Geometry of the trilogarithm

A.B. Goncharov

July 7, 1997

Contents

1 Introduction 1

2 The scissors congruence groups of pairs of simplices in P^{n}(F) 3

3 Main construction 7

4 Configurations of 2n points in ℙ^{n−1}, Grassmannian n-logarithms and motivic co-Lie algebra of a field 22

1 Introduction

The classical n-logarithm is defined by induction as an integral

Li_{n}(z) := ∫_{0}^{z} Li_{n−1}(t)d log t, Li_{1}(z) = − log(1 − z)

So it can be written as an n-dimensional integral

Li_{n}(z) = ∫_{0≤1−t_{1}≤...≤t_{n}} \frac{dt_{1}}{t_{1}} ∧ ... ∧ \frac{dt_{n}}{t_{n}}

Aomoto considered [A] more general integrals where the differential form \frac{dt_{1}}{t_{1}} ∧ ... ∧ \frac{dt_{n}}{t_{n}} is integrated over an arbitrary n-dimensional real simplex in ℂ^{n}. Let me recall this construction in a more formal setting.

Let F be a field. An n-simplex in P^{n}(F) is a collection of hyperplanes L = (L_{0}, ..., L_{n}). It is nondegenerate if the intersection of the hyperplanes L_{i} is empty. A face of a simplex is any nonempty intersection of hyperplanes from L. A pair of simplices is admissible if L and M have no common faces of the same dimension. Now let F = ℂ. Then there is a canonical n-form ω_{L} in ℂP^{n} with logarithmic poles on the hyperplanes L_{i}. If z_{i} = 0 are homogeneous equations of L_{i} then

ω_{L} = d log(z_{1}/z_{0}) ∧ ... ∧ d log(z_{n}/z_{0})

Let Δ_{M} be an n-cycle representing a generator of H_{n}(ℂP^{n}, M).

The Aomoto n-logarithm is a multivalued function on configurations of admissible pairs of simplices (L; M) in ℂP^{n} defined as follows:

Λ_{n}(L; M) := ∫_{Δ_{M}} ω_{L}

The classical n-logarithm corresponds to a very special pair of simplices in ℙ^{n}, see fig.1, so it is a very special case of the Aomoto n-logarithm.

In the present paper, which is a continuation of [BGSV1-2] and [G0-3], we express explicitly the Aomoto trilogarithm by classical trilogarithms and investigate the algebric-geometric structures staying behind: different realisations of the weight three motivic complexes. For the Aomoto dilogarithm a similar problem was solved in [BGSV1-2].

The function Λ_{3}(L; M) is defined on configurations, i.e. projective equivalence classes, of 4 + 4 points in ℙ^{3} (vertices of a pair of tetrahedra), while the classical trilogarithm lives on ℙ^{1}. To build a bridge between these functions we relate each of them with the Grassmannian trilogarithm P_{3}^{G} defined on configurations six points in ℙ^{2} (see section 4.1 for a definition of Grassmannian polylogarithms). Our main geometric construction, the map a_{3} defined in s. 3.3, (see its different versions on fig. 3, 8 and 9), permits to go from a configuration of 4 + 4 points in ℙ^{3} to configurations of 6 points in ℙ^{2}. Then we apply the generalized cross-ratio map r_{3} from [G0-3] to get to ℙ^{1}. The map a_{3} sheds a new light on the key ansatz from [G0-3] leading to the functional equation for the classical trilogarithm, see fig. 10 and the discussion there.

The Aomoto n-logarithm for n > 3 can not be expressed by the classical n-logarithm. However the explicit relation between the Aomoto n-logarithm, which is defined on configurations of 2 × (n + 1) points in ℙ^{n}, and the Grassmannian n-logarithm, which lives on configurations of 2n points in ℙ^{n−1}, should exist for all n. In section 4 we explain how such a relation would give an explicit construction of a certain graded co-Lie algebra G(F)_{•} over ℚ, which should be isomorphic to the Lie algebra of the Galois group of the category of mixed Tate motives over a field F. It would be very interesting to relate our approach with the work of M. Hanamura and R. MacPherson [HM]. The cohomology of this Lie algebra should give the appropriate pieces of Quillen’s K-theory of the field F modulo torsion.

There are several other candidates for the motivic co-Lie algebra, see [BK], [BMS-BGSV], [G5]. However all of them are constructed as Hopf algebras, so we get the co-Lie algebras as the quotient by the decomposable elements. Our approach should lead directly to a co-Lie algebra. The degree 2 and 3 parts of its standard cochain complex are precisely the Bloch-Suslin complex and the weight three motivic complex defined in [G0-3], so G(F)_{•} should be the smallest possible realization of the motivic Lie algebra.

Our results partially generalize the work [BGSV] from ℙ^{2} to ℙ^{3}. We discuss in s. 4 what remains to be done. An application to an explicit construction of the weight four motivic complexes will be discussed elsewhere.

I am extremely grateful to Herbert Gangl who wrote a proof of lemma (3.7), helped me to check coefficients in the formulas and and pointed out a lot of errors in the preliminary version of the paper.

The results of this paper were obtained in May 1992 during my stay in the Max-Planck-Institute (Bonn). The paper was written in MPI later on (when I learned how to draw pictures using computer). I am very grateful to the MPI for hospitality and support.

The work was partially supported by the NSF grant DMS-9500010.

2 The scissors congruence groups of pairs of simplices in P^{n}(F)

1.The scissors congruence groups A_{n}(F). Let me recall some definitions from [BMS], [BGSV]. It is handy to work with configurations of points than with hyperplanes. Let us apply the projective duality P^{n} ⟶ P̂^{n} which transforms the configuration of 2(n + 1) hyperplanes (L_{0}, ..., L_{n}; M_{0}, ..., M_{n}) to a configuration of points (l_{0}, ..., l_{n}; m_{0}, ..., m_{n}). Abusing notations we will denote it also (L; M), where now L = (l_{0}, ..., l_{n}) and M = (m_{0}, ..., m_{n}).

The group A_{n}(F) is generated by configurations of 2(n + 1) points (l_{0}, ..., l_{n}; m_{0}, ..., m_{n}) in P^{n}(F) which are vertices of admissible pairs of simplices subject to the following relations:

1)Nondegeneracy. (L; M) = 0 if (l_{0}, ..., l_{n}) or (m_{0}, ..., m_{n}) belong to a hyperplane.

2) Skew symmetry. (σL; M) = (L; σM) = (−1)^{|σ|}(L; M) for any permutation σ ∈ S_{n+1}.

3)Additivity. For any configuration (l_{0}, ..., l_{n+1})

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

if all the terms are admissible (additivity in L). A similar condition is imposed for (m_{0}, ..., m_{n+1}) (additivity in M).

ˆ3)Dual additivity. For any configuration (l_{0}, ..., l_{n+1})

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

if all the terms are admissible, and a similar condition is imposed for (m_{0}, ..., m_{n+1}). 4) Projective invariance. (gL; gM) = (L; M) for any g ∈ P GL_{n+1}(F).

These relations reflect properties of Aomoto polylogarithms.

The cross-ratio provides a canonical isomorphism

a_{1} : A_{1}(F) ⟶ F^{∗}, a_{1} : (l_{0}, l_{1}; m_{0}, m_{1}) ⟼ r(l_{0}, l_{1}, m_{0}, m_{1})

2. A coproduct on the generic part of A_{n}(F) ([BMS], [BGSV]). Set A_{0} = ℤ. Let A_{n}^{0}(F) ⊂ A_{n}(F) be the subgroup generated by pairs of simplices in generic position. Let us define a coproduct ν : A_{n}^{0} ⟶ ⊕_{k}A_{k}^{0} ⊗ A_{n−k}^{0}. Set for k, n − k > 0

ν = ⊕ν_{n−k,k}, ν_{n−k,k} : A_{n}^{0} :⟶ A_{n−k}^{0} ⊗ A_{k}^{0},
ν_{n−k,k} : (l_{0}, ..., l_{n}; m_{0}, ..., m_{n}) ⟼

∑_{I,J}(−1)^{σ(I,J)}(l_{i_{1}}...l_{i_{k}}|l_{0}, ..., l̂_{i_{1}}, ..., l̂_{i_{k}}, ..., l_{n}; m_{0}, m_{j_{1}}, ..., m_{j_{n−k}})

⊗(m_{j_{1}}...m_{j_{n−k}}|l_{0}, l_{i_{1}}, ..., l_{i_{k}}; m_{0}, ..., m̂_{j_{1}}, ..., m̂_{j_{n−k}}, ..., m_{n})

where I := {0 < i_{1} < ... < i_{k}}, J := {0 < j_{1} < ... < j_{n−k}}. Here σ(I, J) = sign(I, Ī)·sign(J, J̄), where sign(I, Ī) is the sign of the permutation (1, ..., n) → (I, Ī) ( similarly sign(J, J̄)). For example

ν_{2,1} : (l_{0}, ..., l_{3}; m_{0}, ..., m_{3}) ⟼
(l_{3}|l_{0}, l_{1}, l_{2}; m_{0}, m_{2}, m_{3}) ⊗ (m_{2}, m_{3}|l_{0}, l_{3}; m_{0}, m_{1}) + ...

