The Chow group of zero-cycles on certain Ch{â}telet surfaces over local fields

Chow group of 0-cycles with modulus and higher dimensional class field theory

One of the main results of this paper is a proof of the rank one case of an existence conjecture on lisse l-adic sheaves on a smooth variety over a finite field due to Deligne and Drinfeld. The problem is translated into the language of higher dimensional class field theory over finite fields, which describes the abelian fundamental group by Chow groups of zero cycles with moduli. A key ingredient is the construction of a cycle theoretic avatar of refined Artin conductor in ramification theory originally studied by Kazuya Kato.

Un théorème de finitude pour le groupe de Chow des zéro-cycles d'un groupe algébrique linéaire sur un corps p-adique

Let X be a smooth compactification of a connected linear algebraic group over a field k. The Chow group of degree nought zero-cycles on X is a torsion group. When k is a p-adic field, we show that the prime-to-p component of this group is finite. ----- Sei X eine glatte Kompaktifizierung einer zusammenhaengenden linearen Gruppe ueber einem Koerper k. Die Chowgruppe der nulldimensionalen Zyklen von X vom Grad Null ist eine Torsionsgruppe. Wir zeigen : wenn k ein p-adischer Koerper ist, dann ist der prim-zu-p Anteil dieser Gruppe endlich.

在$\mathbb{P}^1 \times \mathbb{P}^1$的循环覆盖上构造高 Chow 环，第二部分

Construction of higher Chow cycles on cyclic coverings of $\mathbb{P}^1 \times \mathbb{P}^1$, Part II

本文研究了在$\mathbb{P}^1 \times \mathbb{P}^1$的循环覆盖上构造高 Chow 环，并证明其生成不可分解部分的子群。

循环覆盖上的高阶Chow环的构造$\mathbb{P}^1 \times \mathbb{P}^1$

Construction of Higher Chow cycles on cyclic coverings of $\mathbb{P}^1 \times \mathbb{P}^1$

本研究针对$\mathbb{P}^1 \times \mathbb{P}^1$的循环覆盖曲面族，构造了$(2, 1)$型高阶Chow环，通过计算超越调节映射的像，证明其在$\mathbb{Z}$上线性无关且生成$\operatorname{rank} \ge 36 \cdot φ(N)$的子群。

零周的Chow-Witt群的双有理不变性

Birational invariance of the Chow-Witt group of zero-cycles

该研究证明了基域上光滑真概形的零周Chow-Witt群具有双有理不变性，为代数几何中相关不变量研究提供了理论结论。

Quintic surface over $p$-adic local fields with infinite $p$-primary torsion in the Chow group of 0-cycles

We construct a quintic surface over p-adic local fields such that there is infinite p-primary torsion in the Chow group of 0-cycles.

A simple construction of indecomposable higher Chow cycles in elliptic surfaces

I construct regulator indecomposable higher Chow cycles in elliptic surfaces satisfying certain conditions. As an application I give an alternative proof of a theorem of Gordon and Lewis, which asserts that there is a real regulator indecomposable cycles in a product of general elliptic curves.

Chow Groups of Abelian Varieties and Beilinson's Conjecture

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.

A filtration on the higher Chow group of zero cycles on an abelian variety

In this paper we extend Gazaki's results on the Chow groups of abelian varieties to the higher Chow groups. We introduce a Gazaki type filtration on the higher Chow group of zero-cycles on an abelian variety, whose graded quotients are connected to the Somekawa type $K$-group. Via the étale cycle map, we will compare this filtration with a filtration on the étale cohomology induced by the Hochschild-Serre spectral sequence. As an application over local fields, we obtain an estimate of the kernel of the reciprocity map.

