leads to new results on nontrivial indecomposable higher Chow cycles in the regulator kernel.
We express the kernel of Griffiths' Abel-Jacobi map by using the inductive limit of Deligne cohomology in the generalized sense (i.e. the absolute Hodge cohomology of A. Beilinson). This generalizes a result of L. Barbieri-Viale and V. Srinivas in the surface case. We then show that the Abel-Jacobi map for codimension 2 cycles and the Albanese map are bijective if a general hyperplane section is a surface for which Bloch's conjecture is proved. In certain cases we verify Nori's conjecture on the Griffiths group. We also prove a weak Lefschetz-type theorem for (higher) Chow groups, generalize a…
In this paper, the Lawson homology and morphic cohomology are defined on the Chow motives. We also define the rational coefficient Lawson homology and morphic cohomology of the Chow motives of finite quotient projective varieties. As a consequence, we obtain a formula for the Hilbert scheme of points on a smooth complex projective surface. Further discussion concerning generic finite maps is given. As a result, we give examples of self-product of smooth projective curves with nontrivial Griffiths groups by using a result of Ceresa.
We investigate the integral cohomology ring and the Chow ring of the classifying space of the complex projective linear group PGL_p, when p is an odd prime. In particular, we determine its additive structure completely, and we reduce the problem of determing its multiplicative structure to a problem in invariant theory.
We study Grothedieck groups of triangulated categories using weight structures, weight complexes, and the corresponding pure (co)homological functors. We prove some general statements on $K_0$ of weighted categories and apply it to Voevodsky motives endowed with so-called Chow weight structures. We obtain certain "motivic substitutes" for smooth compactifications of smooth varieties over arbitrary perfect fields; this enables us to make certain unramified cohomology and Euler characteristic calculations that are closely related to results of T. Ekedahl and B. Kahn.
We determine a subring of the Chow ring and the cohomology of $BPGL_n$, the classifying space of the projective linear group of degree $n$ over complex numbers, and explain a way in which this computation might play a role in the period-index problem. In addition, we show that the Chow ring of $BPGL_n$ is not generated by the Chern classes of linear representations of $PGL_n$.
We determine the Chow ring (with Q-coefficients) of M_6 by showing that all Chow classes are tautological. In particular, all algebraic cohomology is tautological, and the natural map from Chow to cohomology is injective. To demonstrate the utility of these methods, we also give quick derivations of the Chow groups of moduli spaces of curves of lower genus. The genus 6 case relies on the particularly beautiful Brill-Noether theory in this case, and in particular on a rank 5 vector bundle "relativizing" a baby case of a celebrated construction of Mukai, which we interpret as a subbundle of the …
Construction of logarithmic cohomology theories II: On Chow groups
本文通过展示 toric 变形的 Chow 群技术结果，为第一部分提供关键支持。
In this paper, we give a Prym-Tjurin construction for the cohomology and the Chow groups for a cubic hypersurface. On the space of lines meeting a given rational curve, there is the incidence correspondence. This correspondence induces an action on both of the primitive cohomology and the primitive Chow groups. We first show that this action satisfies a quadratic equation. Then the Abel-Jacobi homomorphism induces an isomorphism between the primitive cohomology of the cubic hypersurface and the Prym-Tjurin part of the above action. This also holds for Chow groups with rational coefficients.
We compute the Chow-Witt rings of split quadrics over a field of characteristic not two. We even determine the full bigraded I-cohomology and Milnor-Witt cohomology rings, including twists by line bundles. The results on I-cohomology corroborate the general philosophy that I-cohomology is an algebro-geometric version of singular cohomology of real varieties: our explicit calculations confirm that the I-cohomology ring of a split quadric over the reals is isomorphic to the singular cohomology ring of the space of its real points.