Proposition 2.1 (A(F)_{•}, ν) is a graded coalgebra.

I will need this statement only in the case of degree 3. In this case it is not hard to deduce it from proposition (2.3) below.

Proposition 2.2 P : (L; M) ⟶ (M; L) defines an antiautomorphism of the graded coalgebra (A(F)_{•}, ν).

Proof follows from the definitions.

3. Another formula for ν_{1,n−1} and ν_{n−1,1}. Unfortunately in the definition of the coproduct ν we have to choose vertices l_{0} in L and m_{0} in M first, so the skew-symmetry is not obvious. In the next proposition we give another formula for ν_{1,n−1} and ν_{n−1,1} which is skew-symmetric from the beginning and much more convenient.

Proposition 2.3 (See figure 2)

ν_{1,n−1} : (l_{0}, ..., l_{n}; m_{0}, ..., m_{n}) ⟼

− ∑_{i,j=0}^{n}(−1)^{i+j}Δ(l_{i}, m_{0}, ..., m̂_{j}, ..., m_{n}) ⊗ (l_{i}|l_{0}, ..., l̂_{i}, ..., l_{n}; m_{0}, ..., m̂_{j}, ..., m_{n})

ν_{n−1,1} : (l_{0}, ..., l_{n}; m_{0}, ..., m_{n}) ⟼

− ∑_{i,j=0}^{n}(−1)^{i+j}(m_{j}|l_{0}, ..., l̂_{i}, ..., l_{n}; m_{0}, ..., m̂_{j}, ..., m_{n}) ⊗ Δ(m_{j}, l_{0}, ..., l̂_{i}, ..., l_{n})

Proof. Let us compute

ν_{n−1,1}(l_{0}, ..., l_{n}; m_{0}, ..., m_{n}) =  (1)

∑_{i,j=1}^{n}(−1)^{i+j}(l_{i}|l_{0}, ..., l̂_{i}, ..., l_{n}; m_{0}, ..., m̂_{j}, ..., m_{n})⊗Δ(m_{1}...m̂_{j}...m_{n}|l_{0}, l_{i}; m_{0}, m_{j})

using the formula

r(l_{1}, l_{2}, l_{3}, l_{4}) = \frac{Δ(l_{1}, l_{3})Δ(l_{2}, l_{4})}{Δ(l_{1}, l_{4})Δ(l_{2}, l_{3})}

for the cross-ratio. We will get

∑_{i,j=1}^{n}(−1)^{i+j}(l_{i}|l_{0}, ..., l̂_{i}, ..., l_{n}; m_{0}, ..., m̂_{j}, ..., m_{n}) ⊗ Δ(l_{0}, m_{0}, ..., m̂_{j}, ..., m_{n}) (2)

− ∑_{i,j=1}^{n}(−1)^{i+j}(l_{i}|l_{0}, ..., l̂_{i}, ..., l_{n}; m_{0}, ..., m̂_{j}, ..., m_{n}) ⊗ Δ(l_{i}, m_{0}, ..., m̂_{j}, ..., m_{n})

(3)

− ∑_{i,j=1}^{n}(−1)^{i+j}(l_{i}|l_{0}, ..., l̂_{i}, ..., l_{n}; m_{0}, ..., m̂_{j}, ..., m_{n}) ⊗ Δ(l_{0}, m_{1}, ..., m_{n}) (4)
+ ∑_{i,j=1}^{n}(−1)^{i+j}(l_{i}|l_{0}, ..., l̂_{i}, ..., l_{n}; m_{0}, ..., m̂_{j}, ..., m_{n}) ⊗ Δ(l_{i}, m_{1}, ..., m_{n}) (5)

Applying to (2) dual additivity in L we can rewrite it as

− ∑_{j=1}^{n}(−1)^{j}(l_{0}|l_{1}, ..., l_{n}; m_{0}, ..., m̂_{j}, ..., m_{n}) ⊗ Δ(l_{0}, m_{0}, ...m̂_{j}, ..., m_{n})

(3) is already in the desired shape. Applying to (4) first additivity in M and then dual additivity in L we get

−(l_{0}|l_{1}, ..., l_{n}; m_{1}, ..., m_{n}) ⊗ Δ(l_{0}, m_{1}, ..., m_{n})

Finally applying additivity in M to (5) we get

− ∑_{i=1}^{n}(−1)^{i}(l_{i}|l_{0}, ..., l̂_{i}, ..., l_{n}; m_{1}, ..., m_{n}) ⊗ Δ(l_{i}, m_{1}, ..., m_{n})

So we conclude that (1) is equal to

− ∑_{i,j=0}^{n}(−1)^{i+j}(l_{i}|l_{0}, ..., l̂_{i}, ..., l_{n}; m_{0}, ..., m̂_{j}, ..., m_{n}) ⊗ Δ(l_{i}, m_{0}, ..., m̂_{j}, ..., m_{n})

The considerations for ν_{1,n−1} are similar. The proposition is proved.

3 Main construction

1. The weight two case. Let B_{2}(F) be the quotient of the free abelian group ℤ[F^{∗}\1] generated by the symbols {x} where x ∈ F^{∗}\1 modulo the subgroup R_{2}(F) generated by the ”five term relations”, i.e. by the elements

∑_{i=1}^{5}(−1)^{i}{r(x_{1}, ..., x̂_{i}, ..., x_{5})}, x_{i} ∈ ℙ^{1}(F), x_{i} ≠ x_{j}

Denote by {x}_{2} the image of the generator {x} in B_{2}(F). One can prove (see for example [G1]) that there is a well defined homomorphism

δ_{2} : B_{2}(F) ⟶ Λ^{2}F^{∗}, {x}_{2} ⟼ (1 − x) ∧ x

The complex we get is called the Bloch complex.

In [BGSV1-2] there was defined a homomorphism of complexes

A_{2} ⟶^{ν} A_{1} ⊗ A_{1}
↓ a_{2}  ↓ a_{1} ∧ a_{1}
B_{2} ⟶^{δ_{2}} Λ^{2}F^{∗}

Namely

a_{2}(l_{0}, l_{1}, l_{2}; m_{0}, m_{1}, m_{2}) :=< l_{0}, l_{1}, l_{2}; m_{0}, m_{1}, m_{2} >_{2}:=  (6)

∑_{i,j=0}^{2}(−1)^{i+j}{r(l_{i}|l_{0}, ..., l̂_{i}, ..., l_{2}, m_{0}, ..., m̂_{j}, ..., m_{2})}_{2} ∈ B_{2}(F)

2. The weight three motivic complex related to the classical trilogarithm. Let V_{3} be a three dimensional vector space over a field F. Choose a volume form ω_{3} ∈ detV_{3}^{∗}. For any three vectors l_{1}, l_{2}, l_{3} in V_{3} set

Δ(l_{1}, l_{2}, l_{3}) :=< l_{1} ∧ l_{2} ∧ l_{3}, ω_{3} >

Let us define the generalized cross-ratio of six generic points x_{1}, ..., x_{6} on the plane ℙ^{2} by setting

r_{3}(x_{1}, ..., x_{6}) := \frac{1}{15}Alt_{6}\{\frac{Δ(x̃_{1}, x̃_{2}, x̃_{4})Δ(x̃_{2}, x̃_{3}, x̃_{5})Δ(x̃_{3}, x̃_{1}, x̃_{6})}{Δ(x̃_{1}, x̃_{2}, x̃_{5})Δ(x̃_{2}, x̃_{3}, x̃_{6})Δ(x̃_{3}, x̃_{1}, x̃_{4})}\} ∈ ℤ[F^{∗}\1]

(7)

Here x̃ is a vector projecting to the point x. The ratio does not depend on the choise of these vectors.

