MOTIVIC DOUBLE SHUFFLE ISMAEL SOUDÈRES1

Contents Introduction 1. Integral representation of the double shuffle relations 1.1. Series representation of the stuffle relations 1.2. Integral representation of the shuffle relations 1.3. The stuffle relations in terms of integrals 2. Moduli spaces of curves; double shuffle and forgetful maps 2.1. Shuffle and moduli spaces of curves 2.2. Stuffle and moduli spaces of curves 3. Motivic double shuffle for the "convergent" words 3.1. Framed mixed Tate motives and motivic multiple zeta values 3.2. Motivic Shuffle 3.3. Motivic stuffle References

1 2 2 2 3 6 6 7 8 8 10 12 27

Introduction For a p-tuple k = (k1 , . . . , kp ) of positive integers and k1 > 2, the multiple zeta value ζ(k) is defined as X 1 ζ(k) = . kp k1 n1 >...>np >0 n1 · · · np These values satisfy two families of algebraic (quadratic) relations known as double shuffle relations, or shuffle and stuffle described below. In [GM04] A.B. Goncharov and Y. Manin defined a motivic version of multiple zeta values using certain framed mixed Tate motives attached to moduli spaces of genus 0 curves. In this context, the real multiple zeta values appear naturally as periods of those motives attached to the moduli spaces of curves. They do not prove the double shuffle relations directly, referring instead to previous work by A.B. Goncharov in which, using a different definition of motivic multiple polylogarithms based on (P1 )n rather than moduli spaces, the motivic double shuffle relations are shown via results on variations of mixed Hodge structure. The goal of this article is to give an elementary proof of the double shuffle relations directly for the Goncharov and Manin motivic multiple zeta values. The shuffle relation (Proposition 3.7) is straightforward, but for the stuffle (Proposition 3.26) we use a modification of a method first introduced by P. Cartier for the purpose of proving stuffle for the real multiple zeta values via integrals and blowup sequences. In this article, we will work over the base field Q. Date: July 30, 2008. 1 this work has been partially supported by a Marie Curie Early Stage Training fellowship. 1

ISMAEL SOUDÈRES11

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1. Integral representation of the double shuffle relations 1.1. Series representation of the stuffle relations. The stuffle product of a p-tuple k = (k1 , . . . , kp ) and a q-tuple l = (l1 , . . . , lq ) is defined recursively by the formula: (1)

k ∗ l = (k ∗ (l1 , . . . , lq−1 )) · lq + ((k1 , . . . , kp−1 ) ∗ l) · kp + ((k1 , . . . , kp−1 ) ∗ (l1 , . . . , lq−1 )) · (kp + lq )

and k ∗ () = () ∗ k = k. Here the + is a formal sum, A · a means that we concatenate a at the end of the tuple A and · is linear in A. Let k and l be two such tuples of integers. We will write st(k, l) for the set of the individual terms of the formal sum k ∗ l whose coefficients are all equal to 1, such a generic term is then denoted by σ ∈ st(k, l). In order to multiply two multiple zeta values ζ(k) and ζ(l), we split the summation domain of the product ζ(k)ζ(l) {0 < n1 < . . . < np } × {0 < m1 < . . . < mq } into all the domains that preserve the order of the ni as well as the order of the mj and into the boundary domains where some ni are equal to some mj . We obtain the following well-known proposition, giving the quadratic relations (2) between multiple zeta values known as the stuffle relations: Proposition 1.1. Let k = (k1 , . . . , kp ) and l = (l1 , . . . , lq ) as above with k1 , l1 > 2. Then we have: (2)    X X X 1 1  = ζ(σ). ζ(k)ζ(l) =  kp lq k1 l1 m1 >...>mq >0 m1 · · · mq n1 >...>np >0 n1 · · · np σ∈st(k,l) 1.2. Integral representation of the shuffle relations. To the tuple k, with n = k1 + · · · + kp , we associate the n-tuple: k = ( 0, . . . , 0 , 1, . . . , 0, . . . , 0 , 1) = (εn , . . . , ε1 ) | {z } | {z } k1 −1 times kp −1 times ωk = ωk = (−1)p

(3)

dt1 dtn ∧ ··· ∧ . t1 − ε1 tn − εn

Then, setting ∆n = {0 < t1 < . . . < tn < 1}, direct integration yields: Z ζ(k) = ωk . ∆n

The shuffle product of an n-tuple (e1 , . . . , en ) = e1 ·e and an m-tuple (f1 , . . . , fm ) = f1 · f is defined recursively by: (4)

(e1 , . . . , en )

X

X

X (f , . . . , f 1

m)

= e1 · (e

X (f

1

· f )) + f1 · ((e1 · e)

X f)

and e () = () e = e. Here, as above, the + is a formal sum, b · B means that we concatenate b at the beginning of the tuple B and · is linear in B. Let k and l be two tuples of integers as above. We will write sh(k, l) for the set of the individual terms of the formal sum k l whose coefficients are all equal to 1. Such a generic term is then denoted by σ ∈ sh(k, l) and can be identify with a unique permutation σ ˜ of {1, . . . n + m} such that σ ˜ (1) < . . . < σ ˜ (n) and σ ˜ (n + 1) < . . . < σ ˜ (n + m). The permutation σ ˜ will simply be denoted by σ when the context will be clear enough.

X

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We will put an index σ on any object which naturally depends on a shuffle. The following proposition yields the quadratic relations (5) known as the shuffle relations. Proposition 1.2. Let k = (k1 , . . . , kp ) and l = (l1 , . . . , lq ) with k1 , l1 > 2. Then: Z Z X Z (5) ωl = ωk ωσ . ∆m

∆n

σ∈sh(k,l)

∆n+m

Proof. Let n = k1 + ... + kp and m = l1 + ... + lq . Then we have: Z  Z  Z Z dt1 dtn dtn+m dtn+1 ωl = ··· ··· ωk tn tn+m ∆m ∆ 1 − t1 ∆m 1 − tn+1 ∆n Z n dt1 dtn dtn+1 dtn+m = ··· ··· . 1 − t t 1 − t tn+m 1 n n+1 ∆ The set ∆ = {0 < t1 < . . . < tn < 1} × {0 < tn+1 < . . . < tn+m < 1} can be, up to codimension 1 sets, split into a union of simplices a ∆= ∆σ with ∆σ = {0 < tσ(1) < tσ(2) < ... < tσ(n+m) < 1}, σ∈sh([[1,n]],[[n+1,m]])

where [[a, b]] denotes the ordered sequence of integers from a to b. The integral over ∆ is the sum of the integrals over the individual simplices. But the integral over one of these simplices is, up to the numbering of the variables, X Z ωσ .  exactly one term of the sum σ∈sh(k,l)

∆n+m

1.3. The stuffle relations in terms of integrals. We explain here ideasalready written in articles of Goncharov [Gon02] and in Francis Brown’s Ph.D. thesis [Bro06], showing how to express the stuffle relations (2) in terms of integrals. R 2 dt1 Example. We have ζ(2) = ∆2 dt t2 1−t1 . The change of variables t2 = x1 and t1 = x1 x2 gives: Z Z dx1 x1 dx2 dx1 dx2 ζ(2) = = . x 1 − x x 1 − x1 x2 2 2 1 1 2 [0,1] [0,1] This change of variables is nothing but the blow-up of the point (0, 0) in the projective plane, given in n dimensions by a sequence of blow-ups: (6)

tn = x1 , tn−1 = x1 x2 , . . . , t1 = x1 ...xn .

We will write dn x for dx1 · · · dxn where n is the number of variables under the integral. Using the change of variables (6) for n = 4 we write the Kontsevich forms as follows: Z Z d4 x x1 x2 d4 x ζ(4) = , ζ(2, 2) = [0,1]4 (1 − x1 x2 )(1 − x1 x2 x3 x4 ) [0,1]4 1 − x1 x2 x3 x4 and Z ζ(2)ζ(2) = [0,1]4

1 1 d4 x. (1 − x1 x2 ) (1 − x3 x4 )

For any variables α and β we have the equality: (7)

1 α β 1 = + + . (1 − α)(1 − β) (1 − α)(1 − αβ) (1 − β)(1 − βα) 1 − αβ

This identity will be the key of this section.

