Regulators

A. B. Goncharov

To Steve Lichtenbaum for his 65th birthday

Contents

1 Introduction 1

2 Arakelov motivic complexes 9

3 Grassmannian polylogarithms and Borel’s regulator 15

4 Polylogarithmic motivic complexes 23

5 Regulator maps on the polylogarithmic motivic complexes 28

6 Motivic Lie algebras and Grassmannian polylogarithms 34

1 Introduction

The ζ–function is one of the most deep and mysterious objects in mathematics. During the last two centuries it has served as a key source of new ideas and concepts in arithmetic algebraic geometry. The ζ–function seems to be created to guide mathematicians into the right directions. To illustrate this, let me recall three themes in the 20th century mathematics which emerged from the study of the most basic properties of ζ–functions: their zeros, analytic properties and special values.

1. Weil’s conjectures on ζ–functions of varieties over finite fields inspired Grothendieck’s revolution in algebraic geometry and led Grothendieck to the concept of motives, and Deligne to the yoga of weight filtrations. In fact (pure) motives over ℚ can be viewed as the simplest pieces of algebraic varieties for which the L-function can be defined. Conjecturally the L-function characterizes a motive.

2. Langlands’ conjectures predict that n–dimensional representations of the Galois group Gal(\overline{ℚ}/ℚ) correspond to automorphic representations of GL(n)_{/ℚ}. The relationship between these seemingly unrelated objects was manifested by L-functions: the Artin L-function of the Galois representation coincides with the automorphic L–function of the corresponding representation of GL(n).

3. Investigation of the behavior of L-functions of arithmetic schemes at integer points, culminated in Beilinson’s conjectures, led to the discovery of the key principles of the theory of mixed motives.

In this survey we elaborate on a single aspect of the third theme: regulators. We focus on the analytic and geometric aspects of the story, and explore several different approaches to motivic complexes and regulator maps. We neither touch the seminal Birch - Swinnerton-Dyer conjecture and the progress made in its direction nor do we consider the vast generalization of this conjecture, due Bloch and Kato [BK].

1. Special values of the Riemann ζ-function and their motivic nature. Euler proved the famous formula for the special values of the Riemann ζ-function at positive even integers:

ζ(2n) = (−1)^{k}π^{2k} \frac{2^{2k−1}}{(2k − 1)!}(\frac{−B_{2k}}{2k} )

All attempts to find a similar formula expressing ζ(3), ζ(5), ... via some known quantities failed. The reason became clear only in the recent time: First, the special values ζ(n) are periods of certain elements

ζ^{M}(n) ∈ Ext_{M_{T} (ℤ)}^{1}(ℤ(0), ℤ(n)), n = 2, 3, 4, ...  (1)

where on the right stays the extension group in the abelian category M_{T} (ℤ) of mixed Tate motives over Spec(ℤ) (it has been defined, see [DG]). Second, the fact that ζ^{M}(2n − 1) are non torsion elements should imply, according to a version of Grothendieck’s conjecture on periods, that π and ζ(3), ζ(5), ... are algebraically independent over ℚ. The analytic manifestation of the motivic nature of the special values is the formula

ζ(n) = ∫_{0<t_{1}<...<t_{n}<1} \frac{dt_{1}}{1 − t_{1}} ∧ \frac{dt_{2}}{t_{2}} ∧ ... ∧ \frac{dt_{n}}{t_{n}}  (2)

discovered by Leibniz. This formula presents ζ(n) as a length n iterated integral. The existence of such a formula seems to be a specific property of the L-values at integer points. A geometric construction of the motivic ζ-element (1) using the moduli space M_{0,n+3} is given in Chapter 4.5.

Beilinson conjectured a similar picture for special values of L-functions of motives at integer points. In particular, his conjectures imply that these special values should be periods (in fact a very special kind of periods). For the Riemann ζ-function this is given by the formula (2). In general a period is a number given by an integral

∫_{Δ_{B}} Ω_{A}

where Ω_{A} is a differential form on a variety X with singularities at a divisor A, Δ_{B} is a chain with boundary at a divisor B, and X, A, B are defined over ℚ. So far we can write L-values at integer points as periods only in a few cases. Nevertheless, in all cases when Beilinson’s conjecture was confirmed, we have such a presentation. More specifically, such a presentation for the special values of the Dedekind ζ-function this comes from the Tamagawa Number formula and Borel’s work [Bo2], and in the other cases it is given by Rankin-Selberg type formulas. In general the mechanism staying behind this phenomenon remains a mystery.

Let us now turn to another classical example: the residue of the Dedekind ζ-function at s = 1.

2. The class number formula and the weight one Arakelov motivic complex. Let F be a number field with r_{1} real and r_{2} complex places, so that [F : ℚ] = 2r_{1} + r_{2}. Let ζ_{F} (s) be the Dedekind ζ-function of F. Then according to Dirichlet and Dedekind one has

Res_{s=1}ζ_{F} (s) = \frac{2^{r_{1}+r_{2}}π^{r_{2}}R_{F} h_{F}}{w_{F} |D_{F} |^{1/2}}  (3)

Here w_{F} is the number of roots of unity in F, D_{F} is the discriminant, h_{F} is the class number, and R_{F} is the regulator of F, whose definition we recall below. Using the functional equation for ζ_{F} (s), S. Lichtenbaum [Li1] wrote (3) as

lim_{s→0} s^{−(r_{1}+r_{2}−1)}ζ_{F} (s) = −\frac{h_{F} R_{F}}{w_{F}}  (4)

Let us interpret the right hand side of this formula via the weight one Arakelov motivic complex for Spec(O_{F}), where O_{F} is the ring of integers in F. Let us define the following diagram, where P runs through all prime ideals of O_{F}:

\begin{matrix}ℝ^{r_{1}+r_{2}} & ⟶^{Σ} & ℝ \\ R_{1} ↑ & & ↑ l \\ F^{∗} & ⟶^{div} & ⊕_{P}ℤ \end{matrix}  (5)

In this diagram, the maps are given as follows: If val_{P} is the canonical valuation defined on F by P and |P| is the norm of P, then

div(x) = ∑ val_{P}(x)[P],  l : [P] ⟼ − log |P|,  Σ : (x_{1}, ..., x_{n}) ⟼ Σx_{i}

The regulator map R_{1} is defined by x ∈ F^{∗} ⟼ (log |x|_{σ_{1}}, ..., log |x|_{σ_{r_{1}+r_{2}}} ), where {σ_{1}, ..., σ_{r_{1}+r_{2}}} is the set of all archimedian places of F and | ∗ |_{σ} is the valuation defined σ, (|x|_{σ} := |σ(x)|^{2} for a complex place σ).

The product formula tells us Σ ∘ R_{1} + l ∘ div = 0. Therefore summing up the groups over the diagonals in (5) we get a complex. The first two groups of this complex, placed in degrees [1, 2], form the weight one Arakelov motivic complex Γ_{A}(O_{F} , 1) of O_{F}. There is a map

H^{2}Γ_{A}(O_{F} , 1) ⟶ ℝ

Let H̃^{2}Γ_{A}(O_{F} , 1) be its kernel. Then there is an exact sequence

0 ⟶ \frac{Ker(Σ)}{R_{1}(O_{F}^{∗})} ⟶ H̃^{2}Γ_{A}(O_{F} , 1) ⟶ Cl_{F} ⟶ 0

Further, R_{1}(O_{F}^{∗}) is a lattice in Ker(Σ) = ℝ^{r_{1}+r_{2}−1}, its volume with respect to the measure δ(∑ x_{i})dx_{1}∧ ... ∧ dx_{r_{1}+r_{2}} is R_{F}, and h_{F} = |Cl_{F} |. Therefore

volH̃^{2}Γ_{A}(O_{F} , 1) = R_{F} h_{F}; H^{1}Γ̃_{A}(O_{F} , 1) = μ_{F}

Now the class number formula (4) reads

lim_{s→0} s^{−(r_{1}+r_{2}−1)}ζ_{F} (s) = −\frac{volH̃^{2}Γ_{A}(O_{F} , 1)}{H^{1}Γ_{A}(O_{F} , 1)}  (6)

The right hand side is a volume of the determinant of a complex, see Chapter 2.5. I do not know the cohomological origin of the sign in (6).

3. Special values of the Dedekind ζ-functions, Borel regulators and polylogarithms. B. Birch and J. Tate [T] proposed a generalization of the class number formula for totally real fields using Milnor’s K_{2}–group of O_{F}:

ζ_{F} (−1) = ±\frac{|K_{2}(O_{F})|}{w_{2}(F)}

Here w_{2}(F) is the largest integer m such that Gal(\overline{F} /F) acts trivially on μ_{∞}^{⊗2}. Up to a power of 2, the above formula follows from the Iwasawa main conjecture for totally real fields, proved by B. Mazur and A. Wiles for ℚ [MW], and by A. Wiles [W] in general.

S. Lichtenbaum [Li1] suggested that for ζ_{F} (n) there should be a formula similar to (4) with a higher regulator defined using Quillen’s K-groups K_{∗}(F) of F. Such a formula for ζ_{F} (n), considered up to a non zero rational factor, has been established soon after in the fundamental work of A. Borel [Bo1]-[Bo2]. Let us discuss it in more detail. The rational K–groups of a field can be defined as the primitive part in the homology of GL. Even better, one can show that

K_{2n−1}(F) ⊗ ℚ =^{∼} PrimH_{2n−1}(GL_{2n−1}(F), ℚ)

Let ℝ(n) := (2πi)^{n}ℝ. There is a distinguished class, called the Borel class,

B_{n} ∈ H_{c}^{2n−1}(GL_{2n−1}(ℂ), ℝ(n − 1))

in the continuous cohomology of the Lie group GL_{2n−1}(ℂ). Pairing with this class provides the Borel regulator map

r_{n}^{Bo} : K_{2n−1}(ℂ) ⟶ ℝ(n − 1)

Let X_{F} := ℤ^{Hom(F,ℂ)}. The Borel regulator map on K_{2n−1}(F) is the composition

K_{2n−1}(F) ⟶ ⊕_{Hom(F,ℂ)}K_{2n−1}(ℂ) ⟶ X_{F} ⊗ ℝ(n − 1)

The image of this map is invariant under complex conjugation acting both on Hom(F, ℂ) and ℝ(n−1). So we get the map

R_{n}^{Bo} : K_{2n−1}(F) ⟶ (X_{F} ⊗ ℝ(n − 1))^{+}  (7)

Here + means the invariants under the complex conjugation. Borel proved that for n > 1 the image of this map is a lattice, and the volume R_{n}(F) of this lattice is related to the Dedekind ζ-function as follows:

R_{n}(F) ∼_{ℚ^{∗}} lim_{s→1−n}(s − 1 + n)^{−d_{n}}ζ_{F} (s),

Here a ∼_{ℚ^{∗}} b means a = λb for some λ ∈ ℚ^{∗}, and

d_{n} = dim (X_{F} ⊗ ℝ(n − 1))^{+} = \{ \begin{matrix}r_{1} + r_{2} & : n > 1 odd \\ r_{2} & : n ≥ 2 even \end{matrix}

Using the functional equation for ζ_{F} (s) it tells us about ∼_{ℚ^{∗}} ζ_{F} (n). However Lichtenbaum’s original conjecture was stronger since it was about ζ_{F} (n) itself.

In 1977 S. Bloch discovered [Bl4], [Bl5] that the regulator map on K_{3}(ℂ) can be explicitly defined using the dilogarithm. Here is how the story looks today. The dilogarithm is a multivalued analytic function on ℂℙ^{1} − {0, 1, ∞}:

Li_{2}(z) := − ∫_{0}^{z} log(1 − z)\frac{dz}{z}; Li_{2}(z) = ∑_{k=1}^{∞} \frac{z^{2}}{k^{2}} for |z| ≤ 1

The dilogarithm has a single-valued version, called the Bloch-Wigner function:

L_{2}(z) := Im (Li_{2}(z) + log(1 − z) log |z|)

It vanishes on the real line. Denote by r(z_{1}, ..., z_{4}) the cross-ratio of the four points z_{1}, ..., z_{4} on the projective line. The Bloch-Wigner function satisfies Abel’s five term relation: for any five points z_{1}, ......, z_{5} on ℂℙ^{1} one has

∑_{i=1}^{5}(−1)^{i}L_{2}(r(z_{1}, ..., ẑ_{i}, ..., z_{5})) = 0  (8)

Let H_{3} be the hyperbolic three space. Its absolute is identified with the Riemann sphere ℂℙ^{1}. Let I(z_{1}, ..., z_{4}) be the ideal geodesic simplex with the vertices at the points z_{1}, ..., z_{4} at the absolute. Lobachevsky proved that

volI(z_{1}, ..., z_{4}) = L_{2}(r(z_{1}, ..., z_{4}))

(He got this in a different but equivalent form). Lobachevsky’s formula makes Abel’s equation obvious: the alternating sum of the geodesic simplices with the vertices at z_{1}, ..., ẑ_{i}, ..., z_{5} is empty.

Abel’s equation can be interpreted as follows: for any z ∈ ℂℙ^{1} the function

L_{2}(r(g_{1}z, ..., g_{4}z)), g_{i} ∈ GL_{2}(ℂ)  (9)

is a measurable 3-cocycle of the Lie group GL_{2}(ℂ). The cohomology class of this cocycle is non trivial. The simplest way to see it is this. The function

volI(g_{1}x, ..., g_{4}x), g_{i} ∈ GL_{2}(ℂ), x ∈ H^{3}

provides a smooth 3-cocycle of GL_{2}(ℂ). Its cohomology class is nontrivial: indeed, its infinitesimal version is provided by the volume form in H_{3}. Since the cohomology class does not depend on x, we can take x at the absolute, proving the claim.

Let F be an arbitrary field. Denote by ℤ[F^{∗}] the free abelian group generated by the set F^{∗}. Let R_{2}(F) be the subgroup of ℤ[F^{∗}] generated by the elements

∑_{i=1}^{5}(−1)^{i}{r(z_{1}, ..., ẑ_{i}, ..., z_{5})}

Let B_{2}(F) be the quotient of ℤ[F^{∗}] by the subgroup R_{2}(F). Then one shows that the map

ℤ[F^{∗}] ⟶ Λ^{2}F^{∗}; {z} ⟼ (1 − z) ∧ z

kills the subgroup R_{2}(F), providing a complex (called the Bloch-Suslin complex)

δ_{2} : B_{2}(F) ⟶ Λ^{2}F^{∗}  (10)

By Matsumoto’s theorem Cokerδ_{2} = K_{2}(F). Let us define the Milnor ring K_{∗}^{M}(F) of F as the quotient of the tensor algebra of the abelian group F^{∗} by the two sided ideal generated by the Steinberg elements (1 − x) ⊗ x where x ∈ F^{∗} − 1. The product map in the K-theory provides a map

⊗^{n}K_{1}(F) = ⊗^{n}F^{∗} ⟶ K_{n}(F)

It kills the Steinberg elements, and thus provides a map K_{n}^{M}(F) → K_{n}(F). Set

K_{3}^{ind}(F) := Coker(K_{3}^{M}(F) ⟶ K_{3}(F))

A.A. Suslin [Su] proved that there is an exact sequence

0 ⟶ Tor(F^{∗}, F^{∗})^{∼} ⟶ K_{3}^{ind}(F) ⟶ Kerδ_{2} ⟶ 0  (11)

where Tor(F^{∗}, F^{∗})^{∼} is a nontrivial extension of ℤ/2ℤ by Tor(F^{∗}, F^{∗}).