The group R_{3}(F) of functional equations for the trilogarithm. I will use the following definition, which is a bit ad hoc. R_{3}(F) is the group generated by {x}_{3} − {x^{−1}}_{3}, the ”seven term” relations (containing actually 7! terms)

∑_{i=1}^{7}(−1)^{i}r_{3}(x_{1}, ..., x̂_{i}, ..., x_{7}),  x_{i} ∈ ℙ^{2}(F)  (8)

where the points x_{1}, ..., x_{7} are in generic position in the plane, and Kummer’s functional equation for the trilogarithm:

K(x, y) := −\{\frac{x(1 − y)^{2}}{y(1 − x)^{2}} \} − \{xy\} − \{\frac{x}{y}\} − 2\{1\}+

2(\{\frac{−x(1 − y)}{(1 − x)}\} + \{\frac{x(1 − y)}{y(1 − x)}\} + \{\frac{−y(1 − x)}{1 − y}\} + \{\frac{1 − x}{1 − y}\} + \{y\} + \{x\})

It is may be true that Kummer’s relation follows from the generic seven term relations and the one {x}_{3} − {x^{−1}}_{3}. But I just add them to the list.

Remark. A more natural way to define the group R_{3}(F) is this. We extend the generalized cross-ratio to arbitrary configurations of 6 points on the plane. Then R_{3}(F) is given by the seven term relations for arbitrary configurations of seven points in the plane. Using the main results of [G1] one can show that both definitions lead to the same group. In particular we get Kummer’s relations for a certain degenerate configuration of seven points. To extend the definition of r_{3} we take the “limit value” of the definition (7) using {0}_{3} = {∞}_{3} = 0. If two of the points x_{i} coincide or four of them are on a line then r_{3}(x_{1}, ..., x_{6}) = 0. For the remaining cases see lemma 3.7 below. To make the exposition shorter I will not use this approach.

Set

B_{3}(F) := \frac{ℤ[F^{∗}\1]}{R_{3}(F)}

Denote by {x}_{3} the image in B_{3}(F) of the generator {x}. One can prove ([G1-3]) that there is a well defined homomorphism

δ_{3} : B_{3}(F) ⟶ B_{2}(F) ⊗ F^{∗}, {x}_{3} ⟼ {x}_{2} ⊗ x

We get the weight three motivic complex related to the classical trilogarithm

B_{3}(F) ⟶ B_{2}(F) ⊗ F^{∗} ⟶ Λ^{3}F^{∗}

where {x}_{2} ⊗ y ⟼ (1 − x) ∧ x ∧ y.

Lemma 3.1 Using the notations introduced in (6)

δ_{3} ∘ r_{3}(l_{1}, l_{2}, l_{3}, l_{4}, l_{5}, l_{6}) = −\frac{1}{18}Alt_{6}(< l_{1}, l_{2}, l_{3}; l_{4}, l_{5}, l_{6} >_{2} ⊗Δ(l_{1}, l_{2}, l_{3})) =

(9)

−2 < l_{1}, l_{2}, l_{3}; l_{4}, l_{5}, l_{6} >_{2} ⊗Δ(l_{1}, l_{2}, l_{3}) + 19 other terms

Proof. Let C_{n}(V_{3}) be the free abelian group generated by the configurations of n generic vectors (i.e. n-tuples of vectors in generic position modulo the action of GL(V_{3})) in V_{3}. Let

d : C_{n}(V_{3}) → C_{n−1}(V_{3}), d(l_{1}, ..., l_{n}) := ∑_{i=1}^{n}(−1)^{i−1}(l_{1}, ..., l̂_{i}, ..., l_{n})

Define a homomorphism C_{5}(V_{3}) → B_{2}(F) ⊗ F^{∗} by setting

f_{5}(3)(l_{1}, ..., l_{5}) := \frac{1}{2}Alt_{5}(\{r(l_{1}|l_{2}, ..., l_{5})\}_{2} ⊗ Δ(l_{1}, l_{2}, l_{3}))

According to theorem 2.3 in [G3] the following diagram is commutative:

C_{6}(V_{3}) ⟶^{d}  C_{5}(V_{3})
r_{3} ↓  ↓ f_{5}(3)
B_{3}(F) ⟶^{δ_{3}} B_{2}(F) ⊗ F^{∗}

Therefore

δ_{3} ∘ r_{3}(l_{1}, l_{2}, l_{3}, l_{4}, l_{5}, l_{6}) = −\frac{1}{2}Alt_{6}((l_{1}|l_{2}, l_{3}, l_{4}, l_{5})_{2} ⊗ Δ(l_{1}, l_{2}, l_{3}))

which equals to (9).

3. A homomorphism between the weight three motivic complexes. We have constructed in chapter 2 a complex

A_{3}^{0} ⟶^{ν_{2,1}⊕ν_{1,2}} A_{2} ⊗ A_{1} ⊕ A_{1} ⊗ A_{2} ⟶^{ν_{1,1}⊗Id−Id⊗ν_{1,1}} A_{1} ⊗ A_{1} ⊗ A_{1}

Let us define a homomorphism of complexes

A_{3}^{0} ⟶^{ν} A_{2} ⊗ A_{1} ⊕ A_{1} ⊗ A_{2} ⟶ A_{1} ⊗ A_{1} ⊗ A_{1}
↓ a_{3}  ↓ a_{2} ∧ a_{1}  ↓ ∧^{3}a_{1}
B_{3} ⟶^{δ_{3}}  B_{2}(F) ⊗ F^{∗}  ⟶  Λ^{3}F^{∗}

where a_{2} ∧ a_{1}(x_{2} ⊗ x_{1} + y_{1} ⊗ y_{2}) := x_{2} ⊗ x_{1} − y_{2} ⊗ y_{1} and ∧^{3}a_{1}(x_{1} ⊗ x_{2} ⊗ x_{3}) = x_{1} ∧ x_{2} ∧ x_{3}.

Let x_{1}, x_{2}, x_{3}; y_{1}, y_{2}, y_{3} be six different points on a line. Set

μ_{3}(x_{1}, x_{2}, x_{3}; y_{1}, y_{2}, y_{3}) =

\frac{1}{4} · Alt_{(x_{1},x_{2},x_{3})(y_{1},y_{2},y_{3})}(\{r(x_{1}, y_{2}, x_{2}, y_{1})\}_{3} − \{r(x_{1}, y_{1}, x_{2}, y_{2})\}_{3})

This formula, which contains 18 terms, means simply that m_{3}(x_{1}, x_{2}, x_{3}; y_{1}, y_{2}, y_{3}) is skewsymmetric with respect to x_{1}, x_{2}, x_{3} and y_{1}, y_{2}, y_{3} and a typical term is {r(x_{1}, y_{2}, x_{2}, y_{1})}_{3}.

We set

a_{3} := \frac{1}{6}a′_{3} − \frac{1}{3}a′′_{3}

where a′_{3} and a′′_{3} are defined on generators by the following formula (see fig. 2):

a′_{3}(l_{0}, l_{1}, l_{2}, l_{3}; m_{0}, m_{1}, m_{2}, m_{3}) := ∑_{i,j=0}^{3}(−1)^{i+j}r_{3}(l_{i}|l_{0}, ..., l̂_{i}, ..., l_{3}, m_{0}, ..., m̂_{j}, ..., m_{3})

a′′_{3}(l_{0}, ..., l_{3}; m_{0}, ..., m_{3}) := ∑_{i,j=0}^{3}(−1)^{i+j}·μ_{3}(l_{i}, m_{j}|l_{0}, ..., l̂_{i}, ..., l_{3}; m_{0}, ..., m̂_{j}, ..., m_{3})

Denote by Ã_{3}^{0} the free abelian group generated by the generators of the group A_{3}^{0}, i.e. by pairs of tetrahedra in generic position.

Theorem 3.2 The diagram

Ã_{3}^{0} ⟶^{ν} A_{2} ⊗ A_{1} ⊕ A_{1} ⊗ A_{2}
↓ a_{3}  ↓ a_{2} ∧ a_{1}
B_{3} ⟶^{δ_{3}}  B_{2}(F) ⊗ F^{∗}

is commutative.

Remark. The proof of this theorem is the crucial point of the paper. The commutativity of the diagram is what we really wanted for the homomorphism a_{3}. Surprisingly neither homomorphism a′_{3} nor a′′_{3} make the diagram commutative, even up to a scalar. Only their sum does the job. Moreover, theorem (3.2) ”morally” implies that a_{3} should be a homomorphism of groups. We will prove this later on.

Proof. Because of the skew-symmetry of the formulas for ν (see proposition 2.3 ) it is sufficient to compute for

(a_{2} ∧ a_{1}) ∘ ν(l_{0}, ..., l_{3}; m_{0}, ..., m_{3}) and δ_{3} ∘ a_{3}(l_{0}, ..., l_{3}; m_{0}, ..., m_{3})

the B_{2}(F)-factors of the following elements of F^{∗}:

Δ(m_{0}, m_{1}, m_{2}, m_{3}), Δ(l_{0}, m_{0}, m_{1}, m_{2}), Δ(l_{0}, l_{1}, m_{0}, m_{1}),
Δ(l_{0}, l_{1}, l_{2}, m_{0}), Δ(l_{0}, l_{1}, l_{2}, l_{3})

Since ν(L; M) = ν(M, L) by proposition (2.2), and a_{3}(L; M) = a_{3}(M; L) by lemma (3.5), we see that it is sufficient to consider only the first three of them.

