Bogdan Zavyalov
In the present paper we introduce the property AA of a subsemigroup of the endomorphism semigroup of an abelian variety, which holds for semigroup of endomorphisms of an abelian variety defined over a number field, and show that the orbit of any cycle under a semigroup with property AA in the Chow group $\otimes\Q$ has finite dimensional span.
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Beilinson-Tate cycles on semiabelian varieties
Beilinson沿着Hodge和Tate定理的思路，证明了在q次上同调中，光滑复簇的所有权为2 q的Hodge圈和光滑簇的所有权为2 q的Tate圈都来自高Chow群。对于曲线和半交换簇的乘积，Beilinson-Hodge猜想在以前的文章中被证明。在这里，Beilinson-Hodge和Beilinson-Tate定理都被证明对于由曲线的乘积支配的品种是正确的。我们还表明，较低的重量霍奇循环（分别。泰特循环）在这些情况下是代数的。
Beilinson-Hodge cycles on semiabelian varieties
Beilinson证明了光滑复代数簇的q次上同调上的所有（q，q）型有理圈都应该来自于motivic上同调。这注记的目的是证明这一点时，品种是一个半阿贝尔品种或产品的曲线。证明是基于Mumford-Tate群下的不变量的研究。
On Goncharov's regulator and higher arithmetic Chow groups
本文证明了Goncharov定义的从高阶代数Chow群到Deligne-Beilinson上同调的调节子与Beilinson调节子一致。我们给出了一个直接的比较贡恰罗夫的监管机构的建设布尔戈斯和Feliu。因此，我们表明，高算术周组定义的Goncharov同意，所有的投射算术品种的算术领域，与Burgos和Feliu定义的。
Finite dimensional motives and the Conjectures of Beilinson and Murre
我们将光滑射影簇X（定义于S. Kimura）与Beilinson，Bloch和Murre关于Chow环CH（X）上滤子存在性的猜想.我们证明了（定理14）M（X）的有限维性意味着M（X）的Murre分解的唯一性，直到同构。相反地（定理17），对于X的适当m重积的Murre猜想本身暗示M（X）的有限维数。定理27还证明了：对于具有平凡几何亏格的曲面X，原动机M（X）是有限维的当且仅当X的0-圈的Chow群在Mumford意义下是有限维的，即当且仅当Bloch猜想对X成立。
In the first section of his seminal paper on height pairings, Beilinson constructed an $\ell$-adic height pairing for rational Chow groups of homologically trivial cycles of complementary codimension on smooth projective varieties over the function field of a curve over an algebraically closed field, and asked about an generalization to higher dimensional bases. In this paper we answer Beilinson's question by constructing a pairing for varieties defined over the function field of a smooth variety $B$ over an algebraically closed field, with values in the second $\ell$-adic cohomology group of …
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.
On the Beilinson-Bloch conjecture over function fields
本文研究了函数域上的贝利逊-布洛赫猜想，通过扩展贾恩森的结果并证明塔特猜想，推导出该猜想的特殊情形，并回答了Moonen关于Chow群的问题。
The Beilinson-Bloch conjecture for some non-isotrivial varieties over global function fields
本文证明了非全纯变种上Beilinson-Bloch猜想，并推导出中间Jacobians上的Birch-Swinnerton-Dyer猜想，同时推广了有限域上的Tate猜想。
On the Beilinson-Hodge conjecture for $H^2$ and rational varieties
Beilinson-Hodge猜想证明了循环映射$$H^n_M(X,\Q(n)) \to {\rm Hom}_{MHS}(\Q(-n),H^n(X,\Q))$$对所有正整数$n$和所有光滑复代数簇$X$的满射性。对于$n=2$，如果$X$是有理数，则证明了猜想.
Motivated by the Bloch-Beilinson conjectures, we formulate a certain covariant weak Lefschetz property for Chow groups. We prove this property in some special cases, using Kimura's nilpotence theorem.
We review and simplify A. Beilinson's construction of a basis for the motivic cohomology of a point over a cyclotomic field, then promote the basis elements to higher Chow cycles and evaluate the KLM regulator map on them.
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.
