Cube invariance of higher Chow groups with modulus

Higher Arithmetic Chow Groups

We give a new construction of higher arithmetic Chow groups for quasi-projective arithmetic varieties over a field. Our definition agrees with the higher arithmetic Chow groups defined by Goncharov for projective arithmetic varieties over a field. These groups are the analogue, in the Arakelov context, of the higher algebraic Chow groups defined by Bloch. The degree zero group agrees with the a...

Additive Chow Groups with Higher Modulus and the Generalized De Rham-witt Complex

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(k, n;m), n, m ≥ 1. They prove TH(k, n; 1) ∼= Ω k/Z . In this paper we generalize their result and obtain an isomorphism TH(k, n;m) ∼= WmΩ n−1 k , where W−Ω·k is the generalized de Rham-Witt complex of Hesselholt-Madsen, generalizing the p-typical de Rham-Witt comp...

Extensions of motives and higher Chow groups

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 ...

Higher Chow Groups and Etale Cohomology

The main purpose of the present paper is to relate the higher Chow groups of varieties over an algebraically closed field introduced by S.Bloch [B1] to etale cohomology. We follow the approach suggested by the auther in 1987 during the Lumini conference on algebraic K-theory. The first and most important step in this direction was done in [SV1], where singular cohomology of any qfh-sheaf were c...