Abel’s relation provides a well defined homomorphism

L_{2} : B_{2}(ℂ) ⟶ ℝ; {z}_{2} ⟼ L_{2}(z)

Restricting it to the subgroup Kerδ_{2} ⊂ B_{2}(ℂ), and using (11), we get a map K_{3}^{ind}(ℂ) ⟶ ℝ. Using the interpretation of the cohomology class of the cocycle (9) as a volume of geodesic simplex, one can show that it is essentially the Borel regulator. Combining this with Borel’s theorem we get an explicit formula for ζ_{F} (2) for an arbitrary number field F.

How to generalize this beautiful story? Recall the classical polylogarithms

Li_{n}(z) = ∑_{k=1}^{∞} \frac{z^{k}}{k^{n}} , |z| ≤ 1; Li_{n}(z) = ∫_{0}^{z} Li_{n−1}(z)\frac{dz}{z}

D. Zagier [Z1] formulated a precise conjecture expressing ζ_{F} (n) via classical polylogarithms, see the survey [GaZ]. It was proved for n = 3 in [G1]-[G2], but it is not known for n > 3, although its easier part has been proved in [dJ3], [BD2] and, in a different way, in Chapter 4.4 below.

Most of the ζ_{F} (2) picture has been generalized to the case of ζ_{F} (3) in [G1]-[G2] and [G3]. Namely, there is a single valued version of the trilogarithm:

L_{3}(z) := Re (Li_{3}(z) − Li_{2}(z) log |z| + \frac{1}{6}Li_{1}(z) log^{2} |z|)

It satisfies the following functional equation which generalizes (8). Let us define the generalized cross-ratio of 6 points x_{0}, ..., x_{5} in P^{2} as follows. We present P^{2} as a projectivization of the three dimensional vector space V_{3} and choose the vectors l_{i} ∈ V_{3} projecting to the points x_{i}. Let us choose a volume form ω ∈ detV_{3}^{∗} and set Δ(a, b, c) :=< a ∧ b ∧ c, ω >. Set

r_{3}(x_{0}, ..., x_{5}) := Alt_{6}\{\frac{Δ(l_{0}, l_{1}, l_{3})Δ(l_{1}, l_{2}, l_{4})Δ(l_{2}, l_{0}, l_{5})}{Δ(l_{0}, l_{1}, l_{4})Δ(l_{1}, l_{2}, l_{5})Δ(l_{2}, l_{0}, l_{3})}\} ∈ ℤ[F^{∗}]

Here Alt_{6} denotes the alternation of l_{0}, ..., l_{5}. The function L_{3} extends by linearity to a homomorphism L_{3} : ℤ[ℂ^{∗}] ⟶ ℝ, and there is a generalization of Abel’s equation to the case of the trilogarithm (see [G2] and the appendix to [G3]):

∑_{i=1}^{7}(−1)^{i}L_{3}(r_{3}(x_{1}, ..., x̂_{i}, ...x_{7})) = 0  (12)

We define the group B_{3}(F) as the quotient of ℤ[F^{∗}] by the subgroup generated by the functional equations (12) for the trilogarithm. Then there is complex

B_{3}(F) ⟶^{δ_{3}} B_{2}(F) ⊗ F^{∗} ⟶^{δ_{2}∧Id} Λ^{3}F^{∗}; δ_{3} : {x}_{3} ⟼ {x}_{2} ⊗ x  (13)

There is a map K_{5}(F) ⟶ Kerδ_{3} such that in the case F = ℂ the composition

K_{5}(ℂ) ⟶ Kerδ_{3} ↪ B_{3}(ℂ) ⟶^{L_{3}} ℝ

coincides with the Borel regulator (see [G2] and appendix in [G3]). This plus Borel’s theorem leads to an explicit formula expressing ζ_{F} (3) via the trilogarithm conjectured by Zagier [Z1].

In Chapter 3 we define, following [G4], the Grassmannian n-logarithm function L_{n}^{G}. It is a function on the configurations of 2n hyperplanes in ℂℙ^{n−1}. One of its functional equations generalizes Abel’s equation:

∑_{i=1}^{2n+1}(−1)^{i}L_{n}^{G}(h_{1}, ..., ĥ_{i}, ..., h_{2n+1}) = 0

It means that for a given hyperplane h the function L_{n}^{G}(g_{1}h, ..., g_{2n}h), g_{i} ∈ GL_{n}(ℂ), is a measurable cocycle of GL_{n}(ℂ). Its cohomology class essentially coincides with the restriction of the Borel class B_{n} to GL_{n}(ℂ). To prove this we show that L_{n}^{G} is the boundary value of a certain function defined on configurations of 2n points in the symmetric space SL_{n}(ℂ)/SU(n). Using this we express the Borel regulator via the Grassmannian polylogarithm. We show that for n = 2 we recover the dilogarithm story: L_{2}^{G} coincides with the Bloch-Wigner function, SL_{2}(ℂ)/SU(2) is the hyperbolic space, and the extension of L_{2}^{G} is given by the volume of geodesic simplices. The proofs can be found in [G7].

4. Beilinson’s conjectures and Arakelov motivic complexes. A conjectural generalization of the class number formula (3) to the case of elliptic curves was suggested in the seminal work of Birch and Swinnerton-Dyer. Several years later J.Tate formulated conjectures relating algebraic cycles to the poles of zeta functions of algebraic varieties.

Let X be a regular algebraic variety over a number field F. Generalizing the previous works of Bloch [Bl4] and P. Deligne [D], A.A. Beilinson [B1] suggested a fantastic picture unifying all the above conjectures. Beilinson defined the rational motivic cohomology of X via the algebraic K-theory of X by

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

Here γ is the Adams γ-filtration.

Let us assume that X is projective. For schemes which admit regular models over ℤ, Beilinson [B1] defined a ℚ-vector subspace

H_{Mot/ℤ}^{i}(X, ℚ(n)) ⊂ H_{Mot}^{i}(X, ℚ(n)),  (14)

called integral part in the motivic cohomology. Using alterations, A. Scholl [Sch] extended this definition to arbitrary regular projective schemes over a number field F. For every regular, projective and flat over ℤ model X′ of X the subspace (14) coincides with the image of the map H_{Mot}^{i}(X′, ℚ(n)) ⟶ H_{Mot}^{i}(X, ℚ(n)).

For a regular complex projective variety X Beilinson constructed in [B1] the regulator map to the Deligne cohomology of X:

H_{Mot}^{i}(X, ℚ(n)) ⟶ H_{D}^{i}(X(ℂ), ℤ(n))  (15)

It can be projected to the Deligne cohomology with real coefficients H_{D}^{i}(X(ℂ), ℝ(n)).

Now let X again be a regular projective scheme over a number field F, We will view it as a scheme over ℚ via the projection X → Spec(F) → Spec(ℚ). We define the real Deligne cohomology of X as the following ℝ-vector space:

H_{D}^{i}((X ⊗_{ℚ} ℝ)(ℂ), ℝ(n))^{\overline{F}^{∞}}

where \overline{F}_{∞} is an involution given by the composition of complex conjugation acting on X(ℂ) and on the coefficients. Then, restricting the map (15) to the integral part in the motivic cohomology (14) and projecting the image onto the real Deligne cohomology of X, we get a regulator map

r_{Be} : H_{Mot/ℤ}^{i}(X, ℚ(n)) ⟶ H_{D}^{i}((X ⊗_{ℚ} ℝ)(ℂ), ℝ(n))^{\overline{F}^{∞}}

Beilinson formulated a conjecture expressing the special values ∼_{ℚ^{∗}} L(h^{i−1}(X), n) of the L-function of Grothendieck’s motive h^{i−1}(X) in terms of this regulator map, see [B1], the survey [R] and the book [RSS] for the original version of the conjecture, and the survey by J. Nekovář [N] for a motivic reformulation. For X = Spec(F), where F is a number field, it boils down to Borel’s theorem. A precise Tamagawa Number conjecture about the special values L(h^{i}(X), n) was suggested by Bloch and Kato [BK].

Beilinson [B2] and Lichtenbaum [Li2] conjectured that the weight n integral motivic cohomology of a scheme X should appear as cohomology of some complexes of abelian groups ℤ_{X}^{•}(n), called the weight n motivic complexes of X. One must have

H_{Mot}^{i}(X, ℚ(n)) = H^{i}ℤ_{X}^{•}(n) ⊗ ℚ

Motivic complexes are objects of the derived category. Beilinson conjectured [B2] that there exists an abelian category MS_{X} of mixed motivic sheaves on X, and that one should have an isomorphism in the derived category

ℤ_{X}^{•}(n) = RHom_{MS_{X}} (ℚ(0)_{X}, ℚ(n)_{X})  (16)

Here ℚ(n)_{X} := p^{∗}ℚ(n), where p : X → Spec(F) is the structure morphism, is a motivic sheaf on X obtained by pull back of the Tate motive ℚ(n) over the point Spec(F). This formula implies the Beilinson-Soulé vanishing conjecture:

H_{Mot}^{i}(X, ℚ(n)) = 0 for i < 0 and i = 0, n > 0

Indeed, the negative Ext’s between objects of an abelian category are zero, and we assume that the objects ℚ(n)_{X} are mutually non-isomorphic. Therefore it is quite natural to look for representatives of motivic complexes which are zero in the negative degrees, as well as in the degree zero for n = 0.

Motivic complexes are more fundamental, and in fact simpler objects then rational K–groups. Several constructions of motivic complexes are known.

i) Bloch [Bl1]-[Bl2] suggested a construction of the motivic complexes, called the Higher Chow complexes, using algebraic cycles. The weight n cycle complex appears in a very natural way as a “resolution” for the codimension n Chow groups on X modulo rational equivalence, see s. 2.1 below. These complexes as well as their versions defined by Suslin and Voevodsky played an essential role in the construction of triangulated categories of mixed motives [V], [Lev], [Ha]. However they are unbounded from the left.

ii) Here is a totally different construction of the first few of motivic complexes. One has ℤ_{X}^{•}(0) := ℤ. Let X^{(k)} be the set of all irreducible codimension k subschemes of a scheme X. Then ℤ_{X}^{•}(1) is the complex

O_{X}^{∗} ⟶^{∂} ⊕_{Y ∈X^{(1)}}ℤ; ∂(f) := div(f)

The complex ℚ_{X}^{•}(2) is defined as follows. First, using the Bloch-Suslin complex (10), we define the following complex

B_{2}(ℚ(X)) ⟶^{δ_{2}} Λ^{2}ℚ(X)^{∗} ⟶^{∂_{1}} ⊕_{Y ∈X^{(1)}}ℚ(Y)^{∗} ⟶^{∂_{2}} ⊕_{Y ∈X^{(2)}}ℤ

Then tensoring it by ℚ we get ℚ_{X}^{•}(2). Here ∂_{2} is the tame symbol, and ∂_{1} is given by the divisor of a function on Y. Similarly one can define a complex ℚ_{X}^{•}(3) using the complex (13). Unlike the cycle complexes, these complexes are concentrated exactly in the degrees where they might have nontrivial cohomology. It is amazing that motivic complexes have two so different and beautiful incarnations.

Generalizing this, we introduce in Chapter 4 the polylogarithmic motivic complexes, which are conjectured to be the motivic complexes for an arbitrary field ([G1]-[G2]). Then we give a motivic proof of the weak version of Zagier’s conjecture for a number field F. In Chapter 5 we discuss how to define motivic complexes for an arbitrary regular variety X using the polylogarithmic motivic complexes of its points.

In Chapters 2 and 5 we discuss constructions of the regulator map on the level of complexes. Precisely, we want to define for a regular complex projective variety X a homomorphisms of complexes of abelian groups

a weight n motivic complex of X ⟶ a weight n Deligne complex of X(ℂ)  (17)

In Chapter 2 we present a construction of a regulator map on the Higher Chow complexes given in [G4], [G7]. In Chapter 5 we define, following [G6], a regulator map on polylogarithmic complexes. It is given explicitly in terms of the classical polylogarithms. Combining this with Beilinson’s conjectures we arrive at explicit conjectures expressing the special values of L-functions via classical polylogarithms. If X = Spec(F), where F is a number field, this boils down to Zagier’s conjecture.

The cone of the map (17), shifted by −1, defines the weight n Arakelov motivic complex, and so its cohomology are the weight n Arakelov motivic cohomology of X.

The weight n Arakelov motivic complex should be considered as an ingredient of a definition of the weight n arithmetic motivic complex. Namely, one should exist a complex computing the weight n integral motivic cohomology of X, and a natural map from this complex to the weight n Deligne complex. The cone of this map, shifted by −1, would give the weight n arithmetic motivic complex. The weight one arithmetic motivic complex is the complex Γ_{A}(O_{F} , 1).

In Chapter 6 we discuss a yet another approach to motivic complexes of fields: as standard cochain complexes of the motivic Lie algebras. We also discuss a relationship between the motivic Lie algebra of a field and the (motivic) Grassmannian polylogarithms.

Acknowledgement. This work was partially supported by IHES (Bures sur Yvette), MPI (Bonn), and the NSF grants DMS-0099390. I am grateful to referee for useful comments.

2 Arakelov motivic complexes

In this chapter we define a regulator map from the weight n motivic complex, understood as Bloch’s Higher Chow groups complex [Bl1], to the weight n Deligne complex. This map was defined in [G4], and elaborated in detail in [G7]. The construction can be immediately adopted to the Suslin-Voevodsky motivic complexes.

1. Bloch’s cycle complex [Bl1]. A non degenerate simplex in ℙ^{m} is an ordered collection of hyperplanes L_{0}, ..., L_{m} with empty intersection. Let us choose in ℙ^{m} a simplex L and a generic hyperplane H. Then L provides a simplex in the affine space 𝔸^{m} := ℙ^{m} − H.