附属于平面四次曲线的K3曲面上的高阶周环

Higher Chow cycles on K3 surfaces attached to plane quartics

该研究针对以平面光滑四次曲线为分歧点的K3曲面，利用四次曲线的双切线构造出(2,1)型高阶周环，并证明其在极一般曲面的高阶周群不可分解部分中生成秩2子群。

艾森斯坦K3曲面上的高周环

Higher Chow cycles on Eisenstein K3 surfaces

本文在具3阶非辛自同构的K3曲面族上构造(2,1)型高周环，通过退化论证与调节器映射显式计算，证明其对非常一般成员不可分解，为相关研究提供新构造与结果。

Indecomposable Higher Chow cycles on Jacobians

We construct some natural cycles with trivial regulator in the higher Chow groups of Jacobians. For hyperelliptic curves we use a criterion due to J. Lewis to prove that the cycles we construct are indecomposable, and then use a specialization argument to prove indecomposability for more general curves. Please note that math.AG/9909062 has become an obsolete version of some of the results in the present paper.

Chow Group of 1-cycles of the Moduli of Parabolic Bundles Over a Curve

We study the Chow group of 1-cycles of the moduli space of semistable parabolic vector bundles of fixed rank, determinant and a generic weight over a nonsingular projective curve over $\mathbb{C}$ of genus at least 3. We show that, the Chow group of 1-cycles remains isomorphic as we vary the generic weight. As a consequence, we can give an explicit description of the Chow group in the case of rank 2 and determinant $\mathcal{O}(x)$, where $x\in X$ is a fixed point.

Functional equations of the dilogarithm in motivic cohomology

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…

二次纤维化上 Chow 群的 Hasse 原理

Hasse principle for the Chow groups on quadric fibrations

本文研究了数域上 2 或 3 维二次纤维化的零循环相对 Chow 群，给出了其整体到局部映射为单射的充分条件。

On the Chow group of zero-cycles of a generalized Kummer variety

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.

Kummer曲面族上的高阶Chow环

Higher Chow cycles on a family of Kummer surfaces

本文在二维Kummer曲面族上构造了类型为(2,1)的高阶Chow环，并证明其不可分解部分生成秩至少18的子群，方法结合有限群作用与Picard-Fuchs算子。

Groupe de Chow d'une surface rationnelle sur un corps local

We determine the Chow group of zero-cycles on a rational surface X defined over a finite extension K of the field of p-adic numbers (p a prime) when X is split by an unramified extension of K.

On the Chow group of zero-cycles of Calabi-Yau hypersurfaces

We prove the existence of a canonical zero-cycle on a Calabi-Yau hypersurface X in a complex projective homogeneous variety. More precisely, we show that the intersection of any n divisors on X, n=dim X, is proportional to the class of a point on a rational curve in X.

Deformation theory of the Chow group of zero-cycles

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.

Algebraic cycles and Verra fourfolds

This note is about the Chow ring of Verra fourfolds. For a general Verra fourfold, we show that the Chow group of homologically trivial $1$-cycles is generated by conics. We also show that Verra fourfolds admit a multiplicative Chow-Künneth decomposition, and draw some consequences for the intersection product in the Chow ring of Verra fourfolds.

Torsion in Chow groups of zero cycles of homogeneous projective varieties

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

Relations among Heegner Cycles on families of abelian surfaces

We compute relations of rational equivalence among special codimensional two cycles on families of abelian surfaces using elements of higher chow groups.

Cycle class maps for Chow groups of zero-cycles with modulus

For a quasi-projective smooth scheme X of pure dimension d over a field k and an effective Cartier divisor D on X whose support is a simple normal crossing divisor, we construct a cycle class map from the Chow group of zero-cycles with modulus to the top cohomology of the dth relative Milnor K-sheaf.

Varieties over a finite field with trivial Chow group of 0-cycles have a rational point

Smooth projective varieties $X$ over a finite field $k$ with $CH_0(X\otimes \bar{k(X)})=\mathbb Z$ have a rational point, in particular Fano varieties. We also refer to http://link.springer.de/link/service/journals/00222/tocs.htm where the last version, cleaned from some typos, should be published online.

On the mod $p$ unramified cohomology of varieties having universally trivial Chow group of zero-cycles

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.

