The Classical Polylogarithm

Abstract of a series of lectures given at the workshop on polylogs in Essen, May 1 – 4, 1997

Annette Huber and Jörg Wildeshaus

Contents

I  Motivation  3

II Review of Hodge-theory 5 II.1 Mixed Hodge structures . . . . . . . . . . . . . . . . . . . . . 5 II.2 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 II.3 Everything over ℝ . . . . . . . . . . . . . . . . . . . . . . . . 7

III The Logarithm and the polylogarithm 8 III.1 The logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 III.2 The polylogarithm extension . . . . . . . . . . . . . . . . . . 9 III.3 Geometric origin of Log^{(1)} . . . . . . . . . . . . . . . . . . . . 10

IV Explicit description of the polylog 11 IV.1 Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 IV.2 The local system underlying pol . . . . . . . . . . . . . . . . . 12 IV.3 Extensions of variations of Tate-Hodge structure . . . . . . . 12 IV.4 Explicit shape of pol . . . . . . . . . . . . . . . . . . . . . . . 13

V Cyclotomic elements, and special values 14 V.1 The splitting principle . . . . . . . . . . . . . . . . . . . . . . 14 V.2 Cyclotomic elements . . . . . . . . . . . . . . . . . . . . . . . 15

VI K-theory 15 VI.1 Generalized cohomology . . . . . . . . . . . . . . . . . . . . . 16 VI.2 Motivic cohomology . . . . . . . . . . . . . . . . . . . . . . . 16

VII Motivic polylogarithm 17 VII.1 Geometric origin of Log and pol . . . . . . . . . . . . . . . . 18 VII.2 The motivic splitting principle . . . . . . . . . . . . . . . . . 19

VIII Zagier’s conjecture 20 The purpose of this series of lectures was to give an overview of the central ideas in the proof of the following

Main Theorem. Let d ≥ 2, j ≥ 2, μ_{d}^{0} the set of primitive d-th roots of unity in F := ℚ(μ_{d}). There is a (unique) map of sets

ϵ_{j} : μ_{d}^{0} ⟶ K_{2j−1}(F)_{ℚ} := K_{2j−1}(F) ⊗_{ℤ} ℚ

whose composition with the regulator to Deligne, or absolute Hodge cohomology

r_{H} : K_{2j−1}(F)_{ℚ} ⟶ ( ⨁_{σ:F ↪ℂ} ℂ/(2πi)^{j}ℚ)^{+}

sends ω ∈ μ_{d}^{0} to the element

(−Li_{j}(σω))_{σ} = (− ∑_{k≥1} \frac{σω^{k}}{k^{j}} )_{σ} .

Here, + denotes the invariant part under the joint operation of complex conjugation on the set of embeddings of F into ℂ, and on ℂ/(2πi)^{j}ℚ.

The result is due to Beilinson ([B1], 7.1.5). The original proof, somewhat sketchy, is beautifully reviewed in [Neu]. It relies heavily on a result on the explicit shape of a construction called the “Loday symbol” in Deligne cohomology. This so-called “Crucial Lemma” ([Neu], II.2.4) was subsequently proved in [E], 3.9.

The proof given in the course of this series is different from the original one, and makes use of the classical, or cyclotomic polylogarithm. One of the great advantages of this approach is that the complicated calculations in Deligne cohomology are no longer necessary. In fact, the polylog enjoys a characteristic property called rigidity. One of the aims of the lectures was to emphasize the rôle of rigidity played in the explicit representation of the objects.

Let us remark that the Main Theorem admits an l-adic counterpart: [HW], Corollary 9.7. The statement was conjectured by Bloch and Kato ([BK], Conjecture 6.2), and here the only complete proof is via polylogarithms. That the talks concentrate on the Hodge theoretic aspects of the theory has to do with the speakers’ desire to make the objects as “visible” as possible – to their taste, this requirement is satisfied to a larger degree of satisfaction by the objects of Hodge theory rather than those of the étale world. The main strategy of proof, and the abstract concepts however admit immediate translations to the l-adic setting.

The main ideas of the proof of the Main Theorem, and its l-adic counterpart, appear already in the preprint [B2]. Since then, quite a lot of polylogarithmic literature has been published. The speakers like to think of their talks, and in fact, of this abstract, as a guide through the literature. We hope that it will be useful particularly to those who are new to the field.

We would like to thank the organizers of the workshop, G. Frey, H. Gangl, and H.-G. Rück, for the invitation to Essen. To us, it meant a great opportunity to give a self-contained exposition of some central aspects of the theory.

I Motivation

In this talk, it was tried to indicate the main strategy of proof: Consider, for a number field F, the regulator

r_{H} : K_{2j−1}(F)_{ℚ} ⟶ ( ⨁_{σ:F ↪ℂ} ℂ/(2πi)^{j}ℚ)^{+} .  (*)

The zeroth step is to introduce the concept of “Yoneda extensions in categories of mixed sheaves” in order to reinterpret both sides of (∗). In lecture II, we shall define, for a smooth and separated ℝ-scheme X, a ℚ-linear tensor category

Var(X/ℝ)

of variations of mixed ℚ-Hodge structure over ℝ on X.

The reinterpretation of the right hand side of (∗) acquires the following shape:

Proposition. Let j ≥ 1. There is a canonical isomorphism

Ext_{Var(F ⊗_{ℚ}ℝ/ℝ)}^{1}(ℚ(0), ℚ(j)) ⟶^{∼} ( ⨁_{σ:F ↪ℂ} ℂ/(2πi)^{j}ℚ)^{+}

where + denotes the invariant part under the joint operation of complex conjugation on {F ↪ ℂ}, and on ℂ/(2πi)^{j}ℚ.

According to the motivic folklore, there should be a ℚ-linear tensor category of smooth mixed motivic sheaves MM^{s}(X) on any smooth and separated scheme X over ℚ, together with an exact tensor functor, called the Hodge realization

real_{H} : MM^{s}(X) ⟶ Var (X ×_{ℚ} ℝ/ℝ) .

There should be an isomorphism

Ext_{MM(Spec F)}^{1}(ℚ(0), ℚ(j)) ⟶^{∼} K_{2j−1}(F)_{ℚ}

for j ≥ 1 identifying the regulator r_{H} with the morphism induced by real_{H}. This would give the sheaf theoretical reinterpretation of the left hand side of (∗), and in fact, of r_{H}.

