Integral Regulators for Higher Chow Complexes
Building on Kerr, Lewis, and Müller-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 Kerr-L...
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| citation_txt | Integral Regulators for Higher Chow Complexes / M. Li // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 11 назв. — англ. |
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| description | Building on Kerr, Lewis, and Müller-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 Kerr-Lewis-Müller-Stach, Petras, and Kerr-Yang.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 118, 12 pages
Integral Regulators for Higher Chow Complexes
Muxi LI
University of Science and Technology, Hefei, Anhui, P.R. China
E-mail: limuxi@ustc.edu.cn
Received May 12, 2018, in final form October 31, 2018; Published online November 03, 2018
https://doi.org/10.3842/SIGMA.2018.118
Abstract. Building on Kerr, Lewis and Müller-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 Kerr–Lewis–Müller-Stach, Petras, and Kerr–Yang.
Key words: integral regulator; higher Chow groups; algebraic cycles; Abel–Jacobi map
2010 Mathematics Subject Classification: 14C15; 14C25; 19F27
1 Introduction
Higher Chow groups were introduced by S. Bloch in the mid-80’s as a geometric representation of
algebraic K-theory [1]. 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:
CHp(X,n)⊗Q ' GrPγK
alg
n X ⊗Q. (1.1)
As Bloch showed, these groups come with natural Chern class maps
AJp,nZ : CHp(X,n)→ H2p−n
D (X,Z(p)) (1.2)
to the cohomology of the underlying variety [2], which “interpolate” Griffiths’s Abel–Jacobi
maps on Chow groups (i.e., K0) 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 [4, 5]. By introducing a subcomplex ZpR(X, •) ı
↪→ Zp(X, •)
of cycles in good position with respect to the “wavefront” set of certain currents on
(
P1
)n
, they
are able to construct a map of complexes
ÃJ : ZpR(X, •)→ C2p−•
D (X,Z(p)) (1.3)
agreeing rationally with (1.2). While the explicit formula will be recalled in Section 5, we remark
that CmD (X,Z(p)) consists of triples (T,Ω, R) where T ∈ Cm(X, (2πi)pZ) is a smooth chain of
real codimension m, and Ω ∈ F pDm(X), R ∈ Dm−1(X) are currents. The map (1.3) sends
higher Chow cycles Z to triples (TZ ,ΩZ , RZ), and the relations that make (1.3) a morphism of
complexes are ∂TZ = T∂Z , d[ΩZ ] = Ω∂Z , and d[RZ ] = R∂Z + ΩZ − δTZ (where δTZ is the current
of integration over TZ).
This paper is a contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko
Yui. The full collection is available at http://www.emis.de/journals/SIGMA/modular-forms.html
mailto:limuxi@ustc.edu.cn
https://doi.org/10.3842/SIGMA.2018.118
http://www.emis.de/journals/SIGMA/modular-forms.html
2 M. Li
At first glance, the “KLM formula” (1.3) looks well-adapted to detecting torsion. For exam-
ple, if X = Spec(k) is a point, then a portion of (1.3) takes the form
· · · // ZpR(k, 2p)
∂ //
W 7→ (2πi)pW∩T2p
��
ZpR(k, 2p− 1)
∂ //
Z 7→ 1
(2πi)p−1
∫
Z R2p−1
��
ZpR(k, 2p− 2) //
0
��
· · ·
· · · // Z(p)⊕ 0⊕ 0 �
� // 0⊕ 0⊕ C // 0⊕ 0⊕ 0 // · · · ,
where T2p = R2p
<0 ⊂
(
P1
)2p
and R2p−1 is a certain (2p − 2)-current on
(
P1
)2p−1
. We want
to detect torsion in CHp(k, 2p − 1) by the middle map; denote the image of Z ∈ ker(∂) by
R(Z) ∈ C/Z(p). In particular, if Z1 :=
(
1 − 1/t, 1 − t, t−1
)
t∈P1 ∈ Z2(Q, 3), we find that
R(Z1) = π2/6 ∈ C/(2πi)2Z, in agreement with the known result that CH2(Q, 3) is 24-torsion
(see [10]).
Unfortunately, it appears very difficult to determine whether ı is an integral quasi-isomor-
phism, as expected in [4]. Indeed, the proof in [5] 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”
AJp,nQ : CHp(X,n)→ H2p−n
D (X,Q(p)). (1.4)
It is easy to see the problem: we could have that the class of Z in H2p−1
(
ZpR(k, 2p − 1)
)
and
its ÃJ-image are m-torsion (but nonzero), whilst Z is a boundary in the larger complex (hence
zero in CHp(k, 2p− 1)). That is, there would be some W ∈ Zp(k, 2p) \ZpR(k, 2p) with ∂W = Z,
but only mZ ∈ ∂
(
ZpR(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 ZpR(X,n).
An alternative is to extend KLM to a formula that works on all cycles. Doing this with
one map of complexes on Zp(X, •) is probably too optimistic, as one can’t just wish away the
“wavefront sets” arising from the branch cuts in the {log(zi)}. Our first idea was to try an infinite
family of homotopic maps on nested subcomplexes Zpε (X, •) with union Zp(X, •), by allowing
cycles in good position with respect to “perturbations” of these branch cuts by sufficiently
small nonzero “phase” eiε, 0 < ε < ε. Provided one tunes the branches of log in the regulator
currents accordingly, and the same ε is used for each zi, one gets a morphism of complexes
on the ε-subcomplexes. Since the homotopy class of this morphism is independent of ε, this
approach would define an integral refinement of ÃJ provided the ε→ 0 limit of the “perturbed”
subcomplexes gives all of Zp(X, •). Unfortunately, this is not true: there is a counterexample
involving triples of functions on a curve, see Section 3. So a more subtle approach is required.
