Integral Regulators for Higher Chow Complexes

Building on Kerr, Lewis and Mueller-Stach's work on the rational regulator, we prove the existence of an integral regulator on higher Chow complexes and give an explicit expression. This puts firm ground under some earlier results and speculations on the torsion in higher cycle groups by K-L-M, Petras, and Kerr-Yang.


Introduction
Higher Chow groups were introduced by S. Bloch in the mid-80's as a geometric representation of algebraic K-theory [Bl1]. For X a smooth quasi-projective variety over an infinite field k, Bloch's Grothendieck-Riemann-Roch theorem identifies them rationally with certain graded pieces of K-theory: (1.1) CH p (X, n) ⊗ Q ≃ Gr P γ K alg n X ⊗ Q. As Bloch showed, these groups come with natural Chern class maps (1.2) AJ p,n Z : CH p (X, n) → H 2p−n D (X, Z(p)) to the cohomology of the underlying variety [Bl2], which "interpolate" Griffiths's Abel-Jacobi maps on Chow groups (i.e. K 0 ) and Borel's regulators on the higher K-theory of number fields. While abstractly defined, these maps were successfully computed in many specific cases by Bloch, Beilinson, Deninger, and others. However, an explicit general formula only emerged in the work of Kerr, Lewis and Müller-Stach [KLM, KL] in the early 00's. By introducing a subcomplex Z p R (X, •) ı ֒→ Z p (X, •) of cycles in good position with respect to the "wavefront" set of certain currents on (P 1 ) n , they are able to construct a map of complexes agreeing rationally with (1.2). (The explicit formula shall be recalled in §5).
Unfortunately, it appears very difficult to determine whether ı is an integral quasi-isomorphism, as expected in [KLM]. Indeed, the proof in [KL] that this inclusion of complexes is a Q-quasi-isomorphism makes essential use of Kleiman transversality in K-theory and hence of some form of (1.1). So the KLM formula only induces a "rational regulator" It is easy to see the problem: we could have that the class of Z in H 2p−1 (Z p R (k, 2p − 1)) and its AJ-image are m-torsion (but nonzero), whilst Z is a boundary in the larger complex (hence zero in CH p (k, 2p− 1)). That is, there would be some W ∈ Z p (k, 2p) \ Z p R (k, 2p) with ∂W = Z, but only mZ ∈ ∂ (Z p R (k, 2p)). Moreover, even if we could improve the result on ı (and eliminate this particular worry), it would remain inconvenient to find representative cycles in Z p R (X, n). An alternative is to extend KLM to a formula that works on all cycles. Doing this with one map of complexes on Z p (X, •) is probably too optimistic, as one can't just wish away the "wavefront sets" arising from the branch cuts in the {log(z i )}. Our first idea was to try an infinite family of homotopic maps on nested subcomplexes Z p ε (X, •) with union Z p (X, •), by allowing cycles in good position with respect to "perturbations" of these branch cuts by sufficiently small nonzero "phase" e iǫ , 0 < ǫ < ε. Provided one tunes the branches of log in the regulator currents accordingly, and the same θ is used for each z i , one gets a morphism of complexes on the ε-subcomplexes. Since the homotopy class of this morphism is independent of ǫ, this approach would define an integral refinement of AJ provided the ε → 0 limit of the "perturbed" subcomplexes gives all of Z p (X, •). Unfortunately, this is not true: there is a counterexample involving triples of functions on a curve, see §3. So a more subtle approach is required.
In particular, we need a way to vary phases ǫ i independently for the branches of log(z i ), so as to place weaker demands on our cycles. But this can never lead to a morphism of complexes from Z p (X, •), since this independence would conflict with the way the Bloch differential ∂ intersects cycles with all the facets. On the other hand, one has an explicit Z-homotopy equivalence for the inclusion N p (X, •) ⊂ Z p (X, •) of the normalized cycles, on which the differential restricts to just one facet [Bl3]. In N p (X, •), we now consider the "ε-subcomplex" N p ε (X, •), consisting of cycles which are in good position with respect to the (e iǫ 1 , . . . , e iǫn )-perturbed wavefront set for any (ǫ 1 , . . . , ǫ n ) belonging to B n ε := ǫ ∈ R n 0 < ǫ 1 < ε, 0 < ǫ 2 < e −1/ǫ 1 , . . . , 0 < ǫ n < e −1/ǫ n−1 . Our main technical results are . and induced by the perturbed KLM currents, are integrally homotopic.
These results are proved in § § 4 and 6, respectively. It is now easy to deduce that, taken over all ǫ, these morphisms induce a map of the form (1.2) refining (1.4), see §7. We conclude by indicating several applications of the KLM formula to torsion in §8 due to [KLM], Petras [Pe], Kerr-Yang [KY] which are now validated by our construction, and indicate future work in this direction.
We define the algebraic n-cube (over k) by . . , z n−1 ), and coordinate projections π i : n → n−1 sending (z 1 , . . . , z n ) to (z 1 , . . . ,ẑ i , . . . , z n ). We call the facets of n , and makes Z p (X, •) into a complex, with the higher Chow groups CH p (X, n) given by their homology. For convenience, we shall often use cohomological indexing:

A moving lemma.
We recall the subcomplex from [KLM]. Henceforth we shall take k to be a subfield of C, so we can consider the complex analytic spaces associated to components of a cycle Z. Let c p R (X, n) be the set of all the cycles Z ∈ c p (X, n) whose components (or rather, their analytizations) intersect X × (T z 1 ∩ · · · ∩ T z i ) and It is shown in [KL] that this subcomplex is Q-quasi-isomorphic to the original one: 2.3. Normalized cycles. Higher Chow groups may also be computed by complexes of cycles that have trivial boundary on all but one face.
In this section, we will write down an explicit retraction of Z p (X, •) onto the normalized cycle complex which is homotopic to the identity. The construction is derived from Bloch's manuscript [Bl3], by replacing the notations from (A 1 ) n (using {0, 1} as boundary) by (P 1 \ {1}) n (using {0, ∞} as boundary). In addition, Bloch uses a different definition for the normalized cycles: for any j}; so we need to apply a "conjugation" to the proof in [Bl3] as well. Define Theorem 2.6. The inclusion N p (X, •) ⊂ Z p (X, •) is an integral quasi-isomorphism.
Proof. Any Z ∈ Z p (X, n) may be lifted to c p (X, n), and we may add degenerate cycles to any element of c p (X, n) to force it into Z p ∞,n (X, n). The (well-defined) map given by this process is an isomorphism, and we shall tacitly equate Z p (X, n) and Z p ∞,n (X, n) in what follows. For each integer l ≤ n − 1, define h l : n+1 → n by stabilizes in any degree and so defines an endomorphism φ : Z p (X, •) → Z p (X, •), which is visibly homotopic to the identity.
To determine its image, write (for any Z ∈ Z p (X, n)) where the notation means that we pull back (the equations defining) ) then maps Z ′ to some Z ′′ ∈ Z p ∞,i−2 , and so forth until finally we reach Z p ∞,0 (X, n) = N p (X, n). Since all the ∂ • H l + H l • ∂ are zero on Z p ∞,0 , φ| N gives the identity on normalized cycles.
We have thus constructed a morphism φ : Z p (X, •) → N p (X, •), whose composition with the inclusion N p ֒→ Z p is homotopic to (resp. equal to) the identity on Z p (resp. N p ); thus φ and the inclusion are both quasi-isomorphisms.

Simple perturbations
The Kerr-Lewis moving lemma can only yield a rational regulator due to the passage through K-theory in the proof. Instead, one might consider maps of complexes on a nested family of subcomplexes of Z p R (X, •), given by "perturbing" the conditions defining Z p R (X, •). Though this turns out to be too naive, it is the first step toward a strategy that works.
Begin by defining Z p ε (X, •) to be the subcomplex of Z p (X, n) given by the cycles that intersect X×(T ǫ z 1 ∩· · ·∩T ǫ z i ) and X×(T ǫ z 1 ∩· · ·∩T ǫ z i ∩∂ k n ) properly for all 1 ≤ j ≤ n, 1 ≤ k < n and 0 < ǫ < ε. Here T ǫ z is given by arg(z) = π − ǫ, the "perturbation" of the branch cut of log(z) in the currents defined below.
In order for this nested family of subcomplexes to be any better than Z p R (X, •), we must have that their union gives us the original Z p : Unfortunately, this fails in a very simple case: , and H(z) = iz−1 3+z . Then we have Z = (F (z), G(z), H(z)) z∈P 1 ∈ Z 2 (pt, 3); but for all ε > 0, Z / ∈ Z 2 ε (pt, 3). More precisely, for any ǫ > 0, we have dim R (Z ∩ T ǫ z 1 ∩ T ǫ z 2 ∩ T ǫ z 3 ) = 0, not −1 (i.e. empty) as required for a proper-analytic intersection.
Thus we need to find another way to do the "perturbation", which will be given in the next section.

