POLYLOGARITHMS AND MIXED MOTIVES

Masaki Hanamura

§1. The category of mixed motives D_{finite}(k).

Let k be a field, □^{1} = ℙ_{k}^{1} − {1} and □^{n} = (□^{1})^{n} with coordinates (x_{1}, ⋯, x_{n}). Faces of □^{n} are intersections of codimension one faces, and the latter are divisors of the form □_{i,a}^{n−1} = {x_{i} = a} where a = 0 or ∞. A face of dimension m is canonically isomorphic to □^{m}.

Let Σ_{n} be the permutation group of the set {1, ⋯, n}; it acts on (ℤ/2ℤ)^{n} by permutations. Let G_{n} be the semi-direct product of (ℤ/2ℤ)^{n} and Σ_{n}; an element is of the form (i_{1}, ⋯, i_{n}; τ) with i_{1}, ⋯, i_{n} ∈ ℤ/2ℤ = {±1} and τ ∈ Σ_{n}. Σ_{n} acts on □^{n} by permutations of coordintates, and (ℤ/2ℤ)^{n} acts by: (i_{1}, ⋯, i_{n}) · (x_{1}, ⋯, x_{n}) = (x_{1}^{i_{1}}, ⋯, x_{n}^{i_{n}}). So G_{n} acts of □^{n}. There is a homomorphism sgn : G_{n} → {±1}, (i_{1}, ⋯, i_{n}; τ) ↦ i_{1} ⋯ i_{n} · sgn τ.

Let X be an equi-dimensional variety (or a scheme) over a field k. Let Z^{r}(X, n) be the ℚ-vector space of ℚ-cycles z of codimension r on X × □^{n} such that

(i) each irreducible component of z meets each face X × □^{m} properly, and

(ii) z is alternating with respect to G_{n}, namely, for any σ ∈ G_{n}, σ(z) = sgn(σ)z. The inclusions of codimension one faces δ_{i,a} : □_{i,a}^{n−1} ↪ □^{n} induce the map

∂ = ∑(−1)^{i+a}δ_{i,a}^{∗} : Z^{r}(X, n) → Z^{r}(X, n − 1)

(namely, ∂ sends alternating cycles to alternating ones) and (Z^{r}(X, ·), ∂) is a homology complex. We call this the cycle complex (of codimension r) of X. By definition the (rational) higher Chow groups are the homology groups of this complex:

CH^{r}(X, n) = H_{n}Z^{r}(X, ·) .

Note CH^{r}(X, 0) = CH^{r}(X), the rational Chow group of X.

The following notion will be of frequent use. A C-complex (of abelian groups) consists of (i) Complexes A^{m} = (A^{m,•}, d_{A^{m}}) (each of which is not necessarily bounded) for m ∈ ℤ,

such that for all but finitely many m’s, A^{m} = 0; and (ii) For m < n, maps of graded groups

F^{m,n} : A^{m,•} → A^{n,•−(n−m−1)}

subject to the condition

F^{m,n}∘(−1)^{m} d_{A^{m}} + (−1)^{n} d_{A^{n}} ∘F^{m,n} + ∑_{m<ℓ<n} F^{ℓ,n}∘F^{m,ℓ} = 0

as a map A^{m,•} → A^{n,•−n+m+2}.

Typeset by AMS-TEX

One can associate with it a complex, the total complex T ot (A) = (T ot (A)^{•}, d), defined as:

T ot (A)^{p} = ⨁_{p∈ℤ} A^{m,p−m}

(which is a finite sum), and the differential d is the direct sum, for m, of (−1)^{m} d_{A^{m}} + F^{m,n} : A^{m,•} → ⊕_{n≥m}A^{n,•−(n−m−1)}. The condition in (ii) is equivalent to d being a differential.

Let X and Y be smooth projective. For an element f ∈ Z^{s}(X × Y, ℓ), one has the partially defined map (of graded vector spaces)

f_{∗} : Z^{r}(X, n) − − → Z^{r+s}(Y, n + ℓ), f_{∗}(z) = p_{Y} _{∗}[(f × □^{n}) · (z × Y × □^{ℓ})] .

