ALGEBRAIC CYCLES AND ADDITIVE CHOW GROUPS

SPENCER BLOCH

Abstract. A report on recent results and outstanding problems concerning additive chow groups.

“Mighty oaks from little acorns grow.”

1. Introduction

Additive algebraic K-theory means roughly “replace the algebraic group GL by the lie algebra 𝔤l where ever you see it”, [L]. Given the central role of algebraic cycles in motivic cohomology, one may ask for an algebraic cycle interpretation of additive K-theory. More than a simple restatement of the theory, such a geometric reformulation suggests new problems. What are motivic sheaves over k[t]/(t^{2})? What is the tangent space to the space of motives?

One should, I believe, have the following picture in mind. An algebraic circle is represented by the pair 𝔸^{1}, {0, t} for any t ≠ 0. It is natural, geometrically, to think of the limiting situation t → 0 as represented by

(1.1)  𝔸^{1}, 2(0)

As a simple example, for k a field

(1.2)  H_{M}^{1}(k, ℤ(1)) = P ic(𝔸_{k}^{1}, {0, t}) = 𝔾_{m}(k) = k^{×},

It is natural to write for the corresponding additive group

(1.3)  T H_{M}^{1}(k, ℤ(1)) := P ic(𝔸_{k}^{1}, 2(0)) = 𝔾_{a}(k) = k.

A word of warning. The notation T H_{M} suggests tangent space, but this is perhaps not the precise analogy. A better picture would be a sort of non-semi-stable degeneration, with a group like 𝔾_{m} which is constant in the parameter t degenerating to an additive group for t = 0.

Date: July 23, 2006.

1991 Mathematics Subject Classification. 11G55.

2. Hyperbolic versus Euclidean

Another analogy is the picture of the hyperbolic metric on a disk of expanding radius. As the radius tends to ∞, the metric tends to the euclidean metric. The link with motivic cohomology is summarized by the following 4-term exact sequence

(2.1) \begin{aligned}0 & ⟶ & H_{M}^{1}(ℂ, ℤ(2)) & ⟶ & B_{2}(ℂ) & ⟶^{δ} & ℂ^{×} ⊗ ℂ^{×} & ⟶ & K_{2}(ℂ) & ⟶ & 0 \\ & & ↓^{reg} & & ↓^{vol} \\ & & ℝ & ⟶^{=} & ℝ \end{aligned}

Here B_{2}(ℂ) has generators [x]_{2}, x ∈ ℂ−{0, 1}, and vol[x]_{2} is the hyperbolic volume of the tetrahedron in hyperbolic 3-space with vertices at infinity at the points 0, 1, ∞, x ∈ ℙ^{1}(ℂ). One defines δ[x]_{2} = x⊗(1−x), and the symbol [x]_{2} ∈ B_{2} satisfies the classical 5-term relation.

The additive analogue of 2.1 was worked out in [BE] using a Ktheoretic definition of the additive motivic group T H_{M}^{1}(k, ℤ(2)) involving the relative K-group K_{2}(O_{𝔸^{1},0}, (t^{2})): (2.2)

0 ⟶T H_{M}^{1}(k, ℤ(2)) ⟶ T B_{2}(k) ⟶^{T δ} k ⊗ k^{×} ⟶^{a⊗b↦adb/b} Ω_{k}^{1} ⟶0
↓^{ρ}  ↓^{ρ}
k  ⟶^{=} k

By definition, T B_{2}(k) is a k^{×}-module (not a k-vector space!) with generators ⟨x⟩_{2} which satisfy the 4-term information-theory equation first identified in this context by Kontsevich

(2.3) ⟨x⟩_{2} − ⟨y⟩_{2} + x ⋆ ⟨y/x⟩_{2} + (1 − x) ⋆ ⟨(1 − y)/(1 − x)⟩_{2} = 0

The regulator map ρ in (2.2) is defined by ρ⟨x⟩_{2} = x(1−x). One might hope that ρ⟨x⟩_{2} represents some Euclidean polyhedron with Euclidean volume x(1 − x). The action of k^{×} on the target of ρ is by the cube of the standard character, suggesting an interpretation in terms of volume in ℝ^{3}.