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Setting α = x1 x2 and β = x3 x4 and applying (7), we recover the stuffle relation:  Z x1 x2 x3 x4 ζ(2)ζ(2) = + (1 − x x )(1 − x x x x ) (1 − x x )(1 − x3 x4 x1 x2 ) 4 1 2 1 2 3 4 3 4 [0,1]  1 + d4 x 1 − x1 x2 x3 x4 ζ(2)ζ(2) = ζ(2, 2) + ζ(2, 2) + ζ(4). General case. We will show that the Cartier decomposition (9) below makes it possible to express all the stuffle relations in terms of integrals as in the example above. Let k = (k1 , . . . , kp ) and l = (l1 , . . . , lq ) two tuples of integers with k1 , l1 > 2. As above, if σ is a term of the formal sum k ∗ l, we will write σ ∈ st(k, l). We will put an index σ on any object which naturally depends on a stuffle. Let k = (k1 , . . . , kp ) be as above and n = k1 + · · · + kp . We define fk1 ,...,kp to be the function of n variables defined on [0, 1]n given by: fk1 ,...,kp (x1 , . . . , xn ) =

x1 · · · xk 1 1 1 − x1 · · · xk1 1 − x1 · · · xk1 xk1 +1 · · · xk1 +k2 x1 · · · xk1 +...+kp−1 x1 · · · xk1 +k2 ··· . 1 − x1 · · · xk1 +k2 +k3 1 − x1 · · · xk1 +···+kp

Proposition 1.3. For all p-tuples of integers (k1 , . . . , kp ) with k1 > 2, we have (with n = k1 + · · · + kp ): Z (8) ζ(k1 , . . . , kp ) = fk1 ,...,kp (x1 , . . . , xn )dn x. [0,1]n

Proof. Let ωk be the Kontsevich form associated to a p-tuple (k1 , . . . , kp ) with R n = k1 + · · · + kp , so that ζ(k1 , . . . , kp ) = ∆n ωk . i Applying the variable change (6) to ωk , we see that for each term dt ti , there arises from the t1i a term This gives the result.

1 x1 ···xn−i+1

which cancels with

dti−1 ···

=

x1 ···xn−i+1 dxn−i+2 . ···



To derive the stuffle relations in general using integrals and the functions fk1 ,...,kp , we will use the following notation. Notation. Let k be a sequence (k1 , . . . , kp ), n = k1 + · · · + kp . We have n variables x1 , . . . , x n . • For any sequence a = (a1 , . . . , ar ), we will write a = a1 · · · ar . • The sequence (x1 , . . . , xn ) will be written x. We set x(k, 1) = (x1 , . . . , xk1 ) and x(k, i) = (xk1 +···+ki−1 +1 , . . . , xk1 +···+ki ), Q

so the x is the concatenation of sequences x(k, 1) · · · x(k, p). • The sequence (x1 , . . . , xk1 +···+ki ) = x(k, 1) · · · x(k, i) will be denoted by x(k, 6 i). If k = (k0 , kp ), x0 = x(k, 6 p − 1) will be the sequence (x1 , . . . , xk1 +···+kp−1 ). • If l is a q-tuple with l1 + · · · lq = m and σ ∈ st(k, l), yσ will be the sequence in the variables x1 , . . . , xn , x01 , . . . , x0m in which each group of variables x(k, i) = (xk1 +···+ki−1 +1 , . . . , xk1 +···+ki ) (resp. x0 (l, j) = (x0l1 +···+lj−1 +1 , . . . , x0l1 +···+lj ))

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is in the position of ki (resp. lj ) in σ. Components of σ of the form ki + lj give rise to subsequences like (xk1 +···+ki−1 +1 , . . . , xk1 +···+ki , x0l1 +···+lj−1 +1 , . . . , x0l1 +···+lj ) = (x(k, i), x0 (l, j)). • Following these notations, products x1 · · · xk1 , xk1 +···+ki−1 +1 · · · xk1 +···+ki , Q Q Q x1 · · · xk1 +···+ki will be written respectively x(k, 1), x(k, i), x(k, 6 Q i). As x(k, 6 p − 1) = x0 and x(k, 6 p) = x, products x(k, 6 p − 1) and Q Q Q x(k, 6 p) will be written x0 and x. Q Q Q We remark that for each σ ∈ st(k, l), σ = x x0 . Remark 1.4. Let (k1 , . . . , kp ) = (k0 , kp ) be a sequence of integers. Then: x0 x(k, 6 p − 1) = fk1 ,...,kp−1 (x0 ) Q Q . 1 − x(k, 6 p) 1− x Q

Q

fk1 ,...,kp (x) = fk1 ,...,kp−1 (x(k, 6 p − 1))

Proposition 1.5. Let k = (k1 , . . . , kp ) and l = (l1 , . . . , lq ) be two sequences of weight n and m. Then: (9)

X

fk1 ,...,kp (x(k, 1), . . . , x(k, p)) · fl1 ,...,lq (x0 (l, 1), . . . , x0 (l, q)) =

fσ (yσ ).

σ∈st(k,l)

Proof. We proceed by induction on the depth of the sequence. The recursion formula for the stuffle is given in (1). If p = q = 1 : As we have fn (x(k, 1))fm (x0 (l, 1)) =

1−

Q

using the formula (7) with α =

1 1 1 1 · = Q Q · Q , x(k, 6 1) 1 − x0 (l, 6 1) 1 − x 1 − x0

Q

x and β =

Q

x0 leads to

(10) x x0 fn (x(k, 1))fm (x (l, 1)) = Q Q Q 0 + Q 0 Q Q (1 − x)(1 − x x ) (1 − x )(1 − x0 x) 1 + Q Q . 1 − x x0 Q

0

Q

Inductive step: Let (k1 , . . . , kp ) = (k0 , kp ) and (l1 , . . . , lq ) = (l0 , lq ) be two sequences. By Remark 1.4, the following equality holds fk0 ,kp (x0 , x(k, p))fl0 ,lq (x0 0 , x0 (l, q)) = fk0 (x0 ) Applying the formula (7) with α = the previous equation is equal to fk0 (x0 )fl0 (x0 0 ) · (

Q

x0 ·

Q

x0 0 )

Q

x and β =



Q

1−

Q

x0

Q fl0 (x0 0 )

x

x0 0 Q . 1 − x0 Q

x0 , one sees that the RHS of

Q

(1 −

Q

x Q Q x)(1 − x x0 )

x0 1 + + Q 0 Q 0Q Q Q 0 (1 − x )(1 − x x) (1 − x x ) Q

 .

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Expanding and using the Remark 1.4 we obtain: (11) fk0 ,kp (x0 , x(k, p))fl0 ,lq (x0 0 , x0 (l, q)) =
fk0 ,kp (x)fl0 (x0 0 ) ·

Q

1−

x


x0 x0 x0 0 Q 0 + fk0 (x0 )fl0 ,lq (x0 ) · Q Q x x 1 − x0 x Q

Q

Q

Q

x0 x0 0 + (fk0 (x0 )fl0 (x0 )) · Q Q . 1 − x x0 0

Q

Q

Hence, the product of functions fk1 ,...,kp and fl1 ,...,lq satisfies a recursion formula identical to the formula (1) that defines the stuffle product. Using induction, the proposition follows.  Corollary 1.6 (integral representation of the stuffle). Integrating the statement of the previous proposition over the cube and permuting the variables in each term of the LHS, we obtain: Z Z Z X X n m fσ dn+m x = ζ(σ). ζ(k)ζ(l) = fk d x fl d x = [0,1]n

[0,1]m

[0,1]n+m σ∈st(k,l)

σ∈st(k,l)