Here (f × □^{n}) · (z × Y × □^{ℓ}) denotes the alternation (with respect to the action of G_{n+ℓ}) of the cycle theoretic intersection (f × □^{n}) ⊙ (z × Y × □^{ℓ}), and p_{Y} : X × Y × □^{n+ℓ} → Y × □^{n+ℓ} is the projection.

Similarly there is the partially defined map

Z^{r}(X × Y, ·) ⊗ Z^{s}(Y × Z, ·) − − → Z^{r+s−dim Y} (X × Z, ·)

which sends u ⊗ v ∈ Z^{r}(X × Y, n) ⊗ Z^{s}(Y × Z, m) to

v∘u := the alternation of p_{XZ ∗}[(X × v × □^{n}) ⊙ (u × Z × □^{m})] .

For each smooth projective variety X there is a collection of distinguished subcomplexes of Z^{r}(X, ·) satisfying:

(i) For a distinguished subcomplex Z^{r}(X, ·)′, the inclusion into Z^{r}(X, ·) is a quasi- isomorphism;

(ii) For any cycle f ∈ Z^{s}(X ×Y, ℓ) and a distinguished subcomplex Z^{r+s−dim X}(Y, ·)′ there is a distinguished subcomplex Z^{r}(X, ·)′ on which f_{∗} is defined and induces a map f_{∗} : Z^{r}(X, ·)′ → Z^{r+s−dim X}(Y, · + ℓ)′;

(iii) The intersection of a finite collection of distinguished subcomplexes is again distinguished.

For details on the category D(k) of mixed motives, we refer the reader to [Ha, II]. We will only need the subcategory D_{finite}(k) of mixed motives of finite type; the definitions are briefly recalled.

A finite symbol is a formal sum

⨁_{α∈I}(X_{α}, r_{α})

where X_{α} is a smooth projective variety, I a finite index set and r_{α} ∈ ℤ. We write 0 for the corresponding symbol when I is an empty set.

Define dual, tensor product, and inner Hom of (a) finite symbol(s) as:

(⊕(X_{α}, r_{α}))^{∨} = ⊕(X_{α}, dim X_{α} − r_{α}) .
(⊕(X_{α}, r_{α})) ⊗ (⊕(X′_{α′}, r_{α′}) ) = ⊕(X_{α} × X′_{α′}, r_{α} + r′_{α′}) ;

and

\underline{Hom}(⊕(X_{α}, r_{α}), ⊕(X′_{α′}, r_{α′})) = (⊕(X_{α}, r_{α}))^{∨} ⊗ (⊕(X′_{α′}, r_{α′}) ) .

Define the cycle complex of a finite formal symbol by

Z^{0}(⊕(X_{α}, r_{α}), ·) = ⨁ Z^{r_{α}}(X_{α}, ·) .

One also uses cohomological notation Z^{0}(⊕(X_{α}, r_{α}))^{−n} = Z^{0}(⊕(X_{α}, r_{α}), n).

Note there is a partially defined map

Z^{0}(\underline{Hom}((X_{1}, r_{1}), (X_{2}, r_{2}) ), ·) ⊗ Z^{0}(\underline{Hom}((X_{2}, r_{2})(X_{3}, r_{3}) ), ·)
−− → Z^{0}(\underline{Hom}((X_{1}, r_{1}), (X_{3}, r_{3}) ), ·)

given by the composition of correspondences

u ⊗ v ↦ v ∘ u = p_{13}_{∗}[(u × X_{3}) · (X_{1} × v)] .

By definition, an object of D_{finite}(k) is a set of data K = (K^{m}) = (K^{m}, f^{m,n}) where

(i) For each integer m, K^{m} = ⊕_{α∈I(m)}(X_{α}, r_{α}), a finite symbol.

