Explicit regulator maps on polylogarithmic motivic complexes

A. B. Goncharov

Contents

1 Introduction 1

2 The main result 4

3 Arakelov motivic complexes: examples 13

4 Proofs 18

1 Introduction

1. Regulator maps on the level of complexes. Let X be an algebraic variety. Beilinson [B1] defined the rational motivic cohomology of X via the algebraic K-theory of X by the formula

H_{Mot}^{i}(X, ℚ(n)) := gr_{n}^{γ}K_{2n−i}(X) ⊗ ℚ

Beilinson [B2] and Lichtenbaum [L] conjectured that the weight n motivic cohomology of X should appear as cohomology groups of some complexes, called the weight n motivic complexes of X.

If X is defined over ℂ Beilinson [B1] constructed the regulator map to the Deligne cohomology of X:

H_{Mot}^{i}(X, ℚ(n)) ⟶ H_{D}^{i}(X, ℝ(n))

The regulator map plays key role in the (hypothetical) formulas for special values of L-functions of motives over number fields [B1].

Our point is that the regulator map should be explicitly defined on the level of complexes. So for any algebraic variety over ℂ one should have homomorphisms of complexes

weight n motivic complex of X ⟶ weight n Deligne complex of X (1)

The cone of this map, shifted by −1, defines the Arakelov motivic complex:

RΓ_{X}^{A}(n) := Cone(map (1))[−1]

and so its cohomology should be called the Arakelov motivic cohomology of X. The group H^{2n}(RΓ_{X}^{A}(n)) is canonically isomorphic to a group of codimension n Arakelov cycles on X(ℂ) (see s. 3.2). This isomorphism is transparent for a version of the Deligne complex recalled in s. 3.1. Examples are given in the section 3. (When X is defined over ℚ we should take the cone of the regulator map to the so called real Deligne cohomology of X ⊗_{ℚ} ℝ, see s. 3.2).

Motivic complexes are objects of the derived category. Several candidates for motivic complexes are known, each with its own charm. They should be quasiisomorphic. Explicit regulator maps should be defined for each of them.

First of all there are Bloch’s higher Chow complexes ([Bl]). They satisfy many of the expected properties of motivic complexes. Explicit regulator maps from these complexes to the Deligne complexes were constructed in [G4] using the Chow polylogarithm construction given in [G3].

The goal of the present paper is to define regulator maps for another version of motivic complexes, the polylogarithmic complexes ([G1-2]). Our regulator maps are defined very explicitly via the classical polylogarithm functions with some funny combinations of Bernoulli numbers serving as the coefficients.

Combining this with Beilinson’s conjecture on regulators we get, as a bonus, a precise conjecture on special values of L-functions of varieties over number fields. If the variety in question is spectrum of a number field it reduces precisely to Zagier’s conjecture. So our conjecture is in the same relationship to Beilinson’s conjecture as Zagier’s conjecture [Z] to the Borel theorem [Bo].

The regulator maps for the polylogarithmic complexes of weights n ≤ 3 were constructed in [G1-2] and played an important role in the proofs of Zagier’s and Deninger’s conjectures on ζ-functions at s = 3 (loc. cit., [G5]).

2. Polylogarithmic complexes. Let F be an arbitrary field. In [G1-2] we defined a complex Γ(F; n) of the following shape

B_{n} →^{δ} B_{n−1} ⊗ F^{∗} →^{δ} B_{n−2} ⊗ Λ^{2}F^{∗} →^{δ} … →^{δ} B_{2} ⊗ Λ^{n−2}F^{∗} →^{δ} Λ^{n}F^{∗} (2)

called the polylogarithmic complex. Here the group B_{n} sits in degree 1, and the differential is of degree 1. We conjectured that this complex is quasiisomorphic to the weight n motivic complex of Spec(F), so one should have

H^{i}(Γ(F; n)) ⊗ ℚ =^{?} gr_{n}^{γ}K_{2n−i}(F) ⊗ ℚ  (3)

There is a good evidence this is so for small weights, see s. 1.3 below.

Now let F = ℂ(X) be the field of rational functions on a complex algebraic variety X. Our main results are explicit formulas for the regulator maps from the polylogarithmic complexes to the Deligne complexes of Spec(ℂ(X)), see theorems 2.5 and 2.6. The group B_{n} is directly related to the properties of the classical polylogarithms. In particular there is a homomorphism

B_{n}(ℂ) ⟶ ℝ(n − 1)  (4)

given in terms of the classical polylogarithms. Surprisingly its generalization to a homomorphism of complexes is quite complicated.

Our regulator maps enjoy compatibility with the residues property (condition (d) in theorem 2.2), which would guarantee that they can be extended from the generic point of X to X itself. However there is a serious difficulty in the definition of polylogarithmic complex Γ(X; n) for a general variety X and n > 3, (see p. 240 in [G1]). It would be resolved if homotopy invariance of complexes (2) will be known (see conjecture 1.39 in [G1]). As a result we have unconditional definition of the polylogarithmic complexes Γ(X; n) only in the following cases:

a) X = Spec(F), F is an arbitrary field.

b) X is an regular curve over any field, and n is arbitrary.

c) X is an arbitrary regular scheme, but n ≤ 3.

Compatibility with the residues provides the regulator map on the level of complexes in all these cases, and it would provide it in general if the mentioned above difficulty will be resolved.

3. Comparison with Beilinson’s regulator map. A homomorphism from the motivic cohomology to H^{i}(Γ(X; n)) ⊗ ℚ has been constructed in the following cases:

(1) F is an arbitrary field, n ≤ 3 ([G1-2,5]) and n = 4, i > 1 (to appear).

(2) X is a curve over a number field, n ≤ 3 ([G5]) and n = 4, i > 1 (to appear).

In all these cases we proved that this homomorphism followed by the regulator map on polylogarithmic complexes (when F = ℂ(X) in (1)) coincides with Beilinson’s regulator. This proves the traditionally difficult “surjectivity” property: the image of the regulator map on these polylogarithmic complexes contains the image of Beilinson’s regulator map in the Deligne cohomology.

The results (2) combined with the results of R. de Jeu [RdJ1-2] prove that the image of Beilinson’s regulator map in the Deligne cohomology coincides with the image of the regulator map on polylogarithmic complexes in the case (2).

4. Remarks. We give a detailed proof of the main result. It is a rather involved but direct calculation. However the reader may skip it because there exists a completely different, more conceptual, approach to this result. We outline it in s. 2.8 to show the main theorem in a general framework. The details will be published elsewhere. This approach needs quite elaborate and rather sofisticated machinery, so the elementary proof presented in the last section might be the quickest way to check the main theorem.

Acknowledgment. The author gratefully acknowledges the support of the NSF grant DMS-9800998. I am grateful to Spencer Bloch for discussions on Arakelov motivic complexes.

I am delighted to contribute this paper to the volume related to 20-th anniversary of Spencer Bloch’s Irvine lectures, which always have been a great source of inspiration for me.

2 The main result

1. Classical polylogarithms. Recall their definition:

Li_{1}(z) := − log(1 − z); Li_{n}(z) := ∫_{0}^{z} Li_{n−1}(t)\frac{dt}{t}, n ≥ 2.

They are multivalued analytic functions, but admit the following single-valued cousins:

L̂_{n}(z) := π_{n}(∑_{k=0}^{n−1} β_{k}Li_{n−k}(z) log^{n−k} |z|)

where

π_{n}(a + ib) = \begin{cases}a & n odd \\ ib & n even. \end{cases}

and

β_{k} := \frac{2^{k}B_{k}}{k!} ,  ∑_{k≥0} β_{k}t^{k} = \frac{2t}{e^{2t} − 1}

so β_{2m+1} = 0 for m ≥ 1, and

β_{0} = 1, β_{1} = −1, β_{2} = \frac{1}{3}, β_{4} = −\frac{1}{45}, β_{6} = \frac{2}{945}, ...

These functions were written by Zagier [Z]. Their Hodge-theoretic interpretation was given by Beilinson and Deligne [BD]. For example for the dilogarithm it is the Bloch-Wigner function

L̂_{2}(z) := iL_{2}(z) := π_{2} (Li_{2}(z)) + i arg(1 − z) · log |z|

2. The groups B_{n}(F) and polylogarithmic complexes(see s.1.4 in [G2]). For a set X denote by ℤ[X] the free abelian group generated by symbols {x} where x run through all elements of the set X. Let F be an arbitrary field. We define inductively subgroups R_{n}(F) of ℤ[P_{F}^{1}], n ≥ 1 and set

B_{n}(F) := ℤ[P_{F}^{1}]/R_{n}(F)

By definition R_{1}(F) := ({x} + {y} − {xy}, (x, y ∈ F^{∗}); {0}; {∞}) . Then B_{1}(F) = F^{∗}. Let {x}_{n} be the image of {x} in B_{n}(F). Consider homomorphisms

ℤ[P_{F}^{1}] ⟶^{δ_{n}} \{ \begin{matrix}B_{n−1}(F) ⊗ F^{∗} & : & n ≥ 3 \\ Λ^{2}F^{∗} & : & n = 2 \end{matrix}  (5)

δ_{n} : \{x\} ↦ \{ \begin{matrix}\{x\}_{n−1} ⊗ x & : & n ≥ 3 \\ (1 − x) ∧ x & : & n = 2 \end{matrix}  δ_{n} : \{∞\}, \{0\}, \{1\} ↦ 0  (6)

Set A_{n}(F) := Ker δ_{n} . Any element α(t) = Σn_{i}{f_{i}(t)} ∈ ℤ[P_{F (t)}^{1}] has a specialization α(t_{0}) := Σn_{i}{f_{i}(t_{0})} ∈ ℤ[P_{F}^{1}] at each point t_{0} ∈ P_{F}^{1}.