Let X be a regular scheme over a field. Let I = (i_{1}, ..., i_{k}) and L_{I} := L_{i_{1}} ∩ ... ∩ L_{i_{k}}. Let Z_{m}(X; n) be the free abelian group generated by irreducible codimension n algebraic subvarieties in X × 𝔸^{m} which intersect properly (i.e. the intersection has the right dimension) all faces X × L_{I}. Intersection with the codimension 1 face X × L_{i} provides a group homomorphism ∂_{i} : Z_{m}(X; n) ⟶ Z_{m−1}(X; n). Set ∂ := ∑_{i=0}^{m}(−1)^{i}∂_{i}. Then ∂^{2} = 0, so (Z_{•}(X; n), ∂) is a homological complex. Consider the cohomological complex Z^{•}(X; n) := Z_{2r−•}(X; n). Its cohomology give the motivic cohomology of X:

H_{M}^{i}(X, ℤ(n)) := H^{i}(Z^{•}(X; n))

According to the fundamental theorem of Bloch ([Bl1], [Bl2])

H^{i}(Z^{•}(X; n) ⊗ ℚ) = gr_{n}^{γ}K_{2n−i}(X) ⊗ ℚ

2. The Deligne cohomology and Deligne’s complex. Let X be a regular projective variety over ℂ. The Beilinson-Deligne complex ℝ^{•}(X; n)_{D} is the following complex of sheaves in the classical topology on X(ℂ):

\underline{ℝ}(n) ⟶ O_{X} ⟶ Ω_{X}^{1} ⟶ Ω_{X}^{2} ⟶ ... ⟶ Ω_{X}^{n−1}

Here the constant sheaf \underline{ℝ}(n) := (2πi)^{n}\underline{ℝ} is in the degree zero. The hypercohomology of this complex of sheaves is called the weight n Deligne cohomology of X(ℂ). They are finite dimensional real vector spaces. Beilinson proved [B3] that the truncated weight n Deligne cohomology, which are obtained by putting the weight n Deligne cohomology equal to zero in the degrees > 2n, can be interpreted as the absolute Hodge cohomology of X(ℂ).

One can replace the above complex of sheaves by a quasiisomorphic one, defined as the total complex associated with the following bicomplex:

(\underline{D}_{X}^{0} ⟶^{d} \underline{D}_{X}^{1} ⟶^{d} … ⟶^{d} \underline{D}_{X}^{n} ⟶^{d} \underline{D}_{X}^{n+1} ⟶^{d} …) ⊗ ℝ(n − 1)
↑ π_{n}  ↑ π_{n}
Ω_{X}^{n} ⟶^{∂} Ω_{X}^{n+1} ⟶^{∂}

Here \underline{D}_{X}^{k} is the sheaf of real k-distributions on X(ℂ), that is k-forms with the generalized function coefficients. Further,

π_{n} : \underline{D}_{X}^{p} ⊗ ℂ ⟶ \underline{D}_{X}^{p} ⊗ ℝ(n − 1)

is the projection induced by the one ℂ = ℝ(n − 1) ⊕ ℝ(n) ⟶ ℝ(n − 1), the sheaf \underline{D}_{X}^{0} is placed in degree 1, and (Ω_{X}^{•}, ∂) is the De Rham complex of sheaves of holomorphic forms.

To calculate the hypercohomology with coefficients in this complex we replace the holomorphic de Rham complex by its Doulbeut resolution, take the global sections of the obtained complex, and calculate its cohomology. Taking the canonical truncation of this complex in the degrees [0, 2n] we get a complex calculating the absolute Hodge cohomology of X(ℂ). Let us define, following Deligne, yet another complex of abelian groups quasiisomorphic to the latter complex.

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 D_{cl}^{n,n} is the subspace of closed distributions, and D^{0,0} is in degree 1:

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

The complex C_{D}^{•}(X; n) is a subcomplex of the total complex of this bicomplex provided by the ℝ(n − 1)-valued distributions in the n × n square of the diagram and the subspace D_{ℝ,cl}^{n,n}(n) ⊂ D_{cl}^{n,n} of the ℝ(n)-valued distributions of type (n, n). Notice that \overline{∂}∂ sends ℝ(n − 1)-valued distributions to ℝ(n)-valued distributions. The cohomology of this complex of abelian groups is the absolute Hodge cohomology of X(ℂ), see Proposition 2.1 of [G7]. Now if X is a variety over ℝ, then

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

where \overline{F}_{∞} is the composition of the involution F_{∞} on X(ℂ) induced by the complex conjugation with the complex conjugation of coefficients.

3. The regulator map.

Theorem - Construction 2.1 Let X be a regular complex projective variety. Then there exists canonical homomorphism of complexes

P^{•}(n) : Z^{•}(X; n) ⟶ C_{D}^{•}(X; n)

If X is defined over ℝ then its image lies in the subcomplex C_{D}^{•}(X_{/ℝ}; n).

To define this homomorphism we need the following construction. Let X be a variety over ℂ and f_{1}, ..., f_{m} be m rational functions on X. The form

π_{m}(d log f_{1} ∧ ... ∧ d log f_{m}),

where π_{n}(a + ib) = a if n odd, and π_{n}(a + ib) = ib if n even, has zero periods. It has a canonical primitive defined as follows. Consider the following (m − 1)-form on X(ℂ):

r_{m−1}(f_{1} ∧ ... ∧ f_{m}) :=  (18)

Alt_{m} ∑_{j≥0} c_{j,m} log |f_{1}|d log |f_{2}| ∧ ... ∧ d log |f_{2j+1}| ∧ di arg f_{2j+2} ∧ ... ∧ di arg f_{m}

Here c_{j,m} := ((2j + 1)!(m − 2j − 1)!)^{−1} and Alt_{m} is the operation of alternation:

Alt_{m}F (x_{1}, ..., x_{m}) := ∑_{σ∈S_{m}}(−1)^{|σ|}F (x_{σ(1)}, ..., x_{σ(m)})

So r_{m−1}(f_{1} ∧ ... ∧ f_{m}) is an ℝ(m − 1)-valued (m − 1)-form. One has

dr_{m−1}(f_{1} ∧ ... ∧ f_{m}) = π_{m}(d log f_{1} ∧ ... ∧ d log f_{m})

It is sometimes convenient to write the form (18) as a multiple of

Alt_{m} ∑_{i=1}^{m}(−1)^{i} log |f_{1}|d log f_{2} ∧ ... ∧ d log f_{i} ∧ d log \overline{f}_{i+1} ∧ ... ∧ d log \overline{f}_{m}

Precisely, let A^{i}(M) be the space of smooth i-forms on a real smooth manifold M. Consider the following map

ω_{m−1} : Λ^{m}A^{0}(M) ⟶ A^{m−1}(M)  (19)
ω_{m−1}(φ_{1} ∧ ... ∧ φ_{m}) :=

\frac{1}{m!}Alt_{m}(∑_{k=1}^{m}(−1)^{k−1}φ_{1}∂φ_{2} ∧ ...∂φ_{k} ∧ \overline{∂}φ_{k+1} ∧ ... ∧ \overline{∂}φ_{m})

For example

ω_{0}(φ_{1}) = φ_{1}; ω_{1}(φ_{1} ∧ φ_{2}) = \frac{1}{2}(φ_{1}∂φ_{2} − φ_{2}∂φ_{1} − φ_{1}\overline{∂}φ_{2} + φ_{2}\overline{∂}φ_{1})

Then one easily checks that

dω_{m−1}(φ_{1} ∧ ... ∧ φ_{m}) = ∂φ_{1} ∧ ... ∧ ∂φ_{m} + (−1)^{m}\overline{∂}φ_{1} ∧ ... ∧ \overline{∂}φ_{m}+  (20)

∑_{i=1}^{m}(−1)^{i}\overline{∂}∂φ_{i} ∧ ω_{m−2}(φ_{1} ∧ ... ∧ φ̂_{i} ∧ ... ∧ φ_{m})

Now let f_{i} be rational functions on a complex algebraic variety X. Set M := X^{0}(ℂ), where X^{0} is the open part of X where the functions f_{i} are regular. Then φ_{i} := log |f_{i}| are smooth functions on M, and we have

ω_{m−1}(log |f_{1}| ∧ ... ∧ log |f_{m}|) = r_{m−1}(f_{1} ∧ ... ∧ f_{m})

Denote by D_{X,ℝ}^{∗}(k) = D∗_{ℝ}(k) the subspace of ℝ(k)–valued distributions in D_{X(ℂ)}^{∗}.

Let Y^{0} be the nonsingular part of Y, and i_{Y}^{0} : Y^{0}(ℂ) ↪ Y (ℂ) the canonical embedding.

Proposition 2.2 Let Y be an arbitrary irreducible subvariety of a smooth complex variety X and f_{1}, ..., f_{m} ∈ O^{∗}(Y). Then for any smooth differential form ω with compact support on X(ℂ) the following integral is convergent:

∫_{Y^{0}(ℂ)} r_{m−1}(f_{1} ∧ ... ∧ f_{m}) ∧ i_{Y}^{0} ω

Thus there is a distribution r_{m−1}(f_{1} ∧ ... ∧ f_{m})δ_{Y} on X(ℂ):

< r_{m−1}(f_{1} ∧ ... ∧ f_{m})δ_{Y} , ω >:= ∫_{Y^{0}(ℂ)} r_{m−1}(f_{1} ∧ ... ∧ f_{m}) ∧ i_{Y}^{0} ω

It provides a group homomorphism

r_{m−1} : Λ^{m}ℂ(Y)^{∗} ⟶ D_{X,ℝ}^{m−1}(m − 1)  (21)

Construction. We have to construct a morphism of complexes

... ⟶ Z^{1}(X; n) ⟶ ... ⟶ Z^{2n−1}(X; n) ⟶ Z^{2n}(X; n)
↓ P^{1}(n)  ...  ↓ P^{2n−1}(n)  ↓ P^{2n}(n)
0 ⟶ D0,0_{ℝ}(n − 1) ⟶ ... ⟶ Dn−1,n−1_{ℝ}(n − 1) ⟶^{2\overline{∂}∂} Dn,n_{ℝ}(n)

Here at the bottom stays the complex C_{D}^{•}(X; n).

Let Y ∈ Z^{2n}(X; n) be a codimension n cycle in X. By definition

P^{2n}(n)(Y) := (2πi)^{n}δ_{Y}

Let us construct homomorphisms

P^{2n−k}(n) : Z^{2n−k}(X; n) ⟶ D_{X}^{2n−k−1}, k > 0

Denote by π_{𝔸^{k}} (resp. π_{X}) the projection of X × 𝔸^{k} to 𝔸^{k} (resp. X), and by \overline{π}_{𝔸^{k}} (resp. \overline{π}_{X}) the projection of X(ℂ) × ℂℙ^{k} to ℂℙ^{k} (resp. X(ℂ)).

The pair (L, H) in ℙ^{k} defines uniquely homogeneous coordinates (z_{0} : ... : z_{k}) in ℙ^{k} such that the hyperplane L_{i} is given by equation {z_{i} = 0} and the hyperplane H is {∑_{i=1}^{k} z_{i} = z_{0}}. Then there is an element

\frac{z_{1}}{z_{0}} ∧ ... ∧ \frac{z_{k}}{z_{0}} ∈ Λ^{k−1}ℂ(𝔸^{k})^{∗}  (22)

Let Y ∈ Z^{2n−k}(X; n). Restricting to Y the inverse image of the element (22) by π∗_{𝔸^{k}} we get an element

g_{1} ∧ ... ∧ g_{k} ∈ Λ^{k}ℂ(Y)^{∗}  (23)

Remark. This works if and only if the cycle Y intersects properly all codimension one faces of X × L. Indeed, if Y does not intersect properly one of the faces, then the equation of this face restricts to zero to Y, and so (23) does not make sense.

The element (23) provides, by Proposition 2.2, a distribution on X(ℂ) × ℂℙ^{k}. Pushing it down by (2πi)^{n−k} · \overline{π}_{X} we get the distribution P^{2n−k}(n)(Y):

Definition 2.3 P^{2n−k}(n)(Y) := (2πi)^{n−k} · \overline{π}_{X∗}r_{k−1}(g_{1} ∧ ... ∧ g_{k}).

In other words, the following distribution makes sense:

P^{2n−k}(n)(Y) = (2πi)^{n−k}\overline{π}_{X}_{∗} (δ_{Y} ∧ \overline{π}∗_{𝔸^{k}}r_{k−1} (\frac{z_{1}}{z_{0}} ∧ ... ∧ \frac{z_{k}}{z_{0}} ))

One can rewrite definition 2.3 more explicitly as an integral over Y (ℂ). Namely, let ω be a smooth form on X(ℂ) and i_{Y} : Y ↪ X × ℙ^{k}. Then

< P^{2n−k}(n)(Y), ω >= (2πi)^{n−k} ∫_{Y^{0}(ℂ)} r_{k−1}(g_{1} ∧ ... ∧ g_{k}) ∧ i_{Y^{0}(ℂ)}^{∗}\overline{π}_{X}^{∗}ω

It is easy to check that P^{2n−k}(n)(Y) lies in C_{D}^{2n−k}(X; n). Therefore we defined the maps P^{k}(n). It was proved in chapter [G7] that P^{•}(n) is a homomorphism of complexes.

4. The Higher Arakelov Chow groups. Let X be a regular complex variety. Denote by C̃_{D}^{•}(X; n) the quotient of the complex C_{D}^{•}(X; n) along the subgroup A_{cl,ℝ}^{n,n}(n) ⊂ D_{cl,ℝ}^{n,n}(n) of closed smooth forms. The cone of the homomorphism P^{•}(n) shifted by −1 is the Arakelov motivic complex: :

Ẑ^{•}(X; n) := Cone(Z^{•}(X; n) ⟶ C̃_{D}^{•}(X; n))[−1]

The Higher Arakelov Chow groups are its cohomology:

ĈH^{n}(X; i) := H^{2n−i}(Ẑ^{•}(X; n))

Recall the arithmetic Chow groups defined by Gillet-Soulé [GS] as follows:

ĈH^{n}(X) :=

\frac{\{(Z, g); \frac{\overline{∂}∂}{πi} g + δ_{Z} ∈ A^{n,n}\}}{\{(0, ∂u + \overline{∂}v); (divf, − log |f|), f ∈ ℂ(Y), codim(Y) = n − 1\}}  (24)

Here Z is a divisor in X, f is a rational function on a divisor Y in X,

g ∈ Dn−1,n−1_{ℝ}(n − 1), (u, v) ∈ C_{D}^{2n−2}(X; n) = (D^{n−2,n−1} ⊕ D^{n−1,n−2})_{ℝ}(n − 1)

Proposition 2.4 ĈH^{n}(X; 0) = ĈH^{n}(X).