3. Cycles

We consider algebraic cycles on 𝔸^{1} × (ℙ^{1})^{n}, where 𝔸^{1} has parameter t and the ℙ^{1} have parameters t_{i}, 1 ≤ i ≤ n. We have divisors σ_{i} : t_{i} = 1 and δ_{i} = (t_{i})_{0} − (t_{i})_{∞}. Fix m ≥ 2. A closed subvariety Z ⊂ 𝔸^{1} × (ℙ^{1})^{n} will be said to be congruent to 1 mod t^{m} if scheme-theoretically

(3.1)  Z · {t^{m} = 0} ⊂ ⋃_{i=1}^{n} Z · σ_{i}

We consider algebraic cycles z = ∑ n_{ν}Z_{ν} such that all faces Z_{ν} · δ_{I} (I multi-index) have codim. ≥ #(I) in Z_{ν} and are congruent to 1 mod m. Write

(3.2)  T_{m}CH^{p}(k, q) = T_{m}H_{M}^{2p−q}(k, ℤ(p))

for the resulting higher Chow groups, generated by cycles of codim. p on 𝔸^{1} × (ℙ^{1})^{q−1}. The case of 0-cycles has been worked out by K. Rülling.

Theorem 3.1 (K. Rülling, [R]). T_{m}H_{M}^{n}(k, ℤ(n)) ≅ W_{m−1}Ω^{n−1}, the de Rham-Witt groups built from the “big” Witt ring on k.

Remark 3.2. Rülling uses a slightly different version of the congruence condition (3.1). I have checked casually that his results hold under (3.1), but this should perhaps be verified more carefully.

A cycle-theoretic version of (2.2) involves cycles of dimension 1. This has been considered by J. Park [P]. His main result is the construction of a non-trivial regulator map

(3.3)  ρ_{m,n} : T_{m}CH^{n−1}(k, n) → Ω_{k}^{n−3}.

(Of particular interest is the case n = 3, ρ_{m,3} : T_{m}CH^{2}(k, 3) → k.)

4. An abstract regulator construction

In trying to generalize Park’s regulator construction to cycles of dim. r > 1, one is led to consider meromorphic differential forms

(4.1)  \frac{t_{i_{0}} − 1}{t^{m+1}}dt_{i_{1}}/t_{i_{1}} ∧ … ∧ dt_{i_{r}}/t_{i_{r}}

on 𝔸_{t}^{1} × (ℙ_{t_{i}}^{1})^{n}. The most subtle aspect of his work, the careful control of signs necessary to verify that the regulator he constructs is trivial on boundaries of two-dimensional cycles on 𝔸^{1} × (ℙ^{1})^{n+1}, will not be attempted here; but let me sketch the construction of a generalized regulator on cycles.

Let X be a smooth, projective variety, and let D_{0}, …, D_{r} be effective Cartier divisors on X. Let Z ⊂ X be a closed subvariety of dimension r. We assume all intersections Z · D_{I} are either empty or have the correct dimension. In particular, Z ∩ ⋂_{0}^{r} D_{i} = ∅.

For z ∈ Z a closed point, we have surjections

(4.2)  H_{z}^{r}(Z, Ω_{Z/k}^{r}) ↠ H^{r}(Z, Ω_{Z/k}^{r}) ↠^{deg} k

(The group on the left is local cohomology. We do not assume Z smooth.)