Selmer group associated to the Chow group of certain codimension two cycles

Let $X$ be a surface with geometric genus and irregularity zero which is defined over a number field $K$. Let $\mathscr{X}$ denote a smooth spread of $X$ over the spectrum of a Zariski open subset in the spectrum of the ring of integers and $A^2$ stands for the group of algebraically trivial cycles on schemes modulo rational equivalence. If $j^*: A^2(\mathscr{X})\to A^2(X)$ be the flat pull-back corresponding to the embedding $j:X\hookrightarrow \mathscr{X}$ then we prove that $\im(j^*)(K)/A^2(\mathscr{X})(K)$ is a torsion group. Here $\im(j^*)(K)$, $A^2(\mathscr{X})(K)$ stand for the cycles f…

高维Chow循环、循环三次四维流形与拉格朗日子变种

Higher Chow cycles, cyclic cubic fourfolds and Lagrangian subvarieties

本文研究了holomorphic李代数流形上的高维Chow循环，通过构造显式不可分解的(2,1)和(4,1)循环，首次在holomorphic李代数流形上提供了此类循环的例子，并探讨了循环在拉格朗日子变种上的限制现象。

Transcendence degree of zero-cycles and the structure of Chow motives

We show how the notion of the transcendence degree of a zero-cycle on a smooth projective variety X is related to the structure of the motive M(X). This can be of particular interest in the context of Bloch's conjecture, especially for Godeaux surfaces, when the surface is given as a finite quotient of a suitable quintic in P^3.

Chow-Kuenneth decomposition for 3- and 4-folds fibred by varieties with small Chow group of zero-cycles

Let $k$ be a field and let $Ω$ be a universal domain over $k$. Let $f:X \r S$ be a dominant morphism defined over $k$ from a smooth projective variety $X$ to a smooth projective variety $S$ of dimension $\leq 2$ such that the general fibre of $f_Ω$ has trivial Chow group of zero-cycles. For example, $X$ could be the total space of a two-dimensional family of varieties whose general member is rationally connected. Suppose that $X$ has dimension $\leq 4$. Then we prove that $X$ has a self-dual Murre decomposition, i.e. that $X$ has a self-dual Chow--Kuenneth decomposition which satisfies Murre's…

Tropical cycles and Chow polytopes

The Chow polytope of an algebraic cycle in a torus depends only on its tropicalisation. Generalising this, we associate a Chow polytope to any abstract tropical variety in a tropicalised toric variety. Several significant polyhedra associated to tropical varieties are special cases of our Chow polytope. The Chow polytope of a tropical variety $X$ is given by a simple combinatorial construction: its normal subdivision is the Minkowski sum of $X$ and a reflected skeleton of the fan of the ambient toric variety.

A Hochschild-cyclic approach to additive higher Chow cycles

Over a field of characteristic zero, we introduce two motivic operations on additive higher Chow cycles: analogues of the Connes boundary $B$ operator and the shuffle product on Hochschild complexes. The former allows us to apply the formalism of mixed complexes to additive Chow complexes building a bridge between additive higher Chow theory and additive $K$-theory. The latter induces a wedge product on additive Chow groups for which we show that the Connes operator is a graded derivation for the wedge product using a variation of a Totaro's cycle. Hence, the additive higher Chow groups with t…

Universal triviality of the Chow group of 0-cycles and the Brauer group

We prove that a smooth proper universally CH_0-trivial variety X over a field k has universally trivial Brauer group. This fills a gap in the literature concerning the p-torsion of the Brauer group when k has characteristic p.

0次Suslin同调与带模0-循环周群之比较

The comparison of the 0-th Suslin homology and Chow groups of 0-cycles with modulus

本文证明了在正特征域上，射影正规曲面及任意维数射影光滑概形中，带模零循环周群与0次Suslin同调之间的同构关系。

Higher Chow Cycles on an Abelian Surface and a non-Archimedean analogue of the Hodge-D-conjecture

We construct new indecomposable elements in the higher Chow group CH2(A,1) of a principally polarized Abelian surface over a non Archimedean local field, which generalize an element constructed by Collino. These elements are constructed using a generalization, due to Birkenhake and Wilhelm, of a classical construction of Humbert, along with some recent work of Bogomolov, Hassett and Tschinkel on deformations of rational curves on a K3 surface. They can be used to prove the non-Archimedean Hodge-D-conjecture - namely, the surjectivity of the boundary map in the localization sequence - in the ca…

Strong cycles and intersection products on good moduli spaces

We introduce conjectures relating the Chow ring of a smooth Artin stack $\mathcal{X}$ to the Chow groups of its possibly singular good moduli space $X$. In particular, we conjecture the existence of an intersection product on a subgroup of the full Chow group $A^*(X)$ coming from strong cycles on $\mathcal{X}$.

