Motivic double shuffle

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Submitted on 17 Nov 2008

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Ismaël Soudères

To cite this version:

Ismaël Soudères. Motivic double shuffle. International Journal of Number Theory, World Scientific

Publishing, 2010, 6 (2), pp.339–370. �10.1142/S1793042110002995�. �hal-00308493v3�

ISMAELSOUDÈRES

Contents

Introdu tion 1

1. Integralrepresentationofthedoubleshuerelations 2 1.1. Seriesrepresentationofthestuerelations 2 1.2. Integralrepresentationoftheshuerelations 2 1.3. Thestuerelationsintermsofintegrals 3 2. Modulispa esof urves;doubleshueand forgetfulmaps 6 2.1. Shueandmodulispa esof urves 6 2.2. Stueandmodulispa esof urves 7 3. Motivi shueforthe" onvergent"words 8 3.1. FramedmixedTatemotivesandmotivi multiple zetavalues 8

3.2. Motivi Shue 10

4. Thestue ase 12

4.1. Blow-uppreliminaries 12

4.2. Thespa e

X

n

andsomeofitsproperties 15 4.3. Analternativedenitionformotivi MZV 21

4.4. Motivi Stue 23

Referen es 26

Introdu tion

Fora

p

-tuple

k

= (k

1

, . . . , k

p

)

ofpositiveintegersand

k

1

>

2

,themultiple zeta value

ζ(k)

isdened as

ζ(k) =

X

n

1

>...>n

p

>0

1

n

k

1

1

· · · n

k

p

p

.

Thesevaluessatisfytwofamiliesofalgebrai (quadrati )relationsknownasdouble shue relations,orshueandstue,des ribedbelow.

In [GM04℄A.B. Gon harov andY. Manin dene amotivi versionof multiple zeta values using ertain framed mixed Tate motives atta hed to moduli spa es of genus

0

urves. Inthis ontext, the real multiple zeta valuesappear naturally asperiodsofthose motivesatta hedto themoduli spa esof urves. Theydonot provethedoubleshuerelationsdire tly,referringinsteadtopreviousworkbyA.B. Gon harovinwhi h,usingadierentdenitionofmotivi multiplepolylogarithms basedon

(P

1

₎

n

ratherthanmodulispa es,themotivi doubleshue relationsare shownviaresultsonvariationsofmixedHodgestru ture.

The goal of this arti le is to give an elementary proof of the double shue relations dire tlyfortheGon harovandManinmotivi multiple zetavalues. The

Date:November17,2008.

thisworkhas beenpartiallysupportedbyaMarieCurieEarlyStageTrainingfellowshipfor theAAGnetworkatDurhamUniversity.

shuerelation(Proposition3.8)isstraightforward,butforthestue(Proposition 4.24) we use a modi ation of a method rst introdu ed by P. Cartier for the purposeofprovingstuefor therealmultiple zetavaluesviaintegralsand blow-upsequen es. Inthisarti le,wewillworkoverthebaseeld

Q

.

1. Integral representation of the doubleshuffle relations 1.1. Series representation of the stue relations. The stue produ t of a

p

-tuple

k

_{= (k}

₁

_{, . . . , k}

p

)

anda

q

-tuple

l

_{= (l}

₁

_{, . . . , l}

q

)

isdened re ursivelyby the formula:

(1)

k

∗ l = (k ∗ (l

1

, . . . , l

q−1

)) · l

q

+ ((k

1

, . . . , k

p−1

) ∗ l) · k

p

+ ((k

1

, . . . , k

p−1

) ∗ (l

1

, . . . , l

q−1

)) · (k

p

+ l

q

)

and

k

∗ () = () ∗ k = k

. Herethe

+

isaformalsum,

A · a

meansthatwe on atenate

a

at theend ofthetuple

A

and

·

islinearin

A

.

Let

k

and

l

betwosu htuples of integers. Wewill write

st(k, l)

fortheset of individual termsin theformalsum

k

∗ l

whose oe ientsareallequalto

1

. Su h ageneri termisthendenoted by

σ ∈ st(k, l)

.

Inordertomultiplytwomultiplezetavalues

ζ(k)

and

ζ(l)

,wesplitthe summa-tiondomainoftheprodu t

ζ(k)ζ(l)

{0 < n

1

< . . . < n

p

} × {0 < m

1

< . . . < m

q

}

intoallthedomainsthatpreservetheorderofthe

n

i

aswellastheorderofthe

m

j

and into theboundarydomains where some

n

i

are equalto some

m

j

. Weobtain the following well-known proposition, giving the quadrati relations (2) between multiplezetavaluesknownasthestue relations:

Proposition1.1. Let

k

= (k

1

, . . . , k

p

)

and

l

= (l

1

, . . . , l

q

)

asabovewith

k

1

,

l

1

>

2

. Then we have: (2)

ζ(k)ζ(l) =





X

n

1

>...>n

p

>0

1

n

k

1

1

· · · n

k

p

p









X

m

1

>...>m

q

>0

1

m

l

1

1

· · · m

l

q

q



 =

X

σ∈st(k,l)

ζ(σ).

1.2. Integral representation of the shue relations. To the tuple

k

, with

n = k

1

+ · · · + k

p

, weasso iatethe

n

-tuple:

k = ( 0, . . . , 0

| {z }

k

1

−1

times

, 1, . . . , 0, . . . , 0

| {z }

k

p

−1

times

, 1) = (ε

n

, . . . , ε

1

)

andthedierentialform,introdu edbyKontsevi h

ω

k

= ω

_k

= (−1)

p

dt

1

t

1

− ε

1

∧ · · · ∧

dt

n

t

n

− ε

n

.

(3)

Then, setting

∆

n

= {0 < t

1

< . . . < t

n

< 1}

,dire tintegrationyields:

ζ(k) =

Z

∆

n

ω

k

.

Theshueprodu tofan

n

-tuple

(e

1

, . . . , e

n

) = e

1

·e

andan

m

-tuple

(f

1

, . . . , f

m

) =

f

1

· f

isdened re ursivelyby:

(4)

(e

1

, . . . , e

n

)

X

(f

1

, . . . , f

m

) = e

1

· (e

X

(f

1

· f)) + f

1

· ((e

1

· e)

X

f )

and

e

X

() = ()

X

e = e

. Here,asabove,the

+

isaformalsum,

b · B

meansthat we on atenate

b

atthebeginningofthetuple

B

and

·

islinearin

B

.

Let

k

and

l

be twotuples of integersas above. We will write

sh(k, l)

for the set oftheindividualtermsin theformalsum

k

X

l

whose oe ientsareallequal

to

1

. Su h a generi term is then denoted by

σ ∈ sh(k, l)

and an be identied with aunique permutation

σ

˜

of

{1, . . . n + m}

su h that

σ(1) < . . . < ˜

˜

σ(n)

and

˜

σ(n + 1) < . . . < ˜

σ(n + m)

. Thepermutation

σ

˜

will simplybedenoted by

σ

when the ontextwill be learenough.