Proof. We only need to check that all integrals are convergent. As all the functions are positive on the integration domain, all changes of variable are allowed and we can deduce the convergence of each term from the convergence of the iterated integral representation for the multiple zeta values. Another argument is to remark that the orders of the poles of our functions along a codimension k subvariety is at most k. Then, for each integral, a succession of blow-up ensures that the integral converge.  2. Moduli spaces of curves; double shuffle and forgetful maps 2.1. Shuffle and moduli spaces of curves. Let k and l be as in the previous section, let n = k1 + · · · + kp and m = l1 + · · · + lq . Following the article of Goncharov and Manin [GM04], we will identify a point of M0,j+3 with a sequence (0, z1 , . . . , zj , 1, ∞), the zi being pairwise distinct and distinct from 0, 1 and ∞, and write Φj for the open cell in M0,j+3 (R) which is mapped onto ∆j , the standard simplex, by the map: M0,j+3 → (P1 )j , (0, z1 , . . . , zj , 1, ∞) 7→ (z1 , . . . , zj ). Then we have: Z ζ(k1 , . . . , kp ) = ωk . Φn

Proposition 2.1. Let β be the map defined by M0,n+m+3

−−→

β

M0,n+3 × M0,m+3

(0, z1 , . . . , zn+m , 1, ∞)

7−→

(0, z1 , . . . , zn , 1, ∞) × (0, zn+1 , . . . , zn+m , 1, ∞).

Then, letting ti be the coordinate such that ti (0, z1 , . . . , zn+m , 1, ∞) = zi , we have β ∗ (ωk ∧ ωl ) =

dt1 dtn dtn+1 dtn+m ∧ ··· ∧ ∧ ∧ ··· ∧ . 1 − t1 tn 1 − tn+1 tn+m

Furthermore, if for σ ∈ sh([[1, n]], [[n + 1, n + m]]) we write Φσn+m or Φσ for the open cell of M0,n+m+3 (R) in which the points are in the same order as their indices are in σ, we have a β −1 (Φn × Φm ) = Φσn+m . σ∈sh([[1,n]],[[n+1,n+m]])

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Proof. The first part is obvious. ` In order to show that β −1 (Φn × Φm ) = Φσn+m we have to remember that a cell in M0,n+m+3 (R) is given by a cyclic order on the marked points. Let X = (0, z1 , . . . , zn+m , 1, ∞) be a point in M0,n+m+3 (R) such that β(X) ∈ Φn × Φm . The values of the zi have to be such that (12)

0 < z1 < . . . < zn < 1 (< ∞) and 0 < zn+1 < . . . < zn+m < 1 (< ∞).

However there is no order condition relating, say z1 to zn+1 . So, points on M0,n+m+3 (R) which are in β −1 (Φna × Φm ) are such that the zi are compatible with (12). That is there are in Φσn+m .  σ∈sh([[1,n]],[[n+1,n+m]])


The open embedding β being such that Φn × Φm \ β(β −1 (Φn × Φm )) is of codimension 1, we have the following proposition P Proposition 2.2. The shuffle relation ζ(k)ζ(l) = σ∈sh(k,l) ζ(σ) is a consequence of the following change of variables: Z Z ωk ∧ ωl = β ∗ (ωk ∧ ωl ). β −1 (Φn ×Φm )

Φn ×Φm

Proof. Using the previous proposition, the right hand side of this equality is equal to Z X dtn+m dt1 ∧ ··· ∧ . σ 1 − t tn+m 1 Φn+m σ∈sh([[1,n]],[[n+1,n+m]])

Then we permute the variables and change their names in order to have an integral over Φn+m for each term. This is the same computation we did for the integral over Rn+m in proposition 1.2. dtσ(1) dtσ(n+m) dt1 n+m ∧ · · · ∧ dt As the form 1−t tn+m (resp. 1−tσ(1) ∧ · · · ∧ tσ(n+m) ) does not have any 1 pole on the boundary of Φσn+m (resp. Φn+m ), all the integrals are convergent.  2.2. Stuffle and moduli spaces of curves. In Section 1.3, we introduced cubical coordinates on Ar in order to have an other integral representation of the MZVs. Those cubical coordinates are lifted to local coordinates ui on M0,r+3 , the ui being defined by u1 = tr and ui = tr−i+1 /tr−i+2 for i < r where the ti are the usual (simplicial) coordinates on M0,r+3 . This cubical system is well adapted to express the stuffle relations on the moduli spaces of curves. Proposition 2.3. Let δ be the map defined by δ

M0,n+m+3 −−−−−→ M0,n+3 × M0,m+3 (0, z1 , . . . , zn+m , 1, ∞) 7−→ (0, zm+1 , . . . , zm+n , 1, ∞) × (0, z1 , . . . , zm , zm+1 , ∞). Writing the expression of ωk and ωl in the cubical coordinates, one finds ωk = fk (u1 , . . . , un )dn u and ωl = fl (un+1 , . . . , un+m )dm u where the fk are as in section 1.3. Then, using those coordinates we have δ ∗ (ωk ∧ ωl ) = fk1 ,...,kp (u1 , . . . , un )fl1 ,...,lq (un+1 , . . . , un+m )dn+m u and δ −1 (Φn × Φm ) = Φn+m . Proof. To prove the second statement, let X = (0, z1 , . . . , zn+m , 1, ∞) such that δ(X) ∈ Φn × Φm . Then the values of the zi ’s have to verify (13) 0 < z1 < . . . < zm < zm+1 (< ∞) and 0 < zm+1 < . . . < zn+m < 1 (< ∞).

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These conditions show that 0 < z1 < . . . < zm < zm+1 < . . . < 1 < ∞, so X ∈ Φn+m . To prove the first statement, we claim that δ is expressed in cubical coordinates by (u1 , . . . , un+m ) 7−→ (u1 , . . . , un ) × (un+1 , . . . , un+m ). It is obvious to see that for the left hand factor the coordinates are not changed. For the right hand factor we have to rewrite the expression of the right side in terms of the standard representatives on M0,m+3 . We have (0, z1 , . . . , zm , zm+1 , ∞) = (0, z1 /zm+1 , . . . , zm /zm+1 , 1, ∞) = (0, t1 , . . . , tm , 1, ∞) in simplicial coordinates. This point is given in cubical coordinates on M0,m+3 by (tm , tm−1 /tm , . . . , t1 /t2 ) = (zm /zm+1 , . . . , z1 /z2 ) = (un+1 , . . . , un+m ).  As a consequence of this discussion and the results of Section 1.3, we have the following proposition. Proposition 2.4. Using the Cartier decomposition (9), the stuffle product can be viewed as the change of variables: Z Z ωk ∧ ωl = δ ∗ (ωk ∧ ωl ). Φn ×Φm

δ −1 (Φn ×Φm )

Remark 2.5. We should point out here the fact that the Cartier decomposition "does not lie in the moduli spaces of curves", in the sense that forms appear in the decomposition which are not holomorphic on the moduli space. For example, in the Cartier decomposition of f2,1 (u1 , u2 , u3 )f2,1 (u4 , u5 , u6 ), we see the term u1 u2 u4 u5 du1 du2 du3 du4 du5 du6 (1 − u1 u2 u4 u5 )(1 − u1 u2 u3 u4 u5 u6 ) which is not a holomorphic differential form on M0,6 . However, it is a well-defined convergent form on the standard cell where it is integrated. Changing the numbering of the variables (which stabilises the standard cell) gives the equality with ζ(4, 2) . This example represents the situation in the general case: when simply dealing with integrals, the non-holomorphic forms are not a problem. However, in the context of framed motives they are. 3. Motivic double shuffle for the "convergent" words 3.1. Framed mixed Tate motives and motivic multiple zeta values. This section is a short introduction to the motivic tools we will use to prove the motivic double shuffle. The motivic context is a cohomological version of Voevodsky’s category DMQ [Voe00]. Goncharov developed in [Gon99], [Gon05] and [Gon01] an additional structure on mixed Tate motives, introduced in [BGSV90], in order to select a specific period of a mixed Tate motive. An n-framed mixed Tate motive is a mixed Tate motive M equipped with two non-zero morphisms:
∨ ∨ v : Q(−n) → GrW f : Q(0) → GrW = GrW 2n M 0 M 0 M . On the set of all n-framed mixed motives, we consider the coarsest equivalence relation for which (M, v, f ) ∼ (M 0 , v 0 , f 0 ) if there is a linear map M → M 0 respecting the frames. Let An be the set of equivalence classes and A• be the direct sum of the An . We write [M ; v; f ] for an equivalence class