Restriction, corestriction and the characteristic homomorphism for 0-cycles of degree 0

We show that the characteristic homomorphism of Colliot-Th{é}l{è}ne and Sansuc from the Chow group of 0-cycles of degree~0 is compatible with restriction and corestriction maps.

某些带对合的K3曲面上的高阶周环

Higher Chow cycles on some K3 surfaces with involution

该研究针对2<r<18的情形，在格极化K3曲面族上构造了显式(2,1)型高阶周环，证明其不可分解部分非挠，属中间皮卡秩下首批此类显式例子，构造基于奇异二重平面模型，用退化方法完成证明。

$\mathbb A^1$-connectivity on Chow monoids v.s. rational equivalence of algebraic cycles

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…

The multiplicative group action on singular varieties and Chow varieties

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.

The universal regular quotient of the Chow group of 0-cycles on a singular projective variety

We show the existence of a regular universal quotient as a smooth commutative algebraic group of the Chow group of 0-cycles on a projective reduced variety, and give over the field of complex numbers an analytic description of it. This generalizes the classical theory of the Albanese. The Chow group of 0-cycles is then isomorphic to this smooth algebraic group if and only if it is finite dimensional in the sense of Mumford. This generalizes the classical theorem of Roitman.

On flat pullbacks for Chow groups

It is a fundamental property of the Chow groups of algebraic schemes that they are contra-functorial with respect to flat morphisms between schemes. While the pullback homomorphism is easy to define at the level of algebraic cycles, the crucial step is to show that the pullback of cycles preserves rational equivalence, so that it descends to the Chow groups. The purpose of this note is to give a natural sheaf theoretic proof of the preservation of rational equivalence under flat pullback on cycles.

Hodge-type conjecture for higher Chow groups

Let X be a smooth quasi-projective variety over the algebraic closure of the rational number field. We show that the cycle map of the higher Chow group to Deligne cohomology is injective and the higher Hodge cycles are generated by the image of the cycle map as conjectured by Beilinson and Jannsen, if the cycle map to Deligne cohomology is injective and the Hodge conjecture is true for certain smooth projective varieties over the algebraic closure of the rational number field. We also verify the conjecture on the surjectivity in the case of the complement of a union of general hypersurfaces in…

Nonvanishing of External Products for Higher Chow Groups

Consider an external product of a higher cycle and a usual cycle which is algebraically equivalent to zero. Assume there exists an algebraically closed subfield k such that the higher cycle and its ambient variety are defined over k, but the image of the usual cycle by the Abel-Jacobi map is not. Then we prove that the external product is nonzero if the image of the higher cycle by the cycle map to the reduced Deligne cohomology does not vanish. We also give examples of indecomposable higher cycles on even dimensional hypersurfaces of degree at least four in a projective space such that the la…

Rigidity for relative $0$-cycles

We present a relation between the classical Chow group of relative $0$-cycles on a regular scheme $\mathcal{X}$, projective and flat over an excellent Henselian discrete valuation ring, and the Levine-Weibel Chow group of 0-cycles on the special fiber. We show that these two Chow groups are isomorphic with finite coefficients under extra assumptions. This generalizes a result of Esnault, Kerz and Wittenberg.

