Angelo Vistoli
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.
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We compute generators for the Chow ring of the classifying space of PGL_3 (over the complex numbers) as defined by Totaro. We also find enough relations after inverting 3. We show that this ring is not generated by Chern classes (this is the first example of this kind among classical groups) and prove that Totaro's refined cycle class map to a quotient of complex cobordism of BPGL_3 is surjective
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.
We compute the integral Chow ring of the quotient stack $[(\mathbb{P}^1)^n/PGL_2]$, which contains $\mathcal{M}_{0,n}$ as a dense open, and determine a natural $\mathbb{Z}$-basis for the Chow ring in terms of certain ordered incidence strata. We further show that all $\mathbb{Z}$-linear relations between the classes of ordered incidence strata arise from an analogue of the WDVV relations in $A^\bullet(\overline{\mathcal{M}}_{0,n})$. Next we compute the classes of unordered incidence strata in the integral Chow ring of the quotient stack $[{\rm Sym}^n\mathbb{P}^1/PGL_2]$ and classify all $\math…
We use the Steenrod algebra to study $CH^*BG$, the mod $p$ Chow ring of the classifying space of $G$. We describe a localization property which relates a given $G$ to its elementary abelian subgroups, and we study a number of particular cases, namely symmetric groups and Chevalley groups. It turns out that the Chow rings of these groups are completely determined by the abelian subgroups and their fusion.
Let $A$ be an Azumaya algebra over a field. If $G$ is the group of automorphisms of $A$ and $X$ denotes a projective homogeneous variety under $G$, we construct in a very explicit way and under suitable hypotheses a bundle $\mathcal{V}$ on $S$, where $S$ is a (generalized) Severi-Brauer variety associated to $A$, and a canonical isomorphism between $X$ and a flag bundle on $\mathcal{V}$. This allows to explicitely compute Chow groups of $X$ in terms of the Chow groups of $S$.
On the $\mathrm{PGL}_2$-equivariant intersection theory of $\mathrm{Gr}(2,4)$
本文确定了Gr(2,4)的PGL₂-稳定区域的PGL₂-等变Chow环，并探讨了计算其完整PGL₂-等变Chow环的困难。
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 prove that the class of the classifying stack $B PGL_n$ is the multiplicative inverse of the class of the projective linear group $PGL_n$ in the Grothendieck ring of stacks for $n = 2$ and $n = 3$ under mild conditions on the base field $k$. In particular, although it is known that the multiplicativity relation $\{T\} = \{S\}\cdot\{PGL_n\}$ does not hold for all $PGL_n$-torsors $T \to S$, it holds for the universal $PGL_n$-torsors for said $n$.
We show how the stratification method, introduced by Vezzosi in his study of PGL_3, provides a unified approach to the known computations of the Chow rings of the classifying spaces of GL_n, SL_n, Sp_n, O_n and SO_n.
The Chow--Witt rings of the classifying space of quadratically oriented bundles
该研究计算了秩n二次定向向量丛分类空间${\ m BSL}_n^c$的Chow--Witt环，还证明对应二次定向配边谱${\ m MSL}^c$在对η取逆后等价于MSL，为相关代数拓扑研究提供了新结果。
The Chow Ring Classes of $\mathrm{PGL}_3$ Orbit Closures in $\mathbb{G}(1, 5)$
本文研究平面圆锥曲线束构成的格拉斯曼簇在 PGL(V) 作用下的八个轨道，通过几何描述、枚举技巧和陈类计算，确定了各轨道闭包在周环中的类及其普吕克嵌入次数。
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…
Let $p$ be a prime and $q$ be a power of $p$. We compute the Chow ring of the classifying space of some Chevalley groups $G(\mathbb{F}_q)$, when considered as a finite algebraic group over a field of characteristic $p$ containing $\mathbb{F}_q$. Using specialization from characteristic $0$ to characteristic $p$ we also obtain results over the complex numbers.
We define the Chow ring of the classifying space of a linear algebraic group. In all the examples where we can compute it, such as the symmetric groups and the orthogonal groups, it is isomorphic to a natural quotient of the complex cobordism ring of the classifying space, a topological invariant. We apply this to get torsion information on the Chow groups of varieties defined as quotients by finite groups. This generalizes Atiyah and Hirzebruch's use of such varieties to give counterexamples to the Hodge conjecture with integer coefficients.
We investigate the structure of the Chow ring of the classifying stacks $BT$ of algebraic tori, as it has been defined by B. Totaro. Some previous work of N. Karpenko, A. Merkurjev, S. Blinstein and F. Scavia has shed some light on the structure of such rings. In particular Karpenko showed the absence of torsion classes in the case of permutation tori, while Merkurjev and Blinstein described in a very effective way the second Chow group $A^2(BT)$ in the general case. Building on this work, Scavia exhibited an example where $A^2(BT)_\text{tors}\neq 0$. Here, by making use of a very elementary a…
Let H(d) be the (open) Hilbert scheme of rational normal curves of degree d in P^d. A presentation of the integral Chow ring of H(d) is given via equivariant Chow ring computations. Included also in the paper are algebraic computations of the integral equivariant Chow rings of the algebraic groups O(n), SO(2k+1). The results for S0(3)=PGL(2) are needed for the Hilbert scheme calculation.