In first approximation, the proof of the Main Theorem, and indeed, also of its l-adic counterpart, proceeds in two steps, corresponding to lectures II–V, and VI–VII respectively:

1. Construct

r_{H}∘ϵ_{j} : μ_{d}^{0} ⟶ Ext_{Var(F ⊗_{ℚ}ℝ/ℝ)}^{1}(ℚ(0), ℚ(j))

first.

a) The construction is a priori sheaf theoretical, and uses concepts like Leray spectral sequences. The objects will be characterized by certain universal properties, one consequence of which will be the earlier mentioned rigidity.

b) Via rigidity, it is possible to describe explicitly the objects defined by abstract nonsense. In particular, we get the formula of the Main Theorem for r_{H}∘ϵ_{j}(ω), ω ∈ μ_{d}^{0}.

c) Again via rigidity, it is possible to show that the abstract construction of a) is “geometrically motivated”: the one-extensions r_{H}∘ϵ_{j}(ω) occur as cohomology objects, with Tate coefficients, of certain F -schemes.

2. Because of the present non-availability of a sheaf theoretical machinery on the level of motives, step 1.a) cannot simply be imitated. However, it turns out that 1.c) admits a translation to K-theory, yielding the map

ϵ_{j} : μ_{d}^{0} ⟶ K_{2j−1}(F)_{ℚ} .

Its compatibility with the map r_{H}∘ϵ_{j} under the regulator is then a consequence of the definition.

Let us give a more detailed account of step 1. Again, we owe to Beilinson the insight that instead of treating the r_{H}∘ϵ_{j}(ω), ω ∈ μ_{d}^{0}, d, j ≥ 2 separately, one should construct one object containing all the information.

In lecture III, we are going to define, by some universal property, the logarithmic (pro-)variation Log on 𝔾_{m,ℝ}. We have

Gr_{∗}^{W} Log = ∏_{j≥0} ℚ(j) ,

i.e., Log is a successive extension of ℚ(0) by ℚ(1) by ℚ(2)... The polylogarithmic extension pol is a one-extension, in the category of variations on the ℝ-scheme 𝕌_{ℝ} := ℙ_{ℝ}^{1} \ {0, 1, ∞}, of ℚ(0) by the restriction of Log:

pol ∈ Ext_{Var(𝕌_{ℝ}/ℝ)}^{1} (ℚ(0), Log_{𝕌_{ℝ}}) .

Again, the definition is via a universal property. From it, we deduce rigidity, and are consequently able, in talk IV, to give an explicit description of pol, essentially in terms of (the inverse of) its period matrix. This description will then justify the name “polylogarithm” as the entries of this matrix are essentially given by the higher logarithms.

In lecture V, we establish another characteristic feature of our objects: the so-called splitting principle. Any ω ∈ μ_{d}^{0} induces an embedding

i_{ω} : Spec ℚ(μ_{d}) ↪ 𝕌 .

Theorem. i_{ω}^{∗} Log is canonically split:

i_{ω}^{∗} Log = ∏_{j≥0} ℚ(j) .

Therefore, we may think of i_{ω}^{∗} pol as an element of

∏_{j≥1} Ext_{Var(ℚ(μ_{d})⊗_{ℚ}ℝ/ℝ)}^{1}(ℚ(0), ℚ(j)) = ∏_{j≥1} (⨁_{σ} ℂ/(2πi)^{j}ℚ)^{+} .

From the explicit description of pol, it is straightforward to see that the j-th component of i_{ω}^{∗} pol equals, up to scaling, the element

(−Li_{j}(σω))_{σ} .

In this very precise sense, all the r_{H}∘ϵ_{j}(ω), ω ∈ μ_{d}^{0} are interpolated by pol. The additional data is the action of the fundamental group of 𝕌(ℂ) given by the local system underlying the variation pol. In fact, rigidity is formulated in terms of this local system.

In talk VI, we give a sketch of some of the technical ingredients for a suitable formalism of “relative K-theory”. Lecture VII establishes the geometric realization of Log, and of pol in absolute and motivic cohomology.

II Review of Hodge-theory

We assemble some facts from Hodge theory that are needed in the construction of the polylogarithm. Basically everything (including references) is contained in [BZ].

II.1 Mixed Hodge structures

We are mostly interested in Hodge structures of Tate type, i.e., ones where only Hodge numbers (n, n) for n ∈ ℤ occur.

Lemma II.1.1. There is a natural isomorphism for n > 0

Ext_{MHS}^{1}(ℚ(0), ℚ(n)) ≅ ℂ/(2πi)^{n}ℚ

which assigns to s ∈ ℂ the extension class of E_{s} where E_{sℂ} = ℂ^{2} and E_{sℚ} ⊂ ℂ^{2} is given by

( \begin{matrix}1 & 0 \\ −\frac{s}{(2πi)^{n}} & 1 \end{matrix} ) ( \begin{matrix}ℚ \\ (2πi)^{n}ℚ \end{matrix} ) .

with weight and Hodge filtration in ℂ^{2} given by

W_{i} = \begin{cases}( \begin{matrix}0 \\ 0 \end{matrix} ) & i < −2n, \\ ( \begin{matrix}0 \\ ∗ \end{matrix} ) & −2n ≤ i < 0, \\ ( \begin{matrix}∗ \\ ∗ \end{matrix} ) & 0 ≤ i, \end{cases} F^{p} = \begin{cases}( \begin{matrix}∗ \\ ∗ \end{matrix} ) & p ≤ −n, \\ ( \begin{matrix}∗ \\ 0 \end{matrix} ) & −n < p ≤ 0, \\ ( \begin{matrix}0 \\ 0 \end{matrix} ) & 0 < p. \end{cases}

Proof. E.g. [J], Lemma 9.2, and Remark 9.3.a).

Examples:

1. In particular, we have

Ext_{MHS}^{1}(ℚ(0), ℚ(1)) ≅ ℂ/2πiℚ ⟶^{exp} ℂ^{∗} ⊗ ℚ.

We reinterpret this relation by saying that we assign to each z ∈ 𝔾_{m}(ℂ) a mixed Hodge structure called Log^{(1)}(z), namely E_{log(z)} in the notation of the lemma.

2. For z ∈ ℂ^{∗} we have a long exact sequence in MHS:

0 → \underline{H}^{0}(𝔾_{m}(ℂ)) ⟶^{Δ} \underline{H}^{1}(𝔾_{m}(ℂ) rel \{1\} ⨿ \{z\}) → \underline{H}^{1}(𝔾_{m}(ℂ)) → 0.

(\underline{H} denotes the corresponding singular cohomology as mixed Hodge structure.) Put G^{(1)}(z) = \underline{H}^{1}(𝔾_{m}(ℂ) rel \{1\} ⨿ \{z\}). This is again an element in Ext_{MHS}^{1}(ℚ(0), ℚ(1)). We will see in lecture III that G^{(1)}(z) = Log^{(1)}(z).