In particular, we need a way to vary phases εi independently for the branches of log(zi), so as
to place weaker demands on our cycles. But this can never lead to a morphism of complexes from
Zp(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, •) ⊂ Zp(X, •) of the normalized cycles, on which the differential restricts
to just one facet [3]. 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
(
eiε1 , . . . , eiεn
)
-perturbed wavefront set for
any (ε1, . . . , εn) belonging to Bn
ε :=
{
ε ∈ Rn | 0 < ε1 < ε, 0 < ε2 < e−1/ε1 , . . . , 0 < εn < e−1/εn−1
}
.
Our main technical results are
Theorem 1.1.
⋃
ε>0 N p
ε (X, •) = N p(X, •).
Theorem 1.2. Given ε, ε′ ∈ BN
ε , the corresponding morphisms
Rε, Rε′ : τ≤NN p
ε (X, •)→ C2p−•
D (X,Z(p))
induced by the perturbed KLM currents, are integrally homotopic in degrees • < N .
Integral Regulators for Higher Chow Complexes 3
(Here τ≤N truncates the complex above the N th term.) These results are proved in Sections 4
and 6, respectively. It is now easy to deduce that, taken over all ε, these morphisms induce a map
of the form (1.2) refining (1.4), see Section 7. We conclude by indicating several applications of
the KLM formula to torsion in Section 8 due to [4], Petras [10], Kerr–Yang [7] which are now
validated by our construction, and indicate future work in this direction.
Remark 1.3. In this article, we are working with analytic Deligne cohomology, which is not
the optimal generalization to quasi-projective varieties. It’s more work to define the map to
absolute Hodge cohomology.
2 Higher Chow cycles
2.1 Basic definitions
Definitions in this section follow [4]. Let X be a smooth quasiprojective algebraic variety over
an infinite field k. An algebraic cycle on X is a finite linear combination ΣnV [V ] of subvarieties
V ⊂ X, where nV ∈ Z.
We define the algebraic n-cube (over k) by 2n :=
(
P1\{1}
)n
with face inclusions ρfi : 2
n−1 →
2n (f ∈ {0,∞}) sending (z1, . . . , zn−1) to (z1, . . . , zi−1, f, zi, . . . , zn−1), and coordinate projections
πi : 2
n → 2n−1 sending (z1, . . . , zn) to (z1, . . . , ẑi, . . . , zn). We call
∂2n :=
⋃
i=1,...,n
f=0,∞
(
ρfi
)
∗2
n−1
the facets of 2n, and
∂k2n :=
⋃
i1<···<ik
f1,...,fk=0,∞
(
ρf1i1
)
∗ · · ·
(
ρfkik
)
∗2
n−k
the codimension-k subfaces of 2n.
Definition 2.1. cp(X,n) ⊂ Zp(X × 2n) is the free abelian group on irreducible subvarieties
V ⊂ X ×2n of codimension p such that V meets all faces of X ×2n properly.
Definition 2.2. The degenerate cycles dp(X,n) ⊂ cp(X,n) are defined as
n∑
i=1
π∗i (c
p(X,n− 1)).
Set Zp(X,n) := cp(X,n)/dp(X,n).
The Bloch differential
∂B :=
n∑
i=1
(−1)i
(
ρ0∗i − ρ∞∗i
)
: Zp(X,n)→ Zp(X,n− 1)
makes Zp(X, •) into a complex, with the higher Chow groups CHp(X,n) given by their homology.
For convenience, we shall often use cohomological indexing:
Definition 2.3. CHp(X,n) := H−n{Zp(X,−•)}.
2.2 A moving lemma
We recall the subcomplex from [4]. 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. Write Tzi for the
(codimention 1) geometric chain z−1i (R<0), oriented so that ∂Tzi = (zi) = z−1i (0) − z−1i (∞).
4 M. Li
Let cpR(X,n) be the set of all the cycles Z ∈ cp(X,n) whose components (or rather, their
analytizations) intersect X × (Tz1 ∩ · · · ∩ Tzi) and X ×
(
Tz1 ∩ · · · ∩ Tzi ∩ ∂k2n
)
properly for
all 1 ≤ i ≤ n and 1 ≤ k < n, and dpR(X,n) := cpR(X,n) ∩ dp(X,n). We get a new complex
ZpR(X,n) := cpR(X,n)/dpR(X,n). It is shown in [5] that this subcomplex is Q-quasi-isomorphic
to the original one:
Theorem 2.4 (Kerr–Lewis). ZpR(X, •) '−→ Zp(X, •).
2.3 Normalized cycles
Higher Chow groups may also be computed by complexes of cycles that have trivial boundary
on all but one face.
Definition 2.5. N p(X,n) := {Z ∈ Zp(X,n) | ∂∞i Z = 0 for i < n, ∂0jZ = 0 for any j}, where
∂fiZ :=
(
ρfi
)∗
Z ∈ Zp(X,n− 1).