Let ω be a meromorphic Kähler r-form on X which is regular on X − ⋃_{0}^{r} D_{i}. (For example, if X = 𝔸_{t}^{1} × (ℙ_{t_{i}}^{1})^{n} one might take ω = \frac{t_{i_{0}}−1}{t^{m+1}}dt_{i_{1}}/t_{i_{1}} ∧…∧dt_{i_{r}}/t_{i_{r}} with D_{0} : t = 0 and D_{j} = (t_{i_{j}})_{0} +(t_{i_{j}})_{∞}, 1 ≤ j ≤ r.

Chose i ≠ j ∈ [0, r] and define X(ij) = (D_{i} + D_{j}) ∩ ⋂_{h≠i,j} D_{h} (resp. Z(ij) = X(ij) ∩ Z). The r open sets X − (D_{i} + D_{j}), X − D_{h}, h ≠ i, j cover X−X(ij), and we may view ω as a Cech r−1 cocycle representing a class ω(ij) ∈ H^{r−1}(X − X(ij), Ω_{X/k}^{r}) (resp. by restriction ω_{Z}(ij) ∈ H^{r−1}(Z − Z(ij), Ω_{Z/k}^{r}).)

Z(ij) is a finite set of points which we can write as a disjoint union Z(ij) = Z_{j}(i) ⨿ Z_{i}(j), where Z_{j}(i) ⊂ Z − Z ∩ D_{j}. Write deg_{j}(i) ∈ k for the image of ω_{Z}(ij) under the composition

(4.3) H^{r−1}(Z − Z(ij), ω^{r}) ⟶^{∂} H_{Z(ij)}^{r}(Z, ω^{r}) ⟶^{proj} H_{Z_{j}(i)}^{r}(Z, ω^{r}) ⟶^{(4.2)} k

Clearly, deg_{j}(i) = − deg_{i}(j).

Now take ν ∈ [0, r] with ν, i, j all distinct. Note the sets Z_{j}(ν) and Z_{j}(i) coincide. (They are the intersection of Z with all the D_{h}, h ≠ j.) Furthermore, the open coverings

X − (D_{i} + D_{j}), X − D_{h}, h ≠ i, j (4.4)

X − (D_{ν} + D_{j}), X − D_{h}, h ≠ ν, j

agree upto reordering as open coverings of X − X(ij) − D_{j} = X − X(νj) − D_{j}. It follows that

(4.5)  deg_{i}(j) = − deg_{j}(i) =^{(∗)} ± deg_{j}(ν) = ∓ deg_{ν}(j).

But now, for a fourth index μ, the identity (*) above gives

(4.6)  deg_{i}(j) = ± deg_{ν}(j) = ± deg_{ν}(μ)

We conclude that, upto a sign which depends on i, j and the ordering of the divisors D_{h}, the quantity

(4.7)  deg_{j}(i) ∈ k

is independent of the choice of i, j.

Example 4.1. When Z is a curve (r = 1) the poles of ω|_{Z} are supported on Z ∩ (D_{0} + D_{1}). Our construction then amounts to taking residues along those poles lying on Z ∩ D_{0}.

References

[BE] Bloch, S., Esnault, H., The Additive Dilogarithm, Documenta Math. Extra Volume: Kazuya Kato’s Fiftieth Birthday (2003), pp. 131-155

[BE2] Bloch, S., Esnault, H., An additive version of higher Chow groups, Annales Scientifiques de l’École Normale Supérieure, 4-o série, 36 (2003), 463-477.

[L] Loday, J.-L., Cyclic Homology. Grundlehren der Mathematischen Wissenschaften 301. Springer-Verlag, Berlin, 1998. xx+513 pp.

[P] Park, J., Infinitesimal and Cyclic Aspects of Motives, Thesis, Univ. Chicago, (2006).

[R] Rülling, K., Additive Chow groups with higher modulus and the generalized de Rham-Witt complex, 74 pages (2005), to appear in J. Alg. Geom.

Dept. of Mathematics, University of Chicago, Chicago, IL 60637, USA

  E-mail address: bloch@math.uchicago.edu