MOTIVIC DOUBLE SHUFFLE

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Theorem 3.1 ([Gon05]). A• has a natural structure of graded commutative Hopf algebra over Q. A• is canonically isomorphic to the dual of Hopf algebra of all endomorphisms of the fibre functor of the Tannakian category of mixed Tate motives. In our context, the morphism v of a frame should be linked with some differential form and the morphism f is a homological counterpart of v, that is a real simplex. We give here two technical lemmas that will be used in the next sections. We write [M, v, f ] for the equivalence class of (M, v, f ) in A• . We will speak of framed mixed Tate motives in both cases. We recall that the adition of two framed mixed Tate motives [M, v, f ] and [M 0 , v 0 , f 0 ] is [M, v, f ] ⊕ [M 0 , v 0 , f 0 ] := [M ⊕ M 0 , (v, v 0 ), f + f 0 ]. Lemma 3.2. Let M be a mixed Tate motive. v, v1 , v2 : Q(−n) → GrW 2n M and ∨ f, f1 , f2 : Q(0) → GrW 0 M . We have: [M ; v; f1 + f2 ] = [M ; v; f1 ] + [M ; v; f2 ] and [M ; v1 + v2 ; f ] = [M ; v1 ; f ] + [M ; v2 ; f ] Proof. It follow directly from the definition in [Gon05]. For the first case, it is straightforward to check that the diagonal map ϕ : M → M ⊕ M is compatible with the frames. For the second equality, the map from M ⊕ M to M which sends (m1 , m2 ) to m1 + m2 gives the map between the underlying vector space and respects the frames.  Lemma 3.3. Let M and M 0 be two mixed Tate motives. Let M be framed by W ∨ 0 v : Q(−n) → GrW 2n and f : Q(0) → Gr0 M . Suppose there exists v : Q(−n) → W Gr2n M 0 and ϕ : M 0 → M compatible with v and v 0 . Then f induces a map f 0 : 0∨ and ϕ gives an equality of framed mixed Tate motives [M ; v; f ] = Q(0) → GrW 0 M 0 0 [M ; v ; f ] We recall a classical result, used in [GM04] and described more explicitly in [Gon02] that allows us to build mixed Tate motives from natural geometric situations. In [Gon02], A.B. Goncharov defined a Tate variety as a smooth projective variety M such that the motive of M is a direct sum of copies of the Tate motive Q(m) (for certain m). We say that a divisor D on M provides a Tate stratification on M if all strata of D, including D∅ = M, are Tate varieties. Let M be a smooth variety and X and Y be two normal crossing divisors on M. Let Y X be Y \ (Y ∩ X), which is a normal crossing divisor on M \ X. Lemma 3.4. Let M be a smooth variety of dimension n over Q and X ∪ Y be a normal crossing divisor on M providing a Tate stratification of M. If X and Y share no common irreducible components then there exists a mixed Tate motive: Hn (M \ X; Y X ) such that its different realisations are given by the respective relative cohomology groups. We have the following version given in [GM04]. Corollary 3.5. Let X and Y be two normal crossing divisors on ∂M0,n+3 and suppose they do not share any irreducible components. Then, any choice of nonzero elements n [ωX ] ∈ GrW 2n (H (M0,n+3 \ X));

n ∨ [ΦY ] ∈ GrW 0 (H (M0,n+3 ; Y ))

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defines a framed mixed Tate motive given by  n  H (M0,n+3 \ X; Y X ); [ωX ]; [ΦY ] . The following lemma shows that we have some flexibility inchoosing X and Y for the framed mixed Tate motive Hn (M \ X; Y X ); [ωX ]; [ΦY ] . Lemma 3.6. With the notations of the previous corollary, let X 0 be a normal crossing divisor containing X which still does not share any irreducible component with Y . Then: i  h  n 0 H (M0,n+3 \ X; Y X ); [ωX ]; [ΦY ] = Hn (M0,n+3 \ X 0 ; Y X ); [ωX ]; [ΦY ] . Suppose now that Y 0 is a normal crossing divisor containing Y which does not share any irreducible component with X 0 . Then: i h i h 0 0 Hn (M0,n+3 \ X 0 ; Y X ); [ωX ]; [ΦY ] = Hn (M0,n+3 \ X 0 ; Y 0X ); [ωX ]; [ΦY ] . We are now in a position to introduce Goncharov’s and Manin’s definition of motivic multiple zeta values. Definition 3.1. In particular, let k be a p-tuple with k1 > 2 and let Ak be the divisor of singularities of ωk . Let Bn be the Zariski closure of the boundary of Φn . The motivic multiple zeta value is defined in [GM04] by:  n  H (M0,n+3 \ Ak ; BnAk ); [ωk ]; [Φn ] 3.2. Motivic Shuffle. The map β defined in Proposition 2.1 will be the key to check that the motivic multiple zeta values satisfy the shuffle relations.This map extends continuously to the Deligne-Mumford compactification of the moduli spaces of curves: M0,n+m+3

β

−−−−−→ M0,n+3 × M0,m+3 .

Let ωk and ωl be as in section 2.1, and write Ak and Al for their respective singularity divisors. Let Bn and Bm denote the Zariski closures of the boundary of Φn and Φm respectively. For σ ∈ sh([[1, n]], [[n + 1, n + m]]), let ωσ denote the differential form which corresponds to the shuffled MZV and let Aσ denote its divisor of singularities. Let Bn+m denote the Zariski closure of the boundary of Φn+m and Bσ that of Φσ . The shuffle relations between motivic multiple zeta values are given in the following proposition. Proposition 3.7. We have an equality of framed motives:  n

 Al H M0,n+3 \ Ak ; BnAk ; [ωk ]; [Φn ] · Hm M0,m+3 \ Al ; Bm ; [ωl ]; [Φm ] = X

h   i Aσ Hn+m M0,n+m+3 \ Aσ ; Bn+m ; [ωσ ]; [Φn+m ] .

σ∈sh([[1,n]],[[n+1,n+m]])

Proof. To prove this equality, we need to display a map between the underlying vector spaces which respects the frames. We set A0 the boundary of (M0,n+3 \ Ak ) × (M0,m+3 \ Al ), it is equal to the divisor of singularities of ωk ∧ ωl on M0,n+3 × M0,m+3 . Let A0 = β −1 (A0 ) and let B0 be the Zariski closure of the boundary of Φ0 = −1 β (Φn × Φm ). Let Bn,m be the Zariski closure of the boundary of Φn × Φm . The

MOTIVIC DOUBLE SHUFFLE

11

map β gives a map: (M0,n+m+3 \ A0 ; B0A0 )

  / (M0,n+3 \ Ak ) × (M0,m+3 \ Al ); β(B0 )A0 O

β

α



?  A0 (M0,n+3 \ Ak ) × (M0,m+3 \ Al ); Bn,m .

