The cosimplicial scheme $$Delta^bullet = Δ^0 smallmatrix \to smallmatrix Δ^1 smallmatrix to smallmatrix ...;\quad Δ^n :=\Spec\Big(k[t_0,...c,t_n]/(\sum t_i -t)\Big)$$ was used in B to define higher Chow groups. In this note, we let t tend to 0 and replace Δ^\bullet by a degenerate version $$Q^\bullet = Q^0 smallmatrix to smallmatrix Q^1 smallmatrix to smallmatrix ...;\quad Q^n := \Spec\Big(k[t_0,...c,t_n]/(\sum t_i)\Big) $$ to define an additive version of the higher Chow groups. For a field k, we show the Chow group of 0-cycles on $Q^n$ in this theory is isomorphic to the absolute $(n-1)$-Kähler forms $Ω^{n-1}_k$.
An analogous degeneration on the level of de Rham cohomology associated to ``constant modulus'' degenerations of varieties in various contexts is discussed.
We begin a study of the intersection theory of the moduli spaces of degree two stable maps from two-pointed rational curves to arbitrary-dimensional projective space. First we compute the Betti numbers of these spaces using Serre polynomial and equivariant Serre polynomial methods developed by E. Getzler and R. Pandharipande. Then, via the excision sequence, we compute an additive basis for their Chow rings in terms of Chow rings of nonlinear Grassmannians, which have been described by Pandharipande. The ring structure of one of these Chow rings is addressed in a sequel to this paper.
The additive invariants of an algebraic variety is calculated in terms of those of the fixed point set under the action of additive and multiplicative groups, by using Bialynicki-Birula's fixed point formula for a projective algebraicset with a G_m-action or G_a-action. The method is also generalized to calculate certain additive invariants for Chow varieties. As applications, we obtain the Hodge polynomial of Chow varieties in characteristic zero and the number of points for Chow varieties over finite fields. As applications, we obtain the l-adic Euler-Poincare characteristic for the Chow var…
We prove moving lemma for additive higher Chow groups of smooth projective varieties. As applications, we prove the very general contravariance property of additive higher Chow groups. Using the moving lemma, we establish the structure of graded-commutative differential graded algebra (CDGA) on these groups.
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…
We show how to make the additive Chow groups of Bloch-Esnault, Ruelling and Park into a graded module for Bloch's higher Chow groups, in the case of a smooth projective variety over a field. This yields a a projective bundle formula as well as a blow-up formula for the additive Chow groups of a smooth projective variety. In case the base-field admits resolution of singularieties, these properties allow us to apply the technique of Guille'n and Navarro Aznar to define the additive Chow groups "with log poles at infinity" for an arbitrary finite-type k-scheme X. This theory has the usual propert…
We show that the multivariate additive higher Chow groups of a smooth affine $k$-scheme $\Spec (R)$ essentially of finite type over a perfect field $k$ of characteristic $\not = 2$ form a differential graded module over the big de Rham-Witt complex $\W_mΩ^{\bullet}_{R}$. In the univariate case, we show that additive higher Chow groups of $\Spec (R)$ form a Witt-complex over $R$. We use these structures to prove an étale descent for multivariate additive higher Chow groups.
As an attempt to understand motives over $k[x]/(x^m)$, we define the cubical additive higher Chow groups with modulus for all dimensions extending the works of S. Bloch, H. Esnault and K. Rülling on 0-dimensional cycles. We give an explicit construction of regulator maps on the groups of 1-cycles with an aid of the residue theory of A. Parshin and V. Lomadze.
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 perfect field $k$, we construct a triangulated category of mixed motives over $k[t]/{(t^{m+1})}$. The ext groups in this category are given by higher Chow groups, and additive higher Chow groups.
Motivated by the generalized Bloch conjecture, we formulate a conjecture about the Chow groups of Plücker hypersurfaces in Grassmannians. We prove weak versions of this conjecture.
In this paper we calculate the ring of unstable (possibly non-additive) operations from algebraic Morava K-theory K(n) to Chow groups with $\mathbb{Z}_{(p)}$-coefficients. More precisely, we prove that it is a formal power series ring on generators $c_i:K(n)\rightarrow CH^i \otimes \mathbb{Z}_{(p)}$, which satisfy a Cartan-type formula.
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.
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.