It follows from proposition (2.3) that in (a_{2} ∧ a_{1}) ∘ ν(l_{0}, ..., l_{3}; m_{0}, ..., m_{3}) appears only

< l_{0}|l_{1}, l_{2}, l_{3}; m_{1}, m_{2}, m_{3} >_{2} ⊗Δ(l_{0}, m_{1}, m_{2}, m_{3})  (10)

Step 1. Let us do the computations for δ_{3} ∘ a_{3}(l_{0}, ..., l_{3}; m_{0}, ..., m_{3}). The crucial and most nontrivial case is the term with Δ(l_{0}, l_{1}, m_{0}, m_{1}). If the diagram is commutative it must be zero because of the observation we just made. The only summands in a′_{3}(l_{0}, ..., l_{3}; m_{0}, ..., m_{3}) that give a contribution to this term are:

−r_{3}(l_{0}|l_{1}, l_{2}, l_{3}, m_{0}, m_{1}, m_{2}) + r_{3}(l_{0}|l_{1}, l_{2}, l_{3}, m_{0}, m_{1}, m_{3})+
r_{3}(l_{1}|l_{0}, l_{2}, l_{3}, m_{0}, m_{1}, m_{2}) − r_{3}(l_{1}|l_{0}, l_{2}, l_{3}, m_{0}, m_{1}, m_{3})

Lemma (3.1) shows using the symmetry 1 < − > 4, 2 < − > 5 that the term with Δ(l_{0}, l_{1}, m_{0}, m_{1}) in δ_{3} ∘ a_{3}(l_{0}, ..., l_{3}; m_{0}, ..., m_{3}) is

−2(− < l_{0}|l_{1}, m_{0}, m_{1}; l_{2}, l_{3}, m_{2} >_{2} + < l_{0}|l_{1}, m_{0}, m_{1}; l_{2}, l_{3}, m_{3} >_{2} +

< l_{1}|l_{0}, m_{0}, m_{1}; l_{2}, l_{3}, m_{2} >_{2} − < l_{1}|l_{0}, m_{0}, m_{1}; l_{2}, l_{3}, m_{3} >_{2})⊗Δ(l_{0}, l_{1}, m_{0}, m_{1}) =

2 · Alt_{l_{0}l_{1}}Alt_{m_{2}m_{3}}(< l_{0}|l_{1}, m_{0}, m_{1}; l_{2}, l_{3}, m_{2} >_{2}) ⊗ Δ(l_{0}, l_{1}, m_{0}, m_{1}) (11)

Step 2. The contribution for Δ(l_{0}, l_{1}, m_{0}, m_{1}) in δ_{3} ∘ a′′_{3}(l_{0}, ..., l_{3}; m_{0}, ..., m_{3}) is coming from

δ_{3} ∘ μ_{3}((l_{0}m_{0}|l_{1}, l_{2}, l_{3}; m_{1}, m_{2}, m_{3}) − (l_{0}m_{1}|l_{1}, l_{2}, l_{3}; m_{0}, m_{2}, m_{3})
−(l_{1}m_{0}|l_{0}, l_{2}, l_{3}; m_{1}, m_{2}, m_{3}) + (l_{1}m_{1}|l_{0}, l_{2}, l_{3}; m_{0}, m_{2}, m_{3}))

And the contribution is equal to

Alt_{(l_{0},l_{1});(m_{2},m_{3});(m_{0},m_{1});(l_{2},l_{3})}((l_{0}m_{0}|l_{1}, m_{1}, l_{2}, m_{2})_{2}) ⊗ Δ(l_{0}, l_{1}, m_{0}, m_{1}) =

Alt_{(l_{0},l_{1});(m_{2},m_{3})}(< l_{0}|m_{0}, l_{1}, m_{1}; l_{2}, m_{2}, l_{3} >_{2}) ⊗ Δ(l_{0}, l_{1}, m_{0}, m_{1}) =

Alt_{(l_{0},l_{1});(m_{2},m_{3})}(< l_{0}|l_{1}, m_{0}, m_{1}; l_{2}, l_{3}, m_{2} >_{2}) ⊗ Δ(l_{0}, l_{1}, m_{0}, m_{1})

Comparing the last formula with (11) we see that

\frac{1}{2}a′_{3} − a′′_{3} has contribution 0 ⊗ Δ(l_{0}, l_{1}, m_{0}, m_{1})

Step 3. Now look at the terms of δ_{3}∘a_{3}(l_{0}, ..., l_{3}; m_{0}, ..., m_{3}) with Δ(m_{0}, m_{1}, m_{2}, m_{3}) and Δ(l_{0}, m_{1}, m_{2}, m_{3}).

1) Δ(m_{0}, m_{1}, m_{2}, m_{3}) does not appear in a_{3}(l_{0}, ..., l_{3}; m_{0}, ..., m_{3}).

2i)Using proposition (2.3) we get that Δ(l_{0}, m_{1}, m_{2}, m_{3}) appears in δ_{3} ∘ a′_{3}(l_{0}, ..., l_{3}; m_{0}, ..., m_{3}) as

2 < l_{0}|l_{1}, l_{2}, l_{3}; m_{1}, m_{2}, m_{3} >_{2} ⊗Δ(l_{0}, m_{1}, m_{2}, m_{3})

2ii) Let us compute δ_{3} ∘ a′′_{3}(l_{0}, ..., l_{3}; m_{0}, ..., m_{3}). The nontrivial nontrivial contribution of Δ(l_{0}, m_{1}, m_{2}, m_{3}) is given by

−1/2Alt_{(m_{1},m_{2},m_{3})}μ_{3}(l_{0}m_{1}|l_{1}, l_{2}, l_{3}; m_{0}, m_{2}, m_{3})

and it is equal to

1/2Alt_{(l_{1},l_{2},l_{3})(m_{1},m_{2},m_{3})}(l_{0}m_{1}|l_{1}, m_{2}, l_{2}, m_{3})_{2} ⊗ Δ(l_{0}, m_{1}, m_{2}, m_{3}) =
−2 < (l_{0}|m_{1}, m_{2}, m_{3}; l_{1}, l_{2}, l_{3}) > ⊗Δ(l_{0}, m_{1}, m_{2}, m_{3})

For the last step use (l_{0}m_{1}|l_{1}, m_{2}, l_{2}, m_{3})_{2} = −(l_{0}m_{1}|m_{2}, m_{3}, l_{1}, l_{2})_{2}. So the contribution of \frac{1}{2}a′_{3} − a′′_{3} is

3· < l_{0}|l_{1}, l_{2}, l_{3}; m_{1}, m_{2}, m_{3} >_{2} ⊗Δ(l_{0}, m_{1}, m_{2}, m_{3})

It remains to compare this answer with (10). Theorem (3.2) is proved.

Theorem 3.3 a_{3} is a homomorphism of groups.

We start the proof with

Proposition 3.4 Both a′_{3} and a′′_{3} send relations 2), 3), 4) to zero.

Proof. This is clear for relations 2) and 4). It is also clear that a′′_{3} is additive in L and in M. To check that a′′_{3} sends the dual additivity in L to zero notice that the typical term for a′′_{3}(∑_{i=0}^{3}(−1)^{i} · (l_{i}|l_{0}, ..., l̂_{i}, ..., l_{4}; m_{0}, ..., m_{3})) is

±m̂_{3}(l_{i}l_{j}m_{k}|l_{0}, ..., l̂_{i}, ..., l̂_{j}, ..., l_{4}; m_{0}, ..., m̂_{k}, ..., m_{3})

It is symmetric in l_{i}, l_{j}. So we get 0 after alternation in (l_{0}, ..., l_{4}).

Similarly using the skewsymmetry of r_{3} we immediately see that the map a′_{3} sends dual additivity in L to zero. The additivity in M of a′_{3} is obvious. Therefore the proposition follows immediately from the following lemma.