On functional equations for Chow polylogarithms
研究Chow多重对数的函数方程，证明其边界消失性方程可由斜对称性等简单方程推导，并建立Bloch循环复形类似物的Beilinson-Soule消失猜想类似结果，且经典多重对数函数方程可由此类方程推导。
Quiver varieties and Beilinson-Drinfeld Grassmannians of type A
我们构造Nakajima的A型的共轭类的矩阵和（非Slodowy的）横切片自然产生的仿射格拉斯曼。在完全一般性的情况下，A型的Beilinson-Drinfeld Grassmannian变种嵌入其中。我们的建设提供了一个紧化的Nakajima的Nakajima品种和分解的仿射格拉斯曼到一个不相交的联盟nakajima品种。作为一个应用，我们提供了一个几何版本的斜对称$(GL(m), GL(n))$对偶。
Tempered currents and Deligne cohomology of Shimura varieties, with an application to $\mathrm{GSp}_6$
本文利用缓增电流给出任意Shimura簇Deligne-Beilinson上同调的新描述，并应用于计算Siegel六维簇motivic上同调类的Beilinson高阶调节子像，将其与Spin $L$-函数非临界特殊值联系，验证Beilinson猜想。
The Abel-Jacobi map for higher Chow groups
我们构建了一个映射之间的Bloch的高周群和Deligne的光滑，复杂的拟投射品种的水平上的复同调。对于复杂的投射品种，这导致在一个公式，推广在同一时间的经典格里菲斯阿贝尔-雅可比映射和Borel/Beilinson/Goncharov调节器类型的地图。
We compute the spaces of sections of powers of the determinant line bundle on the spherical Schubert subvarieties of the Beilinson- Drinfeld affine Grassmannians. The answer is given in terms of global Demazure modules over the current Lie algebra.
We answer two questions of Carrell on a singular complex projective variety admitting the multiplicative group action, one positively and the other negatively. The results are applied to Chow varieties and we obtain Chow groups of 0-cycles and Lawson homology groups of 1-cycles for Chow varieties. A short survey on the structure of the Chow varieties is included for comparison and completeness. Moreover, we give counterexamples to Shafarevich's question on the rationality of the irreducible components of Chow varieties.
Chow groups and higher congruences for the number of rational points on proper varieties over finite fields
给定q元有限域k上的光滑基上的簇的一个真族，证明了Chow群CH_i，i= 0，1，.的有限性假设，r导致模q^{r+1}的同余，模q^{r+1}是在基的k-有理点上的所有纤维中的有理点的数目。这些关于Chow群的假设对于许多Fano变种中的低次交的族是成立的，从而导致Ax-Katz定理的广泛推广，以及作者和C. S.拉詹作为一个无条件的应用，我们给出了一个渐近推广的Ax-Katz定理的一个大类的齐性空间中的低度交叉。
We first present an intersection theory of partial differential varieties with quasi-generic differential hypersurfaces. Then based on the generic intersection theory, we define the partial differential Chow form for an irreducible partial differential variety $V$ of Kolchin polynomial $ω_V(t)=(d+1){t+m\choose m}-{t+m-s\choose m}$. And we establish for the partial differential Chow form most of the basic properties of the ordinary differential Chow form. Furthermore, we prove the existence of a type of partial differential Chow varieties.
Special values of zeta functions of varieties over finite fields via higher Chow groups
研究了有限域上奇异簇的zeta函数的特殊值。通过构造一个从高阶Chow群到权同调的同调论的态射，给出了一个新的特殊值公式，我们称之为调节子.我们的调节器是由Gillet和Soule引入的完满域上品种的权复形的概念定义的。我们的特殊值公式的证明的主要思想是利用同调论的权谱序列，其E1项是光滑射影概形的同调群。为了计算特殊值，证明了完满域上任意簇的权复形是有界的。这个有界性结果是已知的吉列和索尔的情况下，基场承认奇点的分解，但不是一般的。
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.
A refined Beilinson-Bloch conjecture for motives of modular forms
我们提出了一个改进版本的Beilinson-Bloch猜想的动机与模形式的偶数重量。这一猜想涉及到相关的p-adic Abel-Jacobi映射的图像的尺寸，以某些组合的Kuga-Sato品种的Heegner循环。我们证明定理的方向的猜想，并在这样做，获得更高的权重类似物的结果椭圆曲线由于达蒙。
Beilinson's Hodge Conjecture for K_1 revisited
设U是光滑拟投射复簇，CH^r（U，1）是Bloch高阶Chow群的特例.扬森是第一个表明循环类映射cl_{r,1}从CH^r（U，1）（张量与Q）到hom_{MHS}（Q（0），H^{2 r-1}（U，Q（r））不是一般的满射，矛盾的早期猜想贝林森。本文给出了Jannsen反例的一个改进，并进一步证明了上述圈类映射在泛点上是满射的。
We show that for a very general principally polarized complex abelian 3-fold, the Chow group of algebraic cycles is infinite modulo every prime number. In particular, this gives the first examples of complex varieties with infinite Chow groups modulo 2.