(ii) For (m, n) with m < n, given f^{m,n} = (f_{αβ}^{m,n}) ∈ Z^{0}(\underline{Hom}(K^{m}, K^{n}) )^{−n+m+1}, which are subject to the conditions:

For f^{m_{k},m_{k+1}} ∈ Z^{0}(\underline{Hom}(K^{m_{k}}, K^{m_{k+1}}) )^{−m_{k+1}+m_{k}+1} (k = 1, 2, ⋯, r) one has

f^{m_{r},m_{r+1}} ∘f^{m_{r},m_{r−1}} ∘ ⋯ ∘f^{m_{1},m_{2}} is defined and ∈ Z^{0}(\underline{Hom}(K^{m_{1}}, K^{m_{r+1}}) )^{−m_{r+1}+m_{1}+r} .

For m < n, one has

(−1)^{n} ∂f^{m,n} + ∑_{m<ℓ<n} f^{ℓ,n} ∘ f^{m,ℓ} = 0 , .

On the left side the compositions of the correspondences are required to be defined.

There is the functor of cycle complexes Z^{0} from D_{finite}(k) to the derived category of ℚvector spaces. To define Z^{0}(K, ·), for each m and α ∈ I(m), take a distinguished subcomplex Z^{0}((X_{α}, r_{α}), ·)′ so that each f_{α,β}^{m,n} induces the map f_{α,β ∗}^{m,n} : Z^{0}((X_{α}, r_{α}), ·)′ → Z^{0}((X_{β}, r_{β}), ·)′. We then let

Z^{0}(K^{m}, ·)′ := ⊕_{α}Z^{0}((X_{α}, r_{α}), ·)′

and have f^{m,n} _{∗} : Z^{0}(K^{m}, ·)′ → Z^{0}(K^{n}, · + (n − m − 1) )′ is defined. We define Z^{0}(K, ·) to be the total complex T ot(Z^{0}(K^{m}, ·), f_{∗}^{m,n}), namely the complex (𝕂, d) with

𝕂^{i} = ⨁_{j≥i} Z^{0}(K^{j}, j − i)′ ,

and

d^{i} = ∑_{j}((−1)^{j}∂_{j} + ∑_{j<ℓ} f_{∗}^{j,ℓ}) .

Let (K, f) and (L, g) be objects in D_{finite}(k). The function cycle complex Hom(K, L)^{•} is defined as follows. Let Z^{0}(\underline{Hom}(K^{m}, L^{m′}), ·)′ be distinguished subcomplexes such that

For u ∈ Z^{0}(\underline{Hom}(K^{m}, L^{m′}), ·), both u∘f^{n,m} and g^{m′,n′} ∘u are defined.

(This is possible since there are only finitely many non-zero f^{n,m}’s and g^{m′,n′}’s.) The cohomological complex to be defined has the group of N-cochains

Hom(K, L)^{N} = ⨁_{−m+m′−p=N} Z^{0}(\underline{Hom}(K^{m}, L^{m′}), p)′

The differential of this complex, which we denote by D, is the sum of the three kinds of maps:

(−1)^{p+m′+n+1} (∘ f^{n,m}) : Z^{0}(\underline{Hom}(K^{m}, L^{m′}), p)′ → Z^{0}(\underline{Hom}(K^{n}, L^{m′}), p + n′ − m′ − 1)′ ,

(−1)^{m′+n′} (g^{m′,n′} ∘) : Z^{0}(\underline{Hom}(K^{m}, L^{m′}), p)′ → Z^{0}(\underline{Hom}(K^{m}, L^{n′}), p + n′ − m′ − 1)′ ,

and

(−1)^{m′} ∂ : Z^{0}(\underline{Hom}(K^{m}, L^{m′}), ·)′ → Z^{0}(\underline{Hom}(K^{m}, L^{m′}), · − 1 )′ .

Given three objects K, L and M, the partially defined composition map

Hom(K, L)_{•} ⊗ Hom(L, M)^{•} − − → Hom(K, M)^{•}

u ⊗ v ↦ v∘u; (v∘u)^{m,n} = ∑_{ℓ∈ℤ} v^{ℓ,n}∘u^{m,ℓ}

satisfies the Leibniz formula

D(v∘u) = Dv∘u + (−1)^{deg v}v∘Du ,

where deg v is the total degree of v in the cohomological complex. There is a quasi-isomorphic subcomplex of Hom(K, L)_{•} ⊗ Hom(L, M)^{•} on which the composition is defined. See [Ha, II, §1].