Lemma 3.5 a′_{3}(L; M) = a′_{3}(M; L); a′′_{3}(L; M) = a′′_{3}(M; L)

Proof. The statement about a′′_{3} is clear from the definition. Let us prove it for a′_{3}. Applying the 7-term relation for the configuration of 7 points in ℙ^{2} (l_{i}|l_{0}, ..., l̂_{i}, ..., l_{3}, m_{0}, ..., m_{3}) we get

a′_{3}(l_{0}, , ..., l_{3}; m_{0}, ..., m_{3}) = − ∑_{0≤i≠j≤3} γ(i, j)r_{3}(l_{i}|l_{0}, ..., l̂_{i}, ..., l̂_{j}, ..., l_{3}, m_{0}, ..., m_{3})

(12)

where γ(i, j) = (−1)^{i+j} if i < j and −(−1)^{i+j} otherwise. Applying the dual 7-term relation to the configuration (l_{0}, ..., l̂_{i}, ..., l_{3}, m_{0}, ..., m_{3}) of 7 points in ℙ^{3} we can rewrite (12) as

− ∑_{i,j=0}^{3}(−1)^{i+j}r_{3}(m_{i}|l_{0}, ..., l̂_{j}, ..., l_{3}, m_{0}, ..., m̂_{i}, ..., m_{3})

It remains to interchange the l- and m- tetrahedra using the skewsymmetry. The lemma is proved.

Neither a′_{3} nor a′′_{3} send the relation 1) to zero. Only their weighted sum a_{3} does the job. To show this we have to prove the proposition 3.6 below.

Let L_{0}, ..., L_{3} be 4 lines and m_{0}, ..., m_{3} 4 points in P^{2}. Let

(L_{0}, ..., L̂_{i}, ..., L_{3}; m_{0}, ..., m̂_{j}, ..., m_{3})

be a pair of triangles where the first triangle is given by its sides and the second one by its vertices (see fig. 4). For example

(L_{1}, L_{2}, L_{3}; m_{1}, m_{2}, m_{3}) = (l_{23}, l_{13}, l_{12}, m_{1}, m_{2}, m_{3})

where l_{ij} := L_{i} ∩ L_{j}, so the first three points are the vertices of the triangle (L_{1}, L_{2}, L_{3}).

Proposition 3.6 Let L_{0}, ..., L_{3} be 4 lines and m_{0}, ..., m_{3} 4 points in P^{2}. Then

∑_{i,j=0}^{3}(−1)^{i+j}r_{3}(L_{0}, ..., L̂_{i}, ..., L_{3}; m_{0}, ..., m̂_{j}, ..., m_{3}) =  (13)

2 ∑_{i,j=0}^{3}(−1)^{i+j}r_{3} · μ_{3}(m_{i}|l_{j0}, ..., l̂_{jj}, ..., l_{j3}; m_{0}, ..., m̂_{i}, ..., m_{3})  (14)

Proof. Applying the 7-term relation to the configuration of 7 points formed by the vertices of the triangle L_{0}, ..., L̂_{j}, ..., L_{3} and m_{0}, ..., m_{3} we rewrite (13) as

−\frac{1}{2}Alt_{(L_{0},...,L_{3})}r_{3}(l_{13}, l_{23}, m_{0}, m_{1}, m_{2}, m_{3})

(We alternate the indices 0, 1, 2, 3 in the l-variables only). It is equal to

−r_{3}(l_{13}, l_{23}, m_{0}, m_{1}, m_{2}, m_{3})+r_{3}(l_{03}, l_{23}, m_{0}, m_{1}, m_{2}, m_{3})−r_{3}(l_{03}, l_{13}, m_{0}, m_{1}, m_{2}, m_{3})

+ 9 other terms

Consider the three intersection points of the line L_{i} with the other lines and add to them the points m_{0}, m_{1}, m_{2}, m_{3}. For instance (l_{03}, l_{13}, l_{23}, m_{0}, m_{1}, m_{2}, m_{3}) is the configuration related to the line L_{3}. Applying the 7-term relation to these configurations we rewrite the previous formula as

−r_{3}(∑_{i=0}^{3}(−1)^{i}(l_{03}, l_{13}, l_{23}, m_{0}, ..., m̂_{i}, ..., m_{3})−∑_{i=0}^{3}(−1)^{i}(l_{10}, l_{20}, l_{30}, m_{0}, ..., m̂_{i}, ..., m_{3})

(15)

+ ∑_{i=0}^{3}(−1)^{i}(l_{21}, l_{31}, l_{01}, m_{0}, ..., m̂_{i}, ..., m_{3})−∑_{i=0}^{3}(−1)^{i}(l_{32}, l_{02}, l_{12}, m_{0}, ..., m̂_{i}, ..., m_{3}))

(16)

Now comes a little trick: we will use the fact that one can extend the generalized cross ratio to certain degenerate configurations of six points such that the seven term relation holds.

Lemma 3.7 Suppose the points l_{1}, l_{2}, l_{3} are on the same line. Then

r_{3}(l_{1}, l_{2}, l_{3}, m_{1}, m_{2}, m_{3}) = −2μ_{3}(l_{1}, l_{2}, l_{3}; m_{1}, m_{2}, m_{3})

Remark. It is easy to check that

δ_{3} ∘ (r_{3} + 2μ_{3})(l_{1}, l_{2}, l_{3}, m_{1}, m_{2}, m_{3}) = 0

This implies that (r_{3} + 2μ_{3})(l_{1}, l_{2}, l_{3}, m_{1}, m_{2}, m_{3}) is a functional equation for the trilogarithm. The following proof verifies that it is already in the group R_{3} defined above (so we do not have to enlarge this group).

Proof. The following proof based on direct computation using {0}_{3} = {∞}_{3} = 0 was provided to me by Herbert Gangl. Consider the following special configuration of 6 points on the plane (fig. 5)

given by the columns of the matrix

C(b, c) := ( \begin{matrix}0 & 0 & 0 & 1 & 1 & 1 \\ 1 & 1 & 0 & 1 & 1 & 0 \\ 0 & c & 1 & b & 1 & 0 \end{matrix} )

It is sufficient to prove the lemma for this configuration, since the general case will follow by the seven term relation.

Set

l̃_{3}{x} := {1 − x^{−1}}_{3} − {1 − x}_{3}

M_{3}(C(b, c)) := l̃_{3}(\{\frac{b − c}{1 − c}\}_{3} − \{b\}_{3} + \{\frac{b(1 − c)}{b − c}\}_{3} − \{1 − c\}_{3} + \{\frac{b − c}{b}\}_{3})

A configuration of six points x_{1}, x_{2}, x_{3}, l_{1}, l_{2}, l_{3} on the plane, three of which, x_{1}, x_{2}, x_{3}, are on a line L, is determined completely by the configuration (x_{1}, x_{2}, x_{3}, n_{1}, n_{2}, n_{3}) of 3+3 points on the line L, where the point n_{1} := l_{2}l_{3} ∩L and so on, see fig. 6.

(Recall that by configuration we always mean its projective equivalence class.)

Let C̃(b, c) be the configuration of six points on the line corresponding to the configuration C(b, c) by this rule. Then

M_{3}(C(b, c)) = −μ_{3}(C̃(b, c))

One has

15 · r_{3}(C(b, c)) = −18\{1 − b\}_{3} − 12\{\frac{b}{1 − c}\}_{3} − 18\{\frac{b − c}{b − 1}\}_{3} − 18\{c\}_{3}

−12\{\frac{1 − b}{c}\}_{3} − 24\{\frac{b(c − 1)}{c(b − 1)}\}_{3} − 6\{\frac{c}{(c − b)(1 − b)}\}_{3} − 6\{\frac{c(b − c)}{b − 1}\}_{3} − 18\{\frac{c − b}{c}\}_{3}

+18\{1 − b^{−1}\}_{3} + 12\{b − c\}_{3} + 18\{\frac{b − 1}{c − 1}\}_{3} + 12\{\frac{b − 1}{c}\}_{3} + 18\{\frac{b}{c}\}_{3}

+24\{\frac{b − c}{c(b − 1)}\}_{3} + 6\{\frac{b(b − 1)}{c(c − 1)}\}_{3} + 18\{1 − c^{−1}\}_{3} + 6\{\frac{(1 − b)(1 − c)}{b · c}\}_{3}

Recall two Kummer relations:

6K(\frac{b}{1 − c}, \frac{c}{1 − b}) = 12\{\frac{b}{1 − c}\}_{3}+12\{\frac{c}{1 − b}\}_{3}+12\{\frac{1 − b}{1 − c}\}_{3}+12\{\frac{c}{c − 1}\}_{3}+12\{\frac{b}{b − 1}\}_{3}

+12\{\frac{c}{b}\}_{3} − 6\{\frac{bc}{(1 − b)(1 − c)}\}_{3} − 6\{\frac{b(b − 1)}{c(c − 1)}\}_{3} − 6\{\frac{c(1 − b)}{b(1 − c)}\}_{3} − 12\{1\}_{3}

and

−6K(b−c, \frac{b − 1}{c}) = −12\{b−c\}_{3}−12\{\frac{b − 1}{c}\}_{3}−12\{c\}_{3}−12\{1−b\}_{3}−12\{\frac{c}{c − b}\}_{3}

−12\{\frac{b − 1}{b − c}\}_{3} + 6\{\frac{(c − b)(1 − b)}{c}\}_{3} + 6\{\frac{c(b − c)}{b − 1}\}_{3} + 6\{\frac{c(b − 1)}{b − c}\}_{3} + 12\{1\}_{3}

Then one computes, adding the three expressions above,

15r_{3}(C(b, c)) + 6K(\frac{b}{1 − c}, \frac{c}{1 − b}) − 6K(b − c, \frac{b − 1}{c}) =

30(−\{1 − b\}_{3} + \{\frac{b}{b − 1}\}_{3} − \{\frac{b − c}{b − 1}\}_{3} + \{\frac{1 − b}{1 − c}\}_{3} − \{\frac{c(1 − b)}{b(1 − c)}\}_{3} + \{\frac{c − b}{c(1 − b)}\}_{3}

−\{c\}_{3} + \{1 − c^{−1}\}_{3} − \{\frac{c − b}{c}\}_{3} + \{\frac{c}{b}\}_{3}) = −30μ_{3}(C(b, c))

which is exactly what we wanted. The lemma, and hence theorem (3.3) is proved.