II.2 Variations

Let X be a smooth complex algebraic variety.

Definition II.2.1. A variation 𝕍 of mixed Hodge structure on X(ℂ) consists of

• a locally constant sheaf 𝕍_{ℚ} of ℚ-vector spaces on X(ℂ), • an increasing filtration W_{∗} of 𝕍_{ℚ} by locally constant sheaves,

• a decreasing filtration F^{∗} of V = 𝕍_{ℚ} ⊗_{ℚ} O_{X} by holomorphic sub-vector bundles,

such that for each x ∈ X, the data induce a mixed Hodge structure on 𝕍_{x} and such that Griffith transversality holds. We denote the category Var(X(ℂ)). A variation is called unipotent if all Gr_{n}^{W} 𝕍 are constant on X, e.g., all variations of Tate-Hodge structure are.

Examples: Log^{(1)} and G^{(1)} are unipotent variations on 𝔾_{m}(ℂ).

Lemma II.2.2.

Ext_{Var(X(ℂ))}^{1}(ℚ(0), ℚ(1)) ≅ O_{hol}^{∗}(X(ℂ)) ⊗ ℚ.

Proof. [W], IV, Theorem 3.7.a).

There is a notion of admissible variations of Hodge structure. Rather than giving the definition, we give their main properties which will suffice for all that follows. We denote the corresponding category Var(X) to stress its algebraic nature.

1. If X is compact, then all variations are admissible.

2. Everything coming from geometry is admissible, e.g., G^{(1)} is.

3. If U is an algebraic variety and X a smooth proper compactification, then ([W], IV, Theorem 3.7.b)):

Ext_{Var(U)}^{1}(ℚ(0), ℚ(1)) ≅ {g ∈ O_{hol}^{∗}(U(ℂ)) ⊗ ℚ | g meromorphic on X(ℂ)}
= O_{alg}^{∗}(U) ⊗ ℚ.

4. If 𝕍 is admissible on X, then all H^{i}(X, 𝕍) carry a canonical mixed Hodge structure. This is a deep result, due to Steenbrink-Zucker in the case of curves, and M. Saito in general.

5. If Y ⊂ X is an immersion of smooth varieties of pure codimension d, U = X \ Y and 𝕍 ∈ Var(X), then there is a natural long exact sequence in MHS:

⋯ → \underline{H}^{i−2d}(Y, 𝕍_{Y} (−d)) → \underline{H}^{i}(X, 𝕍) → \underline{H}^{i}(U, 𝕍_{U}) → \underline{H}^{i+1−2d}(Y, 𝕍_{Y} (−d)) → · · · .

II.3 Everything over ℝ

The reference for the following is [HW], Appendix A.2. Let X be a smooth algebraic variety over ℝ. Then there is a continuous map ι : X(ℂ) → X(ℂ) given by complex conjugation on points. It induces a functor

ι^{∗} : Var(X(ℂ)) → Var(X(ℂ))
(𝕍, W_{∗}, F^{∗}) ↦ (ι^{∗}𝕍, ι^{∗}W_{∗}, ι^{∗}\overline{F}^{∗}).

Definition II.3.1. An admissible variation of mixed Hodge structure defined over ℝ is a pair (𝕍, F_{∞}) where 𝕍 ∈ Var(X_{ℂ}) and F_{∞} : 𝕍 ⟶^{∼} ι^{∗}𝕍 is an involution, i.e., F_{∞}^{−1} = ι^{∗}F_{∞}.

Lemma II.3.2. Let U be a smooth variety over ℝ and X a smooth compactification. Then

Ext_{Var(U/ℝ)}^{1}(ℚ(0), ℚ(1)) = O_{alg}^{∗}(U) ⊗ ℚ.

In particular, we need the case X = Spec K ⊗_{ℚ} ℝ where K is some finite extension of ℚ. Note that then X(ℂ) = ∐_{σ:k→ℂ} point. Hence

Ext_{Var(X/ℝ)}^{1}(ℚ(0), ℚ(n)) = ( ⨁_{σ:K→ℂ} ℂ/(2πi)^{n}ℚ)^{+}

where + denotes the invariant part under the joint operation of complex conjugation on X(ℂ) and ℂ/(2πi)^{n}ℚ.

III The Logarithm and the polylogarithm

The aim of this lecture was to construct two things:

a) a (pro)-object Log in Var(𝔾_{m,ℝ}/ℝ) such that (at least)

Gr_{∗}^{W} Log = ∏_{n≥0} ℚ(n) and Log_{1} = ∏_{n≥0} ℚ(n).

b) on 𝕌_{ℝ} = ℙ_{ℝ}^{1} \ {0, 1, ∞} an element

pol ∈ Ext_{Var(𝕌_{ℝ}/ℝ)}^{1}(ℚ(0), Log_{𝕌}).

There are three possibilities to do this: explicitly (talk IV), geometrically (end of III and VII) and by a universal property. It is this last method that we describe first.

III.1 The logarithm

Theorem III.1.1 (Chen, [BZ] 6.23). Let X be a smooth algebraic variety over ℂ. Let x ∈ X(ℂ) and π = π_{1}(X(ℂ), x). We denote U = ℚ[π] and its augmentation ideal 𝔞. Then the completion Û = \underleftarrow{lim} U/𝔞^{n} carries a (unique) mixed Hodge structure such that

• Û ⊗ Û ⟶^{mult} Û and unity : ℚ(0) → Û are morphisms of mixed Hodge structures;

• there is an isomorphism of mixed Hodge structures

𝔞/𝔞^{2} ← π_{1}(X(ℂ), x)^{ab} ⊗ ℚ ≅ H_{1}(X(ℂ), ℚ).

Our example is X = 𝔾_{m}, x = 1 and hence π = ℤα_{0} with a positively oriented loop around 0. Then 𝔞 is generated by α_{0} − 1. We get Û = ℚ[[e]] where e = log α_{0}. Note that the latter element is defined in the completion. The mixed Hodge structure on 𝔞/𝔞^{2} ≅ H_{1}(ℂ^{∗}, ℚ) ≅ ℚ(1) is in fact pure. All in all

Û = Sym^{∗} 𝔞/𝔞^{2} = ∏_{n≥0} ℚ(n).

Theorem III.1.2 (Hain-Zucker, [BZ] 7.19). Let X be a smooth connected algebraic variety over ℂ and x ∈ X(ℂ). Then there is an equivalence of categories

{admissible unipotent variations on X} ↕

\{ \begin{matrix}V ∈ MHS together with an operation Û ⊗ V → V \\ which is a morphism of MHS. \end{matrix} \}

where we assign to a variation its monodromy representation on the stalk at x.