Definition 2.1. R_{n}(F) is generated by elements {∞}, {0} and α(0) − α(1) where α(t) runs through all elements of A_{n}(F (t)).

Then δ_{n}(R_{n}(F))= 0 ([G1], 1.16). So we get homomorphisms

δ_{n} : B_{n}(F) ⟶ B_{n−1}(F) ⊗ F^{∗}, n ≥ 3; δ_{2} : B_{2}(F) ⟶ Λ^{2}F^{∗}

and finally the mentioned in the introduction complex Γ(F, n):

B_{n} →^{δ} B_{n−1} ⊗ F^{∗} →^{δ} B_{n−2} ⊗ Λ^{2}F^{∗} →^{δ} … →^{δ} B_{2} ⊗ Λ^{n−2}F^{∗} →^{δ} Λ^{n}F^{∗}

where δ : {x}_{p} ⊗ ⋀_{i=1}^{n−p} y_{i} → δ_{p}({x}_{p}) ∧ ⋀_{i=1}^{n−p} y_{i}.

Let F = ℂ. Set L̂_{n}(∑ m_{i}{z_{i}}_{n}) := ∑ m_{i}L̂_{n}(z_{i}). One can prove that L̂_{n}(R_{n}(ℂ))) = 0 ([G2], theorem 1.13). So we are getting homomorphism (4).

3. The residue homomorphism for complexes Γ(F, n) (s. 1.14 in [G1]). Let F = K be a field with a discrete valuation v, the residue field k_{v} and the group of units U. Let u → ū be the projection U → k_{v}^{∗}. Choose a uniformizer π. There is a homomorphism θ : Λ^{n}K^{∗} ⟶ Λ^{n−1}k_{v}^{∗} uniquely defined by the following properties (u_{i} ∈ U):

θ (π ∧ u_{1} ∧ ⋯ ∧ u_{n−1}) = ū_{1} ∧ ⋯ ∧ ū_{n−1}; θ (u_{1} ∧ ⋯ ∧ u_{n}) = 0

It is clearly independent of π. Define a homomorphism s_{v} : ℤ[P_{K}^{1}] ⟶ ℤ[P_{k_{v}}^{1}] by setting s_{v}{x} = {x̄} if x is a unit and 0 otherwise. It induces a homomorphism s_{v} : B_{m}(K) ⟶ B_{m}(k_{v}). Put

∂_{v} := s_{v} ⊗ θ : B_{m}(K) ⊗ Λ^{n−m}K^{∗} ⟶ B_{m}(k_{v}) ⊗ Λ^{n−m−1}k_{v}^{∗}

It defines a morphism of complexes ∂_{v} : Γ(K, n) ⟶ Γ(k_{v}, n − 1)[−1].

4. Main result: a preliminary form. Let A^{i}(η_{X}) be the space of real smooth i-forms at the generic point η_{X} := Specℂ(X) of a complex variety X. Denote by X^{(1)} the set of the codimension one closed irreducible subvarieties in X. Let D be the de Rham differential on distributions, and d the de Rham differential on A^{i}(η_{X}). A typical example:

d(di arg z)= 0; D(di arg z)= 2πiδ(z)  (7)

The difference D − d is the de Rham residue homomorphism. It is defined on distributions smooth at the generic point of X(ℂ). Its value on such a distribution is concentrated on a union of codimension one subvarieties.

Theorem 2.2. There exist a homomorphism of complexes

B_{n}(ℂ(X)) →^{δ} B_{n−1}(ℂ(X)) ⊗ ℂ(X)^{∗} →^{δ} … →^{δ}  ⋀^{n} ℂ(X)^{∗}
↓ r_{n}(1)  ↓ r_{n}(2)  ↓ r_{n}(n)
A^{0}(η_{X})(n − 1) →^{d}  A^{1}(η_{X})(n − 1)  →^{d} ... →^{d} A^{n−1}(η_{X})(n − 1)

such that (a) r_{n}(1)({f}_{n}) = L̂_{n}(f).

(b) dr_{n}(n)(f_{1} ∧ ... ∧ f_{n}) + π_{n}(d log f_{1} ∧ ... ∧ d log f_{n}) = 0.

(c) The differential form r_{n}(m)(∗) defines a distribution on X(ℂ).

(d) The homomorphism r_{n}(·) is compatible with residues:

D ∘ r_{n}(m) − r_{n}(m + 1) ∘ δ = 2πi · ∑_{Y ∈X^{(1)}} r_{n−1}(m − 1) ∘ ∂_{v_{Y}} , m < n

D ∘ r_{n}(n) − π_{n}(d log f_{1} ∧ ... ∧ d log f_{n}) = 2πi · ∑_{Y ⊂X^{(1)}} r_{n−1}(n − 1) ∘ ∂_{v_{Y}}

where v_{Y} is the valuation on the field ℂ(X) defined by a divisor Y.

Remark. This result has been formulated in [G4], see theorem 4.3. The part (d) means that r_{n}(·) sends the residue homomorphism ∂_{v_{Y}} to the

de Rham residue homomorphism.

Here are two examples. Set

α(f, g) := − log |f|d log |g| + log |g|d log |f|

Example 1: n=3. Set

r_{3}(2) : \{f\}_{2} ⊗ g ⟼ L̂_{2}(f)di arg g − \frac{1}{3} log |g| · α(1 − f, f)

Example 2: n=4.

r_{4}(2) : \{f\}_{3} ⊗ g ↦ L̂_{3}(f)di arg g − \frac{1}{3}L̂_{2}(f) log |g| · d log |f|

r_{4}(3) : \{f\}_{2} ⊗ g_{1} ∧ g_{2} ↦ +L̂_{2}(f)di arg g_{1} ∧ di arg g_{2} − \frac{1}{3}α(1 − f, f) ·

(log |g_{1}|di arg |g_{2}| − log |g_{2}|di arg |g_{1}|) + \frac{1}{3}L̂_{2}(f)d log |g_{1}| ∧ d log |g_{2}|

5. The numbers β_{k,p}. Define for any integers p ≥ 1 and k ≥ 0 the numbers

β_{k,p} := (−1)^{p}(p − 1)! ∑_{0≤i≤[ \frac{p−1}{2} ]} \frac{1}{(2i + 1)!}β_{k+p−2i}

For instance

β_{k,1} = −β_{k+1};  β_{k,2} = β_{k+2};  β_{k,3} = −2(\frac{1}{1!}β_{k+3} + \frac{1}{3!}β_{k+1});

β_{k,4} = 3!(\frac{1}{1!}β_{k+4} + \frac{1}{3!}β_{k+2}); β_{k,5} = −4!(\frac{1}{1!}β_{k+5} + \frac{1}{3!}β_{k+3} + \frac{1}{5!}β_{k+1})

One has recursions

2p · β_{k+1,2p} = −β_{k,2p+1} − \frac{1}{2p + 1}β_{k+1};  (2p − 1) · β_{k+1,2p−1} = −β_{k,2p} (8)

These recursions together with β_{k,1} = −β_{k+1} determine the numbers β_{k,p}.

Lemma 2.3. Let m ≥ 1. Then

β_{0,2m} = β_{0,2m+1} = \frac{1}{2m + 1}  (9)

β_{1,2m−1} = −\frac{1}{(2m − 1)(2m + 1)},  β_{1,2m} = 0  (10)

Proof. Let us prove formula

β_{0,2m} := (2m − 1)!(\frac{1}{1!}β_{2m} + \frac{1}{3!}β_{2m−2} + ... + \frac{1}{(2m − 1)!}β_{2}) = \frac{1}{2m + 1} (11)

Since β_{0} = 1, β_{1} = −1 and β_{2m+1} = 0 for m > 0 this formula, as one easily checks, is equivalent to the following one:

∑_{p≥1} \frac{1}{p!}β_{2m+1−p} = 0

The left hand side is the coefficient in t^{2m+1} of the power series

(e^{t} − 1)\frac{2t}{e^{2t} − 1} = \frac{2t}{e^{t} + 1}

The right hand side of the last equation is an almost even function in t: if we denote it by F (t) then F (−t) = F (t) + 2t.

Formula (11) immediately implies that

β_{1,2m−1} := −(2m−2)!(\frac{1}{1!}β_{2m}+\frac{1}{3!}β_{2m−2}+...+\frac{1}{(2m − 1)!}β_{2}) = \frac{1}{(2m − 1)(2m + 1)}

Using β_{2m+1} = 0 for m > 0 and β_{1} = −1 we have

β_{0,2m+1} = −(2m)!(\frac{1}{1!}β_{2m+1}+\frac{1}{3!}β_{2m−1}+...+\frac{1}{(2m − 1)!}β_{1}+\frac{1}{(2m + 1)!}β_{1}) = \frac{1}{2m + 1}

and

β_{1,2m} = (2m − 1)!(\frac{1}{1!}β_{2m+1} + \frac{1}{3!}β_{2m−1} + ... + \frac{1}{(2m − 1)!}β_{3}) = 0

The lemma is proved.