Comparing this lemma with (15), (16) we get the proposition.

Another way to proceed. The right hand side of (13) considered as a function of 4 lines and 4 points on the plane satisfies the 5 term relation with respect to lines as well as points. Using this observation it is sufficient to check the proposition for the degenerate configuration of lines and points when 3 of the lines L_{i} pass through a point and 3 points among the m_{i} are on a line (see fig. 7).

4. Another way to define the homomorphism a_{3}. Let us associate to 8 points (l_{0}, ..., l_{3}; m_{0}, ..., m_{3}) in ℙ^{3} a degenerate configuration of 6 points on the plane, denoted

d(l_{i}, m_{j}||l_{0}, ..., l̂_{i}, ..., l_{3}; m_{0}, ..., m̂_{j}, ..., m_{3})

as follows. Consider the configuration

(l_{i}|l_{0}, ..., l̂_{i}, ..., l_{3}; m_{0}, ..., m_{3})

of 7 points in ℙ^{2}. We construct out of them a degenerate configuration of 6 lines in the plane. Let L_{ij} (resp (M_{ij}) be the line through the points l_{i} and l_{j} (resp. m_{i} and m_{j}). Take the three lines formed by the sides of the triangle (l_{i}|l_{0}, ..., l̂_{i}, ..., l_{3}) and add to them the three lines M_{j0}, ...M̂_{jj}, ..., M_{j3} connecting m_{j} with the other three m-points. For example

d(l_{0}, m_{0}||l_{1}, l_{2}, l_{3}; m_{1}, m_{2}, m_{3}) = (L_{23}, L_{13}, L_{12}, M_{01}, M_{02}, M_{03})

Let C̄_{2n}(F) be the free abelian group generated by the configurations of arbitrary 2n F -points in ℙ^{n−1}. One can define a_{3} as a composition

A_{3}^{0}(F) ⟶^{p_{3}} C̄_{6}(P^{2})/{7 term relations} ⟶^{r_{3}} B_{3}(F)

where (see fig. 8)

p_{3}(l_{0}, l_{1}, l_{2}, l_{3}; m_{0}, m_{1}, m_{2}, m_{3}) := ∑_{i,j=0}^{3}((−1)^{i+j}(l_{i}|l_{0}, ..., l̂_{i}, ..., l_{3}, m_{0}, ..., m̂_{j}, ..., m_{3})

+2 · d(l_{i}, m_{j}||l_{0}, ..., l̂_{i}, ..., l_{3}; m_{0}, ..., m̂_{j}, ..., m_{3}))  (17)

There is a different candidate for the term (17). We define a configuration

d̂(l_{i}, m_{j}||l_{0}, ..., l̂_{i}, ..., l_{3}; m_{0}, ..., m̂_{j}, ..., m_{3})  (18)

of 3+3 points on a line as follows. Consider the planes L_{i} and M_{j} in ℙ^{3} (see fig. 9 for i = 0, j = 0.) Their intersection is a line L_{i} ∩M_{j}. The sides of the triangles (l_{0}, ..., l̂_{i}, ..., l_{3}) and (m_{0}, ..., m̂_{j}, ..., m_{3}) cut this line in 3 + 3 points. This is the configuration (18) we promised to define.

Recall that we can think of a configuration of 3 + 3 points on a line as of a configuration of six points on a plane (see fig. 6). So we can describe configuration (18) by the configuration of points (n_{0}, ..., n̂_{i}, ..., n_{3}, m_{0}, ..., m̂_{j}, ..., m_{3}) on the plane M_{j}, or by a similar configuration on the plane L_{i}.

The definition of (18) given on fig. 9 is projectively dual to the one on fig.

8. This means that if we consider the points l_{i}, m_{j} as planes in the dual space ℙ̂^{3} then (17) corresponds to (18).

Remark. One has (mainly thanks to lemma (3.7))

∑_{i,j=0}^{3}(−1)^{i+j}(d(l_{i}, m_{j}||l_{0}, ..., l̂_{i}, ..., l_{3}; m_{0}, ..., m̂_{j}, ..., m_{3})−  (19)

d̂(l_{i}, m_{j}||l_{0}, ..., l̂_{i}, ..., l_{3}; m_{0}, ..., m̂_{j}, ..., m_{3})) = 0

Let us imagine that we would be able to prove this using only the (possibly degenerate) seven term relations but not lemma (3.7). Then it is straightforward to show that p_{3} sends to zero all the defining relations for the group A_{3} except the degeneracy relation. So we can define the group R_{3} by adding to the (possibly degenerate) seven term relations the image of the degeneracy relations, i.e. the relations from proposition (3.6). Moreover we get a nice free gift: now we can skip lemma (3.7) together with its computational proof. So instead of the mysterious relation (8) for the classical trilogarithm which we think of as a relation

generic configuration of 6 points on ℙ^{2} = ∑ of degenerate configurations

(and which does not follow from the seven term relations between the configurations of six points on the plane), we would have the geometrically natural relations from proposition (3.6). However I was not able to prove (19) without using lemma (3.7). I wish somebody will try.

5. The key ansatz from [G0-3]. Consider the following admissible pair of tetrahedra which represents 0 on A_{3} (see fig 10).

Here l_{0}, m_{0}, m_{1}, m_{2}, m_{3} are on the same plane. The right hand side of the picture illustrates the computation of the homomorphism a_{3}.

Then

a′_{3}(L, M) = (l_{12}, l_{13}, l_{23}, m_{1}, m_{2}, m_{3})

Indeed, all the other terms in (15)-(16) vanish since by definition a degenerate configuration of six points on the plane represents zero in B_{3} if two of them coincide or four are on the same line. The condition that for such (L, M) one has a_{3}(L, M) = (a′_{3} + 2a′′_{2})(L, M) = 0 just means that we get a formula expressing the configuration l_{12}, l_{13}, l_{23}, m_{1}, m_{2}, m_{3} as a sum of configurations like on fig. 11 (each of them corresponds to a generator {x}_{3} of the group B_{3}).

6. A formula for Λ_{3}(L, M). One can define (see [BGSV]) a commutative product map

μ : A_{k} ⊗ A_{l} → A_{k+l}

Then one should have a structure of the Hopf algebra on A_{•} given by the coproduct ν and the product μ (at the present moment there are still some problems with proving that ν ∘ μ = μ ⊗ μ(ν)). Let

P_{n} := ⊕_{k+l=n}μ(A_{k} ⊗ A_{l}) (k > 0, l > 0)

Consider the homomorphism π_{3} : A_{3} → A_{3} given by

π_{3}(x) := x − \frac{1}{2}μ ∘ ν(x) + \frac{1}{3}μ ∘ ν_{1,1,1}(x)

where ν_{1,1,1} : A_{3} → A_{1} ⊗ A_{1} ⊗ A_{1} is the composition

A_{3} ⟶^{ν_{2,1}} A_{1} ⊗ A_{1} ⟶^{ν_{1,1}⊗id} A_{1} ⊗ A_{1} ⊗ A_{1}

Lemma 3.8 π_{3} is a projector and Kerπ_{3} = P_{3}.

See also section 4.3 for a generalization.

Proof. Let us check that P_{3} ⊂ Kerπ_{3}. This implies π_{3}^{2} = π_{3}. Since π_{3} = id on A_{3}/P_{3} we have Kerπ_{3} = P_{3}.

Suppose ν_{1,1}(y_{2}) = ∑_{i} y′_{i} ⊗ y′′_{i} ⊂ A_{1} ⊗ A_{1}. Let x_{1} ∈ A_{1}. Denote by · the product in A_{•}. Then

ν ∘ μ(y_{2} · x_{1}) = ∑_{i}(x_{1} · y′_{i} ⊗ y′′_{i} + y′_{i} ⊗ y′′_{i} · x_{1}) + y_{2} ⊗ x_{1} + x_{1} ⊗ y_{2})

So a simple calculation proves the assertion.