Definition III.1.3 ([W] p.43). Gen_{x}, the generic variation based at x is the variation corresponding to the representation Û⊗Û → Û given by multiplication.

The generic variation has a universal property. Let

Γ : MHS → {ℚ-vector spaces}

be the global section functor, i.e., Γ(H) = Hom_{MHS}(ℚ(0), H) = W_{0}H_{ℚ} ∩ F^{0}H_{ℂ}.

Proposition III.1.4. The pair (Gen_{x}, 1 ∈ Γ(Û)) (pro)-represents the functor

Γ(x^{∗}?) : {unipotent objects in Var(X)} ⟶ ℚ-vector spaces.

Proof. This follows immediately from the Theorem of Hain and Zucker.

We can now identify the object we were after:

Definition III.1.5 ([W] p. 94). We define the logarithmic sheaf on 𝔾_{m} as Log = Gen_{1}. It has the above universal property with respect to the stalk at 1, which is ∏_{i≥0} ℚ(i). Let Log^{(n)} = Log /W_{−2n−2} Log be the quotients of finite length.

In fact Log is easily seen to be defined over ℝ.

III.2 The polylogarithm extension

From the Leray spectral sequence for the composition of functors

Hom_{Var(𝕌/ℝ)}(ℚ(0),?) = Hom_{MHS /ℝ}(ℚ(0), \underline{H}^{0}(𝕌(ℂ),?))

we get the short exact sequence

0 → Ext_{MHS /ℝ}^{1}(ℚ(0), \underline{H}^{0}(𝕌(ℂ), Log_{𝕌})) → Ext_{Var(𝕌/ℝ)}^{1}(ℚ(0), Log_{𝕌})

→ Hom_{MHS /ℝ}(ℚ(0), \underline{H}^{1}(𝕌, Log_{𝕌})) → 0.

Lemma III.2.1.

\underline{H}^{0}(𝕌, Log_{𝕌}) = \underline{H}^{0}(𝔾_{m}, Log) = 0,
\underline{H}^{1}(𝕌, Log_{𝕌}) = ℚ(−1) ⊕ Log_{1}(−1) = ℚ(−1) ⊕ ∏_{k≥0} ℚ(k − 1).

The map \underline{H}^{1}(𝕌, Log_{𝕌}) → Log_{1}(−1) is residue at the point 1.

Hence

Ext_{Var(𝕌/ℝ)}^{1}(ℚ(0), Log_{𝕌}) = Hom_{MHS /ℝ}(ℚ(0), ℚ(−1) ⊕ Log_{1}(−1)) = ℚ .

Definition III.2.2. We define the polylogarithmic extension

pol ∈ Ext_{Var(𝕌/ℝ)}^{1}(ℚ(0), Log_{𝕌})

as the preimage of 1 under the above identification. In terms of the group Hom_{MHS /ℝ}(ℚ(0), ℚ(−1) ⊕ Log_{1}(−1)) it is given by 1 ↦ e ⊗ (2πi)^{−1}.

III.3 Geometric origin of Log^{(1)}

Recall the variation G^{(1)} with fibre \underline{H}^{1}(𝔾_{m}(ℂ) rel \{1\} ⨿ \{z\}, ℚ(1)) at z ∈ ℂ^{∗}. It is unipotent and admissible. At z = 1, the short exact sequence of mixed Hodge structures

0 → ℚ(1)^{2}/Δ(ℚ(1)) → G_{1}^{(1)} → ℚ(0) → 0

has a splitting by \underline{H}^{1}(𝔾_{m}(ℂ) rel \{1\}, ℚ(1)) ≅ ℚ(0). This defines a global section of G_{1}^{(1)}. By the universal property of Log, there is a canonical morphism

ϕ : Log → G^{(1)}

compatible with the projection to ℚ(0).

Proposition III.3.1 ([HW], Theorem 4.11). ϕ factors through an isomorphism Log^{(1)} ⟶^{∼} G^{(1)}.

Proof. The morphism factors for weight reasons. It is enough to check that it induces an isomorphism on the underlying local systems. Note that both objects are 2-dimensional. The image is at least one-dimensional. If is was indeed just one-dimensional, then G^{(1)} would be split and hence constant. So all we have to see is whether G^{(1)} has non-trivial monodromy. This is not hard to do explicitly.

IV Explicit description of the polylog

In this lecture, we use the abstract definition of the polylog to deduce its main characteristic property, the so-called rigidity principle. We then determine the explicit shape of pol.

IV.1 Rigidity

Denote by loc(M) the category of local systems in finite dimensional ℚ-vector spaces on a topological space M, and by

For : Var(X/ℝ) ⟶ loc(X(ℂ))

the forgetful functor.

Theorem IV.1.1 ([B2], 2.1, [W], III, Theorem 2.1). pol is uniquely determined by

For(pol) ∈ Ext_{loc(𝕌(ℂ))}^{1} (ℚ, For(Log_{𝕌})) .

Proof. There is a commutative diagram of boundary morphisms in Leray spectral sequences

Ext_{Var(𝕌/ℝ)}^{1} (ℚ(0), Log_{𝕌}) ⟶ Hom_{MHS /ℝ} (ℚ(0), \underline{H}^{1}(𝕌, Log_{𝕌}))
For↓  ↓^{For}
Ext_{loc(𝕌(ℂ))}^{1} (ℚ, For(Log_{𝕌})) ⟶ Hom_{ℚ} (ℚ, H^{1}(𝕌(ℂ), For(Log_{𝕌})))

By III.2, the upper horizontal map is an isomorphism. Since For is injective on the level of homomorphisms, we see that the left vertical map is injective, too.

Recall from Lemma III.2.1 that

\underline{H}^{1}(𝕌, Log_{𝕌}) = ℚ(−1) ⊕ Log_{1}(−1) = ℚ(−1) ⊕ ∏_{k≥0} ℚ(k − 1) ,

and that \underline{H}^{0}(𝕌, Log_{𝕌}) = 0. It follows as in III.2 that the lower horizontal map of the diagram in the proof of the theorem is an isomorphism, and that it maps the class of For(pol) to the morphism

ℚ ⟶ H^{1}(𝕌(ℂ), For(Log_{𝕌})) = For (ℚ(−1) ⊕ ∏_{k≥0} ℚ(k − 1)) ,

1 ⟼ \frac{1}{2πi} · e .

The l-adic counterpart of this result replaces Var by the category of lisse l-adic sheaves on a smooth scheme over a number field, and loc by the category of lisse l-adic sheaves on the base change by the algebraic closure of the field.