We will put an index

σ

on any obje t whi h naturally depends on a shue. The following proposition yields thequadrati relations (5) known as theshue relations.

Proposition1.2. Let

k

= (k

1

, . . . , k

p

)

and

l

= (l

1

, . . . , l

q

)

with

k

1

,

l

1

>

2

. Then: (5)

Z

∆

n

ω

_k

Z

∆

m

ω

_l

=

X

σ∈sh(k,l)

Z

∆

n+m

ω

σ

.

Proof. Let

n = k

1

+ ... + k

p

and

m = l

1

+ ... + l

q

. Thenwehave:

Z

∆

n

ω

_k

Z

∆

m

ω

_l

=

Z

∆

n

dt

1

1 − t

1

· · ·

dt

n

t

n

 Z

∆

m

dt

n+1

1 − t

n+1

· · ·

dt

n+m

t

n+m



=

Z

∆

dt

1

1 − t

1

· · ·

dt

n

t

n

dt

n+1

1 − t

n+1

· · ·

dt

n+m

t

n+m

.

Theset

∆ = {0 < t

1

< . . . < t

n

< 1} × {0 < t

n+1

< . . . < t

n+m

< 1}

anbe,upto odimension

1

sets, splitintoaunionofsimpli es

a

σ∈sh([[1,n]],[[n+1,m]])

∆

σ

with

∆

σ

= {0 < t

σ(1)

< t

σ(2)

< ... < t

σ(n+m)

< 1},

where

[[a, b]]

denotestheorderedsequen eofintegersfrom

a

to

b

.

Theintegralover

∆

isthesumoftheintegralsovertheindividualsimpli es. But the integral overone of these simpli es is, upto the numberingof the variables, exa tlyonetermofthesum

X

σ∈sh(k,l)

Z

∆

n+m

ω

σ

.



1.3. The stue relations in terms of integrals. We explain here ideas al-readywritteninarti lesofGon harov[Gon02℄andinFran isBrown'sPh.D.thesis [Bro06℄, showinghowtoexpressthestuerelations(2)intermsofintegrals. 1.3.1. Example. Wehave

ζ(2) =

R

∆

2

dt

2

t

2

dt

1

1−t

1

. The hangeofvariables

t

2

= x

1

and

t

1

= x

1

x

2

gives:

ζ(2) =

Z

[0,1]

2

dx

1

x

1

x

1

dx

2

1 − x

1

x

2

=

Z

[0,1]

2

dx

1

dx

2

1 − x

1

x

2

.

This hange ofvariablesis nothingbut theblow-upof thepoint

(0, 0)

in the pro-je tiveplane,givenin

n

dimensionsbyasequen eofblow-ups:

(6)

t

n

= x

1

, t

n−1

= x

1

x

2

, . . . , t

1

= x

1

...x

n

.

We will write

d

n

_x

for

dx

1

· · · dx

n

where

n

is the number of variables under the integral. Usingthe hangeofvariables(6)for

n = 4

wewritetheKontsevi hforms asfollows:

ζ(4) =

Z

[0,1]

4

d

4

_x

1 − x

1

x

2

x

3

x

4

,

ζ(2, 2) =

Z

[0,1]

4

x

1

x

2

d

4

x

(1 − x

1

x

2

)(1 − x

1

x

2

x

3

x

4

)

and

ζ(2)ζ(2) =

Z

[0,1]

4

1

(1 − x

1

x

2

)

1

(1 − x

3

x

4

)

d

4

_x.

Foranyvariables

α

and

β

wehavetheequality: (7)

1

(1 − α)(1 − β)

=

α

(1 − α)(1 − αβ)

+

β

(1 − β)(1 − βα)

+

1

1 − αβ

.

This identitywillbethekeyofthisse tion.

Setting

α = x

1

x

2

and

β = x

3

x

4

andapplying(7),were overthestuerelation:

ζ(2)ζ(2) =

Z

[0,1]

4



x

1

x

2

(1 − x

1

x

2

)(1 − x

1

x

2

x

3

x

4

)

+

x

3

x

4

(1 − x

3

x

4

)(1 − x

3

x

4

x

1

x

2

)

+

1

1 − x

1

x

2

x

3

x

4



d

4

x

ζ(2)ζ(2) = ζ(2, 2) + ζ(2, 2) + ζ(4).

1.3.2. General ase. WewillshowthattheCartierde omposition(9)belowmakes itpossibletoexpressallthestuerelationsintermsofintegralsasintheexample above.

Let

k

= (k

1

, . . . , k

p

)

and

l

= (l

1

, . . . , l

q

)

betwotuplesofintegerswith

k

1

,

l

1

>

2

. As above,if

σ

isatermoftheformalsum

k

∗ l

,wewillwrite

σ ∈ st(k, l)

. Wewill put anindex

σ

onanyobje twhi hnaturallydependsonastue.

Let

k

= (k

1

, . . . , k

p

)

beasaboveand

n = k

1

+ · · · + k

p

. Wedene

f

k

1

,...,k

p

tobe thefun tion of

n

variablesdenedon

[0, 1]

n

givenby:

f

k

1

,...,k

p

(x

1

, . . . , x

n

) =

1

1 − x

1

· · · x

k

1

x

1

· · · x

k

1

1 − x

1

· · · x

k

1

x

k

1

+1

· · · x

k

1

+k

2

x

1

· · · x

k

1

+k

2

1 − x

1

· · · x

k

1

+k

2

+k

3

· · ·

x

1

· · · x

k

1

+...+k

p−1

1 − x

1

· · · x

k

1

+···+k

p

.

Proposition 1.3. For all

p

-tuples of integers

(k

1

, . . . , k

p

)

with

k

1

>

2

, we have (with

n = k

1

+ · · · + k

p

)

: (8)

ζ(k

1

, . . . , k

p

) =

Z

[0,1]

n

f

k

1

,...,k

p

(x

1

, . . . , x

n

)d

n

_x.

Proof. Let

ω

k

be the Kontsevi h form asso iated to a

p

-tuple

(k

1

, . . . , k

p

)

with

n = k

1

+ · · · + k

p

, sothat

ζ(k

1

, . . . , k

p

) =

R

∆

n

ω

k

.

Applyingthe hangeofvariables(6)to

ω

k

,weseethat forea hterm

dt

i

t

i

,there arises fromthe

1

t

i

aterm

1

x

1

···x

n−i+1

whi h an elswith

dt

i−1

···

=

x

1

···x

n−i+1

dx

n−i+2

···

.

This givestheresult.



Toderivethestuerelationsingeneralusingintegralsandthefun tions

f

k

1

,...,k

p

, wewill usethefollowingnotation.