This note concerns hyperkähler fourfolds $X$ having a non-symplectic involution $ι$. The Bloch-Beilinson conjectures predict the way $ι$ should act on certain pieces of the Chow groups of $X$. The main result is a verification of this prediction for Fano varieties of lines on certain cubic fourfolds. This has consequences for the Chow ring of the quotient $X/ι$.
Beilinson's Hodge conjecture for smooth varieties
考虑圈类映射cl_{r,m}：CH^r（U，m; \Q）\to ΓH^{2 r-m}（U，\Q（r）），其中CH^r（U，m; \Q）是光滑复拟拟射影簇U的Bloch高阶Chow群（用\Q张量），H^{2 r-m}（U，\Q（r））是奇异上同调。我们利用Abel-Jacobi映射的核研究了cl_{r,m}的象.当r=m时，我们从Bloch-Kato定理推导出cl_{r,m}在U的泛点的上核对于整系数和有理系数是相同的.
Conjecturally, Fano varieties of K3 type admit a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. We prove this for many of the families of Fano varieties of K3 type constructed by Fatighenti-Mongardi. This has interesting consequences for the Chow ring of these varieties.
We study special cycles on a Shimura variety of orthogonal type over a totally real field of degree $d$ associated with a quadratic form in $n+2$ variables whose signature is $(n,2)$ at $e$ real places and $(n+2,0)$ at the remaining $d-e$ real places for $1\leq e <d$. Recently, these cycles were constructed by Kudla and Rosu-Yott and they proved that the generating series of special cycles in the cohomology group is a Hilbert-Siegel modular form of half integral weight. We prove that, assuming the Beilinson-Bloch conjecture on the injectivity of the higher Abel-Jacobi map, the generating serie…
The main result of this note is a hard Lefschetz theorem for the Chow groups of generalized Kummer varieties. The same argument also proves hard Lefschetz for Chow groups of Hilbert schemes of abelian surfaces. As a consequence, we obtain new information about certain pieces of the Chow groups of generalized Kummer varieties, and Hilbert schemes of abelian surfaces. The proofs are based on work of Shen-Vial and Fu-Tian-Vial on multiplicative Chow-Künneth decompositions.
Cohomological Chow Groups of codimension one of varieties with isolated singularities
本文研究了具有孤立奇点的代数簇的上同调Chow群，通过不同条件下的计算揭示了其结构特性。
Bloch's Conjecture, Deligne Cohomology and Higher Chow Groups
本文利用广义Deligne上同调的归纳极限（即A. Beilinson）。这推广了L. Barbieri-Viale和V. Srinivas在表面情况下。然后，我们表明，阿贝尔Jacobi映射余维2圈和Albanese映射是双射的，如果一般超平面部分是一个表面，Bloch猜想被证明。在某些情况下，我们验证了诺里的猜想格里菲斯群。证明了（高）Chow群的弱Lefschetz型定理，将Beilinson和Levine关于高圈Abel-Jacobi映射的公式推广到光滑非正常情形，给出了椭圆曲面上高圈Abel-Jacobi映射象的超越部分不为零的一个充分条件，并举例说明.
The Chow variety of polynomials that decompose as a product of linear forms has been studied for more than 100 years. Finding equations in the ideal of secant varieties of Chow varieties would enable one to measure the complexity the permanent to prove Valiant's conjecture $\mathbf{VP\neq VNP}$. In this article, I use the method of prolongation to obtain equations for secant varieties of Chow varieties as $GL(V)$-modules.
For a generalized Kummer variety X of dimension 2n, we will construct for each 0 < i < n some co-isotropic subvarieties in X foliated by i-dimensional constant cycle subvarieties. These subvarieties serve to prove that the rational orbit filtration introduced by Voisin on the Chow group of zero-cycles of a generalized Kummer variety coincides with the induced Beauville decomposition from the Chow ring of abelian varieties. As a consequence, the rational orbit filtration is opposite to the conjectural Bloch-Beilinson filtration for generalized Kummer varieties.
Syntomic cohomology and Beilinson's Tate conjecture for $K_2$
本文研究椭圆曲面上分裂乘法纤维的补U的$K_2$的Tate猜想的一个类似.主要结果是给出了标准上同调$H^2(\bar{U},Q_p(2))$的Galois不动部分的秩的一个上界。作为应用，我们给出了p-adic域上的椭圆K3曲面$X$，其中0-圈的Chow群$CH_0(X)$的挠部分是有限的.这将是第一个例子的表面$X$在一个p-adic域的几何亏格是非零的，其中的扭转部分$CH_0(X)$是有限的。
Fatighenti and Mongardi have defined Fano varieties of type S6 as zero loci of a certain vector bundle on the Grassmannian $\hbox{Gr}(2,10)$. These varieties have 3 Hodge structures of K3 type in their cohomology. We show that the Chow ring of these varieties also displays "K3 type" behaviour.