IV.2 The local system underlying pol

Denote by π̃ the fundamental group of 𝕌(ℂ) at the base point \frac{1}{2}. It is free in the generators α_{0} and α_{1}, where α_{i} denotes the class of the positively oriented circle around i.

Define the following representation of π̃:

𝔼 := < 1 >_{ℚ} ⊕ < e^{k}, k ≥ 0 >_{ℚ} ,
α_{0} : 1 ⟼ 1 ,
e^{k} ⟼ e^{k} · exp(e) .
α_{1} : 1 ⟼ 1 + e ,
e^{k} ⟼ e^{k} .

We get an extension of π̃-modules, i.e., of local systems on 𝕌(ℂ)

0 ⟶ For(Log_{𝕌}) ⟶ 𝔼 ⟶ ℚ ⟶ 0

(recall from III.1 that e = log α_{0}).

From the remark following Theorem IV.1.1, one concludes:

Proposition IV.2.1 ([B2], 2.1, [W], IV, Theorem 2.2). The class of the above extension equals pol.

IV.3 Extensions of variations of Tate-Hodge structure

We need to develop a language in which we can describe variations explicitly. The following will be crucial:

Theorem IV.3.1. Let (𝕍_{ℚ}, W_{∗}, F^{∗}) ∈ Var(X(ℂ)) be a variation of THS (Tate-Hodge structure). Then the underlying bifiltered vector bundle

(V, W_{∗}, F^{∗})

is canonically split. V and all F^{p}W_{2p}V are generated by global sections.

Proof. Since Gr_{F^{∗}}^{p} Gr_{n}^{W_{∗}} V = 0 for n ≠ 2p, we have

V = ⨁_{p} F^{p}W_{2p}V .

For any p, we have canonically

F^{p}W_{2p}V ⟶^{∼} Gr_{W_{∗}}^{2p} V ,

which is constant.

This gives our recipe for describing variations of THS: Let (𝕍_{ℚ}, W_{∗}, F^{∗}) be one such.

1. Choose a basis of global sections of V respecting the decomposition

V = ⨁_{p} F^{p}W_{2p}V .

2. Express a basis of ℚ-rational flat sections respecting the weight filtration W_{∗}𝕍_{ℚ} in the basis of 1.

The result will be a lower triagonal matrix. Its entries will in general consist of multivalued functions since sections of 𝕍_{ℚ} will usually only exist on the universal cover of X(ℂ).

Actually, we already applied this recipe: If X is a point, Lemma II.1.1 determines one-extensions of ℚ(0) by ℚ(n). Lemma II.2.2 describes one-extensions of ℚ(0) by ℚ(1) for arbitrary X.

IV.4 Explicit shape of pol

In order to write down a matrix describing pol in the sense of the previous section, we need to define some multivalued functions:

Definition IV.4.1.

Li_{1}(t) := − log(1 − t) ,
Li_{k+1}(t) := ∫_{0}^{t} \frac{Li_{k}(s)}{s}ds , k ≥ 1 ,

Λ_{k} := \frac{1}{(−2πi)^{k}} ∑_{n=1}^{k} \frac{(− log)^{k−n}}{(k − n)!} Li_{n} .

Using IV.2.1, one proves:

Lemma IV.4.2 ([W], IV, Lemma 3.3).

f := 1 + ∑_{k=1}^{∞} Λ_{k} · e^{k}

is a global section of

For(pol) ⊗_{ℚ} O(𝕌(ℂ)) .

So in the basis of global sections (f, e_{0}, e_{1}, …), where

e_{k} : t ⟼ e^{k} · exp (\frac{log(t)}{2πi} · e) = e^{k} + \frac{log(t)}{2πi} · e^{k+1} + …,

the rational structure is described by the following matrix P:

1  0  0  0  ⋯
0  1  0  0  ⋯
\frac{1}{2πi} Li_{1}  −\frac{1}{2πi} log  1  0  ⋯
−\frac{1}{(2πi)^{2}} Li_{2}  \frac{1}{2!} (−\frac{1}{2πi} log)^{2}  −\frac{1}{2πi} log  1  ⋯
\frac{1}{(2πi)^{3}} Li_{3}  \frac{1}{3!} (−\frac{1}{2πi} log)^{3}  \frac{1}{2!} (−\frac{1}{2πi} log)^{2}  −\frac{1}{2πi} log ⋯
⋮  ⋮  ⋮  ⋮

We need to know that if we define F^{0} as the span of F^{0}(Log_{𝕌}) = < e_{0} > and of f (rather than f + some non-zero global section of Log_{𝕌}), then we get an admissible variation of THS on 𝕌. Modulo a shift of the filtrations, this is the content of [W], IV, Theorem 3.5:

Theorem IV.4.3. If we let f be a section of F^{0}, then the data define an admissible variation on 𝕌. Because of rigidity, it equals pol. Therefore, P is the matrix describing pol in the sense of IV.3.

For the description of pol in l-adics, see [B2], 3.3, or [W], IV.4.

V Cyclotomic elements, and special values

The talk given at the workshop concerned itself with two further properties of our objects: the splitting principle, and norm compatibility. Since the latter plays no strategic rôle in the proof of the Main Theorem, we refer to [W], pp. 224-226 for a detailed account.

V.1 The splitting principle

Splitting over roots of unity is a property of the logarithmic sheaf Log rather than of pol. Let ω ∈ 𝔾_{m}(ℂ)_{tors}, and consider the mixed Tate-Hodge structure given by the fibre ω^{∗} Log of Log at ω.

Proposition V.1.1. ω^{∗} Log splits canonically:

ω^{∗} Log = ∏_{j≥0} ℚ(j) .

Proof. One can either employ the universal property III.1.4 of Log to deduce a canonical isomorphism

Log ⟶^{∼} [n]^{∗} Log ,

where [n] : 𝔾_{m} ⟶ 𝔾_{m}, t ⟼ t^{n}. Since 1^{∗} Log is split, so is the fibre of Log at any preimage of 1 under [n].

Or use Lemma II.1.1, and the explicit description of Log.

V.2 Cyclotomic elements

The splitting principle provides us with canonical projections

pr_{ω,j} : ω^{∗} Log ⟶ ℚ(j) ,

for any root of unity ω, and any j ≥ 1. We get an induced map (pr_{ω,j})_{∗} on the level of Ext groups. Proposition V.2.1. For any ω ≠ 1, we have

(pr_{ω,j})_{∗}(ω^{∗} pol) = (−1)^{j} Li_{j}(ω) mod (2πi)^{j}ℚ

in Ext_{MHS}^{1}(ℚ(0), ℚ(j)) = ℂ/(2πi)^{j}ℚ. Proof. Look at the matrix P (ω)!