We study the structure of various invariants of the symmetric powers of a smooth projective curve in terms of that of the Jacobian of the curve. We generalise the results of Macdonald and Collino to various invariants including the Weil-cohomology theory, the higher Chow groups, the additive higher Chow groups and the rational $K$-groups.
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.
We prove a moving lemma for higher Chow groups with modulus, in the sense of Binda-Kerz-Saito, of projective schemes when the modulus is given by a very ample divisor. This provides one of the first cases of moving lemmas for cycles with modulus, not covered by the additive higher Chow groups. We apply this to prove a contravariant functoriality of higher Chow groups with modulus. We use our moving techniques to show that the higher Chow groups of a line bundle over a scheme, with the 0-section as the modulus, vanishes.
We construct a cycle class map from the higher Chow groups of 0-cycles to the relative $K$-theory of a modulus pair. We show that this induces a pro-isomorphism between the additive higher Chow groups of relative 0-cycles and relative $K$-theory of truncated polynomial rings over a regular semi-local ring, essentially of finite type over a characteristic zero field.
The Chow group of zero cycles in the moduli space of stable pointed curves of genus zero is isomorphic to the integer additive group. Let $M$ be monomial in this Chow group. If no two factors of $M$ fulfill a particular quadratic relation, then the monomial can be represented equivalently by a specific tree; otherwise, $M$ is mapped to zero under the stated isomorphism. Starting from this tree representation, we introduce a graphical algorithm for computing the corresponding integer for $M$ under the aforementioned isomorphism. The algorithm is linear with respect to the size of the tree.
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.
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 $A$.
The higher Chow group with modulus was introduced by Binda-Saito as a common generalization of Bloch's higher Chow group and the additive higher Chow group. In this paper, we study invariance properties of the higher Chow group with modulus. First, we formulate and prove "cube invariance," which generalizes $\mathbb{A}^1$-homotopy invariance of Bloch's higher Chow group. Next, we introduce the nilpotent higher Chow group with modulus, as an analogue of the nilpotent algebraic $K$-group, and define a module structure on it over the big Witt ring of the base field. We deduce from the module stru…
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.
We compute $CH^*BG_2$ and $CH^*BSpin_7$, using in particular the stratification method introduced by Vezzosi. We also give some information on the Chow rings of the finite groups $G_2(q)$. This new version includes an appendix, which will be published on its own as an addendum. It explains how one can use our results to produce a counter-example to the conjecture, due to Totaro, which predicted that the Chow ring of a finite group was generated by transfers of Chern classes.
We study the Chow group of zero-cycles on singular varieties using the cdh topology. We define the cdh versions of the zero-cycles and albanese maps. We prove results comparing these groups for a singular variety with the similar groups on the resolution of singularities. We use these to prove some results about the known Chow group of zero-cycles on surfaces and threefolds and some cases of arbitrary dimension.
We prove a moving lemma which implies the contravariance of Bloch-Esnault's additive higher Chow group in smooth affine varieties and Binda-Saito's higher Chow group (taken in the Nisnevich topology) in smooth varieties equipped with effective Cartier divisors. The new ingredients in the moving method are parallel translation {\em with modulus} in the affine space that involves a new integer parameter, and Noether's normalization lemma over a Dedekind base.
This note contains some examples of hyperkähler varieties $X$ having a group $G$ of non-symplectic automorphisms, and such that the action of $G$ on certain Chow groups of $X$ is as predicted by Bloch's conjecture. The examples range in dimension from $6$ to $132$. For each example, the quotient $Y=X/G$ is a Calabi-Yau variety which has interesting Chow-theoretic properties; in particular, the variety $Y$ satisfies (part of) a strong version of the Beauville-Voisin conjecture.
In this paper we prove the conjecture claiming that, over a flexible field, isotropic Chow groups coincide with numerical Chow groups (with ${\Bbb{F}}_p$-coefficients). This shows that Isotropic Chow motives coincide with Numerical Chow motives. In particular, homs between such objects are finite groups and $\otimes$ has no zero-divisors. It provides a large supply of new points for the Balmer spectrum of the Voevodsky motivic category. We also prove the Morava K-theory version of the above result, which permits to construct plenty of new points for the Balmer spectrum of the Morel-Voevodsky $…
Let X be a separated scheme of finite type over a field k and D a non-reduced effective Cartier divisor on it. We attach to the pair (X, D) a cycle complex with modulus, whose homotopy groups - called higher Chow groups with modulus - generalize additive higher Chow groups of Bloch-Esnault, Rülling, Park and Krishna-Levine, and that sheafified on $X_{Zar}$ gives a candidate definition for a relative motivic complex of the pair, that we compute in weight 1. When X is smooth over k and D is such that $D_{red}$ is a normal crossing divisor, we construct a fundamental class in the cohomology of re…
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 $i\leq c-1$.