In this section, we will write down an explicit retraction of Zp(X, •) onto the normalized
cycle complex which is homotopic to the identity. The construction is derived from Bloch’s
manuscript [3], by replacing the notations from
(
A1
)n
(using {0, 1} as boundary) by
(
P1 \ {1}
)n
(using {0,∞} as boundary). In addition, Bloch uses a different definition for the normalized
cycles: N ′p(X, •) := {Z ∈ Zp(X,n) | ∂∞i Z = 0 for i > 1, ∂0jZ = 0 for any j}; so we need to
apply a “conjugation” to the proof in [3] as well.
Define Zp∞,i(X, •) = {Z ∈ cp(X, •) | ∂∞j Z = 0 for j < n − i, ∂0kZ = 0 for any k}. We have
Zp∞,0(X, •) = N p(X, •), and inclusions of complexes Zp∞,0(X, •) ⊆ Zp∞,1(X, •) ⊆ · · · which
stabilize to Zp(X, •) in any degree. More precisely, we have Zp∞,i(X,n) = Zp(X,n) for i ≥ n,
since dp(X,n) ∩ Zp∞,i(X,n) = {0}.
Theorem 2.6. The inclusion N p(X, •) ⊂ Zp(X, •) is an integral quasi-isomorphism.
Proof. Any Z ∈ Zp(X,n) may be lifted to cp(X,n), and we may add degenerate cycles to any
element of cp(X,n) to force it into Zp∞,n(X,n). The (well-defined) map given by this process is
an isomorphism, and we shall tacitly equate Zp(X,n) and Zp∞,n(X,n) in what follows.
For each integer l ≤ n− 1, define hl : �n+1 → �n by
hl(z1, . . . , zn+1) :=
(
z1, . . . , zl,
zl+1zl+2
zl+1+zl+2−1 , zl+3, . . . , zn+1
)
and for Z ∈ Zp(X,n), define H l(Z) := (−1)l
(
hl
)−1
(Z) ∈ Zp(X,n+ 1). If l ≥ n, set H l(Z) = 0.
The map
φ := · · ·
(
Id−
(
∂ ◦H l +H l ◦ ∂
))
◦ (Id−
(
∂ ◦H l−1 +H l−1 ◦ ∂
))
◦
· · · ◦
(
Id−
(
∂ ◦H0 +H0 ◦ ∂
))
stabilizes in any degree and so defines an endomorphism φ : Zp(X, •) → Zp(X, •), which is
visibly homotopic to the identity.
To determine its image, write (for any Z ∈ Zp(X,n))
(
∂ ◦H l
)
Z =
n−l−1∑
k=1
(−1)n−l+k+1∂∞n−k+1Z
(
z1, . . . ,
zl+1zl+2
zl+1+zl+2−1 , . . . , zn
)
+
n∑
k=n−l+1
(−1)n−l+k∂∞n−k+1Z
(
z1, . . . ,
zlzl+1
zl+zl+1−1 , . . . , zn
)
Integral Regulators for Higher Chow Complexes 5
and
(
H l ◦ ∂
)
Z =
n∑
k=1
(−1)n−l+k∂∞n−k+1Z
(
z1, . . . ,
zl+1zl+2
zl+1+zl+2−1 , . . . , zn
)
,
where the notation means that we pull back (the equations defining) ∂∞n−k+1Z via hl (or hl−1):
�n → �n−1. Thus we have
(
∂ ◦H l +H l ◦ ∂
)
Z =
n∑
k=n−l+1
(−1)n−l+k∂∞n−k+1Z
(
z1, . . . ,
zlzl+1
zl+zl+1−1 , . . . , zn
)
+
n∑
k=n−l
(−1)n−l+k∂∞n−k+1Z
(
z1, . . . ,
zl+1zl+2
zl+1+zl+2−1 , . . . , zn
)
.
In particular, for Z ∈ Zp∞,i(X, •), we have
(
d◦H l+H l◦d
)
Z = 0 for l ≤ n−i−2. For l = n−i−1,
we find
Z ′ := Z −
(
∂ ◦H l +H l ◦ ∂
)
Z = Z − ∂∞n−iZ
(
z1, . . . ,
zn−izn−i+1
zn−i+zn−i+1−1 , . . . , zn
)
,
which belongs to Zp∞,i−1. Applying
(
Id−
(
∂◦H l+1+H l+1◦∂
))
then maps Z ′ to some Z ′′ ∈ Zp∞,i−2,
and so forth until finally we reach Zp∞,0(X,n) = N p(X,n). Since all the ∂ ◦ H l + H l ◦ ∂ are
zero on Zp∞,0, φ|N gives the identity on normalized cycles.
We have thus constructed a morphism φ : Zp(X, •) → N p(X, •), whose composition with
the inclusion N p ↪→ Zp is homotopic to (resp. equal to) the identity on Zp (resp. N p); thus φ
and the inclusion are both quasi-isomorphisms. �
As explicit expressions for φ in low dimension, we have
φ(Z(z1, z2)) = Z(z1, z2)− (∂∞1 Z)
(
z1z2
z1+z2−1
)
,
and
φ(Z(z1, z2, z3)) = Z(z1, z2, z3)− (∂∞2 Z)
(
z1,
z2z3
z2+z3−1
)
− (∂∞1 Z)
(
z1z2
z1+z2−1 , z3
)
+ (∂∞1 Z)
(
z1,
z2z3
z2+z3−1
)
.
3 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 ZpR(X, •), given by “perturbing” the conditions defining ZpR(X, •). Though this
turns out to be too naive, it is the first step toward a strategy that works.
