Chern Classes With Modulus

Iwasa, Ryomei; Kai, Wataru

doi:10.1017/nmj.2018.52

Cited by 7 publications

(6 citation statements)

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“…In recent works of Binda-Saito [8] and Kerz-Saito [29], a theory of higher Chow groups with modulus was introduced, which generalizes the construction of the additive higher Chow groups. These groups, denoted CH * (X|D, * ), are designed to study the arithmetic and geometric properties of a smooth variety X with fixed conditions along an effective (possibly non-reduced) Cartier divisor D on it (see [29]), and are supposed to give a cycle-theoretic description of the relative K-groups K * (X, D), defined as the homotopy groups of the homotopy fiber of the restriction map K(X) → K(D) (see [28]).…”

Section: Introductionmentioning

confidence: 99%

Torsion in the 0-cycle group with modulus

Krishna

2018

Alg. Number Th.

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We show, for a smooth projective variety X over an algebraically closed field k with an effective Cartier divisor D, that the torsion subgroup CH 0 (X|D){l} can be described in terms of a relativeétale cohomology for any prime l = p = char(k). This extends a classical result of Bloch, on the torsion in the ordinary Chow group, to the modulus setting. We prove the Roitman torsion theorem (including p-torsion) for CH 0 (X|D) when D is reduced. We deduce applications to the problem of invariance of the prime-to-p torsion in CH 0 (X|D) under an infinitesimal extension of D.

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Section: Introductionmentioning

confidence: 99%

Torsion in the 0-cycle group with modulus

Krishna

2018

Alg. Number Th.

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“…W. Kai [29] established a moving lemma for cycle complexes with modulus which implies an appropriate contravariant functoriality of the Nisnevich version of (1.3) (see Theorem 2.12 for the precise statement). A work by R. Iwasa and W. Kai [27] provides Chern classes from the relative K-groups of the pair (X, D) to the Nisnevich motivic cohomology groups H * M,Nis (X|D, Z( * )), while a construction of F. Binda [3,Theorem 4.4.10] (see also [4]) gives cycle classes from the groups of higher 0-cycles with modulus CH d+n (X|D, n) to the relative K-groups K n (X, D). Other positive results are obtained in [32], [43] and [5].…”

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confidence: 99%

Relative Cycles With Moduli and Regulator Maps

Binda

Saito²

2017

J. Inst. Math. Jussieu

106

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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, those 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 relative differentials for a cycle satisfying the modulus condition, refining El-Zein's explicit construction of the fundamental class of a cycle. This is used to define a natural regulator map from the relative motivic complex of (X, D) to the relative de Rham complex. When X is defined over C, the same method leads to the construction of a regulator map to a relative version of Deligne cohomology, generalizing Bloch's regulator from higher Chow groups.Finally, when X is moreover connected and proper over C, we use relative Deligne cohomology to define relative intermediate Jacobians with modulus J r X|D of the pair (X, D). For r = dim X, we show that J r X|D is the universal regular quotient of the Chow group of 0-cycles with modulus.generalizing Bloch's computation in weight 1, Z X (1) ∼ = O × X [−1], and proving the first of the expected properties of the relative motivic cohomology groups (see Theorem 4.1). 1.7. Let (X, D) be as in 1.6. The second main technical result of this paper, presented in Section 7, is the construction, using the fundamental class in relative differentials, of regulator maps from the relative motivic complex Z X|D (r) to a the relative de Rham complex of XThe map φ dR is compatible with flat pullbacks and proper push forwards of pairs.When X is a smooth algebraic variety over the field of complex numbers, we can use the same technique to define regulator maps to a relative version of Deligne cohomology (see (8.10)) and to Betti cohomology with compact support, where ǫ is the morphism of sites and j : X → X is the open embedding of the complement of D in X, generalizing Bloch's regulator from higher Chow groups to Deligne cohomology, constructed in [7]. This regulator map is further studied in Section 9 in the case of additive Chow groups. Evaluated on suitable cycles, our regulator recovers Bloch-Esnault additive dilogarithm (introduced in [10]) and gives a refinement of [10, Proposition 5.1] (see (9.5)).1.8. Suppose that X is moreover connected and proper over C, and consider the induced maps in cohomology in degree 2r. We have a commutative diagram (see 10.1.1)and in analogy with the classical situation, we call the kernel J r X|D the r-th relative intermediate Jacobian of the pair (X, D). We note that they admit a description in terms of extensions groups Ext 1 in the abelian category of enriched Hodge structures defined by S.Bloch and V.Srinivas in [12].One can show that J r X|D fits into an exact sequen...

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“…Allowing the extra Y turns out useful in our related work [IK16] with Ryomei Iwasa where Y will be the projective space P r .…”

Section: 34mentioning

confidence: 99%

A Moving Lemma for Algebraic Cycles With Modulus and Contravariance

Kai

2019

International Mathematics Research Notices

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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 with modulus in the affine space that involves a new integer parameter, and Noether's normalization lemma over a Dedekind base.

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Chern Classes With Modulus

Abstract: In this paper, we construct Chern classes from the relative $K$ -theory of modulus pairs to the relative motivic cohomology defined by Binda–Saito. An application to relative motivic cohomology of henselian dvr is given.

Cited by 7 publications

References 20 publications

Torsion in the 0-cycle group with modulus

Torsion in the 0-cycle group with modulus

Relative Cycles With Moduli and Regulator Maps

A Moving Lemma for Algebraic Cycles With Modulus and Contravariance

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