Morihiko Saito
We show that Mildenhall's theorem implies that the indecomposable higher Chow group of a self-product of an elliptic curve over the complex number field is infinite dimensional, if the elliptic curve is modular and defined over rational numbers. For the moment we cannot prove even the nontriviality of the indecomposable higher Chow group by a complex analytic method in this case.
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We use the elements in $K$-cohomology groups which are constructed by Flach and Mildenhall to obtain a finiteness result for the torsion part of the Chow group of a self-product of a modular curve.
In this paper we show that the map % $$\partial:CH^2(E_1 \times E_2,1)\otimes \Q \longrightarrow PCH^1(\XX_v)$$ % is surjective, where $E_1$ and $E_2$ are two non-isogenous semistable elliptic curves over a local field, $CH^2(E_1 \times E_2,1)$ is one of Bloch's higher Chow groups and $PCH^1(\XX_v)$ is a certain subquotient of a Chow group of the special fibre $\XX_{v}$ of a semi-stable model $\XX$ of $E_1 \times E_2$. On one hand, this can be viewed as a non-Archimedean analogue of the Hodge-$\D$-conjecture of Beilinson - which is known to be true in this case by the work of Chen and Lewis \c…
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.
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 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 …
Let X be a geometrically split, geometrically irreducible variety over a field F satisfying Rost nilpotence principle. Consider a field extension E/F and a finite field K. We provide in this note a motivic tool giving sufficient conditions for so-called outer motives of direct summands of the Chow motive of X_E with coefficients in K to be lifted to the base field. This going down result has been used S. Garibaldi, V. Petrov and N. Semenov to give a complete classification of the motivic decompositions of projective homogeneous varieties of inner type E_6 and to answer a conjecture of Rost and…
The decomposition theorem for smooth projective morphisms $π:\mathcal{X}\rightarrow B$ says that $Rπ_*\mathbb{Q}$ decomposes as $\oplus R^iπ_*\mathbb{Q}[-i]$. We describe simple examples where it is not possible to have such a decomposition compatible with cup-product, even after restriction to Zariski dense open sets of $B$. We prove however that this is always possible for families of $K3$ surfaces (after shrinking the base), and show how this result relates to a result by Beauville and the author on the Chow ring of $K3$ surfaces $S$. We give two proofs of this result, the second one involv…
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 give a purely cubical argument for the localization theorem for the cubical version of higher Chow groups.
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.
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.
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…
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.
Peterzil and Starchenko have proved the following surprising generalization of Chow's theorem: A closed analytic subset of a complex algebraic variety that is definable in an o-minimal structure, is in fact an algebraic subset. In this paper, we prove a non-archimedean analogue of this result.
Chow's Theorem for Linear Spaces
该研究证明了线性空间线集间保持相交关系的双射必由共线变换或对偶共线变换诱导，后者仅对三维广义射影空间成立。
For a quasiprojective variety S, we define a category CHM(S) of pure Chow motives over S. Assuming conjectures of Grothendieck and Murre, we show that the decomposition theorem holds in CHM(S). As a consequence, the intersection complex of S makes sense as an object IS of CHM(S). In the forthcoming part II we will give an unconditional definition of intersection Chow groups and study some of their properties.
We prove a decomposition theorem for the cohomological Chow group of 0-cycles on the double of a quasi-projective $R_1$-scheme over a field along a closed subscheme, in terms of the Chow groups, with and without modulus, of the scheme. This yields a significant generalization of the decomposition theorem of Binda-Krishna. As applications, we prove a moving lemma for Chow groups with modulus and an analogue of Bloch's formula for 0-cycles with modulus on singular surfaces. The latter extends a previous result of Binda-Krishna-Saito.