By definition

Hom_{D_{finite}(k)}(K, L) = H^{0}Z^{0}(\underline{Hom}(K, L) )^{•} .

The composition of morphisms is induced from the composition of the function complexes. A morphism u : K → L is represented by u^{m,n} ∈ Hom(K^{m}, L^{n})^{−n+m} (non-zero only for m ≤ n) subject to the condition

(−1)^{n}∂u^{m,n} − ∑(−1)^{m+ℓ} u^{ℓ,n}∘f^{m,ℓ} + ∑(−1)^{ℓ+n} g^{ℓ,n}∘u^{m,ℓ} = 0 .

It defines the zero morphism if there exist U^{m,n} ∈ Hom(K^{m}, L^{n})^{−n+m−1} (non-zero only for m ≤ n − 1) such that

u^{m,n} = (−1)^{n}∂U^{m,n} + ∑(−1)^{m+ℓ} U^{ℓ,n}∘f^{m,ℓ} + ∑(−1)^{ℓ+n} g^{ℓ,n}∘U^{m,ℓ} .

We have the following. Let ℚ(r) = (pt, r)[2r], the Tate objects.

(1.1) Theorem. The category D_{finite}(k) has a structure of triangulated category. Moreover

(1) D_{finite}(k) has dual, tensor product, inner Hom, the unit object ℚ, and the Tate objects ℚ(r).

(2) There is a contravariant functor h : (Smooth P roj./k) → D_{finite}(k).

(3) If X is smooth and projective, one has

Hom_{D_{finite}(k)}(ℚ, h(X)(r)[2r − m]) = K_{m}(X)_{ℚ}^{(r)} .

Here the right hand side is an Adams-graded piece of the K-group of X.

(4) There is the cycle complex functor Z^{0} : D_{finite}(k) → D(ℚ).

For the rest we will simply write D(k) for D_{finite}(k).

§2. Etale realization.

For a smooth projective variety X over a field k and ℓ ≠ ch k, we have the ℓ-adic cohomology H^{∗}(X, ℚ_{ℓ}(r)). We use complexes calculating the etale cohomology, which behave well with respect to composition of correspondences. Let X, Y be smooth projective, and D any variety. One can define a complex of ℚ_{ℓ}-vector spaces Hom(X, Y)_{D}(r) satisfying the following properties (cf. [Ha, II, §5] for details in case of Betti cohomology).

(1) H^{i} Hom(X, Y)_{D}(r) = H^{i+2r}(X × Y × D, ℚ_{ℓ}(r)). If the first variety of the pair is pt = Spec k, Hom(pt, X)_{pt}(r) = Γ(X, C^{•}(ℚ_{ℓ}(r)) )[2r] := \underleftarrow{lim} Γ (X, C^{•}(ℤ/ℓ^{ν}(r))) ⊗ ℚ_{ℓ}[2r]. Here C^{•} denotes the Godement resolution.

(2) There is a map of complexes

Hom(X, Y)_{D}(r) ⊗ Hom(Y, Z)_{D}(s) → Hom(X, Z)_{D}(r + s − dim Y)

which gives rise to the composition of correspondences

H^{∗}(X × Y × D, ℚ_{ℓ}(r)) ⊗ H^{∗}(Y × Z × D, ℚ_{ℓ}(r)) → H^{∗}(X × Z, ℚ_{ℓ}(r + s − dim Y)) .

The map is associative.

(3) To a map α : D′ → D, there corresponds to a map of complexes α^{∗} : Hom(X, Y)_{D}(r) → Hom(X, Y)_{D′}(r).

There is also the supported theory. Given a closed set V ⊂ X ×Y ×D, there is a subcomplex Hom(X, Y)_{V,D}(r) ⊂ Hom(X, Y)_{D}(r) satisfying H^{i} Hom(X, Y)_{V,D}(r) = H_{V}^{i+2r}(X××D, ℚ_{ℓ}(r)), and properties analogous to (2) and (3).

Now we take as D the cubical scheme □^{•}. For r, s ∈ ℤ, we define a double complex C ((X, r), (Y, s))^{a,b} as follows.