Proof. Let us look at the very right part of the complex Ẑ^{•}(X; n):

...  ⟶ Z^{2n−1}(X; n) ⟶  Z^{2n}(X; n)
↓ P^{2n−1}(n)  ↓ P^{2n}(n)
(D^{n−2,n−1} ⊕ D^{n−1,n−2})_{ℝ}(n − 1) ⟶^{(∂,\overline{∂})} Dn−1,n−1_{ℝ}(n − 1) ⟶^{2\overline{∂}∂} Dn,n_{ℝ}(n)/An,n_{ℝ}(n)

Consider the stupid truncation of the Gersten complex on X:

∏_{Y ∈X_{n−1}} ℂ(Y)^{∗} ⟶ Z_{0}(X; n)  (25)

It maps to the stupid truncation σ_{≥2n−1}Ẑ^{•}(X; n) of the cycle complex as follows. The isomorphism Z_{0}(X; n) = Z^{2n}(X; n) provides the right component of the map. A pair (Y; f) where Y is an irreducible codimension n − 1 subvariety of X maps to the cycle (y, f(y)) ⊂ X × (ℙ^{1} − {1}). It is well known that such cycles (y, f(y)) plus ∂Ẑ^{2n−2}(X; n) generate Ẑ^{2n−1}(X; n). Computing the composition of this map with the homomorphism P^{•}(n) we end up precisely with formula (24). The proposition is proved.

5. Special values of Dedekind ζ-functions and Arakelov motivic complexes. Let Γ̃_{A}(O_{F} , 1) be the three term complex (5). It consists of locally compact abelian groups. Each of them is equipped with a natural Haar measure. Indeed, the measure of a discrete group is normalized so that the measure of the identity element is 1; the group ℝ has the canonical measure dx; and we use the product measure for the products. We need the following general observation.

Lemma - Definition 2.5 Let

A^{•} = ... ⟶ A_{1} ⟶ A_{2} ⟶ A_{3} ⟶ ...

be a complex of locally compact abelian groups such that i) Each of the groups A_{i} is equipped with an invariant Haar measure μ_{i}. ii) The cohomology groups are compact. iii) Only finite number of the cohomology groups are nontrivial, and almost all groups A_{i} are

discrete groups with canonical measures.

Then there is a naturally defined number R_{μ}A^{•}, and R_{μ}A^{•}[1] = (R_{μ}A^{•})^{−1}.

Construction. Let

0 ⟶ A ⟶ B ⟶ C ⟶ 0  (26)

be an exact sequence of locally compact abelian groups. Then a choice of Haar measure for any two of the groups A, B, C determines naturally the third one. For example Haar measures μ_{A}, μ_{C} on A and C determine the following Haar measure μ_{A,C} on B. Take a compact subset of C and its section K_{C} ⊂ B, and take a compact subset K_{A}. Then μ_{A,C}(K_{A} · K_{C}) := μ_{A}(K_{A})μ_{C}(K_{C}).

For the complex (26) placed in degrees [0, 2] we put R_{μ}(26) = μ_{A,C}/μ_{B}.

Let us treat first the case when A^{•} is a finite complex. Then we define the invariant R_{μ}A^{•} by induction. If A^{•} = A[−i] is concentrated in just one degree, i, that A is compact and equipped with Haar measure μ. We put R_{μ}A^{•} := μ(A)^{(−1)^{i}}. Assume A^{•} starts from A_{0}. One has

Kerf_{0} ⟶ A_{0} ⟶ Imf_{0} ⟶ 0; Imf_{0} ⟶ A_{1} ⟶ A_{1}/Imf_{0} ⟶ 0

Since Kerf_{0} is compact, we can choose the volume one Haar measure on it. This measure and the measure μ_{0} on A_{0} provides, via the first short exact sequence, a measure on Imf_{0}. Similarly using this and the second exact sequence we get a measure on A_{1}/Imf_{0}. Therefore we have the measures on the truncated complex τ_{≥1}A^{•}. Now we define

R_{μ}A^{•} := μ_{0}(A_{0}) · R_{μ}τ_{≥1}A^{•}

If A^{•} is an infinite complex we set R_{μ}A^{•} := R_{μ}(τ_{−N,N}A^{•}) for sufficiently big N. Here τ_{−N,N} is the canonical truncation functor. Thanks to iii) this does not depend on the choice of N. The lemma is proved.

Now the class number formula (4) reads

lim_{s→0} s^{−(r_{1}+r_{2}−1)}ζ_{F} (s) = −R_{μ}Γ̃_{A}(O_{F} , 1)  (27)

Lichtenbaum’s conjectures [Li1] on the special values of the Dedekind ζ-functions can be reformulated in a similar way:

ζ_{F} (1 − n) =^{?} ±R_{μ}Γ_{A}(O_{F} , n); n > 1

It is not quite clear what is the most natural normalization of the regulator map. In the classical n = 1 case this formula needs modification, as was explained in s. 1.1, to take into account the pole of the ζ–function.

Example. The group H^{2}Γ_{A}(O_{F} , 2) sits in the exact sequence

0 ⟶ R_{2}(F) ⟶ H^{2}Γ(O_{F} , 2) ⟶ K_{2}(O_{F}) ⟶ 0

To calculate it let us define the Bloch-Suslin complex for Spec(O_{F}):

B(O_{F} , 2) :  B_{2}(F) ⟶ Λ^{2}F^{∗} ⟶ ∏_{P} k_{P}^{∗}  (28)

Its Arakelov version is the total complex of the following bicomplex, where the vertical map is given by the dilogarithm: {x}_{2} ⟼ (L_{2}(σ_{1}(x), ..., L_{2}(σ_{r_{2}}(x)).

\begin{matrix}ℝ^{r_{2}} \\ ↑ \\ B_{2}(F) & ⟶ & Λ^{2}F^{∗} & ⟶ & ∏_{P} k_{P}^{∗} \end{matrix}  (29)

Then H^{2}Γ_{A}(O_{F} , 2) = H^{2}B_{A}(O_{F} , 2). The second map in (28) is surjective ([Mi], cor. 16.2). Using this one can check that

H^{i}Γ_{A}(O_{F} , 2) = 0 for i ≥ 3

(Notice that H^{1}B_{A}(O_{F} , 2) ≠ H^{1}Γ_{A}(O_{F} , 2)). Summarizing, we should have

ζ_{F} (−1) =^{?} ±R_{μ}Γ_{A}(O_{F} , 2) = ±\frac{volH^{2}Γ_{A}(O_{F} , 2)}{|H^{1}Γ_{A}(O_{F} , 2)|}

For totally real fields it is a version of the Birch-Tate conjecture. It would be very interesting to compare this with the Bloch–Kato conjecture.

3 Grassmannian polylogarithms and Borel’s regulator

1. The Grassmannian polylogarithm [G4]. Let h_{1}, ..., h_{2n} be arbitrary 2n hyperplanes in ℂℙ^{n−1}. Choose an additional hyperplane h_{0}. Let f_{i} be a rational function on ℂℙ^{n−1} with divisor h_{i} − h_{0}. It is defined up to a scalar factor. Set

L_{n}^{G}(h_{1}, ..., h_{2n}) := (2πi)^{1−n} ∫_{ℂℙ^{n−1}} r_{2n−2}(∑_{j=1}^{2n}(−1)^{j}f_{1} ∧ ... ∧ f̂_{j} ∧ ... ∧ f_{2n})

It is skew-symmetric by definition. It is easy to see that it does not depend on the choice of scalar in the definition of f_{i}. To check that it does not depend on the choice of h_{0} observe that

∑_{j=1}^{2n}(−1)^{j}f_{1} ∧ ... ∧ f̂_{j} ∧ ... ∧ f_{2n} = \frac{f_{1}}{f_{2n}} ∧ \frac{f_{2}}{f_{2n}} ∧ ... ∧ \frac{f_{2n−1}}{f_{2n}}

So if we choose rational functions g_{1}, ..., g_{2n−1} such that divg_{i} = h_{i} − h_{2n} then

L_{n}^{G}(h_{1}, ..., h_{2n}) = (2πi)^{1−n} ∫_{ℂℙ^{n−1}} r_{2n−2}(g_{1} ∧ ... ∧ g_{2n−1})

Remark. The function L_{n}^{G} is defined on the set of all configurations of 2n hyperplanes in ℂℙ^{n−1}. However it is not even continuous on this set. It is real analytic on the submanifold of generic configurations.

Theorem 3.1 The function L_{n}^{G} satisfies the following functional equations:

a) For any 2n + 1 hyperplanes h_{1}, ..., h_{2n+1} in ℂℙ^{n} one has

∑_{j=1}^{2n+1}(−1)^{j}L_{n}^{G}(h_{j} ∩ h_{1}, ..., h_{j} ∩ h_{2n+1}) = 0  (30)

b) For any 2n + 1 hyperplanes h_{1}, ..., h_{2n+1} in ℂℙ^{n−1} one has

∑_{j=1}^{2n+1}(−1)^{j}L_{n}^{G}(h_{1}, ..., ĥ_{j}, ..., h_{2n+1}) = 0  (31)

Proof. a) Let f_{1}, ..., f_{2n+1} be rational functions on ℂℙ^{n} as above. Then

dr_{2n−1}( ∑_{j=1}^{2n+1}(−1)^{j}f_{1} ∧ ... ∧ f̂_{j} ∧ ...f_{2n+1}) =  (32)

∑_{j≠i}(−1)^{j+i−1}2πiδ(f_{j})r_{2n−2}(f_{1} ∧ ...f̂_{i} ∧ ... ∧ f̂_{j} ∧ ...f_{2n+1})

(Notice that d log f_{1} ∧ ... ∧ { \atop d log f_{j}} ∧ ... ∧ d log f_{2n+1} = 0 on ℂℙ^{n}). Integrating (32) over ℂℙ^{n} we get a). b) is obvious: we apply r_{2n−1} to zero element. Theorem is proved. There is a natural bijection

P G_{n}^{n−1} := {(n − 1)–planes in ℙ^{2n−1} in generic position to a simplex L}/(𝔾_{m}^{∗})^{2n−1}

< −− > {Configurations of 2n generic hyperplanes in ℙ^{n−1}}

given by intersecting of an (n − 1)–plane h with the codimension one faces of L.

Thus L_{n}^{G} is a function on the torus quotient P G_{n}^{n−1} of the generic part of Grassmannian.

Gelfand and MacPherson [GM] suggested a beautiful construction of the real valued version of 2n-logarithms on P G_{2n}^{2n−1}(ℝ). The construction uses the Pontryagin form. These functions generalize the Rogers dilogarithm.

The defined above functions L_{n}^{G} on complex Grassmannians generalize the Bloch-Wigner dilogarithm. They are related to the Chern classes. It would be very interesting to find a link between the construction in [GM] with our construction.

The existence of the multivalued analytic Grassmannian n-logarithms on complex Grassmannians was conjectured in [BMS]. They were constructed in [HM1]-[HM2] and, as a particular case of the analytic Chow polylogarithms, in [G4].

Recall the following general construction. Let X be a G-set and F a G-invariant function on X^{n} satisfying

∑_{i=1}^{n}(−1)^{i}F (x_{1}, ..., x̂_{i}, ..., x_{n}) = 0

Choose a point x ∈ X. Then there is an (n − 1)-cocycle of the group G:

f_{x}(g_{1}, ..., g_{n}) := F (g_{1}x, ..., g_{n}x)

Lemma 3.2 The cohomology class of the cocycle f_{x} does not depend on x.

Thus thanks to (31) the function L_{n}^{G} provides a measurable cocycle of GL_{n}(ℂ). We want to determine its cohomology class, but a priori it is not even clear that it is non zero. To handle this problem we will show below that the function L_{n}^{G} is a boundary value of a certain function ψ_{n} defined on the configurations of 2n points inside of the symmetric space SL_{m}(ℂ)/SU(n). The cohomology class of SL_{n}(ℂ) provided by this function is obviously related to the so-called Borel class. Using this we will show that the Grassmannian n-logarithm function L_{n}^{G} provides the Borel class, and moreover can be used to define the Borel regulator.

Finally, the restriction of the function L_{n}^{G} to certain special stratum in the configuration space of 2n hyperplanes in ℂℙ^{n−1} provides a single valued version of the classical n–logarithm function, see sections 4-5 below.

2. The function ψ_{n}. Let V_{n} be an n-dimensional complex vector space. Let

ℍ_{n} := { positive definite Hermitian forms in V_{n} } /ℝ_{+}^{∗} = SL_{n}(ℂ)/SU(n)

= { positive definite Hermitian forms in V_{n} with determinant = 1} It is a symmetric space of rank n − 1. For example ℍ_{2} = H_{3} is the hyperbolic 3-space. Replacing positive definite by non negative definite Hermitian forms we get a compactification \overline{ℍ}_{n} of the symmetric space ℍ_{n}.

Let G_{x} be the subgroup of SL_{N}(ℂ) stabilizing the point x ∈ ℍ_{n} . A point x defines a one dimensional vector space M_{x}:

x ∈ ℍ_{n} ⟼ M_{x} := {measures on ℂℙ^{n−1} invariant under G_{x}}

Namely, a point x corresponds to a hermitian metric in V_{n}. This metric provides the Fubini-Studi Kahler form on ℂℙ^{n−1} = P (V_{n}). Its imaginary part is a symplectic form. Raising it to (n − 1)-th power we get the Fubini-Studi volume form. The elements of M_{x} are its multiples.

Let x_{0}, ..., x_{2n−1} be points of the symmetric space SL_{n}(ℂ)/SU(n). Consider the following function

ψ_{n}(x_{0}, ..., x_{2n−1}) := ∫_{ℂℙ^{n−1}} log |\frac{μ_{x_{1}}}{μ_{x_{0}}} |d log |\frac{μ_{x_{2}}}{μ_{x_{0}}} | ∧ ... ∧ d log |\frac{μ_{x_{2n−1}}}{μ_{x_{0}}} |

More generally, let X be an m-dimensional manifold. For any m + 2 measures μ_{0}, ..., μ_{m+1} on X such that \frac{μ_{i}}{μ_{j}} are smooth functions consider the following differential m-form on X:

\overline{r}_{m}(μ_{0} : ... : μ_{m+1}) := log |\frac{μ_{1}}{μ_{0}} |d log |\frac{μ_{2}}{μ_{0}} | ∧ ... ∧ d log |\frac{μ_{m+1}}{μ_{0}} |

Proposition 3.3 The integral ∫_{X} \overline{r}_{m}(μ_{0} : ... : μ_{m+1})satisfies the following properties: 1) Skew symmetry with respect to the permutations of μ_{i}.