We consider proper, algebraic semismall maps f from a complex algebraic manifold X. We show that the topological Decomposition Theorem implies a "motivic" decomposition theorem for the rational algebraic cycles of X and, in the case X is compact, for the Chow motive of X.The result is a Chow-theoretic analogue of Borho-MacPherson's observation concerning the cohomology of the fibers and their relation to the relevant strata for f. Under suitable assumptions on the stratification, we prove an explicit version of the motivic decomposition theorem. The assumptions are fulfilled in many cases of i…
A theorem of Esnault, Srinivas and Viehweg asserts that if the Chow group of 0-cycles of a smooth complete complex variety decomposes, then the top-degree coherent cohomology group decomposes similarly. In this note, we prove a similar statement for Chow groups of arbitrary codimension, provided the variety satisfies the Lefschetz standard conjecture.
A Chow-Type theorem from asymptotic directions
该研究针对闭纯$d$维复解析集，建立了基于射影渐近方向集的度量Chow型代数性刻画，并给出了无穷远切锥可定义且实维数不超过$2d$时复解析集为代数集的应用结论。
We present a proof of Chow's theorem using two results of Errett Bishop retated to volumes and limits of analytic varieties. We think this approach suggested a long time ago in the beautiful book by Gabriel Stolzenberg, is very attractive and easier for students and newcomers to understand, also the theory presented here is linked to areas of mathematics that are not usually associated with Chow's theorem. Furthermore, Bishop's results imply both Chow's and Remmert-Stein's theorems directly, meaning that this approach is more economic and just as profound as Remmert-Stein's proof. At the end o…
Naruki gave an explicit construction of the moduli space of marked cubic surfaces, starting from a toric variety and proceeding with blow ups and contractions. Using his result, we compute the Chow groups and the Chern classes of this moduli space.As an application we relate a recent result of Freitag on the Hilbert polynomial of a certain ring of modular forms to the Riemann-Roch theorem for the moduli space.
On the first Steenrod square for Chow groups
本文构建了同调第一Steenrod平方运算的弱版本，无需假设基域特征，并将Karpenko关于二次型第一Witt指数奇偶性的定理推广到特征2基域情形。
We prove analogs of Whitehead's theorem (from algebraic topology) for both the Chow groups and for the Grothendieck group of coherent sheaves: a morphism between smooth projective varieties whose pushforward is an isomorphism on the Chow groups, or on the Grothendieck group of coherent sheaves, is an isomorphism. As a corollary, we show that there are no nontrivial naive $\mathbb{A}^1$-homotopy equivalences between smooth projective varieties.
Let X be a smooth projective variety of dimension n on which rational and homological equivalence coincide for algebraic p-cycles in the range 0\leq p\leq s. We show that the homologically trivial sector of rational Lawson homology L_pH_k(X,Q)_{hom} vanishes for 0\leq n-p\leq s+2. This is an analogue of a theorem of C. Peters in "dual dimensions". Together with Peters' theorem we get that the natural transformation L_pH_k(X,Q)--> H_k(X,Q) is injective for all p and k when X is a smooth projective variety of dimension 4 and Ch_0(X) =Z.
Extended operational Chow group and Lefschetz (1,1)-theorem
本文通过引入扩展的操作Chow群，研究了从操作Chow群到Hodge (p,p)-类的Bloch-Gillet-Soulé（BGS）周类映射，证明了在有理奇点的正规曲面和孤立奇点情形下，该映射的满射性，从而推广了Lefschetz (1,1)-定理。
Let $\mathbb{M}:=(M(X),p)$ be a direct summand of the motive associated with a geometrically split, geometrically variety over a field $F$ satisfying the nilpotence principle. We show that under some conditions on an extension $E/F$, if $\mathbb{M}$ is a direct summand of another motive $M$ over an extension $E$, then $\mathbb{M}$ is a direct summand of $M$ over $F$.
In this short note, we prove a comparision theorem between Levine-Serpé's equivariant higher Chow groups of an algebraic variety equipped with an action of a finite group and ordinary higher Chow groups of its fixed points. As a consequence, we show that the equivariant motivic spectral sequence degenerates rationally. This yields a Riemann-Roch Theorem for equivariant algebraic $K$-theory.
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.