We study mixed versions of the classical quotient functor from Chow motives to numerical motives. We compare two natural definitions, which turn out to be very different. We investigate fullness, conservativity and exactness of these two functors.
We define an integral Borel-Moore homology theory over finite fields, called arithmetic homology, and an integral version of Kato homology. Both types of groups are expected to be finitely generated, and sit in a long exact sequence with higher Chow groups of zero-cycles.
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…
We prove a dynamical version of the Mordell-Lang conjecture in the context of Drinfeld modules. We use analytic methods similar to the ones employed by Skolem, Chabauty, and Coleman for studying diophantine equations.
There are two infinitesimal (i.e., additive) versions of the $K$-theory of a field $F$: one was introduced by Cathelineau, which is an $F$-module, and another one introduced by Bloch-Esnault, which is an $F^*$-module. Both versions are equipped with a regulator map, when $F$ is the field of complex numbers. In our short paper we will introduce an extended version of Cathelineau's group, and a complex-valued regulator map given by the entropy. We will also give a comparison map between our extended version and Cathelineau's group. Our results were motivated by two unrelated sources: Neumann's w…
In the present paper we introduce and study the notion of an equivariant pretheory: basic examples include equivariant Chow groups, equivariant K-theory and equivariant algebraic cobordism. To extend this set of examples we define an equivariant (co)homology theory with coefficients in a Rost cycle module and provide a version of Merkurjev's (equivariant K-theory) spectral sequence for such a theory. As an application we generalize the theorem of Karpenko-Merkurjev on G-torsors and rational cycles; to every G-torsor E and a G-equivariant pretheory we associate a graded ring which serves as an …
Borel's construction of the regulator gives an injective map from the algebraic $K$-groups of a number field to its Deligne-Beilinson cohomology groups. This has many interesting arithmetic and geometric consequences. The formula for the regulator is expressed in terms of the classical polyogarithm functions. In this paper, we give a survey of the additive dilogarithm and the several different versions of the weight two regulator in the infinitesimal setting. We follow a historical approach which we hope will provide motivation for the definitions and the constructions.
We construct an explicit regulator map from the weigh n Bloch Higher Chow group complexto the weight n Deligne complex of a regular complex projective algebraic variety X. We define the Arakelovweight n motivic complex as the cone of this map shifted by one. Its last cohomology group is (a version of) the Arakelov Chow group defined by H. Gillet. and C.Soule. We relate the Grassmannian n-logarithms (defined as in [G5]) to geometry of the symmetric space for GL_n(C). For n=2 we recover Lobachevsky's formula for the volume of an ideal geodesic tetrahedron via the dilogarithm. Using the relations…
The goal of this paper is to prove a version of the non-abelian localization theorem for the rational equivariant K-theory of a smooth variety $X$ with the action of a linear algebraic group $G$. We then use this to prove a Riemann-Roch theorem which represents the completion of the higher equivariant K-theory of $X$ at various maximal ideals of the representation ring, in terms the equivariant higher Chow groups. This generalizes a result of Edidin and Graham to higher $K$-theory with rational coefficients.
We show that the additive higher Chow groups of regular schemes over a field induce a Zariski sheaf of pro-differential graded algebras, whose Milnor range is isomorphic to the Zariski sheaf of big de Rham-Witt complexes. This provides an explicit cycle-theoretic description of the big de Rham-Witt sheaves. Several applications are derived.
In this paper we prove a finiteness result concerning the Chow group of zero-cycles for varieties over $p$-adic local fields. In this final version, there are several corrections concerning mathematical symbols and reference to related known results.
We prove a moving lemma for the additive and ordinary higher Chow groups of relative $0$-cycles of regular semi-local $k$-schemes essentially of finite type over an infinite perfect field. From this, we show that the cycle classes can be represented by cycles that possess certain finiteness, surjectivity, and smoothness properties. It plays a key role in showing that the crystalline cohomology of smooth varieties can be expressed in terms of algebraic cycles.