Begin by defining Zpε (X, •) to be the subcomplex of Zp(X,n) given by the cycles that intersect
X ×
(
T εz1 ∩ · · · ∩ T
ε
zi
)
and X ×
(
T εz1 ∩ · · · ∩ T
ε
zi ∩ ∂
k2n
)
properly for all 1 ≤ j ≤ n, 1 ≤ k < n and
0 < ε < ε. Here T εz := Teiεz is given by arg(z) = π − ε, 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 ZpR(X, •), we must have
that their union gives us the original Zp:⋃
ε
Zpε (X, •) = Zp(X, •).
Unfortunately, this fails in a very simple case:
6 M. Li
Proposition 3.1. For X = Spec(Q(i)), we have
⋃
ε Z
2
ε (X, 3) ( Z2(X, 3).
Proof. Let F (z) = iz − 1, G(z) = − (1+z)(1+3z)
(1+iz)(1−2z) , and H(z) = iz−1
3+z . Then we have Z =
(F (z), G(z), H(z))z∈P1 ∈ Z2(pt, 3); but for all ε > 0, Z /∈ Z2
ε (pt, 3). More precisely, for any
ε > 0, we have dimR
(
Z ∩ T εz1 ∩ T
ε
z2 ∩ T
ε
z3
)
= 0, not −1 (i.e., empty) as required for a proper-
analytic intersection. To see this, we need to find a value of z for which arg(F ), arg(G), arg(H)
equal to π − ε. Such a value is given by z = tan(ε). �
Thus we need to find another way to do the “perturbation”, which will be given in the next
section.
4 Multiple perturbations
In order to have Zan meet the deformations of {Tzi} (and their intersections) properly – say,
for an example like that in the above proof – we 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 ε := (ε1, . . . , εn) in what follows.
Now we are thinking of T εizi as the location of the jump in the 0-current log(zi); these 0-
currents will appear in the definition of the regulator-currents R
ε
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 Zp(X, •) to C2p−•
D (X,Z(p)). Unfortunately, if Z has boundaries
at more than one facet of �n, say ∂1Z =
(
ρ01
)∗
Z and ∂2Z =
(
ρ02
)∗
Z, the residue terms in
d
[
R
(ε1,...,εn)
Z
]
will take the form R
(ε2,...,εn)
∂1Z
resp. R
(ε1,ε3,...,εn)
∂2Z
. This clearly conflicts with having
D
(
T
ε
Z ,ΩZ , R
ε
Z
)
=
(
T
ε′
∂Z ,Ω∂Z , R
ε′
∂Z
)
for a single choice of ε′, so we shall need to restrict to the
normalized cycles N p(X, •) defined in Section 2.3.
For ε > 0, define Bε as the set of infinite sequences (ε1, ε2, . . .) satisfying
0 < ε1 < ε, 0 < ε2 < exp(−1/ε1), 0 < ε3 < exp(−1/ε2), . . . , (4.1)
and define Bn
ε to comprise the n-tuples ε satisfying (4.1).
Definition 4.1. N p
ε (X, •) := {Z ∈ N p(X, •) |Z intersects X×
(
T ε1z1 ∩· · ·∩T
εi
zi
)
and X×
(
T ε1z1 ∩
· · · ∩ T εizi ∩ ∂
k2n
)
properly ∀ i, k, ε ∈ B•ε}.
Theorem 4.2.
⋃
ε N p
ε (X, •) = N p(X, •).
Proof. Consider the projection (C∗)n →
(
S1
)n ∼= (R/2πZ)n defined by
(
r1e
iε1 , . . . , rne
iεn
)
7→
(ε1, . . . , εn), whose fibers are T ε1z1 ∩ · · · ∩ T
εn
zn . There is also a natural 2n : 1 map
(
S1
)n → (
P1
R
)n
by taking slopes: (ε1, . . . , εn) 7→ (tan ε1, . . . , tan εn). The composite map Θn : (C∗)n →
(
S1
)n →(
P1
R
)n
is real algebraic, sending (x1 + iy1, . . . , xn + iyn) 7→ (y1/x1, . . . , yn/xn).
Now let Z ∈ N p(X,n) be given. Set Z∗ := Z̄ ∩ (X × (C∗)n), and let Z̃∗ be its resolution
of singularities. The intersections of Z∗ with the fibers of Θn
X : X × (C∗)n → X ×
(
P1
R
)n
are
Z∗∩
(
X×
{
T−ε1z1 ∩· · ·∩T
−εn
zn
})
. Write ΘZ for the composition of Z̃∗ → X×(C∗)n with Θn
X . The
set of ε for which these intersections are good is the complement of the non-flat locus ∆ ⊂
(
P1
R
)n
of ΘZ (see [9]). Since the flat locus of an algebraic map is Zariski open, ∆ ⊂
(
P1
R
)n
is a real
subvariety, which is proper by dimension considerations. (That is to say, if all the fibers had
real dimension > 2 dimC(Z∗) − n, then Z∗ would have real dimension > 2 dimC(Z∗), which is
a contradiction.)
Therefore the preimage ∆̃ of ∆ in
(
S1
)n
is real analytic. By the form of the inequalities
in Bε, we know that we can choose an ε > 0 such that Bn
ε ∩ ∆̃ = ∅. (This follows from [8,
Integral Regulators for Higher Chow Complexes 7
Theorem 1] for ∆̃, and the fact that all derivatives of e−1/x limit to 0 at 0.) This means that Z
intersects X ×
(
T ε1z1 ∩ · · · ∩ T
εn
zn
)
properly ∀ ε ∈ B•ε , as desired.