We introduce the the right-hand inclusion α because B0 does not map onto Bn,m via β. The map α induces a map on the motivic relative cohomology groups:   ∗ 0 α Hn+m (M0,n+3 \ Ak ) × (M0,m+3 \ Al ); β(B0 )A −−→   (14) A0 Hn+m (M0,n+3 \ Ak ) × (M0,m+3 \ Al ); Bn,m The frames on the RHS of (14) is given by [Φn × Φm ] and [ωk ∧ ωl ]. Applying ˜ from Q(0) to the −2(n + m) graded lemma 3.3 to (14), [Φn × Φm ] induces a map Φ ˜ = [Φn ×Φm ], part of the LHS of (14). In fact, since α is the identity map, we have [Φ] so [Φn × Φm ] and [ωk ∧ ωl ] give a frames on the LHS of (14) which is compatible with the map α∗ . The map β induces a map on the motivic relative cohomology groups:  β∗  0 (15) Hn+m (M0,n+3 \ Ak ) × (M0,m+3 \ Al ); β(B0 )A −−→ Hn+m (M0,n+m+3 \ A0 ; B0A0 ) On the RHS of (15) the frames given by [ω0 ] where ω0 is β ∗ (ωk ∧ ωl ) and [Φ0 ] = [β −1 (Φn × Φm )] which is compatible with the map β ∗ . Now we can prove the proposition. The Künneth formula gives us a map:

Al −−−−→ Hn M0,n+3 \ Ak ; BnAk ⊗ Hm M0,m+3 \ Al ; Bm   A0 . Hn+m (M0,n+3 \ Ak ) × (M0,m+3 \ Al ); Bn,m By theorem 3.1, this map also respects the frames, so the associated framed mixed Tate motives are equal. By (14), the right-hand motive is equal to the left-hand framed motive of (14) and by (15) this framed mixed Tate motive is equal to the right-hand framed motive of (15). It remains to show that: h i (16) Hn+m (M0,n+m+3 \ A0 ; B0A0 ); [ω0 ]; [Φ0 ] = i Xh Aσ Hn+m (M0,n+m+3 \ Aσ ; Bn+m ); [ωσ ]; [Φn+m ] . σ

S In the LHS of (16), B0 being included in Bsh = σ Bσ , we can replace B0 by Bsh using lemma P 3.6. As [Φ0 ] = σ [Φσ ], lemma 3.2 shows that the LHS of 16 is equal to i Xh A0 Hn+m (M0,n+m+3 \ A0 ; Bsh ); [ω0 ]; [Φσ ] . σ

Using the fact that Bσ ⊂ Bsh and the identity map, lemma 3.6 shows that this framed motive is equal to X  Hn+m (M0,n+m+3 \ A0 ; BσA0 ); [ω0 ]; [Φσ ] . σ

As the divisor of singularities A of ω0 is included in A0 , using lemma 3.6 we can replace A0 by A in this framed motive. Then permuting the points gives an equality of

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  framed motives on each term of the sum, Hn+m (M0,n+m+3 \ A0 ; BσA0 ); [ω0 ]; [Φσ ] , with i h Aσ ); [ωσ ]; [Φn+m ] . Hn+m (M0,n+m+3 \ Aσ ; Bn+m Thus, we obtain the desired formula:  n
 
 Al H M0,n+3 \ Ak ; BnAk ; [ωk ]; [Φn ] · Hm M0,m+3 \ Al ; Bm ; [ωl ]; [Φm ] =  h  i X Aσ Hn+m M0,n+m+3 \ Aσ ; Bn+m ; [ωσ ]; [Φn+m ] . σ∈sh((1,...,n),(n+1,...,n+m))

 3.3. Motivic stuffle. The goal of this section is to be able to translate all the calculations done in Section 1.3 into a motivic context. In order to achieve this goal, we need to define, for all n greater than 2, a variety Xn → An resulting of successive blow-ups of An together with a differential form Ωsk1 ,...,kp for any tuple of integer (k1 , . . . , kp ) (with k1 + · · · kp = n) and any permutation s of [[1, n]]. After defining another but equivalent motivic counterpart of the multiple zeta values, we will show, using a natural map from Xn+m to an open subset of M0,n+m+3 , that the stuffle product is defined at a motivic level. 3.3.1. Blow up preliminaries. Lemma 3.8 (Flag Blowup Lemma; [Uly02].). Let V01 ⊂ V02 ⊂ · · · V0r ⊂ W0 be a flag of smooth subvarieties in a smooth algebraic variety W0 . For k = 1, . . . , r, define k inductively Wk as the blow-up of Wk−1 along Vk−1 , then Vkk as the exceptional i i divisor in Wk and Vk , k 6 i, as the proper transform of Vk−1 in Wk . Then the r preimage of V0 in the resulting variety Wr is a normal crossing divisor Vr1 ∪· · ·∪Vrr . If F is a flag of subvarieties V0i of a smooth algebraic variety W0 as in the previous lemma, the resulting space Ws will be denoted by BlF W0 . Theorem 3.9 ([Hu03]). Let X0 be an open subset of a nonsingular algebraic variety X. Assume that X \ X0 can be decomposed as a finite union ∪i∈I Di of closed irreducible subvarieties such that (1) For all i ∈ I, Di is smooth; (2) for all i, j ∈ I, Di and Dj meet cleanly, that is the scheme-theoretic intersection is smooth and the intersection of the tangeant space TX (Di ) ∩ TX (Dj ) is the tangeant space of the intersection TX (Di ∩ Dj ); (3) for all i, j ∈ I, Di ∩ Dj = ∅ ; or a disjoint union of Dl . The set D = {Di }i∈I is then a poset. Let k be the rank of D. Then there is a sequence of well-defined blow-ups BlD X → BlD6k−1 X → · · · → BlD60 X → X} where BlD60 X → X is the blowup of X along Di of rank 0, and, inductively, BlD6r X → BlD6r−1 X is the blowup of BlD6r−1 X along the proper transforms of Dj of rank r, such that (1) BlD X is smooth; S fi is a divisor with normal crossings; (2) BlD X \ X0 = i∈I D g g (3) For any integer k, D i1 ∪· · ·∪ Dik is non-empty if and only if Di1 ⊂ · · · ⊂ Dik fi and D fj meet if and only if form a chain in the poset D. Consequently, D Di and Dj are comparable. The fact that blow-ups are local constructions yields directly to the following

MOTIVIC DOUBLE SHUFFLE

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Corollary 3.10 (Flags blow-up sequence). Let X and D be as in the previous theorem. Let F1 , . . . , Fk be flags of subvarieties of D such that (1) F1 , . . . , Fk is a partition of D, (2) If D is in some Fi , then for all D0 ∈ D with D0 < D there exists some j 6 i such that D0 ∈ Fj . If Fji denotes the flag of the proper transform of elements of Fji−1 in BlF i−1 (· · · (BlF1 X) · · · ) , i

then BlD X = BlF k−1 (· · · (BlF1 X) · · · ) k

We will denote a such sequence of blow-up by BlFk ,...,F1 X As we want to apply these results in order to have a motivic description of the stuffle product in terms of blow-ups, we need some precisions about what sort of motives arise from the construction of Theorem 3.9. Following the notation of the article [Hu03], in particular using the proof of theorems 1.4, 1.7 and Corollary 1.6, we have the following proposition: Proposition 3.11. Suppose that X and D = ∪Di as in proposition 3.9 are such that X and all the Di are Tate varieties. Let E r+1 be the set of exceptional divisors of BlD6r X → X. Then all possible intersection of strata of Dr+1 ∪ E r+1 are Tate Varieties and so is BlD6r X Proof. Mainly following the proof of theorem 1.7 in [Hu03], we use an induction on r. If r = 0 then BlD60 X → X is the blow up along the disjoint subvarities Di of rank 0. All the exceptional divisors in E 1 are of the form P(NX Di ) (with Di of rank 0) and as the Di are Tate, so are the exceptional divisors. The Blow-up formula (17)

h(XZ ) = H(X)

d−1 M

h(Z)(−i)[−2i]

i=0

tell us that the blow-up of a Tate variety X along some Tate variety Z o f codimesion d is a Tate variety. Then BlD60 X is Tate. More over Let Di1 be an element of D1 , it is the proper transform of an element Di in D of rank bigger than 1. And theorem 1.4 in [Hu03] tells us that Di1 = BlDj ⊂Di ;rank(Dj )=0 Dj and therefore is a Tate variety We now need to show that all intersection of strata of D1 ∩ E 1 is Tate. As the centre of the Blow up were disjoint, elements in E 1 do not intersect. Let Di1 and Dj1 be two elements of D1 being the proper transform of Di and and Dj in D. If Di ∩ Dj = ∅ then the same hold for their proper transform and their is nothing to prove, else Di ∩ Dj = ∪Dl . If the maximal rank of the Dl is 0 then the lemma 2.1 in [Hu03] ensures that the proper transform have an empty intersection. If the maximal rank of the Dl is bigger than 1 the fact that Di and Dj meet cleanly ensures that the proper transform of the intersection is the intersection of the proper transform, that is Di1 ∩ Dj1 = BlDl ⊂Di ∩Dj ;rank(Dl )=0 Di ∩ Dj And the intersection is Tate. Moreover from theorem 1.4 ([Hu03]) we have Di1 ∩ Dj1 = ∪Dl1 . This allow to consider only intersection of the form E 1 ∩ Di1 with E 1 in

