Algebraic cycles and additive dilogarithm

The additive dilogarithm

We define an additive version of the Bloch group of a field, together with an additive dilogarithm and an Artin-Schreier realization.

A survey of the additive dilogarithm

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.

Infinitesimal Chow Dilogarithm

Let $C_{2}$ be a smooth and projective curve over the ring of dual numbers of a field $k.$ Given non-zero rational functions $f,g,$ and $h$ on $C_{2},$ we define an invariant $ρ(f\wedge g \wedge h) \in k.$ This is an analog of the real analytic Chow dilogarithm and the extension to non-linear cycles of the additive dilogarithm. Using this construction we state and prove an infinitesimal version of the strong reciprocity conjecture. Also using $ρ,$ we define an infinitesimal regulator on algebraic cycles of weight two which generalizes Park's construction in the case of cycles with modulus.

Functional equations of the dilogarithm in motivic cohomology

We prove relations between fractional linear cycles in Bloch's integral cubical higher Chow complex in codimension two of number fields, which correspond to functional equations of the dilogarithm. These relations suffice, as we shall demonstrate with a few examples, to write down enough relations in Bloch's integral higher Chow group CH^2(F,3) for certain number fields F to detect torsion cycles. Using the regulator map to Deligne cohomology, one can check the non-triviality of the torsion cycles thus obtained. Using this combination of methods, we obtain explicit higher Chow cycles generatin…

截断多项式环上曲线的无穷小二重对数

Infinitesimal dilogarithm on curves over truncated polynomial rings

本文在截断多项式环上的光滑射影曲线中，利用权3 Bloch复形构造了任意 m<r<2m 情形下的调节子及有理函数不变量，并推广互反定理，以期给出 CH^2(k_m,3) 无穷小部分的完全不变量。

A Hochschild-cyclic approach to additive higher Chow cycles

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…