Proposition 2.4. For any p ≥ 1 one has

β_{n−2,p+1} − (n − 1)β_{n−1,p} − ∑_{k=1}^{n−3} β_{k,p}β_{n−k−1} = 0  (12)

Proof. We will do it by induction on p using the recursion relations (8). If p = 1 the formula we need to prove boils down to the identity

∑_{k=2}^{n−2} β_{i}β_{n−i} + nβ_{n} = 0  (13)

which is easy to check using the generating functions. Let us denote by (∗)_{n,p} the left hand side of (12). Then (∗)_{n,p} + (∗)_{n+1,p−1} = 0. Indeed, one has

(∗)_{n,p} + (∗)_{n+1,p−1} = β_{n−2,p+1} + (p − 1)β_{n−1,p}  (14)

−(n−1)β_{n−1,p}−n(p−1)β_{n,p−1} − ∑_{k=1}^{n−3} β_{k,p}β_{n−k−1}−(p−1) ∑_{k=0}^{n−3} β_{k+1,p−1}β_{n−k−1}

Let us assume first that p is even. Using recursions (8) we write (14) as

−\frac{1}{p + 1}β_{n−1} − β_{n−1,p} − (p − 1)β_{n,p−1} − (p − 1)β_{1,p−1}β_{n−1}

To prove that it is zero we use again recursions as well as the first formula in (10). Now let p be an odd number. Then the recursion relations show that (14) equals to

−β_{n−1,p} − (p − 1)β_{n,p−1} + \frac{n − 1}{p}β_{n} + \frac{1}{p} ∑_{k=1}^{n−3} β_{k+1}β_{n−k−1} − (p − 1)β_{1,p−1}β_{n−1}

Using the identity (14) together with the recursions and the second formula in (10) we see that this expression is also equal to zero. The proposition is proved.

6. Construction of the homomorphism r_{n}(·). Let us define differential 1-forms L̂_{p,q} on ℂP^{1}\{0, 1, ∞} for q ≥ 1 as follows:

L̂_{p,q}(z) := L̂_{p}(z) log^{q−1} |z| · d log |z|, p ≥ 2  (15)
L̂_{1,q}(z) := α(1 − z, z) log^{q−1} |z|

It provides a distribution on ℂP^{1}. Moreover for any rational function f on a complex variety X the 1-form L̂_{p,q}(f) provides a distribution on X(ℂ).

A useful notation. Set

A_{m} {⋀_{i=1}^{2p} d log |g_{i}| ∧ ⋀_{i=2p+1}^{m} di arg g_{j}} :=

Alt_{m} \{\frac{1}{(2p)!(m − 2p)!} ⋀_{i=1}^{2p} d log |g_{i}| ∧ ⋀_{i=2p+1}^{m} di arg g_{j}\}

and

A_{m} {log |g_{1}| · ⋀_{i=2}^{p} d log |g_{i}| ∧ ⋀_{i=p+1}^{m} di arg g_{j}} :=

Alt_{m} \{\frac{1}{(p − 1)!(m − p)!} log |g_{1}| · ⋀_{i=2}^{p} d log |g_{i}| ∧ ⋀_{i=p+1}^{m} di arg g_{j}\}

So A_{m}(F (g_{1}, .., g_{m})) is a weighted alternation (we divide by the order of the stabilizer of the term we alternate).

Now we are ready for the precise formulation of our main result.

Theorem - Construction 2.5. Let f, g_{1}, ..., g_{m} be rational functions on a complex variety X. Then the following formula provides maps satisfying all the properties of theorem 2.2.

r_{n+m}(m + 1) : {f}_{n} ⊗ g_{1} ∧ ... ∧ g_{m} ⟼

L̂_{n}(f) · A_{m} \{∑_{p≥0} \frac{1}{2p + 1} ⋀_{i=1}^{2p} d log |g_{i}| ∧ ⋀_{j=2p+1}^{m} di arg g_{j}\} +  (16)

∑_{k≥1} ∑_{1≤p≤m} β_{k,p}L̂_{n−k,k}(f) ∧ A_{m} {log |g_{1}| ⋀_{i=2}^{p} d log |g_{i}| ∧ ⋀_{j=p+1}^{m} di arg g_{j}} (17)

Here are several examples.

Example 1. m = 1, n is arbitrary.

{f}_{n} ⊗ g ⟼ L̂_{n}(f)di arg g − ∑_{k=1}^{n−1} β_{k+1}L̂_{n−k,k}(f) · log |g|

Example 2. m = 2, n is arbitrary.

\{f\}_{n} ⊗ g_{1} ∧ g_{2} ⟼ L̂_{n}(f) \{di arg g_{1} ∧ di arg g_{2} + \frac{1}{3}d log |g_{1}| ∧ d log |g_{2}|\}

− ∑_{k=1}^{n−1} β_{k+1}L̂_{n−k,k}(f) ∧ (log |g_{1}|di arg g_{2} − log |g_{2}|di arg g_{1})

+ ∑_{k≥1} β_{k+2}L̂_{n−k,k}(f) ∧ (log |g_{1}|d log |g_{2}| − log |g_{2}|d log |g_{1}|)

Example 3. The homomorphism r_{5}(∗).

\{f\}_{4}⊗g ⟼ L̂_{4}(f)di arg g−\frac{1}{3}L̂_{3}(f)d log |f|·log |g|+\frac{1}{45}α(1−f, f) log^{2} |f|·log |g|

\{f\}_{3} ⊗ g_{1} ∧ g_{2} ⟼ L̂_{3}(f) \{di arg g_{1} ∧ di arg g_{2} + \frac{1}{3}d log |g_{1}| ∧ d log |g_{2}|\}

−\frac{1}{3}L̂_{2}(f)d log |f| ∧ (log |g_{1}|di arg g_{2} − log |g_{2}|di arg g_{1})

−\frac{1}{45}α(1 − f, f) log |f| ∧ (log |g_{1}|d log |g_{2}| − log |g_{2}|d log |g_{1}|)

{f}_{2} ⊗ g_{1} ∧ g_{2} ∧ g_{3} ⟼

L̂_{2}(f) · A_{3}(di arg g_{1} ∧ di arg g_{2} ∧ di arg g_{3} + \frac{1}{3}d log |g_{1}| ∧ d log |g_{2}|di arg g_{3})

−α(1 − f, f) ∧ A_{3}(\frac{1}{3} log |g_{1}|di arg g_{2} ∧ di arg g_{3} + \frac{1}{15} log |g_{1}|d log |g_{2}| ∧ d log |g_{3}|)

Remark. The morphism of complexes r_{n}(·) is not defined uniquely by its properties if n > 3. For example if n = 4 we can have a map homotopic to our regulator map by using the homotopy which is given by the homomorphism

B_{2}(ℂ(X)) ⊗ Λ^{2}ℂ(X)^{∗} ⟶ A^{0}(η_{X}), {f}_{3} ⊗ g ⟼ L̂_{2}(f) · log |f| log |g|

(the other components of the homotopy are zero). Indeed, the residue map for it is zero, and it takes values in ℝ(2).

7. Theorem 2.5 from the point of view of the Deligne complex. Recall that a p-distribution on a manifold X is a linear continuous functional on the space of (dim_{ℝ} X − p)-forms with compact support. Denote by D_{X}^{p} the space of all real p-distributions on X.

Let X be a regular variety over ℂ. The n-th Beilinson-Deligne complex \underline{ℝ}_{D}(n)_{X} can be defined as a total complex associated with the following bicomplex of sheaves in classical topology on X(ℂ):

(D_{X}^{0} ⟶^{d} D_{X}^{1} ⟶^{d} … ⟶^{d} D_{X}^{n}  ⟶^{d} D_{X}^{n+1} ⟶^{d} …) ⊗ ℝ(n − 1)
↑ π_{n}  ↑ π_{n}
Ω_{X,log}^{n} ⟶^{∂} Ω_{X,log}^{n+1} ⟶^{∂}

Here D_{X}^{0} placed in degree 1 and (Ω_{X,log}^{•}, ∂) is the de Rham complex of holomorphic forms with logarithmic singularities at infinity. We will denote by \underline{ℝ}_{D}(n)(U) the complex of the global sections.

Theorem 2.2 can be reformulated as follows

Theorem 2.6. Let X be a complex algebraic variety. Set r̃_{n}(i) := r_{n}(i) for i < n and let

r̃_{n}(n) : Λ^{n}ℂ(X)^{∗} ⟶ A^{n−1}(η_{X})(n − 1) ⊕ Ω_{log}^{n}(η_{X})

f_{1} ∧ ... ∧ f_{n} ⟼ r_{n}(n)(f_{1} ∧ ... ∧ f_{n}) + d log f_{1} ∧ ... ∧ d log f_{n}  (18)

Then we get a homomorphism of complexes

r̃_{n}(·) : Γ(ℂ(X); n) ⟶ \underline{ℝ}_{D}(n)(η_{X})  (19)

compatible with the residues as explained in the part (d) of theorem 2.2.