Let $GO(2n)$ be the general orthogonal group scheme (the group of orthogonal similitudes). In the topological category, Y. Holla and N. Nitsure determined the singular cohomology ring $H^*_{\rm sing}(BGO(2n,\mathbb C),\mathbb F_2)$ of the classifying space $BGO(2n,\mathbb C)$ of the corresponding complex Lie group $GO(2n,\mathbb C)$ in terms of explicit generators and relations. The author of the present note showed that over any algebraically closed field of characteristic not equal to $2$, the smooth-étale cohomology ring $H_{\rm sm-ét}^*(BGO(2n),\mathbb F_2)$ of the classifying algebraic st…
We revisit the classical two-dimensional McKay correspondence in two respects: The first one, which is the main point of this work, is that we take into account of the multiplicative structure given by the orbifold product; second, instead of using cohomology, we deal with the Chow motives. More precisely, we prove that for any smooth proper two-dimensional orbifold with projective coarse moduli space, there is an isomorphism of algebra objects, in the category of complex Chow motives, between the motive of the minimal resolution and the orbifold motive. In particular, the complex Chow ring (r…
In the integral cohomology ring of the classifying space of the projective linear group $PGL_n$ (over $\mathbb{C}$), we find a collection of $p$-torsions $y_{p,k}$ of degree $2(p^{k+1}+1)$ for any odd prime divisor $p$ of $n$, and $k\geq 0$. If in addition, $p^2\nmid n$, there are $p$-torsion classes $ρ_{p,k}$ of degree $p^{k+1}+1$ in the Chow ring of the classifying stack of $PGL_n$, such that the cycle class map takes $ρ_{p,k}$ to $y_{p,k}$. We present an application of the above classes regarding Chern subrings.
Let X be a smooth variety over a field k and D an effective divisor whose support has simple normal crossings. We construct an explicit cycle map from the r-th Nisnevich motivic complex of the pair (X,D) to a shift of the r-th relative Milnor K-sheaf of (X,D). We show that this map induces an isomorphism for all i greater or equal the dimension of X between the motivic Nisnevich cohomology of (X,D) in bidegree (i+r,r) and the i-th Nisnevich cohomology of the r-th relative Minor K-sheaf of (X,D). This generalizes the well-known isomorphism in the case D=0. We use this to prove a certain Zariski…
Using Bloch-Ogus theorem and Chern character from K-theory to cyclic homology, we answer a question of Green and Griffiths on extending Bloch formula. Moreover, we construct a map from local Hilbert functor to local cohomology. With suitable assumptions, we use this map to answer a question of Bloch on constructing a natural transformation from local Hilbert functor to cohomological Chow groups.
We discuss the existence of an absolute Chow-Kuenneth decomposition for complete degenerations of families of Abelian threefolds with complex multiplication over a particular Picard Modular Surface studied by Holzapfel. In addition to the work of Gordon, Hanamura and Murre we use Relatively Complete Models in the sense of Mumford-Faltings-Chai of Picard Modular Surfaces in order to describe complete degenerations of families of abelian varieties. We furthermore prove vanishing results for cohomology groups of irreducible representations of certain arithmetic subgroups in SU(2,1) using the non-…
Chow groups with twisted coefficients
本文推广Rost的Chow群定义至非挠性系数，计算多个有限群分类空间的扭曲Chow群，并揭示其与扭曲动机上同调之间的满射但非同构关系。
Motivated by the computation of the BPS-invariants on a local Calabi-Yau threefold suggested by S. Katz, we compute the Chow ring and the cohomology ring of the moduli space of stable sheaves of Hilbert polynomial $4m+1$ on the projective plane. As a byproduct, we obtain the total Chern class and Euler characteristics of all line bundles, which provide a numerical data for the strange duality on the plane.
We determine the integral Chow and cohomology rings of the moduli stack $\mathcal{B}_{r,d}$ of rank $r$, degree $d$ vector bundles on $\mathbb{P}^1$ bundles. We first show that the rational Chow ring $A_{\mathbb{Q}}^*(\mathcal{B}_{r,d})$ is a free $\mathbb{Q}$-algebra on $2r+1$ generators. The isomorphism class of this ring happens to be independent of $d$. Then, we prove that the integral Chow ring $A^*(\mathcal{B}_{r,d})$ is torsion-free and provide multiplicative generators for $A^*(\mathcal{B}_{r,d})$ as a subring of $A_{\mathbb{Q}}^*(\mathcal{B}_{r,d})$. From this description, we see that…
The singular cohomology ring of a uniform matroid: combinatorics and Lefschetz properties
本文研究均匀拟阵奇异上同调环的组合结构与Lefschetz性质，通过Koszul同调构造显式基，推导Hodge数公式并证明其满足拟射影强Lefschetz性质。
We study the deformations of the Chow group of zero-cycles of the special fibre of a smooth scheme over a henselian discrete valuation ring. Our main tools are Bloch's formula and differential forms. As a corollary we get an algebraization theorem for thickened zero-cycles previously obtained using idelic techniques. In the course of the proof we develop moving lemmata and Lefschetz theorems for cohomology groups with coefficients in differential forms.