In this article we generalize the injectivity of the push-forward homomorphism at the level of Chow groups, induced by the closed embedding of $\Sym^m C$ into $\Sym^n C$ for $m\leq n$, where $C$ is a smooth projective curve, to symmetric powers of a smooth projective variety of higher dimension. We also prove the analog of this theorem for product of symmetric powers of smooth projective varieties. As an application we prove the injectivity of the push-forward homomorphism at the level of Chow groups, induced by the closed embedding of self product of theta divisor into the self product of the…
Let X be a smooth projective variety. Starting with a finite set of cycles on powers X^m of X, we consider the Q-vector subspaces of the Q-linear Chow groups of the X^m obtained by iterating the algebraic operations and pullback and push forward along those morphisms X^l -> X^m for which each component X^l -> X is a projection. It is shown that these Q-vector subspaces are all finite-dimensional, provided that the Q-linear Chow motive of X is a direct summand of that of an abelian variety.
In this appendix we tabulate preliminary data used in establishing Theorem 1-14 in the paper ``The Chow rings of generalized Grassmannians''
Motivic Lefschetz theorem for twisted Milnor hypersurfaces
本研究针对Milnor超曲面内扭曲形式的光滑超平面截面，证明其Grothendieck-Chow motive可分解为相关循环代数的Severi-Brauer簇与极大交换子域的motive平移拷贝的直和，证明依赖于Galois作用在等变Chow群上的非平凡性。
Chow Groups: A Structure Theorem, RIEMANN-ROCH without denominators and ARTIN approximation
本文研究分歧正则局部环上的 Chow 群问题，给出该类环的刻画，证明特定开子集上 $(i-1)!\mathbb{A}^i(U)=0$，并建立 Hensel 化与完备化 Chow 群之间的构造性关系。
We relate the notion of finite dimensionality of the Chow motive M(X) of a smooth projective variety X (as defined by S. Kimura) with the Conjectures of Beilinson, Bloch and Murre on the existence of a filtration on the Chow ring CH(X). We show (Theorem 14) that finite dimensionality of M(X) implies uniqueness, up to isomorphism, of Murre's decomposition of M(X). Conversely (Theorem 17), Murre's Conjecture for a suitable m-fold product of X by itself implies finite dimensionality of M(X). We also show (Theorem 27) that, for a surface X with trivial geometrical genus, the motive M(X) is finite …
In this paper, a generic intersection theorem in projective differential algebraic geometry is presented. Precisely, the intersection of an irreducible projective differential variety of dimension d>0 and order h with a generic projective differential hyperplane is shown to be an irreducible projective differential variety of dimension d-1 and order h. Based on the generic intersection theorem, the Chow form for an irreducible projective differential variety is defined and most of the properties of the differential Chow form in affine differential case are established for its projective differ…
Ceci est un rapport sur l'article "A finiteness theorem for zero-cycles over p-adic fields" (arXiv:math/0605165) de Shuji Saito et Kanetomo Sato. ----- This is a survey on the paper "A finiteness theorem for zero-cycles over p-adic fields" (arXiv:math/0605165) by Shuji Saito and Kanetomo Sato.
We show that the higher Chow groups with modulus of Binda-Kerz-Saito for a smooth quasi-projective scheme $X$ is a module over the Chow ring of $X$. From this, we deduce certain pull-backs, the projective bundle formula, and the blow-up formula for higher Chow groups with modulus. We prove vanishing of $0$-cycles of higher Chow groups with modulus on various affine varieties of dimension at least two. This shows in particular that the multivariate analogue of Bloch-Esnault--Rülling computations of additive higher Chow groups of 0-cycles vanishes.