C ((X, r), (Y, s))^{•,b} = Hom(X, Y)_{□^{−b}}(s − r + dim X)^{Alt}

where Alt denotes the alternating part with respect to the action of G_{−b} on □^{−b}. Define

∂ : C^{• b}((X, r), (Y, s)) → C^{• b+1}((X, r), (Y, s))

to be the alternating sum of the maps induced by the face maps □^{−b−1} → □^{−b}. Denote the associated simple complex by C ((X, r), (Y, s)) with differential D = d + (−1)^{b}∂.

The above definitions can be extended to finite symbols. For K and L finite symbols we have the complex C(K, L). Set C(X, r) = C ((pt, 0), (X, r)) = Hom(pt, X)(r), and C(K) = C ((pt, 0), K). There is composition map

C(K, L) ⊗ C(L, M) → C(K, M) , f ⊗ g ↦ g∘f ,

satisfying associativity. An element F ∈ C(K, L)^{n} induces a map F_{∗} : C(K) → C(L)[n]. Given Z ∈ Z^{r}(X, n) define a subcomplex

\begin{aligned}C_{|Z|} (X, r)^{• •} & ⊂ C (X, r)^{• ≥−n} \\ & ⊂ C (X, r)^{• •} \end{aligned}

where C_{|Z|} (X, r)^{a,b} is defined by the support condition with respect to ∪_{δ}|δ^{∗}Z|, δ varying over the face maps δ : □^{−b} → □^{n}. Similarly given f ∈ Hom(K, L)^{−n} there is a subcomplex C_{|f|}(K, L). Here |f| denotes the support of f (we sometimes write just f). An element f ∈ Hom(K, L)^{−n} has cycle class cl(f) ∈ H^{0}C_{|f|}^{• ,b}(K, L).

(2.1) Proposition. Given an object (K^{m}, f^{m,n}) of D(k), there exist, for m < n, elements

F^{m,n} ∈ ⨁_{a+b=−(n−m−1),a≤0} C_{|f|}^{a,b}(K^{m}, K^{n})

such that its (0, −n + m + 1)-component ^{0}F^{m,n} ∈ C_{f}^{•,−(n−m−1)}(K^{m}, K^{n}) satisfies

[^{0}F^{m,n}]_{d} = cl(f^{m,n}) ∈ H^{0}C_{f}^{•,−(n−m−1)}(K^{m}, K^{n})

and one has the relation

(∗)  (−1)^{n}D(F^{m,n}) + ∑_{m<ℓ<n} F^{ℓ,n}∘F^{m,ℓ} = 0

in ⨁_{a+b=−(n−m−2),a≤0} C_{f}^{a,b}(K^{m}, K^{n}) .

We call such (F^{m,n}) a representative of (cl(f^{m,n})).

Choose (F^{m,n}); then the maps

F^{m,n}_{∗} : C(K^{m}) → C(K^{n})[−(n − m − 1)]

satisfy

(−1)^{n}D∘F^{m,n}_{∗} + F^{m,n}_{∗}∘(−1)^{m}D + ∑ F^{ℓ,n}_{∗}∘F^{m,ℓ}_{∗} = 0

where D is the differential of C(K^{m}) or C(K^{n}), namely (C(K^{m}), F^{m,n}_{∗}) is a C-complex. So we have the total complex,

C(K) := T ot (⊕C(K^{m}), D + ∑ F^{m,n}_{∗}) .

It can be shown that C(K) is well-defined independent of the choice of representatives F^{m,n}. By definition H^{∗}(K) = H^{∗}(K, ℚ_{ℓ}) := H^{∗}C(K).

(2.2) Theorem. We have the functor of ℓ-adic etale cohomology

H^{∗}(−, ℚ_{ℓ}) : D(k) → V ec_{ℚ_{ℓ}} .

For K in D(k), we have K ⊗_{k} k̄ in D(k̄); the cohomology

H^{∗}(K ⊗_{k} k̄, ℚ_{ℓ})

is a G(k̄/k)-module. We have the functor to the category of Galois modules

D(k) → (G(k̄/k) − V ec_{ℚ_{ℓ}}) .

§3. Polylogarithmic objects.