Indeed, condition b) of theorem 2.2 just means that the right hand side of (18) is a cycle in the Deligne complex \underline{ℝ}(n)_{D}(η_{X}).

There is a natural Dolbeault resolution of the complex Ω_{log}^{≥n}. Using it in the complex of sheaves \underline{ℝ}_{D}(n)_{X} to replace the subcomplex Ω_{log}^{≥n} we get a complex \underline{ℝ}′_{D}(n)_{X} of fine sheafs on X(ℂ). The property (d) would allow us to extend the homomorphism r̃_{n}(·) to a morphism of complexes

r̃_{n}(·) : Γ(X; n) ⟶ \underline{ℝ}′_{D}(n)(X(ℂ))

For small weights this was explained in detail in [G2].

If X is a variety over ℝ, then

H_{D}^{i}(X_{ℝ}; ℝ(n)) = H_{D}^{i}(X(ℂ), ℝ(n))F̄_{∞}

where F̄_{∞} is the de Rham involution, i.e. the composition of the involution F_{∞} on X(ℂ) induced by the complex conjugation with the complex conjugation of coefficients. If X is a variety over ℚ then the regulator map is defined as

H^{i}Γ(X; n) ⟶ H_{D}^{i}(X ⊗_{ℚ} ℝ/_{ℝ}; ℝ(n))  (20)

Conjecture 2.7. The image of the regulator map (20) in the Deligne cohomology coincides with the image of Beilinson’s regulator map.

If X = Spec(F) where F is a number field then the only nontrivial case is i = 1, and we get a version of Zagier’s conjecture. The next case is when X is a curve over number field, and i = 2. Then we come to conjecture 1.5 in [G5], see also theorem 4.4 in [G5] and s. 7 in [G7].

There is a version of conjecture 2.7 expressing the special value of the corresponding L-function at s = n via the regulator map on the polylogarithmic complex. Its specialization for elliptic curves is conjecture 1.10 in [G7], see also theorem 4.7 in [G5]. For the n = 2 it is a theorem of Bloch for elliptic CM curves and of Beilinson for elliptic curves over ℚ, for n = 3 it was conjectured by Deninger and proved by the author in [G5] for elliptic curves over ℚ, and it is also proved now by the author for n = 4 for elliptic curves over ℚ (to appear).

8. Generalizations. According to the Tannakian formalism the category of mixed ℝ-Hodge-Tate structures is equivalent to the category of graded comodules over a certain Lie coalgebra L_{•}(ℂ) positively graded by the weights. There exists a natural homomorphism of groups

B_{n}(ℂ) ⟶ L_{n}(ℂ)

provided by the Hodge realization of the polylogarithmic motive. One has a homomorphism of ℚ-vector spaces

p_{n} : L_{n}(ℂ) ⟶ ℝ(n − 1)

called the Lie-period map, see [D] for its construction. The composition

B_{n}(ℂ) ⟶ L_{n}(ℂ) ⟶ ℝ(n − 1)

coincides with the homomorphism L̂_{n}, see [BD].

We generalize this picture considering variations of mixed Hodge-Tate structures over complex varieties and working on the level of complexes.

First of all, the category of variations of mixed ℝ-Hodge-Tate structures over η_{X} = Specℂ(X), where X is a complex algebraic variety, is equivalent to the category of graded comodules over a Lie coalgebra L_{•}(η_{X}).

The Hodge realization functor provides canonical homomorphism

l_{n} : B_{n}(ℂ(X)) ⟶ L_{n}(η_{X})  (21)

Applying it componentwise we get a morphism of complexes ([G1-2])

l_{n−k} ⊗ ∧^{k}l_{1} : B_{n−k}(ℂ(X)) ⊗ Λ^{k}ℂ(X)^{∗} ⟶ L_{n−k}(η_{X}) ⊗ Λ^{k}L_{1}(η_{X})

The vector space on the right is a subspace in (Λ^{k+1}L_{•}(η_{X}))_{n} where the subscript n means the weight n part. The standard cochain complex of a Lie coalgebra L_{•} looks as follows

L_{•}(η_{X}) ⟶ Λ^{2}L_{•}(η_{X}) ⟶ Λ^{3}L_{•}(η_{X}) ⟶ ...  (22)

It is a complex of graded ℚ-vector spaces. We get a morphism of complexes

B_{n}(ℂ(X)) →^{δ} B_{n−1}(ℂ(X)) ⊗ ℂ(X)^{∗} →^{δ} … →^{δ} ⋀^{n} ℂ(X)^{∗}
↓  ↓  ↓
L_{n}(η_{X}) →  (Λ^{2}L_{•}(η_{X}))_{n}  → ... → Λ^{n}L_{1}(η_{X})

The bottom line of this diagram is the degree n part of the complex (22). The Lie-period provides a natural homomorphism of ℚ-vector spaces

p_{n} : L_{n}(η_{X}) ⟶ A^{0}(η_{X})(n − 1)  (23)

The following result tells us that it is a beginning of a morphism of complexes.

Theorem 2.8. There exists a morphism of complexes

the weight n part of (22) ⟶ the weight n Deligne complex \underline{ℝ}_{D}(n)(η_{X}) (24)

whose degree 1 component L_{•}(η_{X}) ⟶ A^{0}(η_{X}) coincides with the map (23).

A detailed account on this result and related issues would double the size of this paper. We will do it in a different place.

Combining homomorphism (24) with the previous map of complexes

Γ(η_{X}; n) ⟶ the weight n part of (22)

we get a homomorphism of complexes from theorem 2.6.

3 Arakelov motivic complexes: examples

1. Another version of Deligne’s complexes. We need a complex introduced twenty years ago by Deligne and quasiisomorphic to the complex \underline{ℝ}′_{D}(n)(X(ℂ)).

Let D_{X}^{p,q} = D^{p,q} be the abelian group of complex valued distributions of type (p, q) on X(ℂ). Consider the following cohomological bicomplex, where the group D^{0,0} is placed in degree 1:

D_{cl}^{n,n}
2¯∂∂ ↗
D^{0,n−1} ⟶^{∂} D^{1,n−1} ⟶^{∂} ... ⟶^{∂} D^{n−1,n−1}
¯∂ ↑  ¯∂ ↑  ¯∂ ↑
...  ...  ...  ... ... ...  ...
¯∂ ↑  ¯∂ ↑  ¯∂ ↑
D^{0,1}  ⟶^{∂} D^{1,1}  ⟶^{∂} ... ⟶^{∂} D^{n−1,1}
¯∂ ↑  ¯∂ ↑  ¯∂ ↑
D^{0,0}  ⟶^{∂} D^{1,0}  ⟶^{∂} ... ⟶^{∂} D^{n−1,0}

Let T ot^{•} be the total complex of this bicomplex. It is concentrated in degrees [1, 2n]. The complex C_{D}^{•}(X; ℝ(n)) is subcomplex of T ot^{•} defined as follows. Intersect the part of the complex T ot^{•} coming from the n × n square in the diagram with the complex of ℝ(n − 1)-valued distributions. Consider the subgroup D_{cl,ℝ}^{n,n}(n) ⊂ D_{cl}^{n,n} of the ℝ(n)-valued distributions of type (n, n). They form a subcomplex in T ot^{•} because ¯∂∂ sends ℝ(n − 1)-valued distributions to ℝ(n)-valued distributions. This is the complex C_{D}^{•}(X; ℝ(n)) with the differential denoted by D. It is a truncation of the complex considered by Deligne.

Proposition 3.1. The complex C_{D}^{•}(X; n) is quasiisomorphic to the truncated Deligne complex τ_{≤2n}\underline{ℝ}′_{D}(n)(X(ℂ)).

Proof. See [C] or [G3].

Now if X is a variety over ℝ, then

C_{D}^{•}(X_{ℝ}; ℝ(n)) := C_{D}^{•}(X, n)F̄_{∞} H_{D}^{i}(X_{ℝ}; ℝ(n)) = H^{i}(C_{D}^{•}(X_{ℝ}; ℝ(n)))

2. The Arakelov motivic complexes. By definition the weight n Arakelov motivic complex Γ^{A}(X; n) is defined as follows

RΓ_{X}^{A}(n) := Cone(RΓ_{X}(n) ⟶^{Reg} C_{D}(X_{ℝ},; ℝ(n)))[−1]

where RΓ_{X}(n) is the weight n motivic complex and Reg is the regulator map. It is an object of the derived category. When X is regular scheme over ℚ then

RΓ_{X_{ℚ}}^{A}(n) := Cone(RΓ_{X}(n) ⟶^{Reg} C_{D}(X ⊗_{ℚ} ℝ; ℝ(n)))[−1]

Remark. For the application to special values of L-functions we should take a model of X over ℤ and have a similar construction for finite primes p where it has semistable reduction.