Multiple perturbations
In order to have Z an meet the deformations of {T z i } (and their intersections) properly -say, for an example like that in the above proofwe clearly need to make use of the extra degrees of freedom allowed by perturbing each "branch-cut phase" independently. For convenience, we shall use the multi-index notation ǫ := (ǫ 1 , . . . , ǫ n ) in what follows. Now we are thinking of T ǫ i z i as the location of the jump in the 0current log(z i ); these 0-currents will appear in the definition of the regulator-currents R ǫ Z appearing in the next section. To use these currents to define Abel-Jacobi maps, we will need them to induce morphisms of complexes from a subcomplex of Z p (X, •) to C 2p−• D (X, Z(p)).
Therefore the preimage∆ of ∆ in (S 1 ) n is real analytic. By the form of the inequalities in B ε , we know that we can choose an ε > 0 such that B n ε ∩∆ = ∅. (This follows from the implicit function theorem for ∆, and the fact that all derivatives of e −1/x limit to 0 at 0.) This means that Z intersects X × (T ǫ 1 z 1 ∩ · · · ∩ T ǫn zn ) properly ∀ǫ ∈ B • ε , as desired. Repeating the argument for X × (C * ) i × (P 1 C ) n−i and X × (C * ) i × ({0, ∞}) k ×(P 1 C ) n−i−k , we pick the minimum of the required values of ε, so that Z intersects X ×(T ǫ 1 z 1 ∩· · ·∩T ǫ i z i ) and X ×(T ǫ 1 z 1 ∩· · ·∩T ǫ i z i ∩∂ k n ) properly ∀i, k, ǫ ∈ B • ε , which means Z ∈ N p ε (X, n).

Abel-Jacobi maps
In this section, we'll use the strategy in [KLM] to define the Abel-Jacobi maps on our subcomplexes. 5.1. Definition of Deligne cohomology. The Deligne cohomology group H 2p+n D (X, Z(p)) is given by the n th cohomology of the complex Here D k (X) denotes currents of degree k on X an and C k (X, Z(k)) denotes C ∞ (co)chains of real codimension k and Z(k) = (2πi) k Z coefficients.
The cup product in Deligne cohomology is defined on the chain level by It becomes commutative upon passage to cohomology. (See [We] for a commutative chain-level construction.) Note that

KLM Currents.
Firstly we'll review the currents given in [KLM].
In particular, this means that on cycles belonging to Z p R (X, n)∩N p (X, n), our integral AJ map is given by the KLM formula.

Application to torsion cycles
Recent work of Kerr and Yang [KY] provides explicit representatives for generators of CH n (Spec(k), 2n − 1) where k is an abelian extension of Q, assuming the result we're giving here is correct. We'll check that when n = 2, 3, 4, the cycle given by [KY] satisfies the normal and proper intersection condition thus belongs to Z p R (X, 2p − 1) ∩ N p (X, 2p − 1). For n = 5 and higher cases, a normalization of their given generator is needed.
Let ξ N be an N th root of 1.
We can see that ∂ ∞ 3Z = 0 and ∂ ∞ 4Z = ∂ 0 4Z which can be cancelled by adding (for free) a degenerate cycle, so thatZ is normalized.
Also according to [KY], for N = 2 (k = Q) and n = 4, we have | 1 (2πi) 4 c D (Z )| = 7 1440 , which means it is 1440-torsion. The normalization of higher dimension case could be something to work out in the future.