Notation. Let

k

beasequen e

(k

1

, . . . , k

p

)

,

n = k

1

+ · · · + k

p

. Wehave

n

variables

x

1

, . . . , x

n

.

•

Foranysequen e

a

= (a

1

, . . . , a

r

)

,wewillwrite

Q

_a

= a

1

· · · a

r

.

•

Thesequen e

(x

1

, . . . , x

n

)

willbewritten

x

. Weset

x

(k, 1) = (x

1

, . . . , x

k

1

)

and

x

_{(k, i) = (x}

_k

1

+···+k

i−1

+1

, . . . , x

k

1

+···+k

i

),

sothe

x

isthe on atenationofsequen es

x

(k, 1) · · · x(k, p)

.

•

The sequen e

(x

1

, . . . , x

k

1

+···+k

i

) = x(k, 1) · · · x(k, i)

will be denoted by

x

_{(k, 6 i)}

. If

k

_{= (k}

₀

_{, k}

p

)

,

x

₀

_{= x(k, 6 p − 1)}

willbethesequen e

(x

1

, . . . , x

k

1

+···+k

p−1

).

•

If

l

isa

q

-tuplewith

l

1

+· · ·+l

q

= m

and

σ ∈ st(k, l)

,

y

σ

willbethesequen e inthevariables

x

1

, . . . , x

n

, x

′

1

, . . . , x

′

m

inwhi hea hgroupofvariables

x

_{(k, i) = (x}

k

1

+···+k

i−1

+1

, . . . , x

k

1

+···+k

i

)

(

resp.

x

′

_{(l, j) = (x}

′

l

1

+···+l

j−1

+1

, . . . , x

′

l

1

+···+l

j

))

isin thepositionof

k

i

(resp.

l

j

)in

σ

. Componentsof

σ

oftheform

k

i

+ l

j

giverisetosubsequen eslike

(x

k

1

+···+k

i−1

+1

, . . . , x

k

1

+···+k

i

, x

′

l

1

+···+l

j−1

+1

, . . . , x

′

l

1

+···+l

j

) = (x(k, i), x

′

_{(l, j)).}

•

Followingthese notations,produ ts

x

1

· · · x

k

1

,

x

k

1

+···+k

i−1

+1

· · · x

k

1

+···+k

i

,

x

1

· · · x

k

1

+···+k

i

will be written respe tively

Q

_x

(k, 1)

,

Q

_x

(k, i)

,

Q

_x

(k, 6

i)

. As

x

(k, 6 p − 1) = x

0

and

x

(k, 6 p) = x

,produ ts

Q

_x

(k, 6 p − 1)

and

Q

_x

(k, 6 p)

willbewritten

Q

_x

0

and

Q

_x

. Weremarkthat forea h

σ ∈ st(k, l)

,

Q

σ =

Q

x

Q

x

′

.

Remark 1.4. Let

(k

1

, . . . , k

p

) = (k

0

, k

p

)

beasequen eofintegers. Then:

f

k

1

,...,k

p

(x) = f

k

1

,...,k

p−1

(x(k, 6 p − 1))

Q

_x

(k, 6 p − 1)

1 −

Q

x

_{(k, 6 p)}

= f

k

1

,...,k

p−1

(x

0

)

Q

_x

0

1 −

Q

x

.

Proposition 1.5. Let

k

= (k

1

, . . . , k

p

)

and

l

= (l

1

, . . . , l

q

)

be two sequen es of weight

n

and

m

. Then:

(9)

f

k

1

,...,k

p

(x(k, 1), . . . , x(k, p)) · f

l

1

,...,l

q

(x

′

_{(l, 1), . . . , x}

′

_{(l, q)) =}

X

σ∈st(k,l)

f

σ

(y

σ

).

Proof. Wepro eedbyindu tion onthe depthof thesequen e. There ursion for-mulaforthestueisgivenin (1).

If

p

= q = 1

: Aswehave

f

n

(x(k, 1))f

m

(x

′

(l, 1)) =

1

1 −

Q

x

_{(k, 6 1)}

·

1

1 −

Q

x

′

_{(l, 6 1)}

=

1

1 −

Q

x

·

1

1 −

Q

x

′

,

using theformula(7)with

α =

Q

_x

and

β =

Q

_x

_′

leadsto (10)

f

n

(x(k, 1))f

m

(x

′

(l, 1)) =

Q

_x

(1 −

Q

x

_{)(1 −}

Q

x

Q

x

′

₎

+

Q

_x

_′

(1 −

Q

x

′

_{)(1 −}

Q

x

′

Q

x

₎

+

1

1 −

Q

x

Q

x

′

.

Indu tive step: Let

(k

1

, . . . , k

p

) = (k

0

, k

p

)

and

(l

1

, . . . , l

q

) = (l

0

, l

q

)

be two sequen es. ByRemark 1.4,thefollowingequalityholds

f

k

0

,k

p

(x

0

, x(k, p))f

l

0

,l

q

(x

0

′

_{, x}

′

_{(l, q)) = f}

k

0

(x

0

)

Q

_x

0

1 −

Q

x

f

l

0

(x

0

′

₎

Q

_x

_′

0

1 −

Q

x

′

.

Applying the formula(7) with

α =

Q

_x

and

β =

Q

_x

_′

, one seesthat theRHS of thepreviousequationisequalto

f

k

0

(x

0

)f

l

0

(x

0

′

_{) · (}

Q

_x

0

·

Q

x

′

0

)



Q

_x

(1 −

Q

x

_{)(1 −}

Q

x

Q

x

′

₎

+

Q

_x

_′

(1 −

Q

x

′

_{)(1 −}

Q

x

′

Q

x

₎

+

1

(1 −

Q

x

Q

x

′

₎



.

Expanding andusingRemark1.4weobtain: (11)

f

k

0

,k

p

(x

0

, x(k, p))f

l

0

,l

q

(x

0

′

_{, x}

′

_{(l, q)) =}

f

k

0

,k

p

(x)f

l

0

(x

0

′

₎

_·

Q

_x

Q

_x

′

0

1 −

Q

x

Q

x

′

+ f

k

0

(x

0

)f

l

0

,l

q

(x

′

₎

_·

Q

_x

′

Q

_x

0

1 −

Q

x

′

Q

x

+ (f

k

0

(x

0

)f

l

0

(x

0

′

_{)) ·}

Q

_x

0

Q

x

′

0

1 −

Q

x

Q

x

′

.

Hen e, the produ t of fun tions

f

k

1

,...,k

p

and

f

l

1

,...,l

q

satises a re ursion for-mulaidenti altotheformula(1)deningthestueprodu t. Usingindu tion,the

propositionfollows.