The Chow groups of codimension-p algebraic cycles modulo rational equivalence on a smooth algebraic variety X have steadfastly resisted the efforts of algebraic geometers to fathom their structure. This book explores a "linearization" approach to this problem, focusing on the infinitesimal structure of the Chow groups near their identity elements. This method was adumbrated in recent work of Mark Green and Phillip Griffiths. Similar topics have been explored by Bloch, Stienstra, Hesselholt, Van der Kallen, and others. A famous formula of Bloch expresses the Chow groups as Zariski sheaf cohomol…
We compare various groups of 0-cycles on quasi-projective varieties over a field. As applications, we show that for certain singular projective varieties, the Levine-Weibel Chow group of 0-cycles coincides with the corresponding Friedlander-Voevodsky motivic cohomology. We also show that over an algebraically closed field of positive characteristic, the Chow group of 0-cycles with modulus on a smooth projective variety with respect to a reduced divisor coincides with the Suslin homology of the complement of the divisor. We prove several generalizations of the finiteness theorem of Saito and Sa…
Bloch and Esnault defined additive higher Chow groups with modulus (m+1) on the level of zero cycles over a field k, denoted by TH^n(k,n;m), n,m >0. They prove that TH^n(k,n;1) is isomorphic to the group of absolute Kaehler differentials of degree n-1 over k. In this paper we generalize their result and show that TH^n(k,n;m) is isomorphic to W_mΩ^{n-1}_k, the group of degree n-1 elements in the generalized de Rham-Witt of length m over k. This complex was defined by Hesselholt and Madsen and generalizes the p-typical de Rham-Witt complex of Bloch-Deligne-Illusie. Before we can prove this theor…
For a smooth quasi-projective scheme $X$ over a field $k$ with an action of a reductive group, we establish a spectral sequence connecting the equivariant and the ordinary higher Chow groups of $X$. For $X$ smooth and projective, we show that this spectral sequence degenerates, leading to an explicit relation between the equivariant and the ordinary higher Chow groups. We obtain several applications to algebraic $K$-theory. We show that for a reductive group $G$ acting on a smooth projective scheme $X$, the forgetful map $K^G_i(X) \to K_i(X)$ induces an isomorphism $K^G_i(X)/{I_G K^G_i(X)} \co…
This is a reproduction of my thesis. By using higher K-theory(Chern character, cyclic homology, effacement theorem and etc), we provide an answer to a question by Green- Griffiths which says the tangent sequence to Bloch-Gersten-Quillen sequence is Cousin resolution.
The purpose of this paper is to prove an equivariant Riemann-Roch theorem for schemes or algebraic spaces with an action of a linear algebraic group $G$. For a $G$-space $X$, this theorem gives an isomorphism between a completion of the equivariant Grothendieck group and a completion of equivariant equivariant Chow groups. The key to proving this isomorphism is a geometric description of completions of the equivariant Grothendieck group. Besides Riemann-Roch, this result has some purely $K$-theoretic applications. In particular, we prove a conjecture of Köck (in the case of regular schemes) an…
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…
We prove an extension of the Kato-Saito class field theory for smooth projective schemes over a finite field to schemes with singularities. As an application, we obtain Bloch's formula for the Chow groups of 0-cycles on such schemes. We identify the Chow group of 0-cycles on a normal projective scheme over an algebraically closed field to the Suslin homology of its regular locus. Our final result is a Roitman torsion theorem for smooth quasi-projective schemes over algebraically closed fields. This completes the missing $p$-torsion part in the torsion theorem of Spiess and Szamuely.
Given a proper family of varieties over a smooth base, with smooth total space and general fibre, all over a finite field k with q elements, we show that a finiteness hypothesis on the Chow groups, CH_i, i=0,1,...,r, of the fibres in the family leads to congruences mod q^{r+1} for the number of rational points in all the fibres over k-rational points of the base. These hypotheses on the Chow groups are expected to hold for families of low degree intersections in many Fano varieties leading to a broad generalisation of the theorem of Ax--Katz, as well as results of the author and C. S. Rajan. A…
Effective bounds for the finite number of surjective holomorphic maps between canonically polarized compact complex manifolds of any dimension with fixed domain are proven. Both the case of a fixed target and the case of varying targets are treated. In the case of varying targets, bounds on the complexity of Chow varieties are used.
We define a filtration on the Chow groups of a smooth projective variety X over a field k by using the cycle map into continuous l-adic etale cohomology. The main theorem says that if k is a function field in one variable over a finite field, this filtration for zero cycles is of length at most one modulo the kernel of the cycle map.
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.