VI K-theory

The general motivic philosophy says (among other things): Conjecture VI.0.2 (Beilinson et. al.). For all varieties over ℚ, there is a universal cohomology theory X ↦ h^{∗}(X) with values in an abelian category MM (mixed motives). There is also a universal absolute cohomology theory (motivic cohomology) such that the Leray spectral sequence gives short exact sequences 0 → Ext_{MM}^{1}(ℚ(0), h^{n−1}(X)(j)) → H_{M}^{n}(X, j) → Hom_{MM}(ℚ(0), h^{n}(X)(j)) → 0. Moreover, for smooth varieties X, there should be natural isomorphisms

H_{M}^{i}(X, j) ≅ Gr_{γ}^{j} K_{2j−i}(X)_{ℚ} .

This leads us to define Definition VI.0.3. For smooth X over ℤ, we put H_{M}^{i}(X, j) := Gr_{γ}^{j} K_{2j−i}(X)_{ℚ}. We call this motivic cohomology of X.

We need to extend this definition. We want

• relative motivic cohomology groups,

• motivic cohomology of certain singular varieties,

• localization sequences in this context. We use the approach of Gillet and Soulé [GS] also used in [dJ] by de Jeu. Details can be found in [HW], Appendix B. The following is a quick and very imprecise overview.

VI.1 Generalized cohomology

We recall that the geometric realization functor induces an equivalence of categories between simplicial sets and CW -complexes up to homotopy. On the other hand, the ‘associated complex’ functor gives an equivalence of categories between simplicial abelian groups and cohomological complexes concentrated in negative degrees, both up to homotopy. We sheafify these notions for the Zariski-topology. By this we mean the big or small site of smooth schemes over a fixed smooth ℤ-scheme S_{0}, equipped with the Zariski-topology.

Definition VI.1.1 ([HW], B.1). A space Y is a simplicial sheaf of sets for the Zariski-topology which is pointed by a map ∗ → Y. Here ∗ is the constant simplicial object associated to the constant sheaf S ↦ {∗}. A morphism y : Y → Y′ is called weak equivalence if the induced morphisms on the sheafified homotopy groups are isomorphisms for all choices of base point.

Spaces form a closed model category in the sense of Quillen. This means that they behave ‘like topological spaces’. In particular, we can form suspensions S, cones, loop spaces and form a homotopy category by formally inverting weak equivalence.

If X is a scheme, let X̃ be the constant simplicial object associated to S ↦ X(S) ∪ {∗} pointed by the disjoint copy of ∗.

Definition VI.1.2 ([HW], B.1.1). A space Y is constructed from schemes if all Y_{n} are of the form ∗∪ scheme. We define generalized cohomology of Y with coefficients in a space A by

H_{gen}^{−m}(Y, A) = [S^{m}Y, A] for m ≥ 0

where [·, ·] denotes morphisms in the homotopy category.

In particular, we use A = K, the sheafification of S ↦ ℤ × ℤ_{∞}BGl(S). We speak of K-cohomology.

Proposition VI.1.3 (Brown-Gersten). If Y = X̃ for a scheme X in the site, then H_{gen}^{−m}(X̃, K) = K_{m}(X).

Cohomology of abelian sheaves, e.g., absolute Hodge cohomology, can also be written as generalized cohomology of a space K(A). So one point of generalized cohomology is that it allows to treat K-groups and cohomology of abelian sheaves on an equal footing. Gillet has constructed Chern classes ch_{j} : K → K(A) for good A like the complexes defining absolute Hodge cohomology.

VI.2 Motivic cohomology

Gillet and Soulé have constructed maps λ^{i} : K → K for i ≥ 1 such that H_{gen}^{i}(Y, K) is a turned into a λ-algebra, at least if Y is constructed from schemes. (This is a lie! See [HW], B.2.) Hence we also have a γ-filtration on K-cohomology.

Definition VI.2.1. If Y is constructed from schemes and 2j + i ≤ 0, let

H_{M}^{i}(Y, j) = Gr_{γ}^{j} H_{gen}^{2j+i}(Y, K)_{ℚ}.

The extension of the definition to spaces allows a lot of extra flexibility.

Examples:

1. If T = ⋃_{i∈I} C_{i} where all C_{i} and all ⋂_{i∈I′} C_{i} for subsets I′ of I are smooth over our base S_{0}, then we put

T_{∗} = cosk_{0}(∐_{i∈I} C_{i}/T)  i.e.
T_{0} = ∐_{i∈I} C_{i} and T_{n} = ∐_{I_{n}} ⋂_{i∈I_{n}} C_{i}

where I_{n} runs through all n+1-tuples of elements in I. The big advantage of T_{∗} is that all its components are smooth. By adding a disjoint base point we turn this into a space T̃_{∗} constructed from schemes. We define H_{M}^{k}(T, j) = H_{M}^{k}(T̃_{∗}, j).

2. Let T as before, T ⊂ X where X is also smooth. We put H_{M}^{i}(X rel T, j) = H_{M}^{i}(Cone(T̃_{∗} → X̃), j) where the cone is taken in the category of spaces. By definition it sits in a long exact sequence for relative cohomology.

Theorem VI.2.2 (Soulé, de Jeu, [HW], B.2.16). Let T and X be as in the example. Let Z ⊂ X be smooth of pure codimension d. Suppose that Z intersects all ⋂_{i∈I′} C_{i} transversally. Let U = X \ Z. Then there is a natural long exact sequence

⋯ → H_{M}^{i−2d}(Z rel T ∩ Z, j − d) → H_{M}^{i}(X rel T, j) → H_{M}^{i}(U rel T ∩ U, j) → · · · .

Moreover, it is compatible with the same sequence in absolute Hodge cohomology via the Chern class morphism.

VII Motivic polylogarithm

We now need mixed Hodge modules over ℝ ([HW], Appendix A). The category is denoted MHM(X/ℝ). Whereas admissible variations are the Hodge theoretic version of locally constant sheaves, Hodge modules correspond to (perverse) constructible sheaves.

Definition VII.0.3. Let X be a smooth variety over ℝ and Y ⊂ X with complement j : U → X. We define absolute Hodge cohomology by

H_{abs}^{i}(X/ℝ, ℚ(n)) = Ext_{MHM(X/ℝ)}^{i}(ℚ(0), ℚ(n))
H_{abs}^{i}(X rel Y/ℝ, ℚ(n)) = Ext_{MHM(X/ℝ)}^{i}(ℚ(0), j_{!}ℚ(n)).