2) Homogeneity: if λ_{i} ∈ ℝ^{∗} then

∫_{X} \overline{r}_{m}(λ_{0}μ_{0} : ... : λ_{m+1}μ_{m+1}) = ∫_{X} \overline{r}_{m}(μ_{0} : ... : μ_{m+1})

3)Additivity: for any m + 3 measures μ_{i} on X one has

∑_{i=0}^{m+2}(−1)^{i} ∫_{X} \overline{r}_{m}(μ_{0} : ... : μ̂_{i} : ... : μ_{m+2}) = 0

4) Let g be a diffeomorphism of X. Then

∫_{X} \overline{r}_{m}(g^{∗}μ_{0} : ... : g^{∗}μ_{m+1}) = ∫_{X} \overline{r}_{m}(μ_{0} : ... : μ_{m+1})

3. The Grassmannian polylogarithm L_{n}^{G} is the boundary value of the function ψ_{n} [G7]. Let (z_{0} : ... : z_{n−1}) be homogeneous coordinates in ℙ^{n−1}. Let

σ_{n}(z, dz) := ∑_{i=0}^{n−1}(−1)^{i}z_{i}dz_{0} ∧ ... ∧ d̂z_{i} ∧ ... ∧ dz_{n−1}

ω_{F S}(H) := \frac{1}{(2πi)^{n−1}} \frac{σ_{n}(z, dz) ∧ σ_{n}(\overline{z}, d\overline{z})}{H(z, \overline{z})^{n}}  (33)

This form is clearly invariant under the group preserving the Hermitian form H. In fact it is the Fubini-Studi volume form.

Take any 2n non zero nonnegative definite Hermitian forms H_{0}, ..., H_{2n−1}, possibly degenerate. For each of the forms H_{i} choose a multiple μ_{H_{i}} of the Fubini-Studi form given by formula (33). It is a volume form with singularities along the projectivization of kernel of H_{i}.

Lemma 3.4 The following integral is convergent

ψ_{n}(H_{0}, ..., H_{2n−1}) := ∫_{ℂℙ^{n−1}} log |\frac{μ_{H_{1}}}{μ_{H_{0}}} |d log |\frac{μ_{H_{2}}}{μ_{H_{0}}} | ∧ ... ∧ d log |\frac{μ_{H_{2n−1}}}{μ_{H_{0}}} |

This integral does not change if we multiply one of the Hermitian forms by a positive scalar. Therefore we can extend ψ_{n} to a function on the configuration space of 2n points in the compactification \overline{ℍ}_{n−1}. This function is discontinuous.

One can realize ℂℙ^{n−1} as the smallest stratum of the boundary of ℍ_{n}. Indeed, let ℂℙ^{n−1} = P (V_{n}). For a hyperplane h ∈ V_{n} let

F_{h} := { nonnegative definite hermitian forms in V_{n} with kernel h} /ℝ_{+}^{∗}

The set of hermitian forms in V_{n} with the kernel h is isomorphic to ℝ_{+}^{∗}, so F_{h} defines a point on the boundary of \overline{ℍ}_{n}. Therefore Lemma 3.4 provides a function

ψ_{n}(h_{0}, ..., h_{2n−1}) := ψ_{n}(F_{h_{0}}, ..., F_{h_{2n−1}})  (34)

Applying Lemma 3.2 to the case when X is \overline{ℍ}_{n} and using only the fact that the function ψ_{n}(x_{0}, ..., x_{2n−1}) is well defined for any 2n points in \overline{ℍ}_{n} and satisfies the cocycle condition for any 2n + 1 of them we get

Proposition 3.5 Let x ∈ ℍ_{n} and h is a hyperplane in ℂℙ^{n−1}. Then the cohomology classes of the following cocycles coincide:

ψ_{n}(g_{0}x, ..., g_{2n−1}x) and ψ_{n}(g_{0}h, ..., g_{2n−1}h)

The Fubini-Studi volume form corresponding to a hermitian form from the set F_{h} is a Lebesgue measure on the affine space ℂℙ^{n−1} − h. Indeed, if h_{0} = {z_{0} = 0} then (33) specializes to

\frac{1}{(2πi)^{n−1}} d\frac{z_{1}}{z_{0}} ∧ ... ∧ d\frac{z_{n−1}}{z_{0}} ∧ d\begin{matrix}\overline{z} & 1 \\ \overline{z} & 0 \end{matrix} ∧ ... ∧ d\begin{matrix}\overline{z} & n−1 \\ \overline{z} & 0 \end{matrix}

Using this it is easy to prove the following proposition.

Proposition 3.6 For any 2n hyperplanes h_{0}, ..., h_{2n−1} in ℂℙ^{n−1} one has

ψ_{n}(h_{0}, ..., h_{2n−1}) = (−4)^{−n} · (2πi)^{n−1}(2n)^{2n−1}\binom{2n − 2}{n − 1} · L_{n}^{G}(h_{0}, ..., h_{2n−1})

4. A normalization of the Borel class b_{n}. Denote by H_{c}^{∗}(G, ℝ) the continuous cohomology of a Lie group G. Let us define an isomorphism

γ_{DR} : H_{DR}^{k}(SL_{n}(ℂ), ℚ) ⟶^{∼} H_{c}^{k}(SL_{n}(ℂ), ℂ)

We do it in two steps. First, let us define an isomorphism

α : H_{DR}^{k}(SL_{n}(ℂ), ℂ) ⟶^{∼} A^{k}(SL_{n}(ℂ)/SU(n))^{SL_{n}(ℂ)} ⊗ ℂ

It is well known that any cohomology class on the left is represented by a biinvariant, and hence closed, differential k–form Ω on SL_{n}(ℂ). Let us restrict it first to the Lie algebra, and then to the orthogonal complement su(n)^{⊥} to the Lie subalgebra su(n) ⊂ sl_{n}(ℂ). Let e be the point of ℍ_{n} corresponding to the subgroup SU(n). We identify the ℝ–vector spaces T_{e}ℍ_{n} and su(n)^{⊥}. The obtained exterior form on T_{e}ℍ_{n} is restriction of an invariant closed differential form ω on the symmetric space ℍ_{n}.

Now let us construct, following J. Dupont [Du], an isomorphism

β : A^{k}(SL_{n}(ℂ)/SU(n))^{SL_{n}(ℂ)} ⟶^{∼} H_{c}^{k}(SL_{n}(ℂ), ℝ)

For any ordered m + 1 points x_{1}, ..., x_{m+1} in ℍ_{n} there is a geodesic simplex I(x_{1}, ..., x_{m+1}) in ℍ_{n}. It is constructed inductively as follows. Let I(x_{1}, x_{2}) be the geodesic from x_{1} to x_{2}. The geodesics from x_{3} to the points of I(x_{1}, x_{2}) form a geodesic triangle I(x_{1}, x_{2}, x_{3}), and so on. If n > 2 the geodesic simplex I(x_{1}, ..., x_{k}) depends on the order of vertices.

Let ω be an invariant differential m-form on SL_{n}(ℂ)/SU(n). Then it is closed, and provides a volume of the geodesic simplex:

vol_{ω}I(x_{1}, ..., x_{m+1}) := ∫_{I(x_{1},...,x_{m+1})} ω

The boundary of the simplex I(x_{1}, ..., x_{m+2}) is the alternated sum of simplices I(x_{1}, ..., x̂_{i}, ..., x_{m+2}). Since the form ω is closed, the Stokes theorem yields

∑_{i=1}^{m+2}(−1)^{i} ∫_{I(x_{1},...,x̂_{i},...,x_{m+2})} ω = ∫_{I(x_{1},...,x_{m+2})} dω = 0

This just means that for a given point x the function vol_{ω}I(g_{1}x, ..., g_{m+1}x) is a smooth m-cocycle of the Lie group SL_{n}(ℂ). By Lemma 3.2 cocycles corresponding to different points x are canonically cohomologous. The obtained cohomology class is the class β(ω). Set γ_{DR} := β ∘ α.

It is known that

H_{DR}^{∗}(SL_{n}(ℂ), ℚ) = Λ∗_{ℚ}(C_{3}, ..., C_{2n−1})

where

C_{2n−1} := tr(g^{−1}dg)^{2n−1} ∈ Ω^{2n−1}(SL)

The Hodge considerations shows that [C_{2n−1}] ∈ H_{Betti}^{2n−1}(SL_{n}(ℂ), ℚ(n)).

Lemma 3.7 α(C_{2n−1}) is an ℝ(n − 1)–valued differential form. So it provides a cohomology class

b_{n} := γ_{DR}(C_{2n−1}) ∈ H_{c}^{2n−1}(SL_{n}(ℂ), ℝ(n − 1))

We call the cohomology class provided by this lemma the Borel class, and use it below to construct the Borel regulator.

It is not hard to show that the cohomology class of the cocycle ψ_{n}(g_{0}x, ..., g_{2n−1}x) is a non zero multiple of the Borel class. So thanks to propositions 3.5 and 3.6 the same is true for the cohomology class provided by the Grassmannian n–logarithm. The final result will be stated in Theorem 3.8 below.

5. Construction of the Borel regulator via Grassmannian polylogarithms. Let G be a group. The diagonal map Δ : G ⟶ G × G provides a homomorphism Δ_{∗} : H_{n}(G) ⟶ H_{n}(G × G). Recall that

PrimH_{n}G := {x ∈ H_{n}(G)|Δ_{∗}(x) = x ⊗ 1 + 1 ⊗ x}

Set A_{ℚ} := A ⊗ ℚ. One has

K_{n}(F)_{ℚ} = PrimH_{n}GL(F)_{ℚ} = PrimH_{n}GL_{n}(F)_{ℚ}

where the second isomorphism is provided by Suslin’s stabilization theorem. Let

B_{n} ∈ H_{c}^{2n−1}(GL_{2n−1}(ℂ), ℝ(n − 1))

be a cohomology which goes to b_{n} under the restriction map to GL_{n}. We define the Borel regulator map by restricting the class B_{n} to the subspace K_{2n−1}(ℂ)_{ℚ} of H_{2n−1}(GL_{2n−1}(ℂ), ℚ):

r_{n}^{Bo}(b_{n}) :=< B_{n}, ∗ >: K_{2n−1}(ℂ)_{ℚ} ⟶ ℝ(n − 1)

It does not depend on the choice of B_{n}.

Recall the Grassmannian complex C_{∗}(n)

... ⟶^{d} C_{2n−1}(n) ⟶^{d} C_{2n−2}(n) ⟶^{d} ... ⟶^{d} C_{0}(n)

where C_{k}(n) is the free abelian group generated by configurations, i.e. GL(V)-coinvariants, of k + 1 vectors (l_{0}, ..., l_{k}) in generic position in an n–dimensional vector space V over a field F, and d is given by the standard formula

(l_{0}, ..., l_{k}) ⟼ ∑_{i=0}^{k}(−1)^{i}(l_{0}, ..., l̂_{i}, ..., l_{k})  (35)

The group C_{k}(n) is in degree k. Since it is a homological resolution of the trivial GL_{n}(F)–module ℤ (see Lemma 3.1 in [G2]), there is canonical homomorphism

φ_{2n−1}^{n} : H_{2n−1}(GL_{n}(F)) ⟶ H_{2n−1}(C_{∗}(n))

Thanks to Lemma 3.7 the Grassmannian n–logarithm function provides a homomorphism

L_{n}^{G} : C_{2n−1}(n) ⟶ ℝ(n − 1); (l_{0}, ..., l_{2n−1}) ⟼ L_{n}^{G}(l_{0}, ..., l_{2n−1})  (36)

Thanks to the functional equation (30) for L_{n}^{G} it is zero on the subgroup dC_{2n}(n). So it induces a homomorphism

L_{n}^{G} : H_{2n−1}(C_{∗}(n)) ⟶ ℝ(n − 1);

Let us extend the map φ_{2n−1}^{n} ∘ L_{n}^{G} to a homomorphism from H_{2n−1}(GL_{2n−1}(ℂ)) to ℝ(n − 1) following s. 3.10 of [G2], Consider the following bicomplex:

... ⟶^{d} C_{2n−1}(2n − 1)
↓
... ... ...
↓  ↓
...  ⟶^{d} C_{2n−1}(n + 1) ⟶^{d} ... ⟶^{d}  C_{n+1}(n)
↓  ↓  ↓
... ⟶^{d} C_{2n−1}(n) ⟶^{d} C_{2n−2}(n) ⟶^{d} ... ⟶^{d}  C_{n}(n)

The horizontal differentials are given by formula (35), and the vertical by

(l_{0}, ..., l_{k}) ⟼ ∑_{i=0}^{k}(−1)^{i}(l_{i}|l_{1}, ..., l̂_{i}, ..., l_{k})

Here (l_{i}|l_{1}, ..., l̂_{i}, ..., l_{k}) means projection of the configuration (l_{1}, ..., l̂_{i}, ..., l_{k}) to the quotient V/ < l_{i} >. The total complex of this bicomplex is called the weight n bi–Grassmannian complex BC_{∗}(n).

Let us extend homomorphism (36) to a homomorphism

L_{n}^{G} : BC_{2n−1}(n) ⟶ ℝ(n − 1)

by setting it zero on the groups C_{2n−1}(n + i) for i > 0. The functional equation (31) for the Grassmannian n–logarithm just means that the composition

C_{2n}(n + 1) ⟶ C_{2n−1}(n) ⟶^{L_{n}^{G}} ℝ(n − 1),

where the first map is a vertical arrow in BC_{∗}(n), is zero. Therefore we get a homomorphism

L_{n}^{G} : H_{2n−1}(BC_{∗}(n)) ⟶ ℝ(n − 1)

The bottom row of the Grassmannian bicomplex is the stupid truncation of the Grassmannian complex at the group C_{n}(n). So there is a homomorphism

H_{2n−1}(C_{∗}(n)) ⟶ H_{2n−1}(BC_{∗}(n))  (37)

In [G1]-[G2] we proved that there are homomorphisms

φ_{2n−1}^{m} : H_{2n−1}(GL_{m}(F)) ⟶ H_{2n−1}(BC_{∗}(n)), m ≥ n

whose restriction to the subgroup GL_{n}(F) coincides with the composition

H_{2n−1}(GL_{n}(F)) ⟶^{φ_{2n−1}^{n}} H_{2n−1}(C_{∗}(n)) ⟶^{(37)} H_{2n−1}(BC_{∗}(n)),

Theorem 3.8 The composition

K_{2n−1}(ℂ) ⟶^{∼} PrimH_{2n−1}(GL_{2n−1}(ℂ), ℚ) ⟶^{φ_{2n−1}^{2n−1}} H_{2n−1}(BC_{∗}(n)_{ℚ}) ⟶^{L_{n}^{G}} ℝ(n − 1)

equals to

−(−1)^{n(n+1)/2} · \frac{(n − 1)!^{2}}{n(2n − 2)!(2n − 1)!}r_{n}^{Bo}(b_{n})

6. ℙ^{1} −{0, ∞} as a special stratum in the configuration space of 2n points in ℙ^{n−1} [G4]. A special configuration is a configuration of 2n points

(l_{0}, ..., l_{n−1}, m_{0}, ..., m_{n−1})  (38)

in ℙ^{n−1} such that l_{0}, ..., l_{n−1} are vertices of a simplex in ℙ^{n−1} and m_{i} is a point on the edge l_{i}l_{i+1} of the simplex different from l_{i} and l_{i+1}, as on Figure 1.