We study intersection theoretic problems in the setting of Chow-Witt groups with coefficients in a fixed Milnor-Witt cycle algebra over a perfect field. We prove that the product maps on such groups satisfy the following property: given two points in a regular local scheme with supports which do not intersect properly, their product vanishes. This gives an analogue of Serre's vanishing result for intersection multiplicities.
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.
We prove an Atiyah-Segal isomorphism for the higher $K$-theory of coherent sheaves on quotient Deligne-Mumford stacks over $\C$. As an application, we prove the Grothendieck-Riemann-Roch theorem for such stacks. This theorem establishes an isomorphism between the higher $K$-theory of coherent sheaves on a Deligne-Mumford stack and the higher Chow groups of its inertia stack. Furthermore, this isomorphism is covariant for proper maps between Deligne-Mumford stacks.
Varieties over $\bar{\mathbb{Q}}$ with infinite Chow groups modulo almost all primes
本文证明 Fermat 三次曲线三次积的 Chow 群对所有大于 5 的素数模均无限，从而给出首个在几乎所有素数模下 Chow 群无限的光滑射影簇例子。
The moduli space $\bar{M}_A$ of weighted pointed stable curves of genus zero is stratified according to the degeneration types of such curves. We show that the homology groups of the moduli space $\bar{M}_A$ are generated by the strata of $\bar{M}_A$ and give all additive relations between them. We also observe that the Chow groups $A_i \bar{M}_A$ and the homology groups $H_{2i} \bar{M}_A$ are isomorphic. This generalizes Kontsevich-Manin's and Losev-Manin's theorems to arbitrary weight data $A$.
We adapt for algebraically closed fields $k$ of characteristic greater than $2$ two results of Voisin, on the decomposition of the diagonal of a smooth cubic hypersurface $X$ of dimension $3$ over $\mathbb C$, namely: the equivalence between Chow-theoretic and cohomological decompositions of the diagonal of those hypersurfaces and the fact that the algebraicity (with $\mathbb Z_2$-coefficients) of the minimal class $θ^4/4!$ of the intermediate jacobian $J(X)$ of $X$ implies the Chow-theoretic decomposition of the diagonal of $X$. Using the second result, the Tate conjecture for divisors on sur…
Over any complex cubic hypersurface of dimension at least 2, the Chow group of 1-dimensional cycles is spanned by the lines lying on the hypersurface. The smooth case has already been given several other proofs. -- On montre que sur toute hypersurface cubique complexe de dimension au moins 2, le groupe de Chow des cycles de dimension 1 est engendré par les droites. Le cas lisse est un théorème connu. La démonstration ici donnée repose sur un résultat sur les surfaces géométriquement rationnelles sur un corps quelconque (1983), obtenu via la K-théorie algébrique.
A theorem of Esnault, Srinivas and Viehweg asserts that if the Chow group of 0-cycles of a smooth complete complex variety decomposes, then the top-degree coherent cohomology group decomposes similarly. In this note, we prove that (a weak version of) the converse holds for varieties of dimension at most 5 that have finite-dimensional motive and satisfy the Lefschetz standard conjecture. The proof is based on Vial's construction of a refined Chow-Kunneth decomposition for these varieties.
For $X$ a smooth scheme acted on by a linear algebraic group $G$ and $p$ a prime, the equivariant Chow ring $CH^*_G(X)\otimes \mathbb{F}_p$ is an unstable algebra over the Steenrod algebra. We compute Lannes's $T$-functor applied to $CH^*_G(X)\otimes \mathbb{F}_p$. As an application, we compute the localization of $CH^*_G(X)\otimes \mathbb{F}_p$ away from $n$-nilpotent modules over the Steenrod algebra, affirming a conjecture of Totaro as a special case. The case when $X$ is a point and $n = 1$ generalizes and recovers an algebro-geometric version of Quillen's stratification theorem proved by …
In this note, we would like to propose a suitable extension of the arithmetic Chow group of codimension one, in which the Hodge index theorem holds. We also prove an arithmetic analogue of Bogomolov's instability theorem for rank 2 vector bundles on arbitrary regular projective arithmetic varieties.
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