Below we discuss complexes Γ^{A}(X; n) with the polylogarithmic complex Γ(X; n) serving as the motivic complex RΓ_{X}(n). We will abuse notations by writing (D^{p,q}⊕D^{q,p})_{ℝ}(k) for the subgroup of ℝ(k)-valued currents in D^{p,q}⊕D^{q,p}. The last groups of the complex Γ^{A}(X; n) look as follows (see also s. 3.6 below):

...  ⊕_{Y_{2}∈X^{(n−2)}}K_{2}(Y_{2})  ⟶ ⊕_{Y_{1}∈X^{(n−1)}}ℂ(Y_{1})^{∗} ⟶ ⊕_{Y ∈X^{(n)}}ℤ
...  ↓ r_{n}(n − 2)  ↓ r_{n}(n − 1)  ↓ r_{n}(n)
... (D_{ℝ}^{n−2,n−1} ⊕ D^{n−1,n−2})_{ℝ}(n − 1) ⟶^{∂+\overline{∂}} D_{ℝ}^{n−1,n−1}(n − 1) ⟶^{2\overline{∂}∂} D_{cl,ℝ}^{n,n}(n)

The (2n)-th cohomology group of this complex looks as follows:

\frac{\{codim. n cycle Y, g ∈ D_{cl,ℝ}^{n−1,n−1}(n − 1)\} such that 2\overline{∂}∂g + δ(Y) = 0}{\{div(f), − log |f|δ_{Y_{1}}\}, f ∈ ℂ(Y_{1})^{∗}; \{0; ∂u + \overline{∂}v\}}

(25) where

u ∈ D^{n−2,n−1}; v ∈ D^{n−1,n−2}

This is the group of codimension n Arakelov cycles on X(ℂ). To compare with [S] notice that

dd^{ℂ} = \frac{1}{2πi}\overline{∂}∂, 2\overline{∂}∂ log |f| = δ_{div(f)}

If we modify this definition by replacing the last group D_{cl,ℝ}^{n,n}(n) by its quotient by smooth forms D_{cl,ℝ}^{n,n}(n)/A_{cl,ℝ}^{n,n}(n) we get the group of codimension n Arakelov cycles defined by Gillet and Soulé [S]:

\frac{\{Y ∈ X^{(n)}, g ∈ D_{cl,ℝ}^{n−1,n−1}(n − 1)\} such that 2\overline{∂}∂g + δ(Y) is smooth}{\{div(f), − log |f|δ_{Y_{1}}\}, f ∈ ℂ(Y_{1})^{∗}; \{0; ∂u + \overline{∂}v\}}

(26)

Such a modification does not seem natural in our context, but it worked well in the Gillet-Soulé theory. The reason to have several versions of Arakelov groups is the desire to have a theory of Chern classes with values in Arakelov motivic cohomology (and finally the higher Riemann-Roch theorem) for various versions of the category of vector bundles with some kind of hermitian metric discussed below.

a) Consider holomorphic vector bundles with flat hermitian metrics. They have Chern classes with values in the groups (25). In particular holomorphic line bundles with flat hermitian metrics form an abelian group under the tensor product, denoted P̃ic_{0}(X), which sits in an exact sequence

0 ⟶ ℝ^{π_{0}(X(ℂ))} ⟶ P̃ic_{0}(X) ⟶ Pic_{0}(X) ⟶ 0

Indeed, a flat hermitian hermitian metric in a line bundle over a connected manifold is determined up to a constant. There is an isomorphism

P̃ic_{0}(X) ⟶^{=} H^{2}(Γ^{A}(X; 1))

given by the first Chern class as follows. If s is a section of a holomorphic line bundle over X with a hermitian metric || · || then the pair (log ||s||, div(s)) provides the corresponding class in H^{2}(Γ^{A}(X; 1)). Indeed, according to the Poincaré-Lelong formula for a meromorphic section s of any holomorphic line bundle one has

2\overline{∂}∂ log ||s|| − δ_{div(s)} = c_{1}(L, || · ||)

where on the right is the Chern form related to the hermitian structure on the line bundle, which is zero if and only if the metric is flat.

b) To have Chern classes for holomorphic vector bundles with arbitrary hermitian metrics we are more or less forced to the Gillet-Soulé definition. However in this case the groups of Arakelov cycles are infinite dimensional.

c) Choose a Kahler metric on X(ℂ) and consider only such metrics (harmonic metrics) on holomorphic vector bundles over X(ℂ) whose Chern forms are harmonic with respect to the chosen metric. Then we have Chern classes yet to another modification of the Arakelov group where D_{cl,ℝ}^{n,n}(n) replaced by its quotient by the image of the following map provided by the Hodge theory:

H^{2n}(X(ℂ), ℤ(n)) ⟶ A_{cl,ℝ}^{n,n}(n)

3. The weight one. The regulator map on the weight one motivic complex looks as follows:

ℂ(X)^{∗} ⟶ ⊕_{Y ∈X^{(1)}}ℤ
↓ r_{1}(1)  ↓ r_{1}(2)
D^{0,0}  ⟶^{2\overline{∂}∂} D_{cl,ℝ}^{1,1}(1)
r_{1}(2) : Y_{1} ⟼ 2πi · δ_{Y_{1}}, r_{1}(1) : f ⟼ log |f|

Here the top line is the weight 1 motivic complex, sitting in degrees [1, 2]. Shifting by −1 the total complex associated with this bicomplex we get the weight 1 Arakelov motivic complex Γ^{A}(X; 1).

Here are examples of the regulator maps for the weights n ≤ 3.

4. The weight two. The regulator map on the weight two motivic complexes looks as follows.

B_{2}(ℂ(X)) ⟶^{δ}  Λ^{2}ℂ(X)^{∗}  ⟶^{∂} ⊕_{Y ∈X^{(1)}}ℂ(Y)^{∗} ⟶^{∂} ⊕_{Y ∈X^{(2)}}ℤ
↓ r_{2}(1)  ↓ r_{2}(2)  ↓ r_{2}(3)  ↓ r_{2}(4)
D_{ℝ}^{0,0}(1) ⟶^{D} (D^{0,1} ⊕ D^{1,0})_{ℝ}(1) ⟶^{D}  D_{ℝ}^{1,1}(1)  ⟶^{2\overline{∂}∂} D^{2,2}(2)_{cl,ℝ}

where we set

r_{2}(4) : Y_{2} ⟼ (2πi)^{2} · δ_{Y_{2}}
r_{2}(3) : (Y_{1}, f) ⟼ 2πi · log |f|δ_{Y_{1}}
r_{2}(2) : f ∧ g ⟼ − log |f|di arg g + log |g|di arg f
r_{2}(1) : {f}_{2} ⟼ L̂_{2}(f)

To prove that we get a morphism of complexes we use theorem 2.2. The following argument is needed to check the commutativity of the second square. The de Rham differential of the distribution r_{2}(2)(f ∧ g) is

D(− log |f|di arg g + log |g|di arg f) =
π_{2}(d log f ∧ log g) + 2πi · (log |g|δ(f) − log |f|δ(g))

This does not coincide with r_{2}(3) ∘ ∂(f ∧ g), but the difference is

(D ∘ r_{2}(2) − r_{2}(3) ∘ ∂)(f ∧ g) = π_{2}(d log f ∧ log g) ∈ (D^{0,2} ⊕ D^{2,0})_{ℝ}(1)

Defining the differential D on the second group of the complex C_{D}(X_{ℝ}, ℝ(2)) we take the de Rham differential and throw away from it precisely these components. Therefore the middle square is commutative.

5. The weight three. The weight three motivic complex Γ(X; 3) is the total complex of the following bicomplex: (the first group is in degree 1)

B_{3}(ℂ(X)) ⟶ B_{2}(ℂ(X)) ⊗ ℂ(X)^{∗} ⟶  Λ^{3}ℂ(X)^{∗}
↓  ↓
⊕_{Y_{1}∈X^{(1)}}B_{2}(ℂ(Y_{1})) ⟶ ⊕_{Y_{1}∈X^{(1)}}Λ^{2}ℂ(Y_{1})^{∗}
↓
⊕_{Y_{2}∈X^{(2)}}ℂ(Y_{2})^{∗}
↓
⊕_{Y_{3}∈X^{(3)}}ℂ(Y_{3})^{∗}

The Deligne complex C_{D}(X_{ℝ}, ℝ(3)) looks as follows:

D_{cl,ℝ}^{3,3}(3)
2\overline{∂}∂ ↗
D^{0,2} ⟶^{∂} D^{1,2} ⟶^{∂} D^{2,2}
↑ \overline{∂} ↑ \overline{∂} ↑ \overline{∂}
D^{0,1} ⟶^{∂} D^{1,1} ⟶^{∂} D^{1,2}
↑ \overline{∂} ↑ \overline{∂} ↑ \overline{∂}
D^{0,0} ⟶^{∂} D^{1,0} ⟶^{∂} D^{2,0}