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E 1 and Di1 in D1 . Such an intersection is non empty if and only if E 1 comes from an element Dj of rank 0 in D. Then E 1 ∩ Di1 is P(NDi Dj ) and is a Tate variety. Assume the statement is true for BlD6r−1 X, E r and Dr . By corollary 1.6 in [Hu03], the blow-up BlD6r X → BlD6r−1 X is r (BlD6r X) −→ BlD6r−1 X. BlD60

The centre of the blow-up are the element in Dr of rank r which by assumption are Tate, as BlD6r−1 X, then BlD6r X and the new exceptional divisors are Tate. The other exceptional divisor are proper transform of element in E r and are of the form Eir+1 BlEir ∩Dlr ;rank(Dl )=r Eir with Eir in E r and Dlr in Dr coming from some Dl in D. As by induction hypothesis both Eir and Eir ∩ Dlr are Tate, Eir+1 is a Tate variety. The same argument prove that all element in Dr+1 are Tate. As previously the intersection of two element in Dr+1 is either empty or the proper transform of the intersection of two element in Dr ; again this proper transform is Tate. Theorem 1.4 tells us that the intersection Dir+1 ∩ Djr+1 of two elements of Dr+1 is either empty either the union of some elements Dlr+1 in Dr+1 . Then, to prove that all possible intersections of strata of E r+1 ∪ Dr+1 is Tate it is enough to prove that the intersection of some Dir+1 with any intersection E1r+1 ∩ · · · Ekr+1 is Tate. r If two of the Eir+1 are exceptional divisor of BlD60 (BlD6r X) → BlD6r−1 X then the intersection is empty because the corresponding strata Dir and Djr have an empty intersection (they have been separated at a previous stage). Hence at most one of Eir+1 is an exceptional divisor coming from the last blow-up r+1 and we can suppose that the strata Dir+1 , E1r+1 , . . . , Ek−1 are coming from strata r r r at the previous stage Di , E1 , . . . , Ek−1 . • Suppose that Ekr+1 is the proper transform of an exceptional divisor Ekr in E r . The subvariety Y = Dir ∩ E1r ∩ · · · Ekr is Tate by induction hypothesis and its proper transform is BlDjr ∩Y ;rank(Dj )=r Y which is a Tate variety (Djr ∩ Y is either empty or Tate and Y is Tate). On the other side the proper transform of Y is the intersection Dir+1 ∩ E1r+1 ∩ · · · ∩ Ekr+1 which is therefore Tate. • Suppose that Ekr is the exceptional divisor coming from the blow-up of r BlD6r−1 X along Djr . Let Y be the intersection Dir ∩ E1r ∩ · · · Ek−1 . Then r Dj ∩ Y is either empty or a Tate variety In the first case the intersection Dir+1 ∩ E1r+1 ∩ · · · ∩ Ekr+1 is empty. In the later case we have Dir+1 ∩ E1r+1 ∩ · · · ∩ Ekr+1 = P(NY Y ∩ Djr ) which is Tate.  3.3.2. The space Xn and some of its properties. Let n be an integer greater than 2 and for i ∈ [[1, n]] and let x1 , . . . , xn be the natural coordinates on An . We define the divisors AI , Bi0 , Bi1 , Bn , Dn1 , Dn0 and Dn as follow: • For all subset I of [[1, n]], AI is the divisor defined by Y 1− xi ; i∈I

• for all i ∈ [[1, n]],

Bi0

is the divisor defined by xi = 0;

MOTIVIC DOUBLE SHUFFLE

• • • • •

15

for all i ∈ [[1, n]], Bi1 = A{i} is the divisor defined by 1 − xi = 0; S SS Bn is the union (S i Bi0 ) ( i Bi1 ); Dn1 is the union I⊂[[1,n]] AI ; S 0 Dn0 is the union i B Si ; Dn is the union Dn0 Dn1 .

Remark 3.12. The divisor Bn is the Zariski closure of the boundary of the real cube Cn = [0, 1]n in An (R). As the divisor is not normal crossing, we would like to find a suitable succession b n over Dn . In of blow-up that will allow us to have a normal crossing divisor D order to achieve that we first need the following remark and lemma. Remark 3.13. Let I be a non-empty subset of [[1, n]] and x = (x1 , . . . , xn ) a point in AI , then the normal vector of AI at the point x is X 1 I (18) dxi . nA |x = xi i∈I

Therefore, if I and J are two distinct non-empty subsets of [[1, n]], the intersection of AI and AJ is transverse. Lemma 3.14. let I1 , . . . , Ik (k > 3) be distinct non-empty subsets of [[1, n]] and let x be a point in AI1 ∩ · · · ∩ AIk . AI

AI

A

Suppose that n|x k is in Vect(n|xI1 , . . . , n|x k−1 ) then AI1 ∩ · · · ∩ AIk−1 = AI1 ∩ · · · ∩ AIk . Proof. By assumption, there exists rational numbers α1 , . . . , αk such that AI

A

AI

n|x k = α1 n|xI1 + · · · + αk−1 n|x k−1 . Considering the expression (18), if δI is the characteristic function of I, we find that for all i in [[1, n]] δIk (i) = α1 δI1 (i) + · · · + αk−1 δIk−1 (i). Now let y = (y1 , . . . , yn ) be a point in AI1 ∩ · · · ∩ AIk−1 we have Y Y δI (i) Y Y δI (i) δ (i) yi = yi k = (yi I1 )α1 · · · (yi k−1 )αk−1 = 1 i∈Ik

i∈[[1,n]]

i∈[[1,n]]

i∈[[1,n]]

 Dn1

Lemma 3.15. Let be the poset (for the inclusion) formed by all the possible intersections of divisors AI . Then the poset Dn1 satisfy the condition (1), (2) and (3) of theorem 3.9. Proof. The intersection condition (3) follows from the definition of Dn1 . When those intersections are not empty there are isomorphic to a product of Gm therefore they are smooth. Let S1 and S2 be two elements of Dn1 . To show that S1 and S2 meet cleanly, it is enough to show that the normal bundle of the intersection is spanned by the normal bundles of S1 and S2 , that is NAn (S1 ∩ S2 ) = NAn (S1 ) + NAn (S2 ). As S1 and S2 are intersection of some AI , it is enough show that the normal bundle of AI1 ∩ · · · ∩ AIk is spanned by the normal vector of the AIj and that is ensured by lemma 3.14 and remark 3.13. 