One has

π_{3}(l_{3}(x)) = l_{3}(x) − \frac{1}{2}μ(l_{2}(x) ⊗ x) + \frac{1}{12}μ(l_{1}(x) ⊗ x ⊗ x)  (20)

where l_{i}(x) is the generator of A_{n} corresponding to the classical n-logarithm. For n = 2 and n = 3 it is given by the picture on fig.1. Set

L_{3}(x) := Li_{3}(x) − \frac{1}{2}Li_{2}(x) log x + \frac{1}{12}Li_{1}(x)(log x)^{2}

Theorem 3.9

Λ_{3}(π_{3}(L, M)) = L_{3}(a_{3}(L, M))  (21)

Remark. To get a local coincidence of two multivalued analytic functions in this formula we should choose appropriate cycles of integration. The theorem claims in particular that it is possible to do this.

Proof. The main result of this paper implies that

ν(π_{3}(L, M)) = ν(l_{3} ∘ a_{3}(L, M))

Since the differential of Λ_{3}(L, M) is determined by the coproduct of (L, M) (see the lemma below) this implies that the differentials of both sides of (21) coincide. So the difference between the left and right hand sides is a constant. Considering the additivity relation in L (which has odd number of terms) we deduce that this constant is zero.

Lemma 3.10 Let (L, M) ∈ A_{n}(ℂ) and the A_{n−1} ⊗ A_{1} component of ν(L, M) is ∑ x_{i} ⊗ y_{i}. Then

dΛ_{3}(L, M) = ∑ dΛ_{3}(x)d log y_{i}

This is a particular case of a general fact about the differential of the period of n-framed mixed Tate motives. A direct proof can be given by an explicit calculation using proposition 2.3.

4 Configurations of 2n points in ℙ^{n−1}, Grassmannian n-logarithms and motivic co-Lie algebra of a field

1. Grassmannian polylogarithms. Let me recall the construction of the Grassmannian polylogarithm function P_{n}^{G}(h_{1}, ..., h_{2n}) given in [G]. It is a function on configurations of arbitrary 2n hyperplanes in ℂℙ^{n−1}.

Let f_{1}, ..., f_{m} be m complex-valued functions on a manifold X. We attach to them the following (m − 1)-form. Let c_{j,m} := \frac{1}{(2j+1)!(n−2j−1)!}. Set

r_{m−1}(f_{1}, ..., f_{m}) :=  (22)

\frac{1}{(2πi)^{m}} 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}

Now let f_{i} be a rational function on ℂℙ^{n−1} with the divisor (f_{i}) = h_{i} − h_{2n}, i = 1, ..., 2n − 1. Then

P_{n}^{G}(h_{1}, ..., h_{2n}) := ∫_{ℂℙ^{n−1}} r_{2n−2}(f_{1}, ..., f_{2n−1})

Theorem 4.1 P_{n}^{G}(h_{1}, ..., h_{2n}) satisfies the following functional equations: a) For any 2n + 1 hyperplanes h_{1}, ..., h_{2n+1} in ℂℙ^{n−1} one has

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

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

∑_{i=1}^{2n+1}(−1)^{i}P_{n}^{G}(p_{1} ∩ p_{i}, ..., p_{2n+1} ∩ p_{i}) = 0

c) The function P_{n}^{G}(h_{1}, ..., h_{2n}) is skewsymmetric with respect to h_{1}, ..., h_{2n}.

d) P_{n}^{G}(h_{1}, ..., h_{2n}) = 0 if the intersection of certain 2k hyperplanes among h_{1}, ..., h_{2n} has codimension ≤ k.

Proof. a), b) are proved in [G4] and c) is clear.

d). One can show that up to a (computable) rational number d_{n} one has

∫_{ℂℙ^{n−1}} r_{2n−2}(f_{1}, ..., f_{2n−1}) = d_{n} ·∫_{ℂℙ^{n−1}} log |f_{1}|d log |f_{2}|∧...∧d log |f_{2n−1}| (23)

If h_{2} ∩ ... ∩ h_{2k+1} has codimension k then d log |f_{2}| ∧ ... ∧ d log |f_{2k+1}| = 0, so the integral is zero.

On the other hand since for (L, M) ∈ A_{n}(ℂ) the function Λ_{n}(L, M) is a period of the mixed Hodge structure

H^{n}(ℂℙ^{n}\L, M)

one can define a single valued function L_{n}(L, M) ∈ ℝ as the ℝ-period of this mixed Hodge structure. Then L_{n} : A_{n}(ℂ) → ℝ is a homomorphism of groups.

Problems. 1. Find a complete list of functional equations for the function P_{n}^{G}.

2. Find the relationship between the Aomoto and Grassmannian polylogarithms.

In the next section we will formulate these problems more precisely and explain their importance.

2 Conjectures. I conjecture that one can describe explicitly a certain subgroup

R_{n}^{G} ⊂ C̄_{2n}(ℙ^{n−1})

such that R_{n}^{G}(ℂ) is the group of all functional equations for the Grassmannian n-logarithm function P_{n}^{G}. The subgroup should be given by an explicit ”universally defined” finite list of ”relations”. More precisely, one should have a finite set of varieties R_{i}(n) over ℤ and morphisms s_{i}(n) : ℤ[R_{i}(n)] → ℤ[C̄_{2n}(ℙ^{n−1})] given by finite correspondences over ℤ such that

R_{n}^{G}(F) = ∑ s_{i}(n)(ℤ[R_{i}(n)])

The properties of the subgroups R_{n}^{G}(F) are formulated in the conjecture below. Set

G_{n}(F) := \frac{ℚ[ configurations of any 2n F -points in ℙ^{n−1}]}{R_{n}^{G}(F)}

Then P_{n}^{G} : G_{n}(ℂ) → ℝ, (l_{1}, ..., l_{2n}) ⟼ P_{n}^{G}(l_{1}, ..., l_{2n}).

Let G_{•} := ⊕_{n=1}^{∞}G_{n}. Recall that a structure of a Lie coalgebra on G_{•} is given by homomorphisms

G_{n} ⟶^{δ_{n}} ⊕_{i≤n/2}G_{i} ∧ G_{n−i}

such that if δ := ⊕δ_{n} then

G_{•} ⟶^{δ} Λ^{2}G_{•} ⟶^{δ⊗id−id⊗δ} Λ^{3}G_{•} ⟶ ...

is a complex.

Conjecture 4.2 a) G_{•} has a natural structure of a graded Lie coalgebra over ℚ.

b) The category of graded finite dimensional modules over G_{•}(F) is equivalent to the category of mixed Tate motives over SpecF. In particular one should have

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

Here H_{(n)}^{i} is the degree n part of H^{i}. Set

G_{1}(F) := F^{∗} ⊗ ℚ, G_{2}(F) := B_{2}(F) ⊗ ℚ, G_{3}(F) := B_{3}(F) ⊗ ℚ

Then the first components of the cobracket δ are given by

δ_{2} : B_{2}(F) → Λ^{2}F_{ℚ}^{∗}, δ_{3} : B_{3}(F) → G_{2}(F) ⊗ F_{ℚ}^{∗}

where the homomorphisms δ_{2} and δ_{3} were defined in s.3.1 and 3.2.

The 7-term relation for the generalized cross-ratio r_{3} is rather mysterious. Its analog needed to define the group G_{4}(F) is unknown. However the results of this paper suggest the following strategy. Denote by Ã_{n} the free abelian group generated by the generators of the group A_{n}, i.e. by admissible pairs of simplices. Let us assume that we have defined already the subgroups R_{m}^{G} for m < n.

Conjecture 4.3 There exists a homomorphism a_{n} : Ã_{n} → C̄_{2n} such that a) The following diagram is commutative

Ã_{n} ⟶^{ν}  ⊕_{1≤i≤n}A_{i} ⊗ A_{n−i}
↓ a_{n}  ↓ a_{i} ∧ a_{n−i}
C̄_{2n} ⟶^{δ_{n}} ⊕_{i≤n/2}G_{i}(F) ∧ G_{n−i}(F)

b) L_{n}(L; M) = c_{n}P_{n}^{G}(a_{n}(L, M)) for any (L; M) ∈ A_{n}(ℂ), where c_{n} is a normalization constant.

(See also conjecture 1.42 in [G1]).