Repeating the argument for X × (C∗)i ×
(
P1
C
)n−i
and X × (C∗)i × ({0,∞})k ×
(
P1
C
)n−i−k
,
we pick the minimum of the required values of ε, so that Z intersects X ×
(
T ε1z1 ∩ · · · ∩ T
εi
zi
)
and
X ×
(
T ε1z1 ∩ · · · ∩ T
εi
zi ∩ ∂
k2n
)
properly ∀ i, k, ε ∈ B•ε , which means Z ∈ N p
ε (X,n). �
5 Abel–Jacobi maps
In this section, we’ll use the strategy in [4] to define the Abel–Jacobi maps on our subcomplexes.
5.1 Definition of Deligne cohomology
The Deligne cohomology group H2p+n
D (X,Z(p)) is given by the nth cohomology of the complex
C•+2p
D (X,Z(p)) := {C2p+•(X,Z(p))⊕ F pD•(X)⊕D•−1(X)}
with differential D taking (a, b, c) 7→ (−∂a,−d[b], d[c]− b+ δa), where δa denotes the current of
integration over the chain a. Here Dk(X) denotes currents of degree k on Xan and Ck(X,Z(k))
denotes C∞ (co)chains of real codimension k and Z(k) = (2πi)kZ coefficients.
The cup product in Deligne cohomology is defined on the chain level by
(a, b, c) ∪ (A,B,C) :=
(
a ∩A, b ∧B, c ∧B + (−1)deg(a)δa · C
)
.
It becomes commutative upon passage to cohomology. (See [11] for a commutative chain-level
construction.) Note that
D[(a, b, c) ∪ (A,B,C)] = D(a, b, c) ∪ (A,B,C) + (−1)deg(a)(a, b, c) ∪D(A,B,C).
5.2 KLM currents
Firstly we’ll review the currents given in [4]. The currents on 2n are given by Tn := Tz1 ∩ Tz2 ∩
· · · ∩ Tzn , Ωn = dz1
z1
∧ dz2
z2
∧ · · · ∧ dzn
zn
, and
Rn =
n∑
k=1
(−2πi)k−1δTz1∩Tz2∩···∩Tzk−1
log zk
dzk+1
zk+1
∧ · · · ∧ dzn
zn
.
For currents on X associated to a given Z ∈ ZpR(X,n), let π1 : Z̃ → 2n and π2 : Z̃ → X be the
projections (where Z̃ is a desingularization). Then we have:
ÃJ
p,n
KLM(Z) := (−2πi)p−n(π2)∗(π1)
∗((2πi)nTn,Ωn, Rn).
5.3 Currents on N p
ε (X,n)
Using a similar strategy, for a normalized precycle Z ∈ N p
ε (X,n) and ε ∈ Bn
ε , we send
Z 7→ (−2πi)p−n(π2)∗(π1)
∗((2πi)nT εn,Ωn, R
ε
n) =: Rn,εε (Z), (5.1)
where T
ε
n = T ε1z1 ∩ T
ε2
z2 ∩ · · · ∩ T
εn
zn , ΩZ = dz1
z1
∧ dz2
z2
∧ · · · ∧ dzn
zn
, and
Rεn =
n∑
k=1
(−1)(
k
2)(2πi)kδ
T
ε1
z1
∩T ε2z2 ∩···∩T
εk−1
zk−1
logεk zk
dzk+1
zk+1
∧ · · · ∧ dzn
zn
.
8 M. Li
Here logε(z) is the branch of log with argument in (−π − ε, π − ε]. This is a discontinuous
function with cut at T εz , so that d[logε(z)] = dz
z − 2πiδT εz .
The formula (5.1) induces a map of complexes
R•,εε : N p
ε (X,−•)→ C2p+•D (X,Z(p)). (5.2)
Proposition 5.1. R•,εε is a morphism of complexes.
Therefore we get for each p, n, ε, and ε ∈ Bε Abel–Jacobi maps (induced by these maps of
complexes)
AJp,n,εε : Hn
(
N p
ε (X, •)
)
→ H2p−n
D (X,Z(p)).
6 Homotopies of Abel–Jacobi maps
6.1 Notations
Put
Rzi :=
(
2πiTzi ,
dzi
zi
, log(zi)
)
, Rεzi :=
(
2πiTarg(zi)=π−ε,
dzi
zi
, logε(zi)
)
,
where logε(zi) is taking branch cut at arg(zi) = π − ε. We write Targ(zi)=π−ε as T εzi . Define
Sε,ε′zi :=
(
−2πiθε,ε
′
zi , 0, 0
)
,
where θε,ε
′
zi := ±δ{− arg(zi)∈(ε,ε′)} are 0-currents. (The sign is positive if ε > ε′, negative otherwise.)
Clearly we have DSε,ε
′
zi = Rεzi −R
ε′
zi .
6.2 Homotopy property
In this subsection we will prove the
Theorem 6.1. For any ε, ε′ ∈ BN
ε , R•,εε and R•,ε
′
ε are (integrally) homotopic morphisms of
complexes in degrees • < N .