We show that the graded Chow rings of two birational irreducible symplectic varieties are isomorphic. This lifts a result known for the cohomology algebras to the level of Chow rings, despite the non-injectivity the cycle class map. In the special case of general Mukai flops, we present an alternative approach based on explicit calculations.
The main goal of this paper is to define the so-called Chow weight structure for the category of Beilinson motives over any 'reasonable' base scheme $S$ (this is the version of Voevodsky's motives over $S$ defined by Cisinski and Deglise). We also study the functoriality properties of the Chow weight structure (they are very similar to the well-known functoriality of weights for mixed complexes of sheaves). As shown in a preceding paper, the Chow weight structure automatically yields an exact conservative weight complex functor (with values in $K^b(Chow(S))$). Here $Chow(S)$ is the heart of th…
Equivariant Chow cohomology of toric varieties
证明了环面簇的等变Chow上同调环与相应扇上的整分段多项式函数环自然同构。这给出了一个大类的奇异空间，其局部化在整数系数的等变Chow上同调中成立。我们还计算了复环面预簇和一般复超环面簇的分段多项式函数的等变Chow上同调。
We consider a $10$-dimensional family of Lehn-Lehn-Sorger-van Straten hyperkähler eightfolds which have a non-symplectic automorphism of order $3$. Using the theory of finite-dimensional motives, we show that the action of this automorphism on the Chow group of $0$-cycles is as predicted by the Bloch-Beilinson conjectures. We prove a similar statement for the anti-symplectic involution on varieties in this family. This has interesting consequences for the intersection product in the Chow ring of these varieties.
Asymptotic Chow stability of symmetric reflexive toric varieties
该研究针对环面簇的渐近周稳定性问题，给出非渐近周半稳定的对称自反环面簇例子，证明具正则三角剖分的弱对称自反环面簇的渐近周多稳定性，提出新判别准则并给出相关多面体例子。
On the integral Hodge conjecture for varieties with trivial Chow group
本文构造了全周群有限秩却违反整 Hodge 猜想的复光滑射影簇，并证明此类反例也可定义在数域上。
Stable maps and Chow groups
根据布洛赫-贝林森定理，一个K3曲面X的自同构在超越格上充当单位元，它应该平凡地作用在CH ^2（X）上。我们讨论这个猜想辛对合，并证明它在三分之一的所有情况下。其主要思想是利用特殊的椭圆K3曲面和稳定映射在一般的K3曲面上生成对合不变的覆盖椭圆曲线族。
We give bounds on the order of torsion in the Chow group of zero dimensional cycles for isotropic Grassmannians and Brauer-Severi flag varieties. To do this, we introduce tools to understand the behavior of torsion in Chow groups with coefficients under morphisms of proper varieties
We show that certain Fano eightfolds (obtained as hyperplane sections of an orthogonal Grassmannian, and studied by Ito-Miura-Okawa-Ueda and by Fatighenti-Mongardi) have a multiplicative Chow-Künneth decomposition. As a corollary, the Chow ring of these eightfolds behaves like that of K3 surfaces.
We show how to use equidimensional algebraic correspondences between complex algebraic varieties to construct pull-backs and transforms of certain classes of geometric currents. Using this construction we produce explicit formulas at the level of complexes for a regulator map from the Higher Chow groups of smooth complex quasi-projective algebraic varieties to Deligne-Beilinson cohomology with integral coefficients. A distinct aspect of our approach is the use of Suslin's complex of equidimensional cycles over $ Δ^n $ to compute Bloch's higher Chow groups. We calculate explicit examples involv…
This note is about certain locally complete families of Calabi-Yau varieties constructed by Cynk and Hulek, and certain varieties constructed by Schreieder. We prove that the cycle class map on the Chow ring of powers of these varieties admits a section, and that these varieties admit a multiplicative self-dual Chow-Kuenneth decomposition. As a consequence of both results, we prove that the subring of the Chow ring generated by divisors, Chern classes, and intersections of two cycles of positive codimension injects into cohomology, via the cycle class map. We also prove that the small diagonal…
This article is about hyperkähler fourfolds $X$ admitting a non-symplectic involution $ι$. The Bloch-Beilinson conjectures predict the way $ι$ should act on certain pieces of the Chow groups of $X$. The main result is a verification of this prediction for Fano varieties of lines on certain cubic fourfolds. This has some interesting consequences for the Chow ring of the quotient $X/ι$.