Assuming this we introduce R_{n}^{G}(F) as the image under the map a_{n} of the defining relations for the group A_{n}. Thus δ(R_{n}^{G}) = 0 and we are getting a commutative diagram

A_{n}  ⟶^{ν}  ⊕_{1≤i≤n}A_{i} ⊗ A_{n−i}
↓ a_{n}  ↓ a_{i} ∧ a_{n−i}
G_{n}(F) ⟶^{δ_{n}} ⊕_{i≤n/2}G_{i}(F) ∧ G_{n−i}(F)

The results of s. 3 show how this program works for G_{3}.

Conjecture 4.4 The map a_{n} induces an isomorphism

ā_{n} : A_{n}/P_{n} → G_{n}(F)

For n = 2 this was proved in [BGSV1-2].

For n = 3 one can prove that a_{3}(P_{3}) = 0. To show that ā_{3} is an isomorphism one should construct a homomorphism L_{3} : B_{3}(F) → A_{3}(F) splitting the map a_{3}. The homomorphism L_{3} : ℤ[F^{∗}] → A_{n}, {x}_{3} ⟼ L_{3}{x}_{3} is given by the right hand side of formula (20). Then one should prove that the map L_{3} : ℤ[F^{∗}] → A_{3}/P_{3} is surjective and L_{3}(R_{3}) = 0.

Conjecture (4.4) just means that the dual to the Hopf algebra A_{•} := ⊕A_{n} is isomorphic to the universal enveloping of the Lie algebra G_{•}^{∨}. (Here A → A^{∨} is the duality between the ind and pro ℚ-vector spaces). It seems quite remarkable that the universal enveloping algebra of G_{•}^{∨} admits a completely different description (the only similar situation which comes to mind is Lusztig’s construction of U(N)). So we get two different descriptions of the motivic Lie algebra (reflecting the properties of the Aomoto and Grassmannian polylogarithms). It is even more interesting that there are two more ways of thinking about the same Lie algebra (!): the wonderful ”cycle” construction of Bloch and Kriz [BK] (so far the only one which is completely done), and the construction reflecting the properties of multiple polylogarithms ([G5]). This definitely shows the richness of the subject.

All constructions of different models of the motivic Hopf algebra of the category of mixed Tate motives are based on the following idea. The set of appropriately defined equivalence classes of n-framed mixed Tate motives over F form an abelian group A_{n} and A_{•} := ⊕_{n≥1}A_{n} is a commutative Hopf algebra. It is isomorphic to the fundamental Hopf algebra of the category of mixed Tate motives over F (see [BGSV1-2], [BMS] for the definition of n-framed mixed motives). Consider a universal variation of n-framed mixed Tate motives over a base X_{n}. For any F -point of x ∈ X_{n} we get an element m_{n}(x) ⊂ A_{n}. Universality of the variation means that the map ℤ[X_{n}(F)] → A_{n} is surjective. The kernel of this map is supposed to be described explicitly. So

⊕_{n>0} \frac{ℤ[X_{n}(F)]}{Ker m_{n}}

should have a natural structure of a Hopf algebra, and one needs to determine it. (The variation of mixed Tate motives for the “cycle” Hopf algebra of [BK] can be found in the last section of [G4]). So it is quite interesting that we could get a construction of a co-Lie algebra directly, without constructing first its universal enveloping algebra.

3. A canonical map ℤ[F^{∗}] → A_{n}(F). Let A be a commutative graded Hopf algebra with a product μ and coproduct ν, A_{+} the kernel of the augmentation homomorphism and

ν̃ := ν − (id ⊗ 1 + 1 ⊗ id) : A ⟼ A_{+}^{⊗2}

the restricted coproduct. We define a map of graded vector spaces ν̃_{[k]} : A ⟶ A_{+}^{⊗k} as a composition

A ⟶^{ν̃} A_{+} ⊗ A_{+} ⟶^{ν̃⊗id} A_{+} ⊗ A_{+} ⊗ A_{+} ⟶^{ν̃⊗id} ... ⟶^{ν̃⊗id} A_{+}^{⊗k}

Let μ_{k} : A^{⊗k} → A be the product map. Set

π := ∑_{k≥1} \frac{(−1)^{k−1}}{k} μ_{k} ∘ ν̃_{[k]} = Id − \frac{1}{2}μ ∘ ν̃ + ...

Proposition 4.5 π^{2} = π and Kerπ = P.

Proof. Since π = id (mod P) one needs to show only that P ⊂ Kerπ.

Now return to the Hopf algebra A_{•}(F). Let B_{k} be the Bernoulli numbers.

Proposition 4.6

π(l_{n}(x)) = ∑_{k≥0} \frac{B_{k}}{k!} l_{n−k}(x) · x^{k}

Here x^{m} := x · ... · x (m times) is the product in A_{•}(F) of the element of A_{1} corresponding to x under the canonical isomorphism A_{1}(F) → F^{∗}. Notice that this formula coincides with the formula for the function Λ_{n}(z) in s. 4.1 of [BD]. However we get it in a quite different way.

Proof. Let

l(x, t) := ∑_{k≥1} l_{k}(x)t^{k}

Lemma 4.7 ν̃ : l(x, t) ⟼ l(x, t) ⊗ (e^{x·t} − 1)

This is the generating function for the standard formula

ν̃(l_{k}(x)) = ∑_{1≤i≤k−1} l_{k−i}(x) ⊗ \frac{x^{i}}{i!}

for the coproduct of the classical polylogarithm. Therefore

μ ∘ ν_{[k]} : l(x, t) ⟼ l(x, t) · (e^{x·t} − 1)^{k−1}

So

π : l(x, t) ⟼ − ∑_{k≥1} \frac{(−1)^{k}}{k} l(x, t) · (e^{x·t} − 1)^{k−1} =

= l(x, t) ∑_{k≥1} \frac{(1 − e^{x·t})^{k}}{k} \frac{1}{e^{x·t} − 1} = l(x, t)·\frac{xt}{e^{x·t} − 1} = ∑_{n≥1}(∑_{k≥0} \frac{B_{k}}{k!} l_{n−k}(x)·x^{k})t^{n}

The proposition is proved.

The kernel of the map

L_{n} : ℤ[F^{∗}] ⟼ A_{n}(F), {x} ⟼ π(l_{n}(x))

should coincide with the subgroup of all functional equations for the n-logarithm.

REFERENCES

[A] Aomoto K.: Addition theorem of Abel type for hyperlogarithms, Nagoya math. journal 88 (1982), 55-71.

[BD] Beilinson A.A., Deligne P.: Interpretation motivique de la conjecture de Zagier reliant polylogarithmes et regulateurs Symposium in pure mathematics, 1994, vol 55, part 2, p. 97-122.

[BMS] Beilinson A.A., MacPherson R., Schechtman V.V., Notes on motivic cohomology, Duke math. J., 1987, v. 54. 679-710.

[BGSV1] Beilinson A.A., Goncharov A.A., Schechtman V.V., Varchenko A.N.: Aomoto dilogarithms, mixed Hodge structures and motivic cohomology, the Grothendieck Feschtrift, Birkhauser, vol 86, 1990, p. 135-171.

[BGSV2] Beilinson A.A., Goncharov A.A., Schechtman V.V., Varchenko A.N.: Projective geometry and algebraic K-theory, Leningrad math. J. 2 (1991) 523-576.

[BK] Bloch S., Kriz I.: Mixed Tate motives, Annals of mathematics, 140 (1994) 557-605.

[G0] Goncharov A.B.: The classical trilogarithm, algebraic K-theory of fields and Dedekind zeta function, Bull AMS 29 (1991) 155-161.

[G1] Goncharov A.B.: Geometry of configurations, polylogarithms and motivic cohomology, Advances in Mathematics, 114, N2, (1995) 179-319.

[G2] Goncharov A.B.: Polylogarithms and motivic Galois groups Symposium in pure mathematics, 1994, vol 55, part 2, p. 43 - 96.

[G3] Goncharov A.B.: Deninger’s conjecture on L-functions of elliptic curves at s = 3, Special volume dedicated to Manin’s 60-th birthday, Plenum Press, 1997.

[G4] Goncharov A.B.: Chow polylogarithms and regulators, Math. Res. Letters, 1995, N1, 99-114.

[G5] Goncharov A.B.: Multiple polylogarithms, a paper in preparation.

[HM] Hain R., MacPherson R.: Higher logarithms Illinoise J. math. 34 (1990) 392-475.

[HM] Hanamura M., MacPherson R.: Geometric construction of polylogarithms, Functional analysis on the eve of the 21st century. In honor of I.M. Gelfand, Birkhauser vol 132, (1995), 215-282.

Max-Planck-Instute fur Mathematik, Gottfried Claren str 26, 53225, Bonn Germany, e-mail sasha@mpim-bonn.mpg.de

Dept of Mathematics, Brown University, Providence RI 02912, USA. Dept. of Mathematics, MIT, Cambridge, MA 02139, USA, e-mail sashagon@math.mit.edu