(Recall that the 3 × 3 square in this diagram consists of ℝ(2)-valued distributions). We construct the regulator map from the motivic complex Γ(X; 3) to the Deligne complex C_{D}(X_{ℝ}, ℝ(3)) by setting

r_{3}(6) : Y_{3} ⟼ (2πi)^{3} · δ_{Y_{3}}
r_{3}(5) : (Y_{2}, f) ⟼ (2πi)^{2} · log |f|δ_{Y_{2}}
r_{3}(4) : (Y_{1}, f ∧ g) ⟼ 2πi · (− log |f|di arg g + log |g|di arg f)δ_{Y_{1}}
r_{3}(3) : (Y_{1}, {f}_{2}) ⟼ 2πi · L̂_{2}(f)δ_{Y_{1}}

r_{3}(3) : f_{1}∧f_{2}∧f_{3} ⟼ Alt_{3}(\frac{1}{6} log |f_{1}|d log |f_{2}|∧d log |f_{3}|+\frac{1}{2} log |f_{1}|di arg f_{2}∧di arg f_{3})

r_{3}(2) : \{f\}_{2}⊗g ⟼ L̂_{2}(f)di arg g−\frac{1}{3} log |g|·(− log |1−f|d log |f|+log |f|d log |1−f|)

r_{3}(1) : {f}_{3} ⟼ L̂_{3}(f)

6. The general case. Let d := dimX. Then the complex Γ(X; n) is the total complex of the following bicomplex:

Γ(ℂ(X); n) ⟶ ⊕_{Y_{1}∈X^{(1)}}Γ(ℂ(Y_{1}); n − 1)[−1] ⟶
⊕_{Y_{2}∈X^{(2)}}Γ(ℂ(Y_{2}); n − 2)[−1] ⟶ ... ⟶ ⊕_{Y_{d}∈X^{(d)}}Γ(ℂ(Y_{d}); n − d)[−d]

where the arrows are provided by the residue maps, see [G1], p 239-240.

To define the regulator map from this gadget to the complex C_{D}(X_{ℝ}; ℝ(n)) we specify it for each of Γ(ℂ(Y_{k}); n − k)[−k] where k = 0, ..., d. Namely, we take the constructed in theorem (2.5) homomorphism r_{n−k}(·) for η_{Y_{k}} and multiply it by (2πi)^{n−k}δ_{Y_{n−k}}. Notice that the distribution δ_{Y} depends only on the generic point of a subvariety Y. Its image in the bicomplex described in the section 3.1 is in

⊕D^{p,q}; p ≥ k, q ≥ k, p + q ≤ n + k

which is the triangle symmetric with respect to the diagonal in the n × n square presented on the picture below.

Compatibility with the differentials is a corollary of theorem 2.5 and the following remark. If we would consider the n × n square in the Deligne complex as a part of the Dolbeault complex then we do not get a morphism of complexes. For k < n the descrepency lies in the subgroup

D^{k,n+1} ⊕ D^{n+1,k}  (27)

and come from the elements (Y_{k}, f_{1} ∧ ... ∧ f_{n−k}). However since the restriction of D to the subgroup D^{k,n} ⊕ D^{n,k} equals to the de Rham differential D modulo the components (27) this does not create a problem.

4 Proofs

1. The differential equation for the function L_{n}(z). The following proposition was stated without a proof as formula 1.14 in [G2].

Proposition 4.1. The differential equation for L̂_{n}(f) for n ≥ 3 is:

dL̂_{n}(f) = L̂_{n−1}(f)di arg z − ∑_{k=2}^{n−1} β_{k} · L̂_{n−k,k}(f)  (28)

Proof. Consider the generating series

Li(z; t) := ∑_{n≥1} Li_{n}(z)t^{n−1};  L̂(z; t) := ∑_{n≥1} L̂_{n}(z)t^{n−1}

Lemma 4.2. Differential equations (28) together with the formulas

dL̂_{1}(z) = −d log |1 − z|, dL̂_{2}(z) = − log |1 − z|di arg z + log |z|di arg(1 − z) (29)

are equivalent to the formula

dL̂(z; t) = −d log |1 − z| + log |z|di arg(1 − z) · t + L̂(z; t)di arg z · t (30)

− \{\frac{2| log z| · t}{e^{2 log |z|·t} − 1} + log |z| · t − 1\} (L̂(z; t)\frac{d log |z|}{log |z|} + d log |1 − z|)  (31)

Proof. We check directly that the differential equation (29) appears as the coefficients of (31) in t^{0} and t^{1}. ¿From now on we will work modulo the monomials t^{0} and t^{1}. The second term on the right of (28) (written for n ≥ 3) provides the total contribution of

− ∑_{n≥3}(∑_{k≥2}^{n−2} β_{k}L̂_{n−k}(z) · log^{k−1} |z|d log |z| + β_{n−1}α(1 − z, z) log^{n−2} |z|)t^{n−1} =

− ∑_{2≤k<n} β_{k}L̂_{n−k}(z) log^{k−1} |z|t^{n−1} · d log |z|

− ∑_{n≥3} β_{n−1} log^{n−1} |z|t^{n−1} · d log |1 − z| =

− \{\frac{2 log z · t}{e^{2 log |z|·t} − 1} + log |z| · t − 1\} (L̂(z; t)\frac{d log |z|}{log |z|} + d log |1 − z|)

which is precisely (31). The first term on the right of (28) gives the last term in (30). The lemma is proved.

Let us prove formula (30) - (31). One has for n ≥ 2:

dπ_{n}(Li_{m}(z)) = \frac{1}{2}d(Li_{m}(z) + (−1)^{n−1}\overline{Li_{m}(z)}) =

π_{n}(Li_{m−1}(z))d log |z| + π_{n−1}(Li_{m−1}(z))di arg z  (32)

Define an operator P acting on the generating series for a sequence of holomorphic functions f_{n}(z) by

P (∑_{n≥1} f_{n}(z)t^{n−1}) := ∑_{n≥1} π_{n}(f_{n}) · t^{n−1}

Then

L̂(z; t) = P { ∑_{0≤k<n} β_{k}Li_{n−k}(z) · log^{k} |z| · t^{n−1}} =

P \{Li(z; t) · \frac{2 log |z|t}{e^{2 log |z|t} − 1}\}  (33)

One has

d(\frac{2x}{e^{2x} − 1}) = (\frac{2x}{e^{2x} − 1} − \frac{4x^{2}e^{2x}}{(e^{2x} − 1)^{2}} )\frac{dx}{x} = \frac{2x}{e^{2x} − 1} · (1 − 2x − \frac{2x}{e^{2x} − 1})\frac{dx}{x}

Thus since the second factor in (33) is a real function we get

dL̂(z; t) = d_{1}P (Li(z; t)\frac{2 log |z| · t}{e^{2 log |z|·t} − 1})

+P \{Li(z; t) · \frac{2 log |z| · t}{e^{2 log |z|·t} − 1} · (1 − 2 log |z| · t − \frac{2 log |z| · t}{e^{2 log |z|·t} − 1})\} \frac{d log |z|}{log |z|}

where d_{1} is the differential applied to Li(z; t) only. Since \frac{2x}{e^{2x}−1} + x − 1 is an even function in x we rewrite this as

d_{1}P (Li(z; t)\frac{2 log |z| · t}{e^{2 log |z|·t} − 1})  (34)

−L̂(z; t) · (\frac{2 log |z| · t}{e^{2 log |z|·t} − 1} + log |z| · t − 1)\frac{d log |z|}{log |z|}  (35)

−P \{Li(z; t)\frac{2 log |z| · t}{e^{2 log |z|·t} − 1} · t\} d log |z|  (36)

We handle this as follows.

i) (35) matches half of (30).

ii) The Li_{1}-part of (34) is

d_{1}P (Li_{1}(z)\frac{2 log |z| · t}{e^{2 log |z|·t} − 1}) = −\frac{2 log |z| · t}{e^{2 log |z|·t} − 1}d log |1 − z|

It matches the second half of (31) modulo coefficients in t^{0} and t^{1}.

iii) The rest of (34) equals to

P \{Li(z; t)\frac{2 log |z| · t}{e^{2 log |z|·t} − 1}d log z · t\} =  (37)

P \{L̂(z; t)(z; t)\} di arg z · t + P \{Li(z; t)\frac{2 log |z| · t}{e^{2 log |z|·t} − 1} · t\} d log |z| (38)

Thus the first term in (38) matches the right term in (30), and the second is canceled with (36). The proposition is proved.