Varieties over a finite field with trivial Chow group of 0-cycles have a rational point
有限域$k$上的光滑射影簇$X$有一个有理点，特别是Fano簇.我们还参考了http://link.springer.de/link/service/journals/00222/tocs.htm，其中最后一个版本，清除了一些错别字，应该在网上发布。
Chow-Kunneth decomposition for varieties with low cohomological level
我们证明了一个光滑的投射簇允许Chow-Kunneth分解，如果上同调除了中间度之外最多有一个水平。如果态射只有孤立奇点，底空间是1维的，并且类属纤维满足上述条件，则这可以推广到弱意义下的相对情形。
We explain how complexity of rational points on projective varieties can be interpreted via the theories of Chow forms and Okounkov bodies. Precisely, we study discrete measures on filtered linear series and build on work of Boucksom and Chen, Boucksom-et-al and Ferretti. For example, we show how asymptotic volume and Seshadri constants are related to the theories of Chow weights and Okounkov bodies for very ample linear series. We also consider arithmetic applications within the context of Diophantine approximation. In this direction, we establish Roth-type theorems which can be expressed in …
Beilinson conjecture for finite-dimensional associative algebras
本文提出了Beilinson猜想在调节器映射上的一个非交换推广，从代数K理论推广到Q上代数簇的Deligne上同调。我们还检查了一个婴儿的情况下，广义猜想，即有限维结合代数的情况。
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.
Chow Quotients of Toric Varieties as Moduli of Stable Log Maps
设$X$是射影正规环面簇，$T_0$是$X$的定义环面的秩1子环面.我们表明，规范化的周商$X//T_0$，在Kapranov-Sturmfels-Zelevinsky的意义下，粗略地表示模空间的稳定日志映射到$X$与离散数据由$T_0\subset X$。
Chow varieties are a parameter space for cycles of a given variety of a given codimension and degree. We construct their analog for differential algebraic varieties with differential algebraic subvarieties, answering a question of Gao, Li and Wei. The proof uses the construction of classical algebro-geometric Chow varieties, the theory of characteristic sets of differential varieties, the theory of prolongation spaces, and the theory of differential Chow forms. In the course of the proof several definability results from the theory of algebraically closed fields are required. Elementary proofs…
Absolute Chow-Kuenneth decomposition for rational homogeneous bundles and for log homogeneous varieties
本文研究了Murre关于光滑簇上有理齐次丛$Z\to S$的Chow-Kuenneth分解存在性的猜想，定义在复数上.当$S$具有Chow-Kuenneth分解时，对$Z$表现出Chow-Kuenneth分解。同样的结论也适用于M.布里昂
Let $X$ be a hyperkähler variety, and let $G$ be a group of finite order non-symplectic automorphisms of $X$. Beauville's conjectural splitting property predicts that each Chow group of $X$ should split in a finite number of pieces. The Bloch-Beilinson conjectures predict how $G$ should act on these pieces of the Chow groups: certain pieces should be invariant under $G$, while certain other pieces should not contain any non-trivial $G$-invariant cycle. We can prove this for two pieces of the Chow groups when $X$ is the Hilbert scheme of a $K3$ surface and $G$ consists of natural automorphisms.…
Given a smooth projective variety, a Chow-Künneth decomposition is called multiplicative if it is compatible with the intersection product. Following works of Beauville and Voisin, Shen and Vial conjectured that hyper-Kähler varieties admit a multiplicative Chow-Künneth decomposition. In this paper, based on the mysterious link between Fano varieties with cohomology of K3 type and hyper-Kähler varieties, we ask whether Fano varieties with cohomology of K3 type also admit a multiplicative Chow-Künneth decomposition, and provide evidence by establishing their existence for cubic fourfolds and Kü…
We consider Calabi-Yau $n$-folds $X$ arising from certain hyperplane arrangements. Using Fu-Vial's theory of distinguished cycles for varieties with motive of abelian type, we show that the subring of the Chow ring of $X$ generated by divisors, Chern classes and intersections of subvarieties of positive codimension injects into cohomology. We also prove Voisin's conjecture for $X$, and Voevodsky's smash-nilpotence conjecture for odd-dimensional $X$.
Murre's conjectures and explicit Chow--Kuenneth projectors for some varieties
本文研究了Murre关于有理Chow群的结构的理论，并给出了一些例子，给出了明确的Chow-Kuenneth投影.更确切地说，我们研究的例子是具有nef切丛的簇。对于具有nef切丛的曲面和三重曲面，得到了满足Murre图的显式Chow-Kuenneth投影，并证明了运动Hard Lefschetz定理.
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