It can be shown that this agrees with Beilinson’s ad hoc version ([HW], Theorem A.2.7). Everything done in this talk translates immediately into the l-adic setting.

VII.1 Geometric origin of Log and pol

We now come to a quick tour through [HW]. We consider the following geometric situation with 𝕌 = ℙ^{1} \ {0, 1, ∞} and Z = 1 × 𝕌 ⨿ Δ, V = 𝔾_{m,𝕌} \ Z, Z^{(n)} = 𝔾_{m,𝕌}^{n} \ V^{n}:

\begin{aligned}V & ⟶^{v} & 𝔾_{m,𝕌} & ↩ Z \\ & & ↓ p \\ & & 𝕌 \end{aligned}  \begin{aligned}V^{n} & ⟶^{v^{n}} & 𝔾_{m,𝕌}^{n} & ↩ Z^{(n)} \\ & & ↓ p^{n} \\ & & 𝕌 \end{aligned}

Recall from lecture III that Log_{𝕌}^{(1)} = G^{(1)} = R^{1}p_{∗}v_{!}ℚ(1) where we now use the correct formulation in terms of Hodge modules. Hence

Log_{𝕌}^{(n)} = Sym^{n} Log_{𝕌}^{(1)} = Sym^{n} R^{1}p_{∗}v_{!}ℚ(1) = R^{n}p_{∗}^{n}v_{!}^{n}ℚ(n)^{sgn}

where we have to take the sign-eigenspace with respect to the operation of the symmetric group because the cup-product is anti-symmetric.

Definition VII.1.1. Let G^{(n)} = R^{n}p_{∗}^{n}v_{!}^{n}ℚ(n)^{sgn}.

With this definition we have ([HW], section 4)

Ext_{Var(𝕌/ℝ)}^{1}(ℚ(0), G^{(n)}) = Ext_{MHM(𝔾_{m,𝕌}/ℝ)}^{1}(ℚ(0), R^{n}p_{∗}^{n}v_{!}^{n}ℚ(n)^{sgn})
= Ext_{MHM(𝔾_{m,𝕌}^{n}/ℝ)}^{1}(ℚ(0), v_{!}^{n}ℚ(n)^{sgn})
= H_{abs}^{n+1}(𝔾_{m,𝕌}^{n} rel Z^{(n)}, ℚ(n))^{sgn}.

Note that the corresponding motivic cohomology groups H_{M}^{n+1}(𝔾_{m,𝕌}^{n} rel Z^{(n)}, ℚ(n))^{sgn} are also well-defined!!! Our main tool in the sequel is the Residue sequence.

Proposition VII.1.2 ([HW], after 4.6 and 7.2). Let \overline{Z} = 𝔸_{𝕌}^{1} \ V. There are long exact sequences in motivic and absolute Hodge cohomology, which are also compatible under Chern classes:

⋯ → H_{?}^{i}(𝔸_{𝕌}^{n} rel \overline{Z}^{(n)}, j)^{sgn} → H_{?}^{i}(𝔾_{m,𝕌}^{n} rel Z^{(n)}, j)^{sgn}

→ H_{?}^{i−1}(𝔾_{m,𝕌}^{n−1} rel Z^{(n−1)}, j − 1)^{sgn} → H_{?}^{i+1}(𝔸_{𝕌}^{n} rel \overline{Z}^{(n)}, j)^{sgn} → · · · .

Proof. In the n = 2-case we consider the localization sequences in relative cohomology for the two inclusions 𝔾_{m}^{2} ⊂ 𝔸^{2} \ (0, 0) ⊂ 𝔸^{2}. The effect of the sign-eigenspaces leads to the above form of the sequence.

Definition VII.1.3. The residue maps are given by the map from the residue sequence res_{n} : H_{?}^{n+1}(𝔾_{m,𝕌}^{n} rel Z^{(n)}, n) → H_{?}^{n}(𝔾_{m,𝕌}^{n−1} rel Z^{(n−1)}, n).

Lemma VII.1.4 ([HW], 7.3). H_{?}^{i}(𝔸_{𝕌}^{n} rel \overline{Z}^{(n)}, n) ≅ H_{?}^{i−n}(𝕌, j).

There is an alternative description of the localization sequence in the case of absolute Hodge cohomology. The same constructions that led to the residue sequence, also lead to a sequence of variations on 𝕌 by using H_{𝕌}^{i}(·) = R^{i}p_{∗}(·), namely

0 → ℚ(n)_{𝕌} → H_{𝕌}^{n}(𝔾_{m,𝕌}^{n} rel ℤ^{(n)}, n)^{sgn} ⟶^{res_{n}} H_{𝕌}^{n−1}(𝔾_{m,𝕌}^{n−1} rel ℤ^{(n−1)}, n)^{sgn} → 0.

Note that H_{𝕌}^{n}(𝔾_{m,𝕌}^{n} rel ℤ^{(n)}, n)^{sgn} is just a different way of writing G^{(n)}.

Proposition VII.1.5 ([HW], 4.9, 4.8). The square

G^{(n)}  ⟶^{res_{n}} G^{(n−1)}
≅↓  ↓≅
Sym^{n} G^{(1)} ⟶^{proj} Sym^{n−1} G^{(1)}

commutes. Hence the transition maps of Log are of geometric origin. Moreover the residue sequence in absolute Hodge cohomology is the long exact sequence attached to the short exact sequence of sheaves above.

Up to now everything would have worked for a general base S instead of 𝕌. But now we use the simple form of 𝕌. Its cohomology is Tate and by Borel’s theorem we understand the corresponding motivic cohomology and the regulator very well.

Lemma VII.1.6 ([HW], 8.3). With B = Spec ℤ, the following composition is bijective:

H_{M}^{0}(B, 0) ⟶^{i_{1}} ⨁_{i=0,1} H_{M}^{0}(B, 0) = H_{M}^{1}(𝕌, 1) → H_{M}^{2}(𝔾_{m,𝕌} rel Z, 1).

Call the inverse map res, the total residue.

Theorem VII.1.7 ([HW], Corollary 8.8). We have a commutative square:

\underleftarrow{lim} H_{M}^{n+1}(𝔾_{m,𝕌} rel Z^{(n)}, n)^{sgn} ⟶^{r_{H}} H_{abs}^{1}(𝕌_{ℝ}, Log_{𝕌})
res↓  ^{residue at 1}↓^{from III}
H_{M}^{0}(B,)  ⟶ H_{abs}^{0}(B_{ℝ}, ℚ(0))

The map res on the left is an isomorphism.