2. Proof of theorem 2.5. i) Let us show that the map

{f} ⟼ L̂_{n−k,k}(f)  (39)

provides a group homomorphism B_{n}(ℂ(X)) ⟶ A_{η}^{1}(X(ℂ)). The arguments are similar to the proof of theorem 1.15 in [G2]. Suppose first that n − k > 1. By theorem 1.15 in [G2] the map {f} ⟼ L̂_{n}(f) provides a group homomorphism B_{n}(ℂ(X)) ⟶ A_{η}^{0}(X(ℂ)). Consider the following map

B_{n}(ℂ(X)) ⟶ B_{n−k}(ℂ(X)) ⊗ S^{k}ℂ(X)^{∗}, {f}_{n} ⟼ {f}_{n−k} ⊗ f^{k}  (40)

It can be defined as a composition

{f}_{n} ⟼ {f}_{n−1} ⊗ f ⟼ {f}_{n−2} ⊗ f · f ⟼ ... ⟼ {f}_{n−k} ⊗ f · ... · f

Each of these maps is of type δ ⊗ id and thus is a homomorphism of abelian groups. The composition followed by the homomorphism

B_{n−k}(ℂ(X)) ⊗ S^{k}ℂ(X)^{∗} ⟶ A_{η}^{1}(X(ℂ))

\{f\}_{n−k} ⊗ g_{1} · ... · g_{k} ⟼ L̂_{n−k}(f)\frac{1}{k} d(log |g_{1}| · ... · log |g_{k}|)

leads to the map (39). In the case n−k = 1 we present map (39) as a composition

B_{n}(ℂ(X)) ⟶^{(40)} B_{2}(ℂ(X)) ⊗ S^{n−2}ℂ(X)^{∗} ⟶^{a} A_{η}^{1}(X(ℂ))

a : {f}_{2} ⊗ g_{1} · ... · g_{n−2} ⟼ α(1 − f, f) · log |g_{1}| · ... · log |g_{n−2}|

and use the fact that {f}_{2} ⟼ α(1 − f, f) is a group homomorphism. ii) The properties (a) and (b) in theorem 2.2 are true by the very definitions. iii) It is very easy to see that the property (d) is also true by the very

definitions: it basically reflects the fact that the numerical coefficients in the formula for r_{n+m}(m + 1) do not depend on m and uses formula (7).

iv) It remains to prove that the diagram in theorem 2.2 is commutative. Let us show first that its right square

B_{2}(ℂ(X)) ⊗ Λ^{n−2}ℂ(X)^{∗} ⟶^{δ} Λ^{n}ℂ(X)^{∗}
↓ r_{n}(n − 1)  ↓ r_{n}(n)
A_{η}^{n−2}(X(ℂ))  ⟶^{d} A_{η}^{n−1}(X(ℂ))

is commutative. We change the numeration by putting n := m + 2.

Proposition 4.3.

r_{m+2}(m + 1) : {f}_{2} ⊗ g_{1} ∧ ... ∧ g_{m} ⟼

L̂_{2}(f) · A_{m} \{∑_{p≥0} \frac{1}{2p + 1} ⋀_{i=1}^{2p} d log |g_{i}| ∧ ⋀_{j=2p+1}^{m} di arg g_{j}\}

−α(1 − f, f) ∧ A_{m}(∑_{p≥0} \frac{1}{(2p + 1)(2p + 3)} log |g_{1}| · ⋀_{i=2}^{2p+1} d log |g_{i}| ∧ ⋀_{j=2p+2}^{m} di arg g_{j})

Proof. It follows immediately from the definition and lemma 2.3:

Let us compute

d(r_{m+2}(m + 1)({f}_{2} ⊗ g_{1} ∧ ... ∧ g_{m}))

Using proposition 4.3 we get

(− log |1 − f|di arg f + log |f|di arg(1 − f)) ∧  (41)

A_{m} \{∑_{p≥0} \frac{1}{2p + 1} ⋀_{i=1}^{2p} d log |g_{i}| ∧ ⋀_{j=2p+1}^{m} di arg g_{j}\}

+ 2d log |1 − f| ∧ d log |f|∧

A_{m}(∑_{p≥0} \frac{1}{(2p + 1)(2p + 3)} log |g_{1}| · ⋀_{i=2}^{2p+1} d log |g_{i}| ∧ ⋀_{j=2p+2}^{m} di arg g_{j})  (42)

+ α(1 − f, f) ∧ A_{m}(∑_{p≥0} \frac{1}{2p + 3} ⋀_{i=1}^{2p+1} d log |g_{i}| ∧ ⋀_{j=2p+2}^{m} di arg g_{j})  (43)

We can write the differential form r_{m}(g_{1} ∧ ... ∧ g_{m}) as follows:

−A_{m} \{∑_{p≥0} \frac{1}{2p + 1} log |g_{1}| · ⋀_{i=2}^{2p+1} d log |g_{i}| ∧ ⋀_{j=2p+2}^{m} di arg g_{j}\}  (44)

In particular this makes transparent the property

dr_{m}(g_{1} ∧ ... ∧ g_{m}) + π_{m}(d log g_{1} ∧ ... ∧ log g_{m}) = 0

Now computing

dr_{m+2}((1 − f) ∧ f ∧ g_{1} ∧ ... ∧ g_{m})  (45)

using formula (44) and comparing the result with formulas (41) - (43) we see the following:

1) Formula (41) matches the part of (45) where the contribution of (1−f)∧f is given by − log |1 − f|di arg |f| + log |f|di arg |1 − f|.

2) Formula (43) matches the part of (45) where the contribution of (1−f)∧f is given by − log |1 − f|d log |f| + log |f|d log |1 − f|.

Before we continue any further let us note that

d log |1 − f| ∧ di arg(f) = d log |f| ∧ di arg(1 − f)  (46)

d log |1 − f| ∧ d log |f| = − di arg(1 − f) ∧ di arg f  (47)

Indeed,

0 = d log(1 − f) ∧ d log f = (d log |1 − f| + di arg(1 − f)) ∧ (d log |f| + d arg if)

Therefore using (46) we see the following:

3) the sum of the terms of (45) where the contribution of (1 − f) ∧ f is given either by d log |1 − f| ∧ di arg |f| or by di arg(1 − f)f ∧ d log |f| is zero.

4) It remains to show that (42) matches the part of (45) where the contribution of (1 − f) ∧ f is given by d log |1 − f| ∧ d log |f| or di arg(1 − f)f ∧ di arg f.

Let us substitute

\frac{2}{(2p + 1)(2p + 3)} = \frac{1}{2p + 1} − \frac{1}{2p + 3}

into (42) and split the formula we get into two parts. The first part, denoted (42)_{I}, is the one where the terms appear with the coefficient \frac{1}{2p+1}. The second part, denoted (42)_{II}, is the rest. It corresponds to \frac{1}{2p+3}.

Using (47) to calculate the part of (45) where (1 − f) ∧ f contributes via di arg(1−f)f ∧di arg f we see that it matches with (42)_{I}. The other part, (42)_{II}, matches the part of (45) where (1 − f) ∧ f contributes d log |1 − f| ∧ d log |f|. So we achieved our goal 4). The commutativity of the last square is proved.

Now let us prove the commutativity of the squares different from the last one, i.e. (n > 2)

B_{n}(ℂ(X)) ⊗ Λ^{m}ℂ(X)^{∗} ⟶^{δ} B_{n−1}(ℂ(X)) ⊗ Λ^{m+1}ℂ(X)^{∗}
↓ r_{n+m}(m + 1)  ↓ r_{n+m}(m + 2)
A_{η}^{m}(X(ℂ))  ⟶^{d}  A_{η}^{m+1}(X(ℂ))

One has

dr_{n+m}(m + 1)({f}_{n} ⊗ g_{1} ∧ ... ∧ g_{m})=  (48)

dL̂_{n}(f) ∧ A_{m} \{∑_{p≥0} \frac{1}{2p + 1} ⋀_{i=1}^{2p} d log |g_{i}| ∧ ⋀_{j=2p+1}^{m} di arg g_{j}\}  (49)

+ ∑_{k≥1} ∑_{1≤p≤m} β_{k,p}dL̂_{n−k,k}(f) ∧ A_{m} {log |g_{1}| ⋀_{i=2}^{p} d log |g_{i}| ∧ ⋀_{j=p+1}^{m} di arg g_{j}}

(50)

− ∑_{k≥1} ∑_{1≤p≤m} p · β_{k,p}L̂_{n−k,k}(f) ∧ A_{m} {⋀_{i=1}^{p} d log |g_{i}| ∧ ⋀_{j=p+1}^{m} di arg g_{j}} (51)

Let us compare this expression with

r_{n+m}(m + 2)({f}_{n−1} ⊗ f ∧ g_{1} ∧ ... ∧ g_{m})= (16)_{m+2} + (17)_{m+2}  (52)

where (16)_{m+2} (resp. (17)_{m+2}) stays for the term similar to (16) (resp. (17)) in the definition of r_{n+m}(m + 1) (see theorem-construction 2.5).

Observe that a priori f can contribute to (16)_{m+2} via d log |f| or di arg f, and to (17)_{m+2}via log |f|, d log |f| or di arg f. A careful reader should keep track of these five different cases during the proof.

We split the job in two parts:

A) Check that the diagram is commutative if we pay attention only to the terms where d log |1 − f| or di arg(1 − f) appears. We call such terms A-terms.

B) Then we check that the diagram is commutative modulo the terms with d log |1 − f| or di arg(1 − f).