Figure 1: A special configuration of 8 points in ℙ^{3}.

Proposition 3.9 The set of special configurations of 2n points in ℙ^{n−1} is canonically identified with ℙ^{1}\{0, ∞}.

Construction. Let m̂_{i} be the point of intersection of the line l_{i}l_{i+1} with the hyperplane passing through all the points m_{j} except m_{i}. Let r(x_{1}, ..., x_{4}) be the cross-ratio of the four points on ℙ^{1}. Let us define the generalized cross-ratio by

r(l_{0}, ..., l_{n−1}, m_{0}, ..., m_{n−1}) := r(l_{i}, l_{i+1}, m_{i}, m̂_{i+1}) ∈ F^{∗}

It does not depend on i, and provides the desired isomorphism. Here is a different definition, which makes obvious the fact that the generalized cross-ratio is cyclically invariant. Consider the one dimensional subspaces L_{i}, M_{j} in the (n + 1)–dimensional vector space projecting to l_{i}, m_{j}. Then L_{i}, M_{i}, L_{i+1} belong to a two dimensional subspace. The subspace M_{i} provides a linear map L_{i} ⟶ L_{i+1}. The composition of these maps is a linear map L_{0} ⟶ L_{0}. The element of F^{∗} describing this map is the generalized cross-ratio.

7. Restriction of the Grassmannian n–logarithm to the special stratum. The n– logarithm function Li_{n}(z) has a single-valued version ([Z1])

L_{n}(z) := Re (n : odd) Im (n : even) (∑_{k=0}^{n−1} β_{k} log^{k} |z| · Li_{n−k}(z)) , n ≥ 2

It is continuous on ℂℙ^{1}. Here \frac{2x}{e^{2x}−1} = ∑_{k=0}^{∞} β_{k}x^{k}, so β_{k} = \frac{2^{k}B_{k}}{k!} where B_{k} are Bernoulli numbers. For example L_{2}(z) is the Bloch - Wigner function.

Let us consider the following modification of the function L_{n}(z) proposed by A. M. Levin in [Le]:

L̃_{n}(x) :=  (39)

\frac{(2n − 3)}{(2n − 2)} ∑ ; k even; 0 ≤ k ≤ n − 2 \frac{2^{k}(n − 2)!(2n − k − 3)!}{(2n − 3)!(k + 1)!(n − k − 2)!}L_{n−k}(x) log^{k} |x|

For example L̃_{n}(x) = L_{n}(x) for n ≤ 3, but already L̃_{4}(x) is different from L_{4}(x). A direct integration carried out in Proposition 4.4.1 of [Le] shows that

−(2πi)^{n−1}(−1)^{(n−1)(n−2)/2}L̃_{n}(x) =

∫_{ℂℙ^{n−1}} log |1 − z_{1}| ∏_{i=1}^{n−1} d log |z_{i}| ∧ ∏_{i=1}^{n−2} d log |z_{i} − z_{i+1}| ∧ d log |z_{n−1} − a|

This combined with Proposition 3.1 below implies

Theorem 3.10 The value of function L_{n}^{G} on special configuration (38) equals

−(−1)^{n(n−1)/2}4^{n−1}\binom{2n − 2}{n − 1}^{−1}L̃_{n}(a); a = r(l_{0}, ..., l_{n−1}, m_{0}, ..., m_{n−1})

8. Computation of the Grassmannian n–logarithm. It follows from Theorem 3.10 that L_{2}^{G}(l_{1}, ..., l_{4}) = −2L_{2}(r(l_{1}, ..., l_{4})).

It was proved in Theorem 1.3 of [GZ] that

L_{3}^{G}(l_{0}, ..., l_{5}) = \frac{1}{90}Alt_{6}L_{3}(r_{3}(l_{0}, ..., l_{5}))+

\frac{1}{9}Alt_{6}(log |Δ(l_{0}, l_{1}, l_{2})| log |Δ(l_{1}, l_{2}, l_{3})|Δ(l_{2}, l_{3}, l_{4})|)  (40)

We will continue this discussion in chapter 5.

The functions L_{n}^{G} for n > 3 can not be expressed via classical polylogarithms.

4 Polylogarithmic motivic complexes

1. The groups B_{n}(F) and polylogarithmic motivic complexes ([G1]-[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)

One has

R_{1}(F) := ({x} + {y} − {xy}, (x, y ∈ F^{∗}); {0}; {∞}) ; 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}  (41)

δ_{n} : \{x\} ↦ \{ \begin{matrix}\{x\}_{n−1} ⊗ x & : & n ≥ 3 \\ (1 − x) ∧ x & : & n = 2 \end{matrix}  δ_{n} : \{∞\}, \{0\}, \{1\} ↦ 0  (42)

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 4.1 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 polylogarithmic motivic 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} and B_{n} is in degree 1.

Conjecture 4.2 H^{i}Γ(F, n) ⊗ ℚ = gr_{n}^{γ}K_{2n−i}(F) ⊗ ℚ.

Denote by L̂_{n} the function L_{n}, multiplied by i for even n and unchanged for odd n. There is a well defined homomorphism ([G2], Theorem 1.13):

L̂_{n} : B_{n}(ℂ)) ⟶ ℝ(n − 1);  L̂_{n}(∑ m_{i}{z_{i}}_{n}) := ∑ m_{i}L̂_{n}(z_{i})

There are canonical homomorphisms

B_{n}(F) ⟶ B_{n}(F); {x}_{n} ⟼ {x}_{n}, n = 1, 2, 3.  (43)

They are isomorphisms for n = 1, 2 and expect to be an isomorphism for n = 3, at least modulo torsion.

2. The residue homomorphism for complexes Γ(F, n) (1.14, [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].

3. A variation of mixed Hodge structures on P^{1}(ℂ) − {0, 1, ∞} corresponding to the classical polylogarithm Li_{n}(z) ([D2]). Its fiber H(z) over a point z is described via the period matrix

( \begin{matrix}1 & 0 & 0 & ... & 0 \\ Li_{1}(z) & 2πi & 0 & ... & 0 \\ Li_{2}(z) & 2πi log z & (2πi)^{2} & ... & 0 \\ ... & ... & ... & ... & ... \\ Li_{n}(z) & 2πi\frac{log^{n−1} z}{(n−1)!} & (2πi)^{2} \frac{log^{n−2} z}{(n−2)!} & ... & (2πi)^{n} \end{matrix} )  (44)

Its entries are defined using analytic continuation to the point z along a path γ from a given point in ℂ where all the entries are defined by power series expansions, say the point 1/2.

Here is a more natural way to define the entries. Consider the following regularized iterated integrals along a certain fixed path γ between 0 to z:

Li_{n}(z) = ∫_{0}^{z} \frac{dt}{1 − t} ∘ \underbrace{\frac{dt}{t} ∘ ... ∘ \frac{dt}{t}}_{n−1 times}; \frac{log^{n} z}{n!} = ∫_{0}^{z} \underbrace{\frac{dt}{t} ∘ ... ∘ \frac{dt}{t}}_{n times}

To regularize the divergent integrals we take the lower limit of integration to be ε. Then it is easy to show that the integral has an asymptotic expansion of type I_{0}(ε) + I_{1}(ε) log ε + ... + I_{k}(ε) log^{k} ε, where all the functions I_{i}(ε) are smooth at ε = 0. Then we take I_{0}(0) to be the regularized value.

Now let us define a mixed Hodge structure H(z). Let ℂ^{n+1} be the standard vector space with basis (e_{0}, ..., e_{n}), and V_{n+1} the ℚ-vector subspace spanned by the columns of the matrix (44). Let W_{−2(n+1−k)}V_{n+1} be the subspace spanned by the first k columns, counted from the right to the left. One shows that these subspaces do not depend on the choice of the path γ, i.e. they are well-defined inspite of the multivalued nature of the entries of the matrix. Then W_{•}V_{n+1} is the weight filtration. We define the Hodge filtration F^{•}ℂ^{n+1} by setting F^{−k}ℂ^{n+1} := ⟨e_{0}, ..., e_{k}⟩_{ℂ}. It is opposite to the weight filtration. We get a Hodge-Tate structure, i.e. h^{pq} = 0 unless p ≠ q. One checks that the family of Hodge-Tate structures H(z) forms a unipotent variation of Hodge-Tate structures on P^{1}(ℂ) − {0, 1, ∞}.

Let n ≥ 0. An n-framed Hodge-Tate structure H is a triple (H, v_{0}, f_{n}), where v_{0} : ℚ(0) ⟶ gr_{0}^{W} H and f_{n} : gr_{−2n}^{W}H ⟶ ℚ(n) are nonzero morphisms. A framing plus a choice of a splitting of the weight filtration determines a period of a Hodge-Tate structure. Consider the coarsest equivalence relation on the set of all n-framed Hodge-Tate structures for which M_{1} ∼ M_{2} if there is a map M_{1} → M_{2} respecting the frames. Tnen the set H_{n} of the equivalence classes has a natural abelian group structure. Moreover

H_{•} := ⊕_{n≥0}H_{n}  (45)

has a natural Hopf algebra structure with a coproduct Δ, see the Appendix of [G12].

Observe that Gr_{−2k}^{W}H_{n}(z) = ℚ(k) for −n ≤ k ≤ 0. Therefore H_{n}(z) has a natural framing such that the corresponding period is given by the function Li_{n}(z). The obtained framed object is denoted by Li_{n}^{H}(z). To define it for z = 0, 1, ∞ we use the specialization functor to the punctured tangent space at z, and then take the fiber over the tangent vector corresponding to the parameter z on P^{1}. It is straitforward to see that the coproduct ΔLi_{n}^{H}(z) is computed by the formula

ΔLi_{n}^{H}(z) = ∑_{k=0}^{n} Li_{n−k}^{H}(z) ⊗ \frac{log^{H}(z)^{k}}{k!}  (46)

where log^{H}(z) is the 1-framed Hodge-Tate structure corresponding to log(z).

4. A motivic proof of the weak version of Zagier’s conjecture. Our goal is the following result, which was proved in [dJ3] and in the unfinished manuscript [BD2]. The proof below uses a different set of ideas. It follows the framework described in Chapter 13 of [G8], and quite close to the approach outlined in [BD1], although it is formulated a bit differently, using the polylogarithmic motivic complexes.

Theorem 4.3 Let F be a number field. Then there exists a homomorphism

l_{n} : H^{1}(Γ(F, n) ⊗ ℚ) ⟶ K_{2n−1}(F) ⊗ ℚ

such that for any embedding σ : F ↪ ℂ one has the following commutative diagram

\begin{matrix}H^{1}(Γ(F, n) ⊗ ℚ) & ⟶^{σ_{∗}} & H^{1}(Γ(ℂ, n) ⊗ ℚ) & ⟶^{L_{n}} & ℝ(n − 1) \\ ↓ l_{n} & & & & ↓= \\ K_{2n−1}(F) ⊗ ℚ & ⟶^{σ_{∗}} & K_{2n−1}(ℂ) ⊗ ℚ & ⟶^{r_{Bo}} & ℝ(n − 1) \end{matrix}  (47)

Proof. We will use the following background material and facts:

i) The existence of the abelian tensor category M_{T} (F) of mixed Tate motives over a number field F, satisfying all the desired properties, including Beilinson’s formula expressing the Ext groups via the rational K-theory of F and the Hodge realization functor. See [DG] and the references there.

ii) The formalism of mixed Tate categories, including the description of the fundamental Hopf algebra A_{•}(M) of a mixed Tate category via framed objects in M. See [G12], Section 8. The fundamental Hopf algebra of the category M_{T} (F) is denoted A_{•}(F). For example for the category of mixed Hodge-Tate structures fundamental Hopf algebra is the one H_{•} from (45). Let

Δ : A_{•}(F) ⟶ A_{•}(F)^{⊗2}

be the coproduct in the Hopf algebra A_{•}(F), and Δ′ := Δ−Id⊗1+1⊗Id is the restricted coproduct. The key fact is a canonical isomorphism

KerΔ′ ∩ A_{n}(F) =^{∼} K_{2n−1}(F) ⊗ ℚ  (48)

Since A_{•}(F) is graded by ≥ 0 integers, and A_{0}(F) = ℚ, formula (48) for n = 1 reduces to an isomorphism

A_{1}(F) =^{∼} F^{∗} ⊗ ℚ  (49)

iii) The existence of the motivic classical polylogarithms

Li_{n}^{M}(z) ∈ A_{n}(F),  z ∈ P^{1}(F)  (50)

They were defined in Section 3.6 of [G12] using either a geometric construction of [G9], or a construction of the motivic fundamental torsor of path between the tangential base points given in [DG]. In particular one has Li_{n}^{M}(0) = Li_{n}^{M}(∞) = 0. A natural construction of the elements (50) using the moduli space \overline{M}_{0,n+3} is given in Chapter 4.6 below.

iv) The Hodge realization of the element (50) is equivalent to the framed Hodge-Tate structure Li_{n}^{H}(z) from Chapter 4.3. This fact is more or less straitforward if one uses the fundamental torsor of path on the punctured projective line to define the element Li_{n}^{M}(z), and follows from the general specialization theorem proved in [G9] if one uses the approach of loc. cit. This implies that the Lie-period of Li_{n}^{H}(z) equals to L_{n}(z). Indeed, for the Hodge-Tate structure H(z) assigned to Li_{n}(z) this was shown in [BD1]. v) The crucial formula (Section 6.3 of [G12]):

ΔLi_{n}^{M}(z) = ∑_{k=0}^{n} Li_{n−k}^{M}(z) ⊗ \frac{log^{M}(z)^{k}}{k!}  (51)

where log^{M}(z) ∈ A_{1}(F) is the element corresponding to z under the isomorphism (48), and log^{M}(z)^{k} ∈ A_{k}(F) is its k-th power. It follows from formula (46) using the standard trick based on Borel’s theorem to reduce a motivic claim to the corresponding Hodge one.

vi) The Borel regulator map on K_{2n−1}(F) ⊗ ℚ, which sits via (48) inside of A_{n}(F), is induced by the Hodge realization functor on the category M_{T} (F).