We now define pol_{M}, the motivic polylogarithm simply as res^{−1}(1). By construction, r_{H} pol_{M} = pol.

VII.2 The motivic splitting principle

Let d ≥ 2 and b prime to d. Let C = Spec ℤ[T]/Φ_{d}(T)[1/d]. It embeds canonically into 𝕌. It can be twisted by raising to the b-th power on C. Call the resulting embedding i_{b} : C → 𝕌. The morphism [d + 1] on 𝔾_{m,C} (raising to the d + 1-th power) respects Z_{C}. We can analyze the eigenvalues of this operation and find:

Proposition VII.2.1 ([HW], Lemma 9.3). There is a natural splitting

H_{M}^{n+1}(𝔾_{m,C} rel Z_{C}^{(n)}, n) = ∏_{1≤i≤n} H_{M}^{1}(C, i).

The splittings are compatible in the projective system and they are also compatible with the splitting in absolute Hodge cohomology induced by the splitting principle there.

We can now prove our main theorem from lecture I:

Definition VII.2.2. Let

ϵ : {primitive d-th roots of unity} → H_{M}^{1}(C, n)
T^{b} ↦ (−1)^{n−1} n-component of i_{b}^{∗} pol_{M} .

Clearly r_{H}ϵ(ω) = (−1)^{n−1} n-component of ω^{∗} pol whose explicit value was computed in lecture V.

VIII Zagier’s conjecture

This talk was devoted to the presentation of the main ideas of the article [BD].

The weak version of Zagier’s conjecture, meanwhile a theorem of de Jeu’s ([dJ]) concerns itself with the K-theory of number fields. There is a conjecture for any integer j ≥ 1, and the j-th can only be formulated if the preceding ones are true.

Fix a number field F. One wants to construct a ℚ-vector space L_{j}, a map

{ }_{j} : 𝕌(F) = F^{∗} \ {1} ⟶ L_{j} ,

a homomorphism

d_{j} : L_{j} ⟶ ⋀^{2} (⨁_{l=1}^{j−1} L_{l}) ,

and a monomorphism

φ_{j} : ker(d_{j}) ↪ K_{2j−1}(F)_{ℚ} .

For j = 1, one defines L_{1} := F^{∗} ⊗_{ℤ} ℚ,

{x}_{1} := (1 − x) ⊗ 1 ∈ L_{1} ,

d_{1} := 0, and φ_{1} as the isomorphism between F^{∗} ⊗_{ℤ} ℚ and K_{1}(F)_{ℚ}.

For j ≥ 2, let L̃_{j} be the free ℚ-vector space in the symbols ˜{x}_{j}, x ∈ F^{∗}\{1}. Define

d̃_{j} : L̃_{j} ⟶ L_{j−1} ⊗_{ℚ} L_{1} ⟶ ⋀^{2} (⨁_{l=1}^{j−1} L_{l})

by sending the symbol ˜{x}_{j} to {x}_{j−1} ∧ x.

The conjecture predicts a map

φ̃_{j} : ker(d̃_{j}) ⟶ K_{2j−1}(F)_{ℚ} .

It also predicts the explicit shape of the composition

r_{H}∘φ̃_{j} : ker(d̃_{j}) ⟶ ( ⨁_{σ:F ↪ℂ} ℂ/(2πi)^{j}ℝ)^{+} .

If the conjecture holds, one sets

L_{j} := L̃_{j}/ ker(φ̃_{j}) .

In the talk, it was explained, following [BD], section 2, that the conjecture follows from the motivic folklore, explained in lecture I, plus the existence of “pol ∈ MM^{s}(𝕌)”. More precisely, if S ∈ ker(d̃_{j}), then φ̃_{j}(S) is obtained by a linear combination of “coefficients” of the value of pol at points x ∈ 𝕌(F).

The material covered in lectures II-VII can be seen as a description of φ̃_{j} on a very particular kind of elements of ker(d̃_{j}), namely those symbols concentrated on roots of unity.

References

[B1] A.A. Beilinson, ”Higher regulators and values of L-functions”, J. Sov. Math. 30 (1985), 2036–2077.

[B2] A.A. Beilinson, ”Polylogarithm and Cyclotomic Elements”, typewritten preprint, MIT 1989.

[BD] A.A. Beilinson, P. Deligne, “Interprétation motivique de la conjecture de Zagier reliant polylogarithmes et régulateurs”, in U. Jannsen, S.L. Kleiman, J.–P. Serre (eds.), “Motives”, Proceedings of the Research Conference on Motives held July 20 – August 2, 1991, in Seattle, Washington, Proc. of Symp. in Pure Math. 55, Part II, AMS 1994, 97–121.

[BK] S. Bloch, K. Kato, “L–functions and Tamagawa Numbers of Motives”, in P. Cartier et al. (eds.), “The Grothendieck Festschrift”, Volume I, Birkhäuser 1990, 333–400.

[BZ] J.-L. Brylinski, S. Zucker, “An overview of recent advances in Hodge theory”, in “Several complex variables VI. Complex manifolds”, Encycl. Math. Sci. 69, pp. 39-142, Springer 1990.

[E] H. Esnault, “On the Loday Symbol in the Deligne–Beilinson Cohomology”, K–Theory 3 (1989), 1–28.

[GS] H. Gillet, C. Soulé, “Filtrations on higher algebraic K–theory”, unpublished.

[HW] A. Huber, J. Wildeshaus, “Classical motivic polylogarithm according to Beilinson and Deligne”, Preprint 1996, available under http://wwwmath.uni-muenster.de/math/inst/reine/inst/deninger

[HZ] R.M. Hain, S. Zucker, ”Unipotent variations of mixed Hodge structure”, Inv. math. 88 (1987), 83–124.

[J] U. Jannsen, “Mixed Motives and Algebraic K-Theory”, LNM 1400, Springer-Verlag 1990.

[dJ] R. de Jeu, “Zagier’s Conjecture and Wedge Complexes in Algebraic K–theory”, Comp. Math. 96 (1995), 197–247.

[Neu] J. Neukirch, “The Beilinson Conjecture for Algebraic Number Fields”, in M. Rapoport, N. Schappacher, P. Schneider (eds.), “Beilinson’s Conjectures on Special Values of L–Functions”, Perspectives in Mathematics 4, Academic Press 1988, 193–247.

[W] J. Wildeshaus, ”Realizations of Polylogarithms”, LNM 1650, Springer-Verlag 1997.

Annette Huber huber@math.uni-muenster.de Jörg Wildeshaus wildesh@math.uni-muenster.de Math. Institut, Einsteinstr. 62, 48 149 Münster, Germany