For a fixed N ∈ N, consider
(
P1
)N
with subsets
YI,f := ∩i∈I{zi = f(i)} ∼=
(
P1
)N−|I|
,
where I denotes subsets of {1, . . . , N} (with complement Ic) and f : I → {0,∞} ranges over
the 2|I| possible functions. Write ρif(i) for the inclusion of YI{i},f |I{i} in YI,f , sgnI(i) = |{i′ ∈
I | i′ ≤ i}|, sgn(0) = 0, and sgn(∞) = 1. Consider the double complex
Ea,b :=
⊕
I,f
|I|=N−a
C2a+b
D (YI,f ),
with differentials δ : Ea,b → Ea+1,b and D : Ea,b → Ea,b+1 given by
δ = 2πi
∑
i∈I
(−1)sgnI(i)+sgn(f(i))
(
ρif(i)
)
∗
and the direct sum of Deligne differentials, respectively. (The YI,f ’s in the ath column are
∼=
(
P1
)a
, and called “a-faces”.) Put D = D+ (−1)bδ for the differential on the associated simple
complex skE•,• :=
⊕
a+b=k E
a,b.
Integral Regulators for Higher Chow Complexes 9
Fix an n ∈ {0, 1, . . . , N}. For each subset J = {j1, . . . , jn} ⊂ {1, . . . , N} and f : Jc → {0,∞},
we have an element
RεJn := ((2πi)nT
εJ
n ,Ωn, R
εJ
n ) ∈ CnD(YJc,f ),
where εJ = {εj1 , . . . , εjn}. Taken together, these yield an
(
N
n
)
·2N−n-tupleRε2,n :=
{
RεIcn
}
I,f
|I|=N−n
∈ En,−n. Define
Rε2 :=
(
Rε2,0, . . . ,R
ε
2,n
)
∈
N⊕
n=0
En,−n = s0E•,•.
Proposition 6.2. Rε2 is a 0-cocycle in the simple complex.
Proof. According to [4, equations (5.2), (5.3), (5.4)], generally we have (for Rε2,n ∈ (CnD(YI,f )),
without the loss of generality we take I = {1, . . . , n}c):
DRεn =
(
−(2πi)n
n∑
k=1
(−1)k
((
ρ0k
)
∗T
{ε1,...,ε̂k,...,εn}
n−1 −
(
ρ∞k
)
∗T
{ε1,...,ε̂k,...,εn}
n−1
)
,
− 2πi
n∑
k=1
(−1)kΩ(z1, . . . , ẑk, . . . , zn)δ(zk),
− 2πi
n∑
k=1
(−1)kR{ε1,...,ε̂k,...,εn}(z1, . . . , ẑk, . . . , zn)δ(zk)
)
= −(−1)n−1δ
({
R{ε1,...,ε̂k,...,εn}n−1
}
k=1,...,n; f(k)=0,∞
)
,
where for the δ in the last line, we only consider the components mapping into CnD(YI,f ). This
tells us that
DRεn + (−1)n−1δ
({
R{ε1,...,ε̂k,...,εn}n−1
}
k=1,...,n; f(k)=0,∞
)
= 0
for any n; thus each component of DRε2 is 0, and so Rε2 ∈ Ker(D) is a 0-cocycle. �
Remark 6.3. While wedge products and Leibniz formulas for the extension derivative are not
generally valid for currents, they are valid in the setting of exterior products (which is what we
use here), see [11, Appendix B].
For ε, ε′, consider the following (−1)-cochain in E•,•:
Sε,ε′2 :=
{
S ε̂,ε̂′ :=
n∑
k=1
(−1)k−1Rεm1
z1 ∪ · · · ∪ R
εmk−1
zk−1 ∪ S
εmk ,ε
′
mk
zk ∪R
ε′mk+1
zk+1 ∪ · · · ∪ R
ε′mn
zn
}
n,m,i
.
It satisfies the following key property:
Proposition 6.4. DSε,ε
′
2 = Rε2 −Rε
′
2.
Proof. On any given n-“face” (∼=
(
P1
)n
) we have
DSε,ε′n =
n∑
k=1
k−1∑
l=1
(−1)l−1(−1)k−1Rε1z1 ∪ · · · ∪DR
εl
zl
∪ · · · ∪ Sεk,ε
′
k
zk ∪ · · · ∪ Rε′nzn
+
n∑
k=1
n∑
l=k+1
(−1)l(−1)k−1Rε1z1 ∪ · · · ∪ S
εk,ε
′
k
zk ∪ · · · ∪DRε
′
l
zl ∪ · · · ∪ Rε
′
n
zn
+
n∑
k=1
Rε1z1 ∪ · · · ∪DS
εk,ε
′
k
zk ∪ · · · ∪ Rε′nzn .
10 M. Li
Noting that DRεzi = −2πi(δ(zi), δ(zi), 0) =: −2πi∆(zi) (which commutes with other triples)
and DSε,ε
′
zi = Rεzi −R
ε′
zi , we can rewrite this expression by applying the telescoping method and
rearranging the order of the summation. Denoting (−1)k−1l :=
{
(−1)k−1, l > k,
(−1)k−2, l < k,
DSε,ε′n = 2πi
n∑
k=1
n∑
l=1, l 6=k
(−1)l(−1)k−1l ∆(zl)R
ε1
z1 ∪ · · · ∪ S
εk,ε
′
k
zk ∪ · · · ∪ Rε′nzn
(with the lth term omitted, either before k or after k)
+
n∑
k=1
Rε1z1 ∪ · · · ∪
(
Rεkzk −R
ε′k
zk
)
∪ · · · ∪ Rε′nzn
= 2πi
n∑
l=1
(−1)l∆(zl)
n∑
k=1, k 6=l
(−1)k−1l Rε1z1 ∪ · · · ∪ S
εk,ε
′
k
zk ∪ · · · ∪ Rε′nzn +Rεn −Rε
′
n
= −(−1)nδ
({
S{ε1,...,ε̂l,...,εn},{ε
′
1,...,ε̂
′
l,...,ε
′
n}
n−1
}
l=1,...,n;f(l)=0,∞
)
+Rεn −Rε
′
n .