We compute the Chow ring of the classifying space $BSO(2n,\C)$ in the sense of Totaro using the fibration $Gl(2n)/SO(2n) \to BSO(2n) \to BGl(2n)$ and a computation of the Chow ring of $Gl(2n)/SO(2n)$ in a previous paper. We find this Chow ring is generated by Chern classes and a characteristic class defined by Edidin and Graham which maps to $2^{n-1}$ times the Euler class under the usual class map from the Chow ring to ordinary cohomology. Moreover, we show this class represents $1/2^{n-1}(n-1)!$ times the $n^{th}$ Chern class of the representation of SO(2n) whose highest weight vector is twi…
Chow-Witt Rings of Classifying Spaces of Products of Multiplicative and Cyclic Groups
本文基于 di Lorenzo 和 Mantovani 的策略，计算了单位根分类空间及其与乘法群乘积的全 Chow-Witt 环，并进一步确定了射影空间乘积和 BG_m × BG_m 的全 I^j-上同调与 Chow-Witt 环。
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 give the first examples of finite groups G such that the Chow ring of the classifying space BG depends on the base field, even for fields containing the algebraic closure of Q. As a tool, we give several characterizations of the varieties which satisfy Kunneth properties for Chow groups or motivic homology. We define the (compactly supported) motive of a quotient stack in Voevodsky's derived category of motives. This makes it possible to ask when the motive of BG is mixed Tate, which is equivalent to the motivic Kunneth property. We prove that BG is mixed Tate for various "well-behaved" fin…
Chow classes of orbit closures of skew-symmetric matrix pencils
研究斜对称矩阵束轨道闭包的周环类，以六维复向量空间\(V\)相关的\(G\)为研究对象，通过Jordan - Kronecker标准型分类\(\operatorname{PGL}(V)\)在\(G\)上的\(11\)个轨道，计算其在舒伯特基中的周环类。
Let $G$ be a smooth connected reductive group over a field $k$ and $Γ$ be a central subgroup of $G$. We construct Eilenberg-Moore-type spectral sequences converging to the Hodge and de Rham cohomology of $B(G/Γ)$. As an application, building upon work of Toda and using Totaro's inequality, we show that for all $m\geq 0$ the Hodge and de Rham cohomology algebras of the classifying stacks $B\mathrm{PGL}_{4m+2}$ and $B\mathrm{PSO}_{4m+2}$ over $\mathbb{F}_2$ are isomorphic to the singular $\mathbb{F}_2$-cohomology of the classifying space of the corresponding Lie group. From this we obtain a full…
The Chow Ring of the 2-Sylow Subgroup of $\mathrm{GL}(4,2)$
本文计算$\mathrm{Syl}_2(\mathrm{GL}(4,2))$的motivic étale分类空间的模2 Chow环，并提出一种源自Totaro且已自动化的通用策略，以快速计算更多同类Chow环。
If $G$ is a presheaf of groupoids on a small site, and $A$ is a sheaf of abelian groups, we prove that the sheaf cohomology group $H^2 (BG, A)$ is in bijection with a set of central extensions of $G$ by $A$. We use this result to study the motivic cohomology of the Nisnevich classifying space $BG$, when $G$ is a presheaf of groups on the smooth Nisnevich site over a field, and particularly when $G = PGL_{n}$. Finally, we show that, when $p$ is an odd prime, the Chow ring of the classifying space of $PGL_{p}$ injects into the motivic cohomology of the Nisnevich classifying space $BPGL_{p}$, ove…
Chow-Witt rings of Grassmannians
本文计算了一般线性群分类空间的Chow-Witt环，涉及非平凡对偶性，并以此推导出有限Grassmannian的Chow-Witt环，所得公式与实Grassmannian整上同调环公式高度相似。
We introduce a certain compactification of the space of projective configurations i.e. orbits of the group $PGL(k)$ on the space of $n$ - tuples of points in $P^{k-1}$ in general position. This compactification differs considerably from Mumford's geometric invariant theory quotient. It is obtained by considering limit position (in the Chow variety) of the closures of generic orbits. The same result will be obtained if we study orbits of the maximal torus on the Grassmannian $G(k,n)$. We study in detail the closures of the torus orbits and their "visible contours" which are Veronese varieties i…
Equivariant Chow-Witt groups and moduli stacks of elliptic curves
本文引入等变Chow-Witt群定义商栈的Chow-Witt群，计算稳定与光滑椭圆曲线模栈的Chow-Witt环，并给出新生成元的几何解释及$μ_{2n}$分类栈的相应结果。
The irreducibility of the moduli space of pointed stable curves with dormant $\mathrm{PGL}_2^{(N)}$-oper
本文研究素特征下带 dormant $\text{PGL}_2^{(N)}$-oper 的带点稳定曲线模空间，通过引入基于 $p^N$-曲率特征多项式的高阶 Hitchin-Mochizuki 态射，证明了该模空间的不可约性。
We prove that the moduli spaces of twisted $\mathrm{SL}_n$ and $\mathrm{PGL}_n$-Higgs bundles on a smooth projective curve have the same (stringy) class in the Grothendieck ring of rational Chow motives. On the level of Hodge numbers this was conjectured by Hausel and Thaddeus, and recently proven by Groechenig, Ziegler and the second author. To adapt their argument, which relies on p-adic integration, we use a version of motivic integration with values in rational Chow motives and the geometry of Néron models to evaluate such integrals on Hitchin fibers.