修正对角闭链的消失性结果 II：志村曲线

Vanishing results for the modified diagonal cycles II: Shimura curves

本文证明了志村曲线三重积的Chow群中修正对角闭链的消失性结果，并构造了具有平凡自同构群且修正对角闭链消失的曲线例子。

A cycle class map from Chow groups with modulus to relative $K$-theory

Let $\bar{X}$ be a smooth quasi-projective $d$-dimensional variety over a field $k$ and let $D$ be an effective Cartier divisor on it. In this note, we construct cycle class maps from (a variant of) the higher Chow group with modulus of the pair $(\bar{X},D)$ in the range $(d+n, n)$ to the relative $K$-groups $K_n(\bar{X}, D)$ for every $n\geq 0$.

Infinitesimal Chow Dilogarithm

Let $C_{2}$ be a smooth and projective curve over the ring of dual numbers of a field $k.$ Given non-zero rational functions $f,g,$ and $h$ on $C_{2},$ we define an invariant $ρ(f\wedge g \wedge h) \in k.$ This is an analog of the real analytic Chow dilogarithm and the extension to non-linear cycles of the additive dilogarithm. Using this construction we state and prove an infinitesimal version of the strong reciprocity conjecture. Also using $ρ,$ we define an infinitesimal regulator on algebraic cycles of weight two which generalizes Park's construction in the case of cycles with modulus.

关于曲线与阿贝尔簇乘积的零维 Chow 群的滤波

A Filtration of the Chow Group of Zero-Cycles for a Product of Curves and an Abelian Variety

本文提出了一种针对曲线与阿贝尔簇乘积的零维 Chow 群的滤波方法，并通过 Somekawa K-群描述了其生成元和关系。

Indecomposable Higher Chow Cycles on Low Dimensional Jacobians

Title: Indecomposable Higher Chow Cycles on Low Dimensional Jacobians Authors: Alberto Collino Comments: AMS-TeX, 10 pages Subj-class: Algebraic Geometry MSC-class: 14C30 ;19E15 There is a basic indecomposable higher cycle K in Bloch's higher Chow group CH^{g}(J(C),1) on the Jacobian J(C) of a general hyperelliptic curve C of genus g. Consider K(t) the translation of K associated with a point t in C, we prove that in general K - K(t) is indecomposable if the genus is at least 3. Our tool is Lewis' condition for indecomposability. We show next that on the jacobian J(C) of a general curve C of g…

Zero-cycles on double EPW sextics

The Chow rings of hyperKähler varieties are conjectured to have a particularly rich structure. In this paper, we focus on the locally complete family of double EPW sextics and establish some properties of their Chow rings. First we prove a Beauville-Voisin type theorem for zero-cycles on double EPW sextics; precisely, we show that the codimension-4 part of the subring of the Chow ring of a double EPW sextic generated by divisors, the Chern classes and codimension-2 cycles invariant under the anti-symplectic covering involution has rank one. Second, for double EPW sextics birational to the Hilb…

Chow簇的Chow群

Chow groups of Chow varieties

本文研究Chow簇的有理Chow群，计算了指定范围的有理Chow群，验证其与有理奇异同调群的同构性，证明Chow幺半群自然完备化的对应同构性，建立稳定性并确定固定次数闭链空间的部分Chow群。

${\bf C}^*$-extensions of tori, higher Chow groups and applications to incidence equivalence relations for algebraic cycles.

Let X be a smooth projective variety of dimension n. If $p+q=n+1$ then Bloch has defined a ${\bf G}_m$-biextension E over the product of the Chow groups $CH^p_0(X)$ and $CH^q_0(X)$ of homologically trivial cycles. We prove that E is the pullback of the Poincare biextension over the product of intermediate Jacobians in characteristic zero. This is used to study various equivalence relations for algebraic cycles. In particular we reprove Murres result that Griffiths conjecture holds for codimension two cycles, i.e. every codim. two cycle algebraically and incidence equivalent to zero has torsion…

Algebraic cycles on a generalized Kummer variety

We compute explicitly the Chow motive of any generalized Kummer variety associated to any abelian surface. In fact, it lies in the rigid tensor subcategory of the category of Chow motives generated by the Chow motive of the underlying abelian surface. One application of this calculation is to show that the Hodge conjecture holds for arbitrary products of generalized Kummer varieties. As another application, all numerically trivial 1-cycles on arbitrary products of generalized Kummer varieties are smash-nipotent.