This tells us
DSε,ε′n + (−1)nδ
({
S{ε1,...,ε̂l,...,εn},{ε
′
1,...,ε̂
′
l,...,ε
′
n}
n−1
}
l=1,...,n;f(l)=0,∞
)
= Rεn −Rε
′
n (6.1)
holds on each n-face of
(
P1
)n
. Thus DSε,ε
′
2 = Rε2 −Rε
′
2 holds. �
Proof of Theorem 6.1. For Z ∈ N p
ε (X,n), ∂Z only lives in X × ∂∞n 2n by definition. Thus
in (6.1), the only relevant term in the braces is S{ε1,...,εn−1},{ε′1,...,ε′n−1}
n−1 . Therefore Proposition 6.4
implies at once that Rn,εε (Z) − Rn,ε
′
ε (Z) = DSε,ε
′
ε (Z) + Sε,ε
′
ε (∂Z), so that R•,εε ∼ R•,ε
′
ε as clai-
med. �
7 The integral Abel–Jacobi map
Recall our map of complexes from (5.2), with nth term
Rn,εε : N p
ε (X,n)→ C2p−n
D (X,Z(p)).
According to our result from the last section, we know that for ε, ε′ ∈ BN
ε , Rn,εε ∼ Rn,ε
′
ε ; that
is to say, they induce the same homomorphism after taking cohomology:
Corollary 7.1. All the ε ∈ Bε induce the same map:
AJp,nε : Hn
(
N p
ε (X, •)
)
→ H2p−n
D (X,Z(p)).
Moreover, for ε′ < ε and ε ∈ Bε′ ⊂ Bε, the following diagram commutes:
N p
ε (X, •) �
� ı //
R•,εε ''
N p
ε′ (X, •)
R•,ε
ε′ww
C2p−•
D (X,Z(p)),
Integral Regulators for Higher Chow Complexes 11
which is straightforward from the definition. By taking homology, we have that the following
diagram commutes as well:
Hn
(
N p
ε (X, •)
) [ı] //
AJp,nε ((
Hn
(
N p
ε′ (X, •)
)
AJp,n
ε′vv
H2p−n
D (X,Z(p)).
In order to get the integral Abel–Jacobi map, we need the following result:
Theorem 7.2. CHp(X,n) ∼= lim−→ε
Hn
(
N p
ε (X, •)
)
.
Proof. Since N p
ε (X, •) ⊂ N p(X, •), we have Hn
(
N p
ε (X, •)
)
maps to Hn(N P (X, •)) =
CHP (X,n) for every ε, hence there exists a natural map lim−→ε
Hn
(
N p
ε (X, •)
)
→ CHp(X,n). Since⋃
N p
ε (X, •) = N p(X, •), this map is surjective. To show it is injective, consider ξ ∈ CHp(X,n),
and ξ̃, ξ̃′ be two representations of ξ in the following sequence:
Hn
(
N p
ε (X, •)
)
→ Hn
(
N p
ε′ (X, •)
)
→ · · · → CHp(X,n).
We need to show that ξ̃ and ξ̃′ will eventually merge at some ε, that is to say,
⋃
∂N p
ε (X,n+1) =
∂N p(X,n+ 1), which directly comes from the property of normalized cycle and
⋃
N p
ε (X, •) =
N p(X, •). �
Thus we have a well-defined map
AJp,nZ : CHp(X,n)→ H2p−n
D (X,Z(p))
given by AJp,nZ := lim−→ε
AJp,nε . Precisely, for Z ∈ CHp(X,n), if Z̃ ∈ Ker(∂) ⊂ N p
ε (X,n) is
any choice of class mapping to Z and ε any choice of element of Bε, AJp,nZ (Z) := Rn,εε
(
Z̃
)
is
independent of the choices. Thus we have an explicit expression for the integral Abel–Jacobi
map:
AJp,nZ (Z) = lim
ε→0
Rn,εε
(
Z̃
)
.
Moreover, for Z̃ a representative in ZpR(X,n) ∩N p(X,n), we know that Z̃ lies in N p
ε for some
ε > 0, and
lim
ε→0
Rn,εε
(
Z̃
)
= R
(
Z̃
)
.
since we have the same map of the level of cohomology for every ε. In particular, this means
that on cycles belonging to ZpR(X,n) ∩ N p(X,n), our integral AJ map is given by the KLM
formula.
8 Application to torsion cycles
Recent work of Kerr and Yang [7] provides explicit representatives for generators of CHn(Spec(k),
2n − 1) where k is an abelian extension of Q. We’ll check that when n = 2, 3, the cycle given
by [7] satisfies the normal and proper intersection condition thus belongs to ZpR(X, 2p − 1) ∩
N p(X, 2p−1). For n = 4, this is taken up in [6]; while for n ≥ 5 finding a normalized generator
is a future task.
Let ξN be an N th root of 1.
12 M. Li
Proposition 8.1. The cycles given in [7, equations (4.1) and (4.2)] lie in ZnR(Q(ξN ), 2n− 1) ∩
N n(Q(ξN ), 2n− 1).
The ZnR part is given in [7, Remark 3.3]. The N n part is visible from the boundary com-
putations in [7, Sections 4.1 and 4.2]. See [6, Section 5] for torsion calculations arising from
Proposition 8.1.
