Equivalence of various definitions of arithmetic Chow groups

Question

If I understand correctly, $n$-th arithmetic Chow group of arithmetic variety $X$ is defined as a quotient of the group of pairs of the form $(\sum\limits_in_iZ_i, g)$ where $Z := \sum\limits_in_iZ_i$ is an algebraic $n$-cycle in X, and $g$ is an $n$-current on $X(\mathbb{C})$ satisfying the equation $\partial \bar{\partial} g - \delta_Z = \omega$, for some smooth form $\omega$, and $\delta_Z(\alpha)=\sum\limits_{i}n_i\int\limits_{Z_i}\alpha$. I do not want to discuss the group structure on the set of such pairs, see Manin and Panchishkin's book for details.

Now, Goncharov, in his paper "Polylogarithms, regulators and Arakelov motivic complexes" defines something he calls "Arakelov Chow groups," and the definition seems to differ significantly, as it involves complexes of "algebraic simplices," which are just hyperplanes in generic position. He gives a proof (proposition 2.14, page 25) that the two are equivalent, but I have a rather hard time following it.

There is yet another definition of arithmetic Chow groups, "in terms of a Deligne complex of differential forms with logarithmic singularities along infinity," that I know nothing about. nLab remarks that if X is proper, the third definition and the first definition are equivalent.

Now, my question is this: can anyone give a motivation for the equivalence of the three definitions, or a reference where the connection between the three is discussed?

EDIT: I have found a great blog post in Russian that seems to explain the connection between the second and the third types of Chow groups in the question. However, the Arakelov-logarithmic connection is still just as puzzling.

Answer 1

I will start discussing the relationship between first and third definition.

The key point to understand the different definitions of arithmetic Chow groups is to understand the equation $\partial \bar \partial g-\delta_Z=\omega$. In this equation appears the second order differential operator $\partial \bar \partial$.

There is a cohomology theory, real Deligne-Beilinson cohomology, that contains information about the relationship between the real structure and the Hodge filtration of the cohomology of a complex manifold, where the operator $\partial \bar \partial$ appears naturally as the differential at certain degree. See "Burgos Gil, J. I.; Arithmetic Chow rings and Deligne-Beilinson cohomology. J. Algebraic Geom. 6, 2, (1997)" for a construction of a complex that computes Deligne Beilinson cohomology. It is related to the complex used by Goncharov.

Thus we may write the previous equation as $d_D g -\delta_Z=\omega$ where $d_D$ is a differential in a exotic complex that computes Deligne-Beilinson cohomology.

The next step is to notice that the current $g$ can always be represented by a differential form on $X\setminus |Z|$ that has logarithmic singularities along $Z$. This is proved in the original paper by Gillet and Soulé. Differential forms on $X$ with logarithmic singularities along $|Z|$ allow us to compute the cohomology of $X\setminus |Z|$ with its Hodge structure. Hence real Deligne Beilinson cohomology of $X\setminus |Z|$. From now on we assume that $g$ is the current associated to a differential form $g'$ with logarithmic singularities along $Z$.

If $p$ is the codimension of $Z$, the cycle $Z$ defines a class in $H^{2p}_D(X,\mathbb{R}(p))$ (real Deligne Beilinson cohomolgy). But more preciselly defines a class in cohomology with support $H^{2p}_{|Z|,D}(X,\mathbb{R}(p))$. The class of $Z$ with support in $|Z|$ can be represented in two different ways. With the current $\delta_Z$ or with a pair of differential forms $(\omega,g')$, where $\omega$ is smooth on the whole $X$ and $g'$ is smooth on $X\setminus |Z|$ and has logarithmic singularities along $|Z|$.

The condition $d_D g -\delta_Z=\omega$ is equivalent to the condition

"$(\omega,g')$ represent the class of $Z$ in real Deligne-Beilinson cohomology with support on $|Z|$"

A proof of this equivalence is given in Burgos Gil, J. I.; Green forms and their product. Duke Math. J. 75, 3, 529-574 (1994)

Hence we have replaced a differential equation by a cohomological condition. As pointed out by Myshkin this opens the door to an abstract definition of arithmetic Chow groups: Let H be a cohomology theory that has classes with support for cycles and classes of rational functions with some compatibility conditions. Let $\mathcal{C}$ be a particular choice of complexes that compute said cohomology. Then we can define arithmetic Chow groups with values in $\mathcal{C}$. The properties of the obtained arithmetic groups will depend on the properties of the complex.

This abstract point of view is worked out in Burgos Gil, J. I.; Kramer, J.; Kühn, U.; Cohomological arithmetic Chow rings. J. Inst. Math. Jussieu 6, 1, 1-172 (2007).

Examples:

1) Usual complex of differential forms: The obtained groups are isomorphic to the ones defined by Gillet and Soulé.

2) Differential forms with logarithmic singularities at infinity: We obtain groups with better Hodge teoretical properties.

3) Currents: The obtained arithmetic groups are fully covariant.

4) Forms with log log singularities at infinity: We obtain groups that, on one hand receive characteristic classes from certain vector bundles with singular metrics, namely the ones appearing when studying Shimura varieties. On the other hand they have well defined arithmetic degree.

Thus the main motivation of the third definition is "flexibility"

With respect to the second definition it is a completely different beast. The classical Chow groups has been extended by Bloch to higher Chow groups. This is the analogue at the level of cycles of the extension from $K_0$ to higher $K$-theory. The aim of Goncharov is to construct an explicit regulator from the complex of higher cycles to a complex that defines Deligne Beilinson cohomology, and then use this map to define higher arithmetic Chow groups as the cohomology of the cone of this map. He proves that the degree zero part of his construction agrees with classical arithmetic Chow groups by writing explicitely the cohomology of the cone.

There are two caveats in Goncharov's definition: first it is only defined for varieties over a field, not over an arithmetic ring limiting its usefulness. This is because the theory of higher Chow groups over a ring is not well developed yet.

Second there is an error in the construction. In fact the map $\mathcal{P}(n)$ in Theorem-Construction 2.3 is not a morphism of complexes. This error is solved in http://arxiv.org/abs/1502.05459.

Answer 2

The general idea is that for an arithmetic variety $X$, different choices of the pair $(H^\bullet,\mathcal{C})$, where $H^\bullet$ is a cohomology theory and $\mathcal{C}$ is a complex, yield different arithmetic Chow groups with different properties and information about $X$.

In that context, a higher arithmetic Chow group will be anything that looks like this

$$\widehat{\mathrm{CH}^p}(X,\mathcal{C}) = \widehat{\mathrm{Z}^p}(X,\mathcal{C})/ \widehat{\mathrm{Rat}^p}(X,\mathcal{C})$$

for the rational equivalence classes of codimension $p$ cycles.

Note that different choices of complexes for a fixed cohomology need not be equal.

Some apparently distinct defintions might give equivalent groups, as in the cases you mention, but the reason is that they are computing essentially the same data about the variety. That's what Goncharov proves in the proposition in that paper.

I think that it is expected that some refined adelic arithmetic Chow group should contain all the information accessible by such a groups, but at the moment that is just wishful thinking.

(I'll edit later to include some references)

(Hopefully someone more versed in these issues can give a more concrete answer to your question)