Part A). Here is the contribution from (49) - (51). Recall that n ≥ 3. ¿From (49) using (28) we get:

β_{n−1} · log^{n−1} |f|d log |1 − f| ∧ A_{m} \{∑_{p≥0} \frac{1}{2p + 1} ⋀_{i=1}^{2p} d log |g_{i}| ∧ ⋀_{j=2p+1}^{m} di arg g_{j}\}

(53)

¿From (50):

∑_{k=1}^{n−3} ∑_{1≤p≤m} −β_{k,p}β_{n−k−1} log^{n−2} |f|d log |1 − f| ∧ d log |f| ∧  (54)

A_{m} {log |g_{1}| ⋀_{i=2}^{p} d log |g_{i}| ∧ ⋀_{j=p+1}^{m} di arg g_{j}}  (55)

+ ∑_{1≤p≤m} β_{n−2,p}di arg(1 − f) ∧ d log |f| · log^{n−2} |f| ∧ (55)  (56)

− ∑_{1≤p≤m}(n − 1)β_{n−1,p}d log |1 − f| ∧ d log |f| ∧ log^{n−2} |f| ∧ (55) (57)

¿From (51):

∑_{1≤p≤m} pβ_{n−1,p} log^{n−1} |f| · d log |1 − f| ∧  (58)

A_{m} {⋀_{i=1}^{p} d log |g_{i}| ∧ ⋀_{j=p+1}^{m} di arg g_{j}}  (59)

The contribution of (16)_{m+2} is zero. The factor f from {f}_{n−1}⊗f ∧g_{1}∧...g_{m} can contribute into (17)_{m+2} via log |f|, di arg f and d log |f|.

The terms where f contributes via log |f| are

∑_{k≥1} ∑_{1≤p≤m} β_{k,p+1}L̂_{n−1−k,k}(f) log |f| ∧ (59)  (60)

Since we are concerned only with the A-terms, this boils down to

∑_{1≤p≤m} β_{n−2,p+1}L̂_{1,n−2}(f) log |f| ∧ (59) =  (61)

− ∑_{1≤p≤m} β_{n−2,p+1} log^{n−1} |f|d log |1 − f| ∧ (59)

Using formulas (8) we see that it matches (53) + (58).

The terms where f contributes into (17)_{m+2} via di arg f are

− ∑_{k≥1} ∑_{1≤p≤m} β_{k,p}L̂_{n−1−k,k}(f)di arg f ∧ (55)  (62)

Its contribution modulo B-terms is

− ∑_{1≤p≤m} β_{n−2,p} log^{n−2} |f|d log |1 − f| ∧ di arg f ∧ (55)  (63)

Using (46) we see that it matches (56).

Similar considerations for terms where f contributes into (17)_{m+2} via d log |f| minus ((57) + (54)) leads to the left hand side of (12), which is zero according to proposition 2.4. The part A) is proved.

Part B). We will write X =^{A} Y if X − Y = 0 modulo the A-terms.

a) Notice that the contribution of f to (17)_{m+2} via d log |f| is zero since L̂_{n−k,k}(f) ∧ d log |f| = 0 modulo A-terms.

b) The contribution to (49) of the term L̂_{n−1}(f)di arg f in formula (28) for dL̂_{n}(f) matches the part of term (16)_{m+2} where f contributes via di arg f.

c) One has

dL̂_{n−k,k}(f) =^{A} −L̂_{n−k−1,k}(f) ∧ di arg f  (64)

Thus (50) matches the part of (17)_{m+2} where f contributes via di arg f. d) The contribution to (49) of the other term in dL̂_{n}(f) modulo A is

− ∑_{p≥0} ∑_{k=2}^{n−2} \frac{β_{k}}{2p + 1}L̂_{n−k,k−1}(f) ∧  (65)

A_{m} {log |f| · ⋀_{i=1}^{2p} d log |g_{i}| ∧ ⋀_{j=2p+1}^{m} di arg g_{j}}  (66)

The only other terms in (48) with factor (66) are the terms of (51) where k > 1 and the number of d log |g_{i}|’s is even, i.e.

− ∑_{k≥1} 2pβ_{k,2p}L̂_{n−k,k−1}(f) ∧ (66)  (67)

On the other hand, in (52) the terms with factor (66) are

∑_{k≥1} β_{k,2p+1}L̂_{n−1−k,k}(f) ∧ (66)  (68)

which are precisely the terms of (17)_{m+2} where f contributes via log |f|, and p in β_{k,p} is odd.

Thus out of (65), (67) and (68) we see that L̂_{n−k,k−1}(f) ∧ (66) appears in our check of the commutativity of the diagram with factor

β_{k−1,2p+1} + \frac{β_{k}}{2p + 1} + 2pβ_{k,2p} =^{(8)} 0

e) The term of (16)_{m+2} where f contributes via d log |f|, i.e.

∑_{p≥0} \frac{1}{2p + 1}L̂_{n−1}(f) · A_{m+1} \{d log |f| ∧ ⋀_{i=1}^{2p−1} d log |g_{i}| ∧ ⋀_{j=2p}^{m} di arg g_{j}\} (69)

matches the term (51) with k = 1, p: odd, i.e.

− ∑_{p≥0}(2p − 1)β_{1,2p−1}L̂_{n−1,1}(f) ∧ A_{m} { ⋀_{i=1}^{2p−1} d log |g_{i}| ∧ ⋀_{j=2p}^{m} di arg g_{j}} (70)

Indeed, according to (10) (2p − 1)β_{1,2p−1} = −\frac{1}{2p+1}.

f) All the terms of (51) with k = 1 and even p are zero since β_{1,2p} = 0 by (10).

g) The terms of (51) with odd p and k > 1 look as follows:

∑_{p≥0} ∑_{k≥2}(2p + 1)β_{k,2p+1}L̂_{n−k,k−1}(f) ∧  (71)

A_{m+1} {log |f| ⋀_{i=1}^{2p+1} d log |g_{i}| ∧ ⋀_{j=2p+2}^{m} di arg g_{j}}  (72)

They match with the terms of (17)_{m+2} where f contributes via log |f| and p is even, see (60).

The the part B) of the commutativity proof is now complete. To double check that we list all the possibilities for contribution of f in (52), and indicate for each of them the part of the proof where it was handled.

Contribution of f to (16)_{m+1}:

via d log |f|: see e); via di arg f: see b).

Contribution of f to (17)_{m+2}: via log |f|, p odd: see d); via log |f|, p even: see g); via d log |f|: see a); via di arg f: see c). Contribution of f to (49): see b) and d). Contribution of f to (50): see c). Contribution of f to (51): k = 1, p odd: see e); k > 1, p odd: see g); k = 1, p even: see f); k > 1, p even: see d). The main theorem is proved.

References

[B1] Beilinson A.A, Higher regulators and values of L-functions, VINITI, 24 (1984), 181–238.

[B2] Beilinson A.A, Height pairing between algebraic cycles, in K-theory and algebraic geometry, Springer Lect. Notes 1289, 1-25.

[BD] Beilinson A.A., Deligne P, Interpretation motivique de la conjecture de Zagier in Symp. in Pure Math., v. 55, part 2, 1994, 23-41.

[Bl] Bloch S, Algebraic cycles and higher K-theory, Adv. in Math., (1986), v. 61, 267–304.

[Bl2] Bloch S, Function theory of polylogarithms, Structural properties of polylogarithms, L. Math. Surveys Monogr. 37, 275-285, AMS Providence.

[Bo] Borel A, Cohomologie de SL_{n} et valeurs de fonctions zêta aux points entiers, Ann. Sc. Norm. Sup. Pisa 4 (1977), 613–636.

[Co] Consani, C., Double complexes and Euler L-factors, Composition Math., 111, (1998), N3, 323-358.

[D] Deligne, P.: Structures de Hodge mixed réelles in Symp. in Pure Math., v. 55, part 1, 1994, 509-516.

[G1] Goncharov A.B,Geometry of configurations, polylogarithms and motivic cohomology, Adv. in Math., 114, (1995), 197–318.

[G2] Goncharov A.B., Polylogarithms and motivic Galois groups, in Symp. in Pure Math., v. 55, part 2, 1994, 43 - 97.

[G3] Goncharov A.B., Chow polylogarithms and regulators, Math. Res. Letters, 2, (1995), 99-114.

[G4] Goncharov A.B., Geometry of polylogarithms and regulators, To appear.

[G5] Goncharov, A.B.: Deninger’s conjecture on special values of Lfunctions of elliptic curves at s = 3, J. Math. Sci. 81 (1996), N3, 2631-2656 Also avalable as e-preprint, MPI 96–6, http://www.mpimbonn.mpg.de/

[G6] Goncharov, A.B.: Geometry of trilogarithm and the motivic Lie algebra of a field, Proceedings of the conference “Regulators in arithmetic and geometry”, Progress in Mathematics, Birkhäuser (1999).

[G7] Goncharov, A.B.: Polylogarithms in Arithmetic and geometry, Proceedings of the ICM-94 (Zurich), vol. 1, 374-387. (1999).

[RdJ1] de Jeu, Rob: On K_{4}^{(3)} of curves over number fields, Invent. Math. 125 (1996), N3, 523-556.

[RdJ2] de Jeu, Rob: Towards the regulator formulas for curves over number fields. To appear in Compositio math.

[L] Lichtenbaum S, Values of zeta functions at non-negative integers, Lect. Notes in Math., 1068, Springer Verlag, 1984, 127-138.

[Z] Zagier D, Polylogarithms, Dedekind zeta functions and the algebraic Ktheory of fields, Arithmetic Algebraic Geometry (G.v.d.Geer, F.Oort, J.Steenbrink, eds.), Prog. Math., Vol 89, Birkhauser, Boston, 1991, pp. 391–430.

Department of mathematics, Brown University, Providence, RI 02912. email sasha@math.brown.edu