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Applying the construction of theorem 3.9 with D = Dn1 and X = An leads to a pn variety Xn → An , which result from successive blow-up of all the strata of Dn1 such c1 of D1 is a normal crossing divisor. We will write D c1 to mean that the preimage D n n n 1 the preimage of Dn . c0 be the proper transform in Xn of the divisor D0 . Then Lemma 3.16. Let D n n S c1 D c0 is a normal crossing divisor. bn = D D n n c0 (resp. B 0 ) be the intersection in Proof. Let I be a non-empty subset of [[1, n]], B I I c1 , . . . , S ck be strata of D cn such that the Xn (resp. An ) of divisors {xi = 0}. And let S 1 b intersection of the Si is non-empty. We want to show that there is a neighbourhood c0 T S c1 T · · · T S ck such that V ∩ D b n is normal crossing. By theorem 3.9, the V of B I 1 b Si are coming from strata of Dn , S1 ⊂ · · · ⊂ Sk . As the intersection of the Sbi ’s c0 is non-empty, the intersection of B 0 with S is non-empty. There exists with B 1 I I I1 , . . . , Il T non-empty subsets of [[1, n]] such that S1 = AI1 ∩ · · · ∩ AIl . As BI0 S1 is non-empty, we have \ [ [ I (I1 · · · Il ) = ∅. T Then, in An , we have a neighbourhood V0 of BI0 S1 isomorphic to a product Ad × A|I| with d = n − |I|: Ad ∪ ˜1 D d

×

A|I| ∪ S ˜0 i∈I Bi ,

˜ 0 is the hyperplane corresponding to {xi = 0} inside A|I| . where B i c0 in Xn , it becomes isomorphic to Xd × A|I| with Lifting this neighbourhood to V cd of D cd ×A|I| . b 1 ⊂ Xd . Then, for any Sbi there is a stratum S b 1 such that V c0 ∩ Sbi ' S D i i d d cd ’s give a normal crossing divisor in X by Theorem 3.9, V c0 gives the As the S d i c0 T S c1 T · · · T S ck such that V ∩ D b n is a normal crossing divisor neighbourhood of B I in Xn .  bn denote the closure of preimage of Bn and A bn be the divisor Definition 3.2. Let B bn \ B bn . D bn and B bn do not share any irreducible components Remark 3.17. The divisors A and are both normal crossing divisors. bn be the preimage of Cn = [0, 1]n in Xn and C bn its closure. Then B bn is Let C bn and there is a non-zero class the Zariski closure of the boundary of C (19)

n bn ] ∈ GrW b [C 0 H (Xn , Bn ).

If I is a subset of [[1, n]], we define FI and GI to be the functions Q GI : (x1 , . . . , xn ) 7−→ i∈I Qxi FI : (x1 , . . . , xn ) 7−→ 1 − i∈I xi . Definition 3.3. A flag F of [[1, n]] is a collection of non-empty subsets Ij of [[1, n]] such that I1 ( . . . ( Ir . The length of the flag F is the integer r and we may say that F is an r-flag of [[1, n]]. A flag on length n will be a maximal flag. A distinguished r-flag (F, i1 , . . . ip ) will be a flag F of length r together with element i1 < . . . < ip of [[1, r]].

MOTIVIC DOUBLE SHUFFLE

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Definition 3.4. Let (F, i1 , . . . ip ) be a distinguished r-flag of [[1, n]]. Let ΩF i1 ,...,ip denote the differential form of Ω•log (An \ Dn ) defined by ΩF i1 ,...ip =

r ^

d log(gj )

j=1

where



FIj if j ∈ {i1 , . . . , ip } GIj otherwise Let k = (k1 , . . . , kp ) be a tuple of positive integers with k1 > 2 such that k1 + · · · + kp = n and s be a permutation of [[1, n]]. We define a differential form Ωk,s ∈ Ωnlog (An \ Dn ) by gj =

Ωk,s = fk1 ,...,kn (xs(1) , . . . , xs(n) ) dx1 ∧ · · · ∧ dxn . Remark 3.18. Let k and s be as in the previous definition. We associate to the pair (k, s) the maximal distinguished flag (Fk , i1 , . . . , ip ) defined by Ii = {s(1), . . . , s(i)} and ij = k1 + · · · + kj for j running from 1 to p. Then we can see that there exists an integer rs such that k Ωk,s = (−1)rs ΩF i1 ,...ip . b n of, Definition 3.5. We shall write ωiF1 ,...,ip and ωk,s for the pull back on Xn \ D respectively, the forms ΩI1 ,...,Ip and Ωk,s . Proposition 3.19. If (F, i1 , . . . , ip ) is a maximal flag of [[1, n]] such that i1 > 2 and ip = n then: F • The divisor of singularities AF i1 ,...,ip of Ωi1 ,...,ip is AIi1 ∪ · · · ∪ AIip . F bF b • The divisor of singularities A i1 ,...,ip of ωi1 ,...,ip lies in An .Thus, the divisor bn . of singularities of ωk,s lies in A Moreover, if (F, i1 , . . . , ip ) and (F 0 , i01 , . . . , i0q ) are two distinguished flags of length ip and i0q with |i1 | > 2, |i01 | > 2 and Iip , Ii0q being a partition of [[1, n]], then the 0 b divisor of singularities of ω F ∧ ω F0 0 lies in An . i1 ,...,ip

i1 ,...,iq

Let (F, i1 ( . . . ( ip ) be a flag as in the previous proposition. It is straightforward to see that AF Ii1 ,...Iip is AIi1 ∪· · ·∪AIip The following lemma from Goncharov can easily be modify to fit into our situation. Lemma 3.20 ([Gon02][lemma 3.8]). Let Y be a normal crossing divisor in a smooth b −→ X be the blow-up of an irreducible variety X and ω ∈ Ωnlog (X \ Y ). Let p : X variety Z. Suppose that the generic point of Z is different from the generic points b of strata of Y . Then p∗ ω does not have a singularity at the special divisor of X. That is: Lemma 3.21. Let Y be a normal crossing divisor in An and ω ∈ Ωnlog (An \ Y ). Let pn : Xn → An be the map of our previous construction. Suppose that the generic points of the strata of Bn that are blow-up in the construction of Xn are different from the generic points of strata of Y . Then p∗n ω does not have singularities at the bn . corresponding exceptional divisors in B It is enough to check that the divisor of singularities of ΩF i1 ,...,iP is a a normal crossing divisor and that none of its strata is a blown up strata of Bn . The divisor of singularities of ΩF i1 ,...,ip is AIi1 ∪ · · · ∪ AIip and to show it is a normal crossing divisor it is enough to show that the normal vectors of the AIij at any intersection of some of them are linearly independent. The normal vector

ISMAEL SOUDÈRES11

18

P of AIij is i∈Ii 1/xi dxi and as we have I1 ( I2 ( . . . ( Ip , they are linearly j independent. We now have to show that none of the strata of Bn that are blown up in the construction of Xn are exactly some strata of AI1 ∪· · · AIp . Let S be such a strata of Bn of codimension k. The strata S is defined by the equations xr1 = 1, . . . , xrk = 1. If IS denotes the set {r1 , . . . , rk } then, for any subset I of [[1, n]], S is included in AI if and only I is included in IS . As Ii ⊂ Ii0 for i < i0 , if S is included in a strata SA of AI1 ,...,Ip , that strata is of the form AIi1 ∪ · · · ∪ AAij with j < k because |I1 | 6 2. As a consequence, SA is of codimension at most k − 1 and S can not be a strata of AF i1 ,...,ip . We use the same argument in the case of two distinguished flags as in the lemma and the proposition 3.19 is proved. bn does not intersect the boundary of C bn in Xn (R). Proposition 3.22. The divisor A bn containing an interProof. Let S be an irreducible codimension 1 stratum of B bn . As, the divisor An intersect bn strata with the boundary of C section of some A the boundary of the real cube Cn only on strata of Bn that are of codimension at least 2, S have to be such that pn (S) is a stratum of Bn of codimension at least 2. Using the symmetry, with respect to the standard coordinates on An , we can suppose that pn (S) is defined in those coordinates by xk = xk+1 = . . . = xn . Starting from An and blowing up first the point x1 = x2 = . . . = xn = 1, then the edge x2 = x3 = . . . = xn = 1 and after that the plane x3 = x4 = . . . = xn = 1 en → An . There are natural local coordinates and so on, we obtain a variety pen : X e (s1 , . . . , sn )) on Xn such that the coordinates on An defined by zi = 1 − xi satisfy: z1 = s1 ,

z2 = s1 s2 ,

...,

zi = s1 s2 · · · si ,

...,

zn = s1 s2 · · · sn .