This puts some earlier results on firm ground as well, such as O. Petras’s result in [10] that
Z :=
(
1− 1/t, 1− t, t−1
)
+
(
1− ξ5/t, 1− t, t−5
)
+
(
1− ξ̄5/t, 1− t, t−5
)
generates CH2
(
Q
(√
5, 3
))
and (since we have R(Z) = Li2(1) + 5(Li2(ξ5) + Li2(ξ̄5)) = 7π2/30) is
120-torsion.
Acknowledgements
This work was supported by the National Science Foundation [DMS-1361147; PI: Matt Kerr].
The author would like to thank his advisor Matt Kerr for great help and discussions, J. McCarthy
for graciously supplying the counterexample in Section 3, J. Lewis for his interest in this work,
and the referee for their great help on improving the exposition.
References
[1] Bloch S., Algebraic cycles and higher K-theory, Adv. Math. 61 (1986), 267–304.
[2] Bloch S., Algebraic cycles and the Bĕılinson conjectures, in The Lefschetz Centennial Conference, Part I
(Mexico City, 1984), Contemp. Math., Vol. 58, Amer. Math. Soc., Providence, RI, 1986, 65–79.
[3] Bloch S., Some notes on elementary properties of higher chow groups, including functoriality properties and
cubical chow groups, Preprint, available at http://www.math.uchicago.edu/~bloch/publications.html.
[4] Kerr M., Lewis J.D., Müller-Stach S., The Abel–Jacobi map for higher Chow groups, Compos. Math. 142
(2006), 374–396, math.AG/0409116.
[5] Kerr M., Lewis J.D., The Abel–Jacobi map for higher Chow groups. II, Invent. Math. 170 (2007), 355–420,
math.AG/0611333.
[6] Kerr M., Li M., Two applications of the integral regulator, arXiv:1809.04114.
[7] Kerr M., Yang Y., An explicit basis for the rational higher Chow groups of abelian number fields, Ann.
K-Theory 3 (2018), 173–191, arXiv:1608.07477.
[8] Lion J.-M., Rolin J.-P., Théorème de préparation pour les fonctions logarithmico-exponentielles, Ann. Inst.
Fourier (Grenoble) 47 (1997), 859–884.
[9] Nowak K.J., Flat morphisms between regular varieties, Univ. Iagel. Acta Math. 35 (1997), 243–246.
[10] Petras O., Functional equations of the dilogarithm in motivic cohomology, J. Number Theory 129 (2009),
2346–2368, arXiv:0712.3987.
[11] Weißschuh T., A commutative regulator map into Deligne–Beilinson cohomology, Manuscripta Math. 152
(2017), 281–315, arXiv:1410.4686.
https://doi.org/10.1016/0001-8708(86)90081-2
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https://doi.org/10.1112/S0010437X05001867
https://arxiv.org/abs/math.AG/0409116
https://doi.org/10.1007/s00222-007-0066-x
https://arxiv.org/abs/math.AG/0611333
https://arxiv.org/abs/1809.04114
https://doi.org/10.2140/akt.2018.3.173
https://doi.org/10.2140/akt.2018.3.173
https://arxiv.org/abs/1608.07477
https://doi.org/10.5802/aif.1583
https://doi.org/10.5802/aif.1583
https://doi.org/10.1016/j.jnt.2009.04.009
https://arxiv.org/abs/0712.3987
https://doi.org/10.1007/s00229-016-0867-6
https://arxiv.org/abs/1410.4686
1 Introduction
2 Higher Chow cycles
2.1 Basic definitions
2.2 A moving lemma
2.3 Normalized cycles
3 Simple perturbations
4 Multiple perturbations
5 Abel–Jacobi maps
5.1 Definition of Deligne cohomology
5.2 KLM currents
5.3 Currents on Np(X,n)
6 Homotopies of Abel–Jacobi maps
6.1 Notations
6.2 Homotopy property
7 The integral Abel–Jacobi map
8 Application to torsion cycles
References
|
| id | nasplib_isofts_kiev_ua-123456789-209839 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T15:51:31Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Li, M. 2025-11-27T14:47:24Z 2018 Integral Regulators for Higher Chow Complexes / M. Li // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 11 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 14C15; 14C25; 19F27 arXiv: 1805.04646 https://nasplib.isofts.kiev.ua/handle/123456789/209839 https://doi.org/10.3842/SIGMA.2018.118 Building on Kerr, Lewis, and Müller-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 Kerr-Lewis-Müller-Stach, Petras, and Kerr-Yang. This work was supported by the National Science Foundation [DMS-1361147; PI: Matt Kerr]. The author would like to thank his advisor, Matt Kerr, for his great help and discussions, J. McCarthy for graciously supplying the counterexample in Section 3, J. Lewis for his interest in this work, and the referee for their great help on improving the exposition. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Integral Regulators for Higher Chow Complexes Article published earlier |
| spellingShingle | Integral Regulators for Higher Chow Complexes Li, M. |
| title | Integral Regulators for Higher Chow Complexes |
| title_full | Integral Regulators for Higher Chow Complexes |
| title_fullStr | Integral Regulators for Higher Chow Complexes |
| title_full_unstemmed | Integral Regulators for Higher Chow Complexes |
| title_short | Integral Regulators for Higher Chow Complexes |
| title_sort | integral regulators for higher chow complexes |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209839 |
| work_keys_str_mv | AT lim integralregulatorsforhigherchowcomplexes |