Corollary 1.6(integralrepresentationofthestue). Integratingthestatementof the previous propositionoverthe ubeandpermuting the variables inea hterm of the RHS, weobtain:

ζ(k)ζ(l) =

Z

[0,1]

n

f

k

d

n

x

Z

[0,1]

m

f

l

d

m

x =

Z

[0,1]

n+m

X

σ∈st(k,l)

f

σ

d

n+m

x =

X

σ∈st(k,l)

ζ(σ).

Proof. Weonlyneedto he kthatallintegralsare onvergent. Asallthefun tions arepositiveontheintegrationdomain,allvariable hangesareallowed,andwe an dedu ethe onvergen eofea htermfromthe onvergen eoftheiteratedintegral representationforthemultiplezetavalues.

Anotherargumentistoremarkthattheordersofthepolesofourfun tionsalong a odimension

k

subvariety anbeatmost

k

. Then,forea hintegral,asu ession ofblow-upsensuresthattheintegral onverges.



2. Modulispa esof urves; doubleshuffle andforgetful maps 2.1. Shue and moduli spa es of urves. Let

k

and

l

be asin the previous se tion,let

n = k

1

+· · ·+k

p

and

m = l

1

+· · ·+l

q

. Followingthearti leofGon harov andManin[GM04℄,wewillidentify apointof

M

0,j+3

,themodulispa eof urves of genus

0

with

j + 3

marked points, with asequen e

(0, z

1

, . . . , z

j

, 1, ∞)

, the

z

i

being pairwise distin t and distin t from

0

,

1

and

∞

, and write

Φ

j

for theopen ell in

M

0,j+3

(R)

whi h is mapped onto

∆

j

, the standardsimplex, by the map:

M

0,j+3

→ (P

1

)

j

,

(0, z

1

, . . . , z

j

, 1, ∞) 7→ (z

1

, . . . , z

j

)

. Thenwehave:

ζ(k

1

, . . . , k

p

) =

Z

Φ

n

ω

k

.

Proposition2.1. Let

β

bethe map denedby

M

0,n+m+3

β

−−→

M

0,n+3

× M

0,m+3

(0, z

1

, . . . , z

n+m

, 1, ∞)

7−→

(0, z

1

, . . . , z

n

, 1, ∞) × (0, z

n+1

, . . . , z

n+m

, 1, ∞).

Then, letting

t

i

bethe oordinatesu hthat

t

i

(0, z

1

, . . . , z

n+m

, 1, ∞) = z

i

,wehave

β

∗

(ω

k

∧ ω

l

) =

dt

1

1 − t

1

∧ · · · ∧

dt

n

t

n

∧

dt

n+1

1 − t

n+1

∧ · · · ∧

dt

n+m

t

n+m

.

Furthermore, if for

σ ∈ sh([[1, n]], [[n + 1, n + m]])

wewrite

Φ

σ

n+m

for the open ell of

M

0,n+m+3

(R)

inwhi h the pointsare inthe same order astheir indi es are in

σ

,we have

β

−1

(Φ

n

× Φ

m

) =

a

σ∈sh([[1,n]],[[n+1,n+m]])

Φ

σ

n+m

.

Proof. Therstpartisobvious. In order to showthat

β

−1

_(Φ

n

× Φ

m

) =

`

Φ

σ

n+m

we haveto remember that a ell in

M

0,n+m+3

(R)

is given by a y li order on the marked points. Let

X =

(0, z

1

, . . . , z

n+m

, 1, ∞)

be a point in

M

0,n+m+3

(R)

su h that

β(X) ∈ Φ

n

× Φ

m

. Thevaluesofthe

z

i

havetobesu hthat

(12)

0 < z

1

< . . . < z

n

< 1 (< ∞)

and

0 < z

n+1

< . . . < z

n+m

< 1 (< ∞).

Howeverthereisnoorder onditionrelating,say

z

1

to

z

n+1

.

So,pointson

M

0,n+m+3

(R)

whi harein

β

−1

_(Φ

n

× Φ

m

)

aresu hthatthe

z

i

are ompatible with(12). That is,theyarein

a

σ∈sh([[1,n]],[[n+1,n+m]])

Φ

σ

n+m

.



As

Φ

n

× Φ

m

\ β(β

−1

_(Φ

n

× Φ

m

))

is of odimension

1

, we havethe following proposition.

Proposition2.2. The shuerelation

ζ(k)ζ(l) =

P

σ∈sh(k,l)

ζ(σ)

isa onsequen e of the following hange ofvariables:

Z

Φ

n

×Φ

m

ω

k

∧ ω

l

=

Z

β

−1

_(Φ

n

×Φ

m

)

β

∗

(ω

k

∧ ω

l

).

Proof. Usingthepreviousproposition,therighthandsideofthisequalityisequal to

X

σ∈sh([[1,n]],[[n+1,n+m]])

Z

Φ

σ

n+m

dt

1

1 − t

1

∧ · · · ∧

dt

n+m

t

n+m

.

Thenwepermutethevariablesand hangetheirnamesinordertohaveanintegral over

Φ

n+m

for ea h term. This is the same omputation we did for the integral over

R

n+m

in proposition1.2. Astheform

dt

1

1−t

1

∧ · · · ∧

dt

n+m

t

n+m

(resp.

dt

σ(1)

1−t

σ(1)

∧ · · · ∧

dt

σ(n+m)

t

σ(n+m)

)doesnothaveany polesontheboundaryof

Φ

σ

n+m

(resp.

Φ

n+m

),alltheintegralsare onvergent.



2.2. Stue and moduli spa es of urves. In Se tion 1.3, in order to have an integral representation of the stue produ t, we re alled, using the integral over a simplex and a hange of variables, a ubi al representation of the MZVs (integralovera ube).Weusehereasimilar hangeofvariablestointrodu eanother system of lo al oordinateson

M

0,r+3

, theDeligne-Mumford ompa ti ation of themodulispa eof urves. Following[Bro06℄,wewillspeakof ubi al oordinates. Those ubi al oordinates,

u

i

,aredenedonanopensubsetof

M

0,r+3

by

u

1

= t

r

and

u

i

= t

r−i+1

/t

r−i+2

for

i < r

wherethe

t

i

aretheusual(simpli ial) oordinates on

M

0,r+3

. This ubi alsystemis welladapted toexpressthestue relationson themodulispa esof urves.

Proposition2.3. Let

δ

bethe map denedby

M

0,n+m+3

−−−−−→ M

δ

0,n+3

× M

0,m+3

(0, z

1

, . . . , z

n+m

, 1, ∞) 7−→ (0, z

m+1

, . . . , z

m+n

, 1, ∞) × (0, z

1

, . . . , z

m

, z

m+1

, ∞).