In the yi -coordinates the stratum xj = xj+1 = . . . = xn = 1 is yj = yj+1 = . . . = en is given by sj = 0. yn = 0 and its preimage in X For any permutation s of [[1, n]] we could apply the same construction, that is blowing the point xs(1) = xs(2) = . . . = xs(n) = 1 then the edge xs(2) = xs(3) = ens → An . The preimage of Dn . . . = xs(n) = 1 and so on, and have a variety pesn : X ens . e ns \ B esn is D ens will denote the preimage of Bn and A e ns , B ens will be denoted by D in X bn does not intersect the boundary of C bn in Xn (R) it is enough to To prove that A esn does not intersect, in X ens (R), the boundary of show that for any permutation s A the preimage of Cn . By symmetry, it is enough to show it when s is the identity en . Let C en be the preimage of Cn in X en . map and then in the case of X Let AI be a codimension 1 stratum of An , I being the set {i0 , . . . , ip } and suppose eI of the preimage of AI \ Bn that i0 < . . . < ip .We want to show that the closure A en does not intersect the boundary of C en . The k-th symmetric function will be in X denoted by σk with the following convention σ0 = 1,

σk (X1 , . . . , Xl ) = 0 if l > k

The stratum AI is defined in the xi -coordinates by 1 − xi0 · · · xip = 0 and in the zi coordinates by (20)

0=

p+1 X

(−1)k−1 σk (yi0 , yi1 . . . , yip ).

k=1

eI with the si coordinates, we define the Before giving an explicit expression of A set J0 as {1, . . . , i0 } and the sets J1 , . . . , Jp by Jk = {i0 + 1, i0 + 2, . . . , ik }

MOTIVIC DOUBLE SHUFFLE

19

for all k in [[1, p]] Q For any subset J of [[1, n]], ΠJ s will denote the product j∈J sj and we have the following relations yi0 = ΠJ0 s

and

∀k ∈ [[1, p]],

yik = ΠJ0 sΠJk s

The LHS of the equation (20) can be written, using the change of variables yi = s1 · · · si as p+1 X

(21)

(−1)k−1 σk (ΠJ0 s, ΠJ0 sΠJ1 s, . . . , ΠJ0 sΠJp s).

k=1

For any indeterminate λ one have, for any k, σk (λ, λX1 , λX2 , . . . , λXp ) = λk (σk−1 (X1 . . . , Xp ) + σk (X1 , . . . , Xp )). Then the expression (21) is equal to  Π s 1 + σ1 (ΠJ1 s, . . . , ΠJp s) J0

+

p−1 X

(−1)k (ΠJ0 s)k σk (ΠJ1 s, . . . , ΠJp s) + σk+1 (ΠJ1 s, . . . , ΠJp s)





k=1

+ (−1)p σp (ΠJ1 s, . . . , ΠJp s)



eI in the si -coordinates is then The expression of A (22) 0 = 1 + σ1 (ΠJ1 s, . . . , ΠJp s) +

p−1 X

(−1)k (ΠJ0 s)k σk (ΠJ1 s, . . . , ΠJp s) + σk+1 (ΠJ1 s, . . . , ΠJp s)





k=1

+ (−1)p σp (ΠJ1 s, . . . , ΠJp s) en is given, in the si coordinates, by s1 ∈ [0, 1] and for any i ∈ [[1, n]] The closure of C eI with codimension 1 s1 · · · si ∈ [0, 1]. It is enough to look the intersection of A e strata of the boundary of Cn . Suppose that si0 = 0 for some i0 ∈ J0 then the LHS of (22) become 1 + σ1 (ΠJ1 s, . . . , ΠJp s) eI does not intersect which is strictly positive if for any i, si > 0. So the divisor A any component of the form si0 = 0 for i0 in J0 . Then, we can suppose that si 6= 0 for all i ∈ J0 in order to study the intersection eI with the boundary of C en and the LHS of (22) can be written of A   p p Y Y

1  J0 Jj  1 − 1 − Π sΠ s + 1 − ΠJ0 sΠJj s . J 0 Π s j=1 j=1 Suppose that a point x = (s1 , . . . , sn ) with si > 0 for all i in J0 , lies in the closure e That is, for all i in [[1, n]] the product s1 s2 · · · sn is between 0 and 1 which of C. means all the product ΠJ0 sΠJj s are between 0 and 1 for j in [[1, p]] and then one

ISMAEL SOUDÈRES11

20

find the following inequalities   p Y
1 1 1 − ΠJ0 sΠJj s  6 J0 , 0 6 J0 1 − s Π s Π j=1 06

p Y


1 − ΠJ0 sΠJj s 6 1.

j=1

eI does not intersect the boundary Both term can not be equal to 0 together, thus A e of Cn the si being strictly positive or i in J0 and the proposition is proved.  Both propositions 3.19 and 3.22 lead to the following theorem and to an alternative definition for motivic multiple zeta values. Theorem 3.23. Let k = (k1 , . . . , kp ) be a tuple of integers with k1 > 2 and k1 + bs be the divisor of singularities . . .+kp = n and let s be a permutation of [[1, n]]. Let A k of the differential form ωks . Then there exist a mixed Tate motive s

bs ; B bnAbk ). Hn (Xn \ A k bn of the real n-dimensional cube in Xn The differential form ωks and the preimage C give two non zero elements  ∨ s s n n bs b Abk bn ] ∈ GrW bs b Abk [ωks ] ∈ GrW and [C 2n H (Xn \ Ak ; Bn ) 0 H (Xn \ Ak ; Bn ) The periods of the n-framed mixed Tate motive i h s bn ] bs ; B bnAbk ); [ω s ], [C ζ f r.,M (k, s) = Hn (Xn \ A k k is equal to ζ(k1 , . . . , kn ). Moreover, let (F, i1 , . . . , ip ) and (F 0 , i01 , . . . , i0q ) are two distinguished flags of length ip and i0q with |i1 | > 2, |i01 | > 2 and Iip , Ii0q being a partition of [[1, n]] and 0 F F0 bF |F A i1 ,...,lp |i01 ,...,i0q be the divisor of singularities of ωi1 ,...,ip ∧ ωi01 ,...,i0q . There exists an n-framed mixed Tate motive ζ

f r.,M

(F, i1 , . . . , ip |F

0

, i01 , . . . , i0q )

bF |F A

0

0 0

0

bF |F b i1 ,...,lp |i1 ,...,iq ), = H (Xn \ A i1 ,...,lp |i0 ,...,i0 ; Bn n

1

q

0 bn ]. the frames being given by [ωiF1 ,...,ip ∧ ωiF0 ,...,i0 ] and [C 1

q

b n is a Proof. We want to apply theorem 3.6 in [Gon02] to our particular case. As D b normal crossing divisor and as proposition 3.22 ensures that An does not intersect bn ],using Proposition 3.19, the only thing that remained to show is that we have [C a Tate stratification of Xn which is ensured by Lemma 3.25. bn is The computation of the period follows from the fact that integrating over C the same as integrating over the real cube.  The following lemma is the key to prove Lemma 3.25. Lemma 3.24. Let I1 , . . . , Ir be r subsets of [[1, n]] and X the intersection AI1 ∩ · · · ∩ AIr ⊂ An . Then, X is a Tate variety. Proof. We can assume that the equations defining the AIi are independent. If |I1 ∪ · · · ∪ Ir | = a < n then X is isomorphic to (A0I1 ∩ · ·Q · ∩ A0Ir ) × An−a ⊂ Aa × An−a 0 where the AIi are defined by the same equations, 1 − j∈Ii xj = 0, that define AIi but view in Aa instead of An .

MOTIVIC DOUBLE SHUFFLE

21

Thus, using the Künneth formula, it is enough to prove the lemma when I1 ∪ · · · ∪ Ir = [[1, n]]. We will now construct two finite morphisms Gn−r × m

r−1 Y

f

g

n−r {xdn−k ak = 1} − →X− → Gm .

k=1

Let k¯ be an algebraically closed field. The system of equations Ei

1−

:

Y

xj = 0

j∈Ii

can be reduce in the following way. The variable xn is by assumption in some Ii , and we can assume without loss of generality that i = 1, so E1 can be written xn = Q

1

j∈I1

xj

Y

=

β

(1)

(1)

xj j

(1)

0 6 |βj | 6 1 and βj

∈ Z.

j