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。
We give lower bounds for the rank of the first homology group of the real points of the Deligne-Mumford-Knudsen compactification of stable n-pointed curves of genus 0,which coincides with the Chow quotient (RP^1)^n//PGL(2,R).The study has connections with spectral geometry and combinatorics.
In the present article we investigate ordinary and equivariant Rost motives. We provide an equivariant motivic decomposition of the variety X of full flags of a split semisimple algebraic group over a smooth base scheme, study torsion subgroup of the Chow group of twisted forms of X, define some equivariant Rost motives over a field and ordinary Rost motives over a general base scheme, and relate equivariant Rost motives with classifying spaces of some algebraic groups.
We study the homotopy theory of the classifying space of the complex projective linear groups to prove that purity fails for $PGL_p$-torsors on regular noetherian schemes when $p$ is a prime. Extending our previous work when $p=2$, we obtain a negative answer to a question of Colliot-Thélène and Sansuc, for all $PGL_p$. We also give a new example of the failure of purity for the cohomological filtration on the Witt group, which is the first example of this kind of a variety over an algebraically closed field.
The Chow ring of the universal Picard stack over the hyperelliptic locus
本文确定了超椭圆轨迹上万有 Picard 层的有理 Chow 环及其生成元与关系，并计算了整 Picard 群，证明了其 Brauer 类阶由 $d-g$ 奇偶性决定。
Let $k$ be a field of characteristic zero, and let $X$ be a projective variety embedded into a projective space over $k$. For two natural numbers $r$ and $d$ let $C_{r,d}(X)$ be the Chow scheme parametrizing effective cycles of dimension $r$ and degree $d$ on the variety $X$. An effective $r$-cycle of minimal degree on $X$ gives rise to a chain of embeddings of $C_{r,d}(X)$ into $C_{r,d+1}(X)$, whose colimit is the connective Chow monoid $C_r^{\infty }(X)$ of $r$-cycles on $X$. Let $BC_r^{\infty }(X)$ be the motivic classifying space of this monoid. In the paper we establish an isomorphism bet…
On the $PGL_2(q)$-orbits of lines of $PG(3,q)$ and binary quartic forms
本文研究了有限域上射影3维空间中直线在三次挠线线性对称群下的轨道分类问题，通过将超平面与二元四次型空间建立等变联系，解决了二元四次型及一般直线的轨道分类开放问题。
Integral Weyl Invariants in Chow Characteristic Images of Spin and Special Clifford Groups
本文研究了旋群和特殊克利福德群的积分韦尔不变量在柴可夫特征图像中的分类，发现唯一非例外的代数类出现在 $\Spin(10)$ 和 $\Gamma^+(7)$ 中。
Integral Motivic Cohomology of $BSO_{4}$
本文计算了 $BSO_{4}$ 的积分动机上同调，借鉴 Guillot 和 Yagita 的方法，为 $BG_{2}$ 的类似计算奠定基础。
Arithmetic liftings and 2d TQFT for dormant opers of higher level
本文研究正特征下休眠算符的模空间，构造了紧化模空间，证明了n=2时的平展性并导出典范对角提升，还发现次数函数构成2d TQFT，提供了计数这类算符和二阶微分方程的有效方法。
Let GO(2n) be the general orthogonal group (the group of similitudes) over any algebraically closed field of characteristic not equal to 2. We determine the etale cohomology ring with mod 2 coefficients of the algebraic stack BGO(2n). In the topological category, Y. Holla and N. Nitsure determined the singular cohomology ring of the classifying space BGO(2n) of the complex Lie group GO(2n) in terms of explicit generators and relations. We extend their results to the algebraic category. The chief ingredients in this are (i) an extension to etale cohomology of an idea of Totaro, originally used …
For a complex algebraic variety $X$, we show that triviality of the sheaf cohomology group $H^0(X,\mathcal{H}^3)$ occurring on the second page of the Bloch-Ogus spectral sequence follows from a condition on the integral Chow group $CH^2X$ and the integral cohomology group $H^3(X, Z)$. In the case that $X$ is an appropriate approximation to the classifying stack $BG$ of a finite $p$-group $G$, this result states that the group $G$ has no degree three cohomological invariants. As a corollary we show that the nonabelian groups of order $p^3$ for odd prime $p$ have no degree three cohomological in…
Quantum cohomology, Hitchin systems, and Fourier--Mukai comparison
本文证明稳定Higgs丛模空间上除子的等变量量子积由Steinberg对应给出，并证明Fourier-Mukai变换与量子积相容，建立了量子上同调与Hitchin系统的联系。
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