Having this background, we proceed as follows. Namely, let

L_{•}(F) := \frac{A_{>0}(F)}{A_{>0}(F)^{2}}

be the fundamental Lie coalgebra of M_{T} (F). Its cobracket δ is induced by Δ. Projecting the element (50) into L_{n}(F) we get an element l_{n}^{M}(z) ∈ L_{n}(F) such that

δl_{n}^{M}(z) = l_{n−1}^{M}(z) ∧ z  (52)

Consider the following map

l̃_{n} : ℚ[P^{1}(F)] ⟶ L_{n}(F), {z} ⟼ l_{n}^{M}(z)

Proposition 4.4 The map l̃_{n} kills the subspace R_{n}(F), providing a well defined homomorphism

l_{n} : B_{n}(F) ⟶ L_{n}(F); {z}_{n} ⟶ l̃_{n}(z)

Proof. We proceed by the induction on n. The case n = 1 is self-obvious. Suppose we are done for n − 1. Then there is the following commutative diagram:

ℚ[F] ⟶^{δ_{n}}  B_{n−1}(F) ⊗ F^{∗}
↓ l̃_{n}  ↓ l_{n−1} ∧ Id
L_{n}(F) ⟶^{δ} ⊕_{k≤n/2}L_{n−k}(F) ∧ L_{k}(F)

Indeed, its commutativity is equivalent to the basic formula (52).

Let x ∈ P^{1}(F). Recall the specialization at x homomorphism

s_{x} : B_{n}(F (T)) ⟶ B_{n}(F), {f(T)}_{n} ⟼ {f(x)}_{n}

It gives rise to the specialization homomorphism

s′_{x} : B_{n−1}(F (T)) ⊗ F (T)^{∗} ⟶ B_{n−1}(F) ⊗ F^{∗},

\{f(T)\}_{n−1} ⊗ g(T) ⟼ \{f(x)\}_{n−1} ⊗ \frac{g(T)}{(T − x)^{v_{x}(g)}} (x)

(Use the local parameter T^{−1} when x = ∞).

Now let

α(T) ∈ Ker(δ_{n} : ℚ[F (T)] → B_{n−1}(F (T)) ⊗ F (T)^{∗})

Using Li_{n}^{M}(0) = Li_{n}^{M}(∞) = 0, for any x ∈ P^{1}(F) one has δ(l̃_{n}(α(x))) = 0. Thus

l̃_{n}(α(x)) ∈ K_{2n−1}(F) ⊗ ℚ ⊂ L_{n}(F)

Let us show that this element is zero. Given an embedding σ : F ↪ ℂ, write σ(α(T)) = ∑_{i} n_{i}{f_{i}^{σ}(T)}. Applying the Lie-period map to this element and using v) we get ∑_{i} n_{i}L_{n}f_{i}^{σ}(z). By Theorem 1.13 in [G2] the condition on α(T) implies that this function is constant on ℂP^{1}. Thus the difference of its values at σ(x_{1}) and σ(x_{2}), where x_{0}, x_{1} ∈ P^{1}(F), is zero. On the other hand thanks to v) and vi) it coincides with the Borel regulator map applied to the corresponding element of K_{2n−1}(F) ⊗ ℚ. Thus the injectivity of the Borel regulator map proves the claim. So l̃_{n}(α(x_{0}) − α(x_{1})) = 0. The proposition is proved.

Proposition 4.4 implies that we get a homomorphism of complexes

B_{n}(F) ⟶^{δ}  B_{n−1}(F) ⊗ F^{∗}
↓ l_{n}  ↓ l_{n−1} ∧ Id
L_{n}(F) ⟶^{δ} ⊕_{k≤n/2}L_{n−k}(F) ∧ L_{k}(F)

The theorem follows immediately from this. Indeed, it remains to check commutativity of the diagram (47), and it follows from v) and vi).

If we assume the existence of the hypothetical abelian category of mixed Tate motives over an arbitrary field F, the same argumentation as above (see Chapter 6.1) implies the following result: one should have canonical homomorphisms

H^{i}(Γ(F, n)) ⊗ ℚ ⟶ gr_{n}^{γ}K_{2n−i}(F) ⊗ ℚ

The most difficult part of Conjecture 4.2 says that this maps are supposed to be isomorphisms.

In the next section we define a regulator map on the polylogarithmic motivic complexes. Combined with these maps, it should give an explicit construction of the regulator map.

5. A construction of the motivic ζ-elements (1). The formula (2) leads to the motivic extension ζ^{M}(n) as follows ([GM]). Recall the moduli space \overline{M}_{n+3} parametrising stable curves of genus zero with n+3 marked points. It contains as an open subset the space M_{n+3} parametrising the (n+3)- tuples of distinct points on ℙ^{1} modulo Aut(ℙ^{1}). Then the complement ∂M_{n+3} := \overline{M}_{n+3} − M_{n+3} is a normal crossing divisor, and the pair (M_{n+3}, ∂M_{n+3}) is defined over ℤ. Let us identify sequences (t_{1}, ..., t_{n}) of distinct complex numbers different from 0 and 1 with the points (0, t_{1}, ..., t_{n}, 1, ∞) of M_{n+3}(ℂ). Let us consider the integrand in (2) as a holomorphic form on M_{n+3}(ℂ). Meromorphically extending it to \overline{M}_{n+3} we get a differential form with logarithmic singularities Ω_{n}. Let A_{n} be its divisor. Similarly, embed the integration simplex 0 < t_{1} < ... < t_{n} < 1 into the set of real points of \overline{M}_{n+3}(ℝ), take its closure Δ_{n} there, and consider the Zariski closure B_{n} of its boundary ∂Δ_{n}. Then one can show that the mixed motive

H^{n}(\overline{M}_{n+3} − A_{n}, B_{n} − (A_{n} ∩ B_{n}))  (53)

is a mixed Tate motive over Spec(ℤ). Indeed, it is easy to prove that its l-adic realization is unramified outside l, and is glued from the Tate modules of different weights, and then refer to [DG]. The mixed motive (53) comes equipped with an additional data, framing, given by non-zero morphisms

[Ω_{n}] : ℤ(−n) ⟶ gr_{2n}^{W}H^{n}(\overline{M}_{n+3} − A_{n}, B_{n} − (A_{n} ∩ B_{n})),  (54)
[Δ_{n}] : gr_{0}^{W} H^{n}(\overline{M}_{n+3} − A_{n}, B_{n} − (A_{n} ∩ B_{n})) ⟶ ℤ(0),  (55)

There exists the minimal subquotient of the mixed motive (53) which inherits non-zero framing. It delivers the extension class ζ^{M}(n). Leibniz formula (2) just means that ζ(n) is its period.

Figure 2: Constructing ζ^{M}(2)

Example. To construct ζ^{M}(2), take the pair of triangles in P^{2} shown on the left of Figure 2. The triangle shown by the punctured lines is the divisor of poles of the differential d log(1 − t_{1}) ∧ d log t_{2} in (2), and the second triangle is the algebraic closure of the boundary of the integration cycle 0 ≤ t_{1} ≤ t_{2} ≤ 1. The corresponding configuration of six lines is defined uniquely up to a projective equivalence. Blowing up the four points shown by little circles on Figure 2 (they are the triple intersection points of the lines), we get the moduli space \overline{M}_{0,5}. Its boundary is the union of the two pentagons, A_{2} and B_{2}, projecting to the two triangles on P^{2}. Then ζ^{M}(2) is a subquotient of the mixed Tate motive H^{2}(\overline{M}_{0,5} − A_{2}, B_{2} − (A_{2} ∩ B_{2})) over Spec(ℤ). Observe that an attempt to use a similar construction for the pair of triangles in P^{2} fails since there is no non-zero morphism [Δ_{2}] in this case. Indeed, there are two vertices of the B-triangle shown on the left of Figure 2 lying at the sides of the A-triangle, and therefore the chain 0 ≤ t_{1} ≤ t_{2} ≤ 1 does not give rise to a relative class in H_{2}(P^{2}(ℂ) − A, B − (A ∩ B)).

6. A geometric construction of the motivic classical polylogarithm Li_{n}^{M}(z). The above construction is easily generalized to the case of the classical polylogarithm. Let A_{n}(z) be the divisor of the meromorphic differential form

Ω_{n}(z) := \frac{dt_{1}}{z^{−1} − t_{1}} ∧ \frac{dt_{2}}{t_{2}} ∧ ... ∧ \frac{dt_{n}}{t_{n}}

extended as a rational form to \overline{M}_{0,n+3}. Suppose that F is a number field and z ∈ F. Then there is the following mixed Tate motive over F:

H^{n}(\overline{M}_{n+3} − A_{n}(z), B_{n} − (A_{n}(z) ∩ B_{n}))  (56)

One checks that the vertices (that is the zero-dimensional strata) of the divisor B_{n} are disjoint with the divisor A_{n}(z). Using this we define a framing ([Ω_{n}(z)], [Δ_{n}]) on the mixed motive (56) similar to the one (54) - (55). The geometric condition on the divisor A_{n}(z) is used to show that the framing morphism [Δ_{n}] is non-zero. We define Li_{n}^{M}(z) as the framed mixed Tate motive (56) with the framing ([Ω_{n}(z)], [Δ_{n}]). A similar construction for the multiple polylogarithms was worked out in the Ph. D. thesis of Q. Wang [Wa].

It follows from a general result in [G9] that the Hodge realization of Li_{n}^{M}(z) is equivalent to the framed mixed Hodge structure Li_{n}^{H}(z). If n = 2 the two corresponding mixed Hodge structures are isomorphic ([Wa]), while in general they are not isomorphic but equivalent as framed mixed Hodge structures.

5 Regulator maps on the polylogarithmic motivic complexes

In this chapter we define explicitly these regulator maps via classical polylogarithm functions following [G6] and [G2]. This implies how the special values of ζ-functions of algebraic varieties outside of the critical strip should be expressed using the classical polylogarithms.

1. 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} = −2β_{k+3} − \frac{1}{3}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}  (57)

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

Let m ≥ 1. Then one can show that

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

2. The regulator map on the polylogarithmic motivic complexes in the case F = ℂ(X), where X is a complex algebraic variety X. 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  (58)
L̂_{1,q}(z) := α(1 − z, z) log^{q−1} |z|

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

Let f, g_{1}, ..., g_{m} be rational functions on a complex variety X. Set

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}\} +  (59)

∑_{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}}  (60)

Proposition 5.1 The differential form r_{n+m}(m + 1)({f}_{n} ⊗ g_{1} ∧ ... ∧ g_{m}) defines a distribution on X(ℂ).

Example 1. r_{n}(1)({f}_{n}) = L̂_{n}(f). Example 2. r_{n}(n)(g_{1} ∧ ... ∧ g_{n}) = r_{n−1}(g_{1} ∧ ... ∧ g_{n}). Example 3. m = 1, n is arbitrary. Then

r_{n+1}(2) : {f}_{n} ⊗ g ⟼ L̂_{n}(f)di arg g − ∑_{k=1}^{n−1} β_{k+1}L̂_{n−k,k}(f) · log |g|

Example 4. m = 2, n is arbitrary.

r_{n+2}(3) : \{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}|)

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. Let D be the de Rham differential on distributions on X(ℂ), and d the de Rham differential on A^{i}(η_{X}). For example:

d(di arg z)= 0; D(di arg z)= 2πiδ(z)

Recall the residue homomorphisms defined in Chapter 4.2.

Theorem 5.2 a) The maps r_{n}(·) provide 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)

b) The maps r_{n}(m) are compatible with the 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.

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

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

Here \underline{D}_{X}^{0} is 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 5.2 can be reformulated as follows. Set r̃_{n}(i) := r_{n}(i) for i < n and

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}  (61)

Theorem 5.3 Let X be a complex algebraic variety. Then there is a homomorphism of complexes

r̃_{n}(·) : Γ(ℂ(X); n) ⟶ \underline{ℝ}_{D}(n)(Specℂ(X))  (62)

compatible with the residues as explained in the part b) of Theorem 5.2.

3. The general case. Let X be a regular projective variety over a field F. Let d := dimX. Then the complex Γ(X; n) should be defined as the total complex of the following bicomplex:

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

where the arrows are provided by the residue maps, see [G1], p 239-240. The complex Γ(X; n) ⊗ ℚ should be quasiisomorphic to the weight n motivic complex.

However there is a serious difficulty in the definition of the complex Γ(X; n) for a general variety X and n > 3, ([G1], p. 240 ). It would be resolved if homotopy invariance of the polylogarithmic complexes were known (Conjecture 1.39 in [G1]). As a result we have an 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.

Now let F be a subfield of ℂ. Having in mind applications to arithmetic, we will restrict ourself by the case when F = ℚ. Assuming we are working with one of the above cases, or assuming the above difficulty has been resolved, let us define the regulator map

Γ(X; n) ⟶ C_{D}(X(ℂ); ℝ(n))

We specify it for each of Γ(ℚ(Y_{k}); n − k)[−k] where k = 0, ..., d. Namely, we take the homomorphism r_{n−k}(·) for Spec(ℚ(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. Then Theorem 5.2, and in particular its part b), providing compatibility with the residues property, guarantee that we get a homomorphism of complexes. Here are some examples.

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

ℚ(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].

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

B_{2}(ℚ(X)) ⟶^{δ}  Λ^{2}ℚ(X)^{∗}  ⟶^{∂} ⊕_{Y ∈X^{(1)}}ℚ(Y)^{∗} ⟶^{∂} ⊕_{Y ∈X^{(2)}}ℤ
↓ r_{2}(1)  ↓ r_{2}(2)  ↓ r_{2}(3)  ↓ r_{2}(4)
D0,0_{ℝ}(1) ⟶^{D} (D^{0,1} ⊕ D^{1,0})_{ℝ}(1) ⟶^{D}  D1,1_{ℝ}(1)  ⟶^{2\overline{∂}∂} D_{cl,ℝ}^{2,2}(2)

where D is the differential in C_{D}(X, ℝ(2))

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

The Deligne complex C_{D}(X, 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}

We construct the regulator map Γ(X; 3) ⟶ C_{D}(X, 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. Classical polylogarithms and special values of ζ-functions of algebraic varieties. We conjecture that the polylogarithmic motivic complex Γ(X; n) ⊗ ℚ should be quasiisomorphic to the weight n motivic complex, and Beilinson’s regulator map under this quasiisomorphism should be equal to the defined above regulator map on Γ(X; n). This implies that the special values of ζ-functions of algebraic varieties outside of the critical strip should be expressed via classical polylogarithms.

The very special case of this conjecture when X = Spec(F) where F is a number field is equivalent to Zagier’s conjecture [Z1].

The next interesting case is when X is a curve over a number field. The conjecture in this case was elaborated in [G3], see also [G8] for a survey. For example if X is an elliptic curve this conjecture suggests that the special values L(X, n) for n ≥ 2 are expressed via certain generalized Eisenstein– Kronecker series. For n = 2 this was discovered of S. Bloch [Bl4], and for n = 3 it implies Deninger’s conjecture [De].