We use Burgos theory of arithmetic Chow groups to exhibit a realization of the degree zero part of the polylogarithm on abelian schemes in Deligne-Beilinson cohomology.
The Chow Characteristic Image of \(\Spin(10)\) via the Affine Cone over the Spinor Variety
研究\(\Spin(10)\)的积分周环限制映射像，核心方法是构造类\(c_2c_3c_5\)，主要贡献是确定了模\(2\)和积分的周环特征标像，为相关研究提供关键结果。
The paper discusses four approaches to the biextension of Chow groups and their equivalences. These are the following: an explicit construction given by S.Bloch, a construction in terms of the Poincare biextension of dual intermediate Jacobians, a construction in terms of K-cohomology, and a construction in terms of determinant of cohomology of coherent sheaves. A new approach to J.Franke's Chow categories is given. An explicit formula for the Weil pairing of algebraic cycles is obtained.
We compute the Chow-Witt rings of the classifying spaces for the symplectic and special linear groups. In the structural description we give, contributions from real and complex realization are clearly visible. In particular, the computation of cohomology with $\mathbf{I}^j$-coefficients is done closely along the lines of Brown's computation of integral cohomology for special orthogonal groups. The computations for the symplectic groups show that Chow-Witt groups are a symplectically oriented ring cohomology theory. Using our computations for special linear groups, we also discuss the question…
We prove relations between fractional linear cycles in Bloch's integral cubical higher Chow complex in codimension two of number fields, which correspond to functional equations of the dilogarithm. These relations suffice, as we shall demonstrate with a few examples, to write down enough relations in Bloch's integral higher Chow group CH^2(F,3) for certain number fields F to detect torsion cycles. Using the regulator map to Deligne cohomology, one can check the non-triviality of the torsion cycles thus obtained. Using this combination of methods, we obtain explicit higher Chow cycles generatin…
We prove that the Chow motives of twisted derived equivalent K3 surfaces are isomorphic, not only as Chow motives (due to Huybrechts), but also as Frobenius algebra objects. Combined with a recent result of Huybrechts, we conclude that two complex projective K3 surfaces are isogenous (i.e. their second rational cohomology groups are Hodge isometric) if and only if their Chow motives are isomorphic as Frobenius algebra objects; this can be regarded as a motivic Torelli-type theorem. We ask whether, more generally, twisted derived equivalent hyper-Kaehler varieties have isomorphic Chow motives a…
In this work we compute the Chen--Ruan cohomology and the stringy Chow ring of the moduli spaces of smooth and stable $n$-pointed curves of genus 1. We suggest a definition for an Orbifold Tautological Ring in genus 1, which is both a subring of the Chen--Ruan cohomology and of the stringy Chow ring.
We construct an analog of the intrinsic normal cone of Behrend-Fantechi in the equivariant motivic stable homotopy category over a base-scheme B and construct a fundament class in E-cohomology for any cohomology theory E in SH(B). For affine B, a perfect obstruction theory gives rise to a virtual fundamental class in a twisted Borel-Moore E-homology for arbitrary E. This includes motivic cohomology (homotopy invariant) K-theory algebraic cobordism and the oriented Chow groups of Barge-Morel and Fasel. In the case of motivic cohomology, we recover the constructions of Behrend-Fantechi, with val…
Based on the Basis theorem of Bruhat--Chevalley [C] and the formula for multiplying Schubert classes obtained in [D\QTR{group}{u}] and programed in [DZ$_{\QTR{group}{1}}$], we introduce a new method computing the Chow rings of flag varieties (resp. the integral cohomology of homogeneous spaces). The method and results of this paper have been extended in [DZ$_{3}$, DZ$_{4}$] to obtain the integral cohomology rings of all complete flag manifolds, and to construct the integral cohomologies of Lie groups in terms of Schubert classes.