Writing the expression of

ω

k

and

ω

l

in ubi al oordinates, one nds

ω

k

=

f

k

(u

1

, . . . , u

n

)d

n

u

and

ω

l

= f

l

(u

n+1

, . . . , u

n+m

)d

m

_u

where the

f

k

are as in se -tion1.3. Then,using those oordinates wehave

δ

∗

(ω

k

∧ ω

l

) = f

_k

₁

_,...,k

_p

(u

₁

, . . . , u

_n

)f

_l

₁

_,...,l

_q

(u

_n+1

, . . . , u

_n+m

)d

n+m

u

and

Proof. Toprovethese ondstatement,let

X = (0, z

1

, . . . , z

n+m

, 1, ∞)

besu hthat

δ(X) ∈ Φ

n

× Φ

m

. Thenthevaluesofthe

z

i

'shavetoverify (13)

0 < z

1

< . . . < z

m

< z

m+1

(< ∞)

and

0 < z

m+1

< . . . < z

n+m

< 1 (< ∞).

These onditions show that

0 < z

1

< . . . < z

m

< z

m+1

< . . . < 1 < ∞

, so

X ∈ Φ

n+m

.

Toprovetherststatement,we laimthat

δ

isexpressedin ubi al oordinates by

(u

1

, . . . , u

n+m

) 7−→ (u

1

, . . . , u

n

) × (u

n+1

, . . . , u

n+m

).

Itisobvioustoseethatforthelefthandfa torthe oordinatesareun hanged. For therighthandfa torwehavetorewritetheexpressionoftherightsideintermsof thestandardrepresentativeson

M

0,m+3

. Wehave

(0, z

1

, . . . , z

m

, z

m+1

, ∞) = (0, z

1

/z

m+1

, . . . , z

m

/z

m+1

, 1, ∞) = (0, t

1

, . . . , t

m

, 1, ∞)

in simpli ial oordinates. Thispointisgivenin ubi al oordinateson

M

0,m+3

by

(t

m

, t

m−1

/t

m

, . . . , t

1

/t

2

) = (z

m

/z

m+1

, . . . , z

1

/z

2

) = (u

n+1

, . . . , u

n+m

).



As a onsequen eof thisdis ussion andthe resultsofSe tion 1.3, wehavethe followingproposition.

Proposition 2.4. Using the Cartier de omposition (9), the stue produ t an be viewedasthe hangeof variables:

Z

Φ

n

×Φ

m

ω

k

∧ ω

l

=

Z

δ

−1

_(Φ

n

×Φ

m

)

δ

∗

(ω

k

∧ ω

l

).

Remark 2.5. We should point out here the fa t that the Cartier de omposition "doesnotlieinthemodulispa esof urves",inthesensethatformsappearinthe de omposition whi h are not holomorphi onthe moduli spa e. For example, in theCartierde ompositionof

f

2,1

(u

1

, u

2

, u

3

)f

2,1

(u

4

, u

5

, u

6

)

,weseetheterm

u

1

u

2

u

4

u

5

du

1

du

2

du

3

du

4

du

5

du

6

(1 − u

1

u

2

u

4

u

5

)(1 − u

1

u

2

u

3

u

4

u

5

u

6

)

whi hisnotaholomorphi dierentialformon

M

0,6

. However,itisawell-dened onvergentform on the standard ell where it is integrated. Changing the num-beringof thevariables(whi h stabilises thestandard ell) givestheequalitywith

ζ(4, 2)

. This example represents the situation in the general ase: when simply dealingwithintegrals,thenon-holomorphi forms arenotaproblem. However,in the ontextofframed motives,theyare.

3. Motivi shuffle forthe " onvergent" words

3.1. Framed mixed Tate motives and motivi multiple zeta values. This se tionisashortintrodu tiontothemotivi toolswewillusetoprovethemotivi double shue. The motivi ontext is a ohomologi al version of Voevodsky's ategory

DM

Q

[Voe00℄. Gon harovdevelopedin[Gon99℄,[Gon05℄and[Gon01℄an additional stru ture onmixed Tatemotives,introdu ed in [BGSV90℄, in order to sele taspe i periodofamixedTatemotive.

An

n

-framed mixed Tate motiveis amixedTatemotive

M

equipped withtwo non-zeromorphisms:

v : Q(−n) → Gr

W

2n

M

f : Q(0) → Gr

W

0

M

∨

= Gr

W

0

M

∨

.

Onthesetofall

n

-framedmixedTatemotives,we onsiderthe oarsestequivalen e relation under whi h

(M, v, f ) ∼ (M

′

_{, v}

′

_{, f}

′

₎

if there is a linear map

M → M

respe tingtheframes. Let

A

n

bethesetof equivalen e lassesand

A

•

thedire t sumofthe

A

n

. Wewrite

[M ; v; f ]

foranequivalen e lass.

Theorem 3.1([Gon05℄).

A

•

has a natural stru tureof graded ommutative Hopf algebraover

Q

.

A

•

is anoni allyisomorphi tothedualoftheHopfalgebraofallendomorphisms of the brefun torof theTannakian ategoryof mixedTate motives.

Inour ontext,themorphism

v

ofaframeshouldbelinkedwithsomedierential form andthemorphism

f

isahomologi al ounterpartof

v

,thatisarealsimplex. Wegivehere two te hni al lemmas that will beused in the nextse tions. We write

[M, v, f ]

for the equivalen e lass of

(M, v, f )

in

A

•

. By a slight abuse of notation, we will speak of framed mixed Tate motivesrefering to both

(M, v, f )

and

[M, v, f ]

.

We re all that the addition of two framed mixed Tate motives

[M, v, f ]

and

[M

′

_{, v}

′

_{, f}

′

_]

isgivenby

[M, v, f ] ⊕ [M

′

, v

′

, f

′

] := [M ⊕ M

′

, (v, v

′

), f + f

′

].

Lemma 3.2. Let

M

be a mixed Tate motive.

v, v

1

, v

2

: Q(−n) → Gr

W

2n

M

and

f, f

1

, f

2

: Q(0) → Gr

W

0

M

∨

. Wehave:

[M ; v; f

1

+ f

2

] = [M ; v; f

1

] + [M ; v; f

2

]

and

[M ; v

1

+ v

2

; f ] = [M ; v

1

; f ] + [M ; v

2

; f ].

Proof. It follows dire tly from the denition in [Gon05℄. Forthe rst ase, it is straightforwardto he k that the diagonal map

ϕ : M → M ⊕ M

is ompatible withtheframes. Forthese ondequality,themapfrom

M ⊕ M

to

M

whi hsends

(m

1

, m

2

)

to

m

1

+ m

2

gives the map between the underlying ve tor spa es and

respe tstheframes.



Lemma 3.3. Let

M

and

M

′

be two mixed Tate motives. Let

M

be framed by

v : Q(−n) → Gr

W

2n

and

f : Q(0) → Gr

W

0

M

∨

. Supposethere exists

v

′

_{: Q(−n) →}

Gr

W

2n

M

′

and

ϕ : M

′

_{→ M}

ompatible with

v

and

v

′

. Then

f

indu es a map

f

′

_{: Q(0) → Gr}

W

0

M

′∨

and if

f

′

is non zero, then

ϕ

gives an equality of framed mixedTate motives

[M ; v; f ] = [M ; v

′

_{; f}

′

_]

.

