Relative Cycles With Moduli and Regulator Maps

Abstract

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...

Cube invariance of higher Chow groups with modulus

“…Remark 2.9. For effective modulus pairs, the above definition coincides with the one introduced by Binda-Saito in [6]. The case q ≥ 1 is by definition.…”

“…Remark 5.9. When X is effective, the relative motivic cohomology is introduced in [6]. If moreover X • is smooth, the relative motivic cohomology is contravariantly functorial with respect to coadmissible left fine (not necessarily flat) morphisms of modulus pairs ( [8]).…”

Zero cycles with modulus and zero cycles on singular varieties

“…When D red is a strict normal crossing divisor, the complex j ! (Z) → I D → Ω 1 (X,D) is used in [5] to construct a universal regular quotient of CH 0 (X|D) deg 0 . One consequence of Proposition 8.1 is that it provides a cohomological proof that the Albanese variety with modulus of § 8.2.1 coincides with the one constructed in [5] when X is a surface and D red is strict normal crossing.…”

Mixed Hodge Structures With Modulus

“…Explicitely, this is given as follows (see [14, §5.2]). Let X Y be a proper C-curve that is obtained by collapsing Y into a single (usually singular) point (see [26,Chapter IV,[3][4]). Let G(X, Y ) be the generalized Jacobian of X with modulus Y in the sense of Rosenlicht-Serre [26], or, which amounts to the same, the Picard scheme Pic 0 (X Y ) of X Y .…”