We define the category of mixed Tate motives to be the triangulated subcategory generated by ℚ(r), r ∈ ℤ.

We give two types of algebraic cycles, each parametrized by a ∈ k^{∗}. For a ∈ k^{∗}, f_{a} := Alt{t = a} ∈ Z^{1}(pt, 1), the alternation of the cycle {t = a}. For a ∈ k^{∗} − {1} and r ≥ 2, using cubical coordinates and parameters t_{1}, ⋯, t_{r−2},

V_{a}^{r} := [t_{1}, t_{2}, ⋯, t_{r−1}, 1 − t_{1}, 1 − \frac{t_{2}}{t_{1}} , ⋯, 1 − \frac{t_{r−1}}{t_{r−2}} ]

and C_{a}^{r} := AltV_{a}^{r} ∈ Z^{r}(pt, 2r − 1). This was considered by Totaro (r = 2) and Bloch (r ≥ 3). Note

∂C_{a}^{r} = \{ \begin{matrix}f_{1−a}∘f_{a} & if r = 2 \\ C_{a}^{r−1}∘f_{a} & if r ≥ 3 \end{matrix}

Define the object

L(a) := [(pt, 0) →^{f_{a}} (pt, 1)]

where (pt, 0), (pt, 1) are placed in degrees 0 and 2, respectively. More precisely, L(a) = (L^{m}, f^{m,n}) consists of

L^{2m} = (pt, m) for m = 0, 1, and L^{n} = 0 otherwise;
f^{0,2} = f_{a}, f^{m,n} = 0 otherwise.

For p ≥ 1 and a ∈ k^{∗} − {1}, define the object K_{p}(a) (called the polylogarithmic object of weight p) by

K^{2m} = (pt, m) for m = 0, 1, ⋯, p, K^{n} = 0 otherwise;
f^{0,2} = f_{1−a}, f^{2m,2m+2} = f_{a} for m = 1, ⋯, p − 1,
f^{0,2m} = C_{a}^{m} for m = 2, ⋯, p and f^{m,n} = 0 otherwise.

(3.1) Proposition. (1) H^{ν}(L(a) ⊗ k̄, ℚ_{ℓ}) = 0 for ν ≠ 0. There is an exact sequence

0 → ℚ_{ℓ}(1) → H^{0}(L(a) ⊗ k̄, ℚ_{ℓ}) → ℚ_{ℓ}(0) → 0

whose extension class is [a] ∈ H^{1}(G(k̄/k), ℚ_{ℓ}(1)).

(2) H^{ν}(K_{p}(a) ⊗_{k} k̄, ℚ_{ℓ}) = 0 for ν ≠ 0; The cohomology H^{0}(K_{p}(a) ⊗_{k} k̄, ℚ_{ℓ}) has a filtration W_{•}(the weight filtration), H^{0} = W_{0} ⊃ W_{−2} ⊃ ⋯ ⊃ W_{−2p} ⊃ W_{−2p−2} = 0 such that Gr_{−2q}^{W} = ℚ_{ℓ}(q) for q = 0, ⋯, p and the extension class of the exact sequence

0 → Gr_{−2q−2}^{W} = ℚ_{ℓ}(q + 1) → W_{−2q}/W_{−2q−4} → Gr_{−2q}^{W} = ℚ_{ℓ}(q) → 0

is [1 − a] for q = 0 and [a] for q = 1, ⋯, p − 1.

References

[Bl] S. Bloch, Algebraic cycles and the Lie algebra of mixed Tate motives, J. Amer. Math. Soc. 4 (1991), 771–791.

[Ha I] M. Hanamura, Mixed motives and algebraic cycles, I, Math. Res. Letters 2 (1995), 811–821. [Ha II] M. Hanamura, Mixed motives and algebraic cycles, II. [Ha III] M. Hanamura, Mixed motives and algebraic cycles, III, Math. Res. Letters 6 (1999), 61–82.

[To] B. Totaro, Milnor K-theory is the simplest part of algebraic K-theory, K-theory 6 (1992), 177–189.

[Ha, II] is available from www.math.tohoku.ac.jp/ h̃anamura

Masaki Hanamura

Mathematical Institute, Tohoku University