We re all a lassi al result, used in [GM04℄ and des ribed more expli itly in [Gon02℄, that allowsus to buildmixed Tatemotivesfrom naturalgeometri situ-ations. In[Gon02℄, A.B.Gon harovdened aTate variety asasmoothproje tive variety

M

su hthatthemotiveof

M

isadire tsumof opiesoftheTatemotive

Q(m)

(for ertain

m

). Wesaythatadivisor

D

on

M

providesaTatestrati ation on

M

ifallstrataof

D

,in luding

D

∅

= M

,areTatevarieties.

Let

M

be asmooth varietyand

X

and

Y

betwo normal rossing divisors on

M

. Let

Y

X

denote

Y \ (Y ∩ X

),whi hisanormal rossingdivisoron

M \ X

. Lemma 3.4. Let

M

be asmooth variety of dimension

n

over

Q

and

X ∪ Y

be a normal rossing divisor on

M

providing a Tate strati ation of

M

. If

X

and

Y

share no ommon irredu ible omponents thenthereexistsamixedTate motive:

H

n

_{(M \ X; Y}

X

₎

su h that its dierent realisations are given by the respe tive relative ohomology groups.

Corollary 3.5. Let

X

and

Y

be two normal rossing divisors on

∂M

0,n+3

and suppose they do not share any irredu ible omponents. Then, any hoi e of non-zeroelements

[ω

X

] ∈ Gr

W

2n

(H

n

(M

0,n+3

\ X));

[Φ

Y

] ∈ Gr

W

0

(H

n

(M

0,n+3

; Y ))

∨

denes a framed mixedTate motive givenby



H

n

(M

0,n+3

\ X; Y

X

); [ω

X

]; [Φ

Y

]



.

Thefollowinglemma showsthat wehavesomeexibilityin hoosing

X

and

Y

fortheframedmixed Tatemotive



H

n

(M \ X; Y

X

_{); [ω}

X

]; [Φ

Y

]



. Lemma3.6. Withthe notationofLemma3.4, let

X

′

beanormal rossingdivisor ontaining

X

whi h still doesnot shareany irredu ible omponent with

Y

,

X

′

_{∪ Y}

beinganormal rossing divisor. Then:



H

n

_{(M \ X; Y}

X

_{); [ω}

X

]; [Φ

Y

]



=

h

H

n

_{(M \ X}

′

_{; Y}

X

′

_{); [ω}

X

]; [Φ

Y

]

i

.

Supposenowthat

Y

′

isanormal rossingdivisor ontaining

Y

whi hdoesnotshare any irredu ible omponentwith

X

′

,

X

′

_{∪ Y}

′

beinganormal rossingdivisor. Then:

h

H

n

(M \ X

′

; Y

X

′

); [ω

X

]; [Φ

Y

]

i

=

h

H

n

(M \ X

′

; Y

′X

′

); [ω

X

]; [Φ

Y

]

i

.

We are now in a position to introdu e Gon harov and Manin's denition of motivi multiplezetavalues.

Denition 3.7. In parti ular, let

k

be a

p

-tuple with

k

1

>

2

and let

A

k

be the divisorofsingularitiesof

ω

k

. Let

B

n

betheZariski losureoftheboundaryof

Φ

n

. Themotivi multiplezetavalueisdened in[GM04℄by:



H

n

_(M

0,n+3

\ A

k

; B

_n

A

k

); [ω

k

]; [Φ

n

]



.

3.2. Motivi Shue. The map

β

dened in Proposition 2.1 will be the key to he kthat themotivi multiplezetavaluessatisfytheshuerelations. Thismap extends ontinuouslytotheDeligne-Mumford ompa ti ationofthemodulispa es of urves:

M

0,n+m+3

β

−−−−−→ M

0,n+3

× M

0,m+3

.

Let

ω

k

and

ω

l

beasinse tion2.1,andwrite

A

k

and

A

l

fortheirrespe tivedivisors of singularities. Let

B

n

and

B

m

denote the Zariski losures of the boundary of

Φ

n

and

Φ

m

respe tively. For

σ ∈ sh([[1, n]], [[n + 1, n + m]])

, let

ω

σ

denote the dierential form whi h orresponds to the shued MZV and let

A

σ

denote its divisor of singularities. Let

B

n+m

denote the Zariski losure of the boundaryof

Φ

n+m

and

B

σ

that of

Φ

σ

n+m

. Theshue relationsbetween motivi multiple zeta valuesaregiveninthefollowingproposition.

Proposition3.8. Wehave anequality offramed motives:



H

n

M

0,n+3

\ A

k

; B

A

_n

k

; [ω

k

]; [Φ

_n

]



·



H

m

M

0,m+3

\ A

l

; B

A

_m

l

; [ω

l

]; [Φ

_m

]



=

X

σ∈sh([[1,n]],[[n+1,n+m]])

h

H

n+m



M

0,n+m+3

\ A

σ

; B

A

n+m

σ



; [ω

σ

]; [Φ

n+m

]

i

.

Proof. To prove this equality, we need to display a map between the underlying ve torspa eswhi hrespe tstheframes.

Let

A

′

be the boundary of

(M

0,n+3

\ A

k

) × (M

0,m+3

\ A

l

)

, it is equalto the divisorofsingularitiesof

ω

k

∧ ω

l

on

M

0,n+3

× M

0,m+3

.

Let

A

0

= β

−1

_(A

′

₎

and let

B

0

be the Zariski losure of the boundaryof

Φ

0

=

β

−1

_(Φ

n

× Φ

m

)

. Let

B

n,m

betheZariski losureoftheboundaryof

Φ

n

× Φ

m

. The map

β

indu esamap:

(M

0,n+m+3

\ A

0

; B

0

A

0

)

β

//



(M

0,n+3

\ A

k

) × (M

_0,m+3

\ A

l

); β(B

₀

)

A

′





(M

0,n+3

\ A

k

) × (M

0,m+3

\ A

l

); B

A

′

n,m



.

?

α

OO

Weintrodu e theverti alin lusion

α

be ause

B

0

doesnot maponto

B

n,m

via

β

. Themap

α

indu esamaponthemixed Tatemotives:

(14)

H

n+m



(M

0,n+3

\ A

k

) × (M

_0,m+3

\ A

l

); β(B

₀

)

A

′



_α

∗

−−→

H

n+m



(M

0,n+3

\ A

k

) × (M

_0,m+3

\ A

l

); B

A

′

n,m



.