A homomorphism from K-theory to H^{i}(Γ(X; n))⊗ ℚ has been constructed in the following cases:

(1) F is an arbitrary field, n ≤ 3 ([G1], [G2], [G5]]) 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 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 main ingredient of the proof is an explicit construction of the motivic cohomology class

H^{2n}(BGL_{•}, Γ(∗; n))

The results (2) combined with the results of R. de Jeu [dJ1]- [dJ2] 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).

7. Coda: Polylogarithms on curves, Feynman integrals and special values of Lfunctions. The classical polylogarithm functions admit the following generalization. Let X be a regular complex algebraic curve. Let us assume first that X is projective. Let us choose a metric on X(ℂ). Denote by G(x, y) the Green function provided by this metric. Then one can define real-valued functions G_{n}(x, y) depending on a pair of points x, y of X(ℂ) with values in the complex vector space S^{n−1}H_{1}(X(ℂ), ℂ)(1), see [G10], Chapter 9.1. By definition G_{1}(x, y) is the Green function G(x, y). It has a singularity at the diagonal, but provides a generalized function. For n > 1 the function G_{n}(x, y) is well-defined on X(ℂ) × X(ℂ). Let X = E be an elliptic curve. Then the functions G_{n}(x, y) are translation invariant, and thus reduced to a signle variable functions: G_{n}(x, y) = G_{n}(x − y). The function G_{n}(x) is given by the classical Kronecker-Eisenstein series. Finally, adjasting this construction to the case when X = ℂ^{∗}, we get a function G_{n}(x, y) invariant under the shifts on the group ℂ^{∗}, i.e. one has G_{n}(x, y) = G_{n}(x/y). The function G_{n}(z) is given by the single-valued version (39) of the classical n-logarithm function.

Let us extend the function G_{n}(x, y) by linearity to a function G_{n}(D_{1}, D_{2}) depending on a pair of divisors on X(ℂ). Although the function G_{n}(x, y) depends on the choice of metric on X(ℂ), the restriction of the function G_{n}(D_{1}, D_{2}) to the subgroups of the degree zero divisors is independent of the choice of the metric.

Let X be a curve over a number field F. In the Section 9.1 of [G10] we proposed a conjecture which allows to express the special values L(Sym^{n−1}H^{1}(X), n) via determinants whose entries are given by the values of the function G_{n}(D_{1}, D_{2}), where D_{1}, D_{2} are degree zero divisors on (X(\overline{F})) invariant under the action of the Galois group Gal(\overline{F} /F). In the case when X is an elliptic curve it boils down to the so-called elliptic analog of Zagier’s conjecture, see [Wil], [G11]. It has been proved for n = 2 at [GL], and a part of this proof (minus surjectivity) can be transformed to the case of arbitrary X.

It was conjectured in the Section 9.3 of [G10] that the special values L(Sym^{n−1}H^{1}(X), n + m − 1), where m ≥ 1, should be expressed similarly via the special values of the depth m multiple polylogarithms on the curve X, defined in the Section 9.2 of loc. cit. One can show that in the case when X = E is an elliptic curve, and n = m = 2, this conjecture reduces to Deninger’s conjecture [De] on L(E, 3), which has been proved in [G3] An interesting aspect of this story is that the multiple polylogarithms on curves are introduced via mathematically well defined Feynman integrals. This seems to be the first application of Feynman integrals in number theory.

6 Motivic Lie algebras and Grassmannian polylogarithms

1. Motivic Lie coalgebras and motivic complexes. Beilinson conjectured that for an arbitrary field F there exists an abelian ℚ–category M_{T} (F) of mixed Tate motives over F. This category is supposed to be a mixed Tate category, see Section 8 of [G12] for the background. Then the Tannakian formalism implies that there exists a positively graded Lie coalgebra L_{•}(F) such that the category of finite dimensional graded comodules over L_{•}(F) is naturally equivalent to the category M_{T} (F).

Moreover L_{•}(F) depends functorially on F. This combined with Beilinson’s conjectural formula for the Ext groups in the category M_{T} (F) imply that the cohomology of this Lie coalgebra are computed by the formula

H_{(n)}^{i}(L_{•}(F)) = gr_{n}^{γ}K_{2n−i}(F) ⊗ ℚ  (63)

where H_{(n)}^{i} means the degree n part of H^{i}. This conjecture provides a new point of view on algebraic K-theory of fields, suggesting and explaining several conjectures and results, see ch. 1 [G2].

The degree n part of the standard cochain complex

L_{•}(F) ⟶^{δ} Λ^{2}L_{•}(F) ⟶^{δ∧Id−Id∧δ} Λ^{3}L_{•}(F) ⟶ ...

of the Lie coalgebra L_{•}(F) is supposed to be quasiisomorphic to the weight n motivic complex of F, providing yet another point of view on motivic complexes. For example the first four of the motivic complexes should look as follows:

L_{1}(F);  L_{2}(F) ⟶ Λ^{2}L_{1}(F);  L_{3}(F) ⟶ L_{2}(F) ⊗ L_{1}(F) ⟶ Λ^{3}L_{1}(F)
L_{4}(F) ⟶ L_{3}(F) ⊗ L_{1}(F) ⊕ Λ^{2}L_{2}(F) ⟶ L_{2}(F) ⊗ Λ^{2}L_{1}(F) ⟶ Λ^{4}L_{1}(F)

Comparing this with the formula (63), the shape of complexes (10) and (13), and the known information relating their cohomology with algebraic K-theory we conclude the following. One must have

L_{1}(F) = F^{∗} ⊗ ℚ; L_{2}(F) = B_{2}(F) ⊗ ℚ

and we expect to have an isomorphism

B_{3}(F) ⊗ ℚ ⟶^{∼} L_{3}(F)

Moreover the motivic complexes (10) and (13) are simply the degree n parts of the standard cochain complex of the Lie coalgebra L_{•}(F). More generally, the very existence for an arbitrary field F of the motivic classical polylogarithms (50) implies that one should have canonical homomorphisms

l_{n} : B_{n}(F) ⟶ L_{n}(F)

These homomorphisms are expected to satisfy the basic relation 52. Therefore the maps l_{n} give rise to a canonical homomorphism from the weight n polylogarithmic complex of F to the degree n part of the standard cochain complex of L_{•}(F).

\begin{matrix}B_{n}(F) & ⟶ & B_{n−1}(F) ⊗ F^{∗} & ⟶ ... ⟶ & Λ^{n}F^{∗} \\ ↓ l_{n} & & ↓ l_{n−1} ∧ l_{1} & & ↓= \\ L_{n}(F) & ⟶ & Λ_{(n)}^{2}L_{•}(F) & ⟶ ... ⟶ & Λ^{n}L_{1}(F) \end{matrix}  (64)

where Λ_{(n)}^{2} denotes the degree n part of Λ_{(n)}^{2}. For n = 1, 2, 3, these maps, combined with the ones (43), lead to the maps above. For n > 3 the map of complexes (64) will not be an isomorphism. The conjecture that it is a quisiisomorphism is equivalent to the Freeness conjecture for the Lie coalgebra L_{•}(F), see [G1]-[G2].

Therefore we have two different points of view on the groups B_{n}(F) for n = 1, 2, 3: according to one of them they are the particular cases of the groups B_{n}(F), and according to the other they provide an explicit computation of the first three of the groups L_{n}(F). It would be very interesting to find an explicit construction of the groups L_{n}(F) for n > 3 generalizing the definition of the groups B_{n}(F). More specifically, we would like to have a “finite dimensional” construction of all vector spaces L_{n}(F), i.e. for every n there should exist a finite number of finite dimensional algebraic varieties X_{n}^{i} and R_{n}^{j} such that

L_{n}(F) = Coker (⊕_{j}ℚ[R_{n}^{j}(F)] ⟶ ⊕_{i}ℚ[X_{n}^{i}(F)])

Such a construction would be provided by the scissor congruence motivic Hopf algebra of F ([BMS], [BGSV]). However so far the problem in the definition of the coproduct in loc. cit. for non generic generators has not been resolved. A beautiful construction of the motivic Lie coalgebra of a field F was suggested by Bloch and Kriz in [BKr]. However it is not finite dimensional in the above sense.

2. Grassmannian approach of the motivic Lie coalgebra. We suggested in [G5] that there should exist a construction of the Lie coalgebra L_{•}(F) such that the variety X_{n} is the variety of configurations of 2n points in P^{n−1} and the relations varieties R_{n}^{j} are provided by the functional equations for the motivic Grassmannian n-logarithm. Let us explain this in more detail.

Let L̃_{n}(F) be the free abelian group generated by 2n-tuples of points (l_{1}, ..., l_{2n}) in P^{n−1}(F) subject to the following relations:

1)(l_{1}, ..., l_{2n}) = (gl_{1}, ..., gl_{2n}) for any g ∈ P GL_{n}(F).

2) (l_{1}, ..., l_{2n}) = (−1)^{|σ|}(l_{σ(1)}, ..., l_{σ(2n)}) for any permutation σ ∈ S_{2n}.

3) for any 2n + 1 points (l_{0}, ..., l_{2n}) in P^{n−1}(F) one has

∑_{i=0}^{2n}(−1)^{i}(l_{0}, ...l̂_{i}, ..., , l_{2n}) = 0

4) for any 2n + 1 points (l_{0}, ..., l_{2n}) in P^{n}(F) one has

∑_{i=0}^{2n}(−1)^{i}(l_{i}|l_{0}, ...l̂_{i}, ..., , l_{2n}) = 0

We conjecture that L_{n}(F) is a quotient of L̃_{n}(F). It is a nontrivial quotient already for n = 3. Then to define the Lie coalgebra L_{•}(F) one needs to produce a cobracket

δ : L_{n}(F) ⟶ ⊕_{i}L_{i}(F) ∧ L_{n−i}(F)

Here is how to do this in the first nontrivial case, n = 4.

Let us define a homomorphism

L̃_{4}(F) ⟶^{δ} B_{3}(F) ⊗ F^{∗} ⊕ B_{2}(F) ∧ B_{2}(F)

by setting δ = (δ_{3,1}, δ_{2,2}) where

δ_{3,1}(l_{1}, ..., l_{8}) := −\frac{1}{9}Alt_{8}((r_{3}(l_{1}|, l_{2}, l_{3}, l_{4}; l_{5}, l_{6}, l_{7}) + \{r(l_{1}l_{2}|l_{3}, l_{6}, l_{4}, l_{5})\}_{3}−

−{r(l_{1}l_{2}|l_{3}, l_{5}, l_{4}, l_{6})}_{3}) ∧ Δ(l_{5}, l_{6}, l_{7}, l_{8})) ∈ B_{3}(F) ∧ F^{∗}
δ_{2,2}(l_{1}, ..., l_{8}) :=

\frac{1}{7} · Alt_{8}(\{r_{2}(l_{1}, l_{2}|l_{3}, l_{4}, l_{5}, l_{6})\}_{2} ∧ \{r_{2}(l_{3}, l_{4}|l_{1}, l_{2}, l_{5}, l_{7})\}_{2}) ∈ Λ^{2}B_{2}(F)

These formulae are obtained by combining the definitions at p. 136-137 and p 156 of [G5]. The following key result is Theorem 5.1 in loc. cit.

Theorem 6.1 a) The homomorphism δ kills the relations 1) - 4).

b) The following composition equals to zero:

L̃_{4}(F) ⟶^{δ} B_{3}(F) ⊗ F^{∗} ⊕ B_{2}(F) ∧ B_{2}(F) ⟶^{δ} B_{2}(F) ⊗ Λ^{2}F^{∗}

Here the second differential is defined by the Leibniz rule, using the differentials in complexes (10) and (13).

Taking the ”connected component” of zero in Kerδ (like we did in s. 1 ch. 4 above, or in [G1]) we should get the set of defining relations for the group L_{4}(F). An explicit construction of them is not known.

4. The motivic Grassmannian tetralogarithm. Let F be a function field on a complex variety X. There is a morphism of complexes

B_{3}(F) ⊗ F^{∗} ⊕ B_{2}(F) ∧ B_{2}(F) ⟶^{δ} B_{2}(F) ⊗ Λ^{2}F^{∗} ⟶^{δ}  Λ^{4}F^{∗}
↓ R_{4}(2)  ↓ R_{4}(3)  ↓ R_{4}(4)
A^{1}(SpecF)  ⟶^{d} A^{2}(SpecF) ⟶^{d} A^{3}(SpecF)

extending the homomorphism r_{4}(∗) from Chapter 5. Namely, R_{4}(∗) = r_{4}(∗) for ∗ = 3, 4 and R_{4}(2) = (r_{4}(2), r′_{4}(2)) where

r′_{4}(2) : Λ^{2}B_{2}(F) ⟶ A^{1}(SpecF)

\{f\}_{2} ∧ \{g\}_{2} ↦ \frac{1}{3} · (L̂_{2}(g) · α(1 − f, f) − L̂_{2}(f) · α(1 − g, g))

It follows from Theorem 6.1 that the composition R_{4}(2) ∘ δ(l_{1}, ..., l_{8}) is a closed 1-form on the configuration space of 8 points in ℂP^{3}. One can show (Proposition 5.3 in [G5]) that integrating this 1-form we get a single valued function on the configuration space, denoted L_{4}^{M} and called the motivic Grassmannian tetralogarithm. It would be very interesting to compute the difference L_{4}^{M} − L_{4}^{L}, similarly to the formula (40) in the case n = 3. We expect that is expressed as a sum of products of functions L_{3}, L_{2} and log | ∗ |.

Theorem 6.2 a) There exists a canonical map

K_{7}^{[3]}(F) ⊗ ℚ ⟶ Ker(L̃_{4}(F) ⟶^{δ} B_{3}(F) ⊗ F^{∗} ⊕ Λ^{2}B_{2}(F))_{ℚ}

b) In the case F = ℂ the composition K_{7}(ℂ) ⟶ L̃_{4}(ℂ) ⟶^{L_{4}^{M}} ℝ coincides with a nonzero rational multiple of the Borel regulator map.

The generalization of the above picture to the case n > 4 is unknown. It would be very interesting at least to define the motivic Grassmannian polylogarithms via the Grassmannian polylogarithms.

For n = 3 the motivic Grassmannian trilogarithm L_{3}^{M} is given by the first term of the formula (40). It is known ([GZ]) that L_{3}^{L} does not provide a homomorphism L_{3}(ℂ) ⟶ ℝ since the second term in (40) does not have this property. Indeed, the second term in (40) vanishes on the special configuration of 6 points in P^{2}, but does not vanish at the generic configuration. On the other hand the defining relations in L_{3}(F) allow to express any configuration as a linear combination of the special ones.

We expect the same situation in general: for n > 2 the Grassmannian n–logarithms should not satisfy all the functional equations for the motivic Grassmannian n–logarithms. It would be very interesting to find a conceptual explanation of this surprising phenomena.

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