Chow-Witt rings and topology of flag varieties
本文利用子商丛的 Pontryagin 类计算 A 型部分旗簇的 Witt 层上同调环，证明了实旗流形整上同调挠均为 2-挠，并描述了旗簇的 Chow-Witt 环及其枚举应用。
Multiplicative Chow-Künneth decomposition and homology splitting of configuration spaces
本文构造了光滑固有簇点构型空间上同调的分裂，证明了超椭圆Torelli群作用平凡及Leray谱序列立即退化，并据此对$M_{2,n}$和$H_{g,n}$的上同调进行了详细计算。
We consider the "homotopy coniveau tower" for an arbitrary cohomology theory on smooth varieties over a field or a Dedekind domain. This tower is a generalization of the construction used by Bloch-Lichtenbaum and Friedlander-Suslin in their studies of the spectral sequence from motivic cohomology to K-theory. Our main result is that an application of the classical Chow's moving lemma and some categorical constructions make the homotopy coniveau tower strictly functorial when the base is a field, and functorial in the homotopy category when the base is a Dedekind domain.
The main goal of this paper is to break up motivic cohomology into smaller pieces as suggested by the conjectural Bloch-Beilinson filtrations for the Chow groups.
We prove that, given a symmetrically distinguished correspondence of a suitable complex abelian variety (which include any abelian variety of dimension atmost 5, powers of complex elliptic curves, etc.) which vanishes as a morphism on a certain quotient of its middle singular cohomology, then it vanishes as a morphism on the deepest part of a particular filtration on the Chow group of 0-cycles of the abelian variety. As a consequence, we prove that given an automorphism of such an abelian variety, which acts as the identity on a certain quotient of its middle singular cohomology, then it acts …
In this paper we study actions of reductive groups on affine spaces. We prove that there is a fan structure on the space of characters of the group, which parameterizes the possible invariant quotients. In the second half of the paper we study various geometrical properties of the quotients: we compute their Chow ring and the cohomology ring (in case we work over the complex numbers), both with rational coefficients. Actually, the cycle map turns out to be an isomorphism. Further, we construct families of such quotients over reduced base schemes, and compute again the Chow and the cohomology r…
Finding coherent relations to define non Abelian cohomology is a thriller which entertains the mathematical community since fifty one years. The purpose of this paper is to simplify the attempt to beat it defined by the author which used the notion of sequences of fibred categories and to apply the resulting theory to higher divisors and Chow theory.
Due to the work of many authors in the last decades, given an algebraic orbifold (smooth proper Deligne-Mumford stack with trivial generic stabilizer), one can construct its orbifold Chow ring and orbifold Grothendieck ring, and relate them by the orbifold Chern character map, generalizing the fundamental work of Chen-Ruan on orbifold cohomology. In this paper, we extend this theory naturally to higher Chow groups and higher algebraic K-theory, mainly following the work of Jarvis-Kaufmann-Kimura and Edidin-Jarvis-Kimura.
The Oka principle for étale Chow groups
本文证明偶数阶上同调群可由底层环描述，即每个余维数的 étale Chow 群与 Gelfand 谱的偶数整值上同调群典范同构。
We show that the cohomology of the perfect cone (also called first Voronoi) toroidal compactification of the moduli space of complex principally polarized abelian varieties stabilizes, in close to the top degree. Moreover, we show that this stable cohomology is purely algebraic, and we compute it in degree up to 13. Our explicit computations and stabilization results apply in greater generality to various toroidal compactifications and partial compactifications, and in particular we show that the cohomology of the matroidal partial compactification stabilizes (in low degree). For degree up to …
Hilbert-Poincaré series of matroid Chow rings and intersection cohomology
本文研究了拟阵Chow理论与Kazhdan-Lusztig框架下四个对象的Hilbert级数，发展了多项式与环级数间的平行关系，利用赋值特性提出快速计算方法，并证明了均匀拟阵的实根性及多项式系数最大化等猜想，将单峰性强化为$\gamma$-正性。
We prove an unconditional (but slightly weakened) version of the main result of our earlier paper with the same title, which was, starting from dimension $4$, conditional to the Lefschetz standard conjecture. Let $X$ be a variety with trivial Chow groups, (i.e. the cycle class map to cohomology is injective on $CH(X)_\mathbb{Q}$). We prove that if the cohomology of a general very ample hypersurface $Y$ in $X$ is ``parameterized by cycles of dimension $c$'', then the Chow groups $CH_{i}(Y)_\mathbb{Q}$ are trivial for $i\leq c-1$.
Torsion higher Chow cycles modulo $\ell$
本文通过高阶Chow群在精细非分支上同调上的作用的内射性，构造了首个光滑复射影簇的例子，其高阶Chow群包含无穷多个模$n$线性无关的$n$阶挠循环，且维数和余维数的界最优，关键工具为态射上同调。
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