TheframesontheRHSof(14)aregivenby

[Φ

n

× Φ

m

]

and

[ω

k

∧ ω

l

]

. Applying lemma3.3to(14),

[Φ

n

× Φ

m

]

indu esamap

Φ

˜

from

Q(0)

tothe

−2(n + m)

graded partoftheLHSof(14).Infa t,sin e

α

istheidentitymap,wehave

[ ˜

Φ] = [Φ

n

×Φ

m

]

, so

[Φ

n

× Φ

m

]

and

[ω

k

∧ ω

l

]

giveframes onthe LHSof (14)whi hare ompatible with themap

α

∗

.

Themap

β

indu esamaponthemixedTatemotives: (15)

H

n+m



_(M

0,n+3

\ A

k

) × (M

0,m+3

\ A

l

); β(B

0

)

A

′



β

∗

−−→

H

n+m

(M

0,n+m+3

\ A

0

; B

0

A

0

).

On the RHS of (15) the frames are given by

[ω

0

]

where

ω

0

is

β

∗

_(ω

_k

_{∧ ω}

_l

₎

and

[Φ

0

] = [β

−1

(Φ

n

× Φ

m

)]

,whi h are ompatiblewiththemap

β

∗

.

Nowwe an provetheproposition. TheKünnethformulagivesamap:

H

n

M

0,n+3

\ A

k

; B

_n

A

k

⊗ H

m

M

0,m+3

\ A

l

; B

_m

A

l

−−−−→

H

n+m



(M

0,n+3

\ A

k

) × (M

0,m+3

\ A

l

); B

A

′

n,m



.

Bytheorem3.1,thismapalsorespe tstheframes,sotheasso iatedframedmixed Tatemotivesareequal. By(14),

h

H

n+m



(M

0,n+3

\ A

k

) × (M

_0,m+3

\ A

l

); B

A

′

n,m



; [ω

k

⊗ ω

l

]; [Φ

_n

× Φ

_m

]

i

isequalto

h

H

n+m



(M

0,n+3

\ A

k

) × (M

_0,m+3

\ A

l

); β(B

₀

)

A

′



; [ω

k

⊗ ω

l

]; [Φ

_n

× Φ

_m

]

i

,

whi h,using (15),isequalto

h

H

n+m

(M

0,n+m+3

\ A

0

; B

0

A

0

); [ω

0

]; [Φ

0

]

i

.

It remainsto showthat (16)

h

H

n+m

(M

0,n+m+3

\ A

0

; B

0

A

0

); [ω

0

]; [Φ

0

]

i

=

X

σ

h

H

n+m

(M

0,n+m+3

\ A

σ

; B

n+m

A

σ

); [ω

σ

]; [Φ

n+m

]

i

.

In the LHSof (16),

B

0

being in ludedin

B

sh

=

S

σ

B

σ

, we anrepla e

B

0

by

As

[Φ

0

] =

σ

[Φ

σ

n+m

]

,lemma3.2showsthat theLHSof16isequalto

X

σ

h

H

n+m

(M

0,n+m+3

\ A

0

; B

A

sh

0

); [ω

0

]; [Φ

σ

n+m

]

i

.

Using the fa t that

B

σ

⊂ B

sh

and an in lusion map, lemma 3.6 shows that this framed motiveisequalto

X

σ



H

n+m

(M

0,n+m+3

\ A

0

; B

A

σ

0

); [ω

0

]; [Φ

σ

n+m

]



.

As the divisorof singularities

A

of

ω

0

is in luded in

A

0

, using lemma3.6 we an repla e

A

0

by

A

inthisframedmotive. Thenpermutingthepointsgivesanequality offramed motivesonea htermofthesum,



H

n+m

(M

0,n+m+3

\ A

0

; B

σ

A

0

); [ω

0

]; [Φ

σ

n+m

]



,

with

h

H

n+m

(M

0,n+m+3

\ A

σ

; B

A

n+m

σ

); [ω

σ

]; [Φ

n+m

]

i

.

Thus,weobtainthedesiredformula:



H

n

M

0,n+3

\ A

k

; B

A

_n

k

; [ω

k

]; [Φ

_n

]



·



H

m

M

0,m+3

\ A

l

; B

_m

A

l

; [ω

l

]; [Φ

_m

]



=

X

σ∈sh((1,...,n),(n+1,...,n+m))

h

H

n+m



M

0,n+m+3

\ A

σ

; B

n+m

A

σ



; [ω

σ

]; [Φ

n+m

]

i

.



4. The stuffle ase

The goalof this se tion is to be able to translate all the al ulations done in Se tion 1.3intoamotivi ontext. Inordertoa hievethisgoal,weneedtodene, for all

n

greaterthan

2

, avariety

X

n

→ A

n

resultingfrom su essiveblow-upsof

A

n

together with adierential form

Ω

s

k

1

,...,k

p

foranytuple of integer

(k

1

, . . . , k

p

)

(with

k

1

+ · · · k

p

= n

) and any permutation

s

of

[[1, n]]

. We use the

X

n

to give another denition of the motivi multiple zeta value whi h we show is a tually equal to Gon harov-Manin's. Then, using anatural map from

X

n+m

to anopen subsetof

M

0,n+m+3

,weusethisnewdenition toprovethatthemotivi multiple zetavaluessatisfythestuerelation.

4.1. Blow-up preliminaries.

Lemma 4.1 (Flag Blow-up Lemma; [Uly02℄.). Let

V

1

0

⊂ V

0

2

⊂ · · · V

0

r

⊂ W

0

be a ag of smooth subvarieties in a smooth algebrai variety

W

0

. For

k = 1, . . . , r

, deneindu tively

W

k

astheblow-upof

W

k−1

along

V

k

k−1

,then

V

k

k

astheex eptional divisor in

W

k

and

V

i

k

,

k 6 i

, as the proper transform of

V

i

k−1

in

W

k

. Then the preimageof

V

r

0

intheresultingvariety

W

r

isanormal rossingdivisor

V

1

r

∪· · ·∪V

r

r

. If

F

is a ag of subvarieties

V

i

0

in a smooth algebrai variety

W

0

as in the previouslemma,theresultingspa e

W

s

willbedenotedby

Bl

F

W

0

.

Theorem4.2([Hu03℄). Let

X

0

beanopensubsetofanonsingularalgebrai variety

X

. Assume that

X \ X

0

an be de omposed as a nite union

∪

i∈I

D

i

of losed irredu ible subvarietiessu hthat

(1) forall

i ∈ I

,

D

i

issmooth;

(2) forall

i, j ∈ I

,

D

i

and

D

j

meet leanly,that isthes heme-theoreti inter-se tionissmoothandtheinterse tionofthetangentspa e

T

X

(D

i

)∩T

X

(D

j

)

isthetangent spa e ofthe interse tion

T

X

(D

i

∩ D

j

)

;

The set

D = {D

i

}

i∈I

is then a poset. Let

k

be the rank of

D

. Then there is a sequen eof well-denedblow-ups

Bl

D

X → Bl

D6k−1

X → · · · → Bl

D60

X → X

where

Bl

D60

X → X

is the blow-up of

X

along

D

i

of rank

0

, and, indu tively,

Bl

D6r

X → Bl

D6r−1

X

is the blow-up of

Bl

D6r−1

X

along the proper transforms of

D

j

of rank

r

,su hthat

(1)

Bl

D

X

issmooth; (2)

Bl

D

X \ X

0

=

S

i∈I

D

f

i

isadivisor withnormal rossings; (3) foranyinteger

k

,

D

g

i

1

∩ · · · ∩ g

D

i

k

isnon-emptyifandonlyif,upto number-ing,

D

i

1

⊂ · · · ⊂ D

i

k

form a hain in the poset

D

. Consequently,

D

f

i

and

f

D

j

meetifandonly if

D

i

and

D

j

are omparable.

Thefa tthatblow-upsarelo al onstru tionsyieldsdire tlythefollowing orol-lary.

Corollary 4.3 (Flagsblow-up sequen e). Let

X

and

D

be asinthe previous the-orem. Let

F

1

, . . . , F

k

be agsof subvarietiesof

D

su hthat

(1)

F

1

, . . . , F

k

isapartitionof

D

, (2) if

D

is in some

F

i

, then for all

D

′

_{∈ D}

with

D

′

_{< D}

there exists some

j 6 i

su hthat

D

′

_{∈ F}

j

. If

F

i

j

denotes the agof the proper transformof elements of

F

i−1

j

in

Bl

F

i−1

i

(· · · (Bl

F

1

X) · · · ) ,

then

Bl

D

X = Bl

F

k−1

k

(· · · (Bl

F

1

X) · · · ) .

Wewill denotesu hasequen eof blow-upsby

Bl

F

k

,...,F

1

X.

As wewantto applytheseresultsin order tohaveamotivi des riptionof the stueprodu tintermsofblow-ups,weneedsomemorepre iseinformationabout what sort of motives arise from the onstru tion of Theorem 4.2. Following the notationofthearti le[Hu03℄,inparti ularusingtheproofoftheorems1.4,1.7and Corollary1.6,wededu ethefollowingproposition:

Proposition 4.4. Suppose that

X

and

D = ∪D

i

as in proposition 4.2 are su h that

X

andallthe

D

i

areTatevarieties. Let

E

r+1

bethe setofex eptional divisors of

Bl

D

6r

X → X

. Then allpossibleinterse tionsofstrata of

D

r+1

_{∪ E}

r+1

are Tate Varietiesandsois

Bl

D

6r

X

.

Proof. Mainlyfollowingtheproofoftheorem1.7in[Hu03℄,weuseanindu tionon

r

.

If

r = 0

then

Bl

D

60

X → X

istheblow-up alongthedisjointsubvarieties

D

i

of rank

0

.

All theex eptionaldivisorsin

E

1

areoftheform

P(N

X

D

i

)

(with

D

i

ofrank

0

), andasthe

D

i

areTate,soaretheex eptionaldivisors.

Theblow-upformula

h(X

Z

) = H(X)

d−1

M

i=0

h(Z)(−i)[−2i]

(17)

tellusthattheblow-upofaTatevariety

X

alongsomeTatevariety

Z

of odimen-sion

d

is aTatevariety. Then

Bl

D

60

X

is Tate. Moreover,if

D

1

i

isan elementof

D

1

Theorem1.4in [Hu03℄tellsusthat

D

1

i

= Bl

D

j

⊂D

i

;rank(D

j

)=0

D

j

andisthereforea Tatevariety.

Wenowneed to show that all interse tions of strataof

D

1

_{∩ E}

1

are Tate. As

Bl

_D

60

X → X

isablow-up alongdisjointsubvarieties,theex eptionaldivisorsare disjointandwe on ludethatelementsin

E

1

donotinterse t. Let

D

1

i

and

D

1

j

be twoelementsof

D

1

; they arethe proper transformsof

D

i

and

D

j

in

D

. If

D

i

∩ D

j

= ∅

then,thesameholdfortheirpropertransformsand there isnothingto prove. Otherwise,byassumption,

D

i

∩ D

j

isaadisjointunion

∪D

l

. If themaximal rankof the

D

l

is

0

then Lemma 2.1 in [Hu03℄ ensuresthat the propertransformshaveanemptyinterse tion. Ifthemaximal rankofthe

D

l

is greater than

1

the fa t that

D

i

and

D

j

meet leanly ensures that the proper transformoftheinterse tionistheinterse tionofthepropertransforms,that is

D

1

i

∩ D

1

j

= Bl

D

l

⊂D

i

∩D

j

;rank(D

l

)=0

D

i

∩ D

j

and theinterse tion is Tate be ause

D

i

∩ D

j

is a disjoint union of

D

l

whi h are Tate. Moreoverfromtheorem1.4([Hu03℄)wehave

D

1

i

∩ D

1

j

= ∪D

1

l

. Thuswe an onsideronlyinterse tionsoftheform

E

1

_{∩ D}

1

i

with

E

1

in

E

1

and

D

1

i

in

D

1

. Su h aninterse tionisnonemptyifandonlyif

E

1

omesfromanelement

D

j

ofrank

0

in

D

with

D

j

⊂ D

i

. Then

E

1

_{∩ D}

1

i

is

P(N

D

i

D

j

)

andisaTatevariety. Assumethe statementis truefor

Bl

D

6r−1

X

,

E

r

and

D

r

. By orollary1.6 in [Hu03℄, theblow-up

Bl

D

6r

X → Bl

D

6r−1

X

is

Bl

D

r

60

(Bl

D

6r

X) −→ Bl

D

6r−1

X.

Thisisablow-upalongelementsin

D

r

ofrank

r

whi hbyassumptionareTate, as

Bl

D

6r−1

X

. Then,

Bl

D

6r

X

andthenewex eptionaldivisorsareTate. Theother ex eptionaldivisorsarepropertransformsofelementsin

E

r

andareoftheform

E

r+1

_i

= Bl

E

r

i

∩D

r

l

;rank(D

l

)=r

E

r

i

with

E

r

i

in

E

r

and

D

r

l

in

D

r

oming from some

D

l

in

D

. As by the indu tion hypothesis both

E

r

i

and

E

r

i

∩ D

r

l

are Tate,

E

r+1

i

is a Tate variety. The same argumentprovesthatallelementsin

D

r+1

areTate. Aspreviouslytheinterse tion oftwoelementsin

D

r+1

iseitheremptyorthepropertransformoftheinterse tion oftwoelementsin

D

r

,againthispropertransformisTate. Theorem1.4tellsusthattheinterse tion

D

r+1

i

∩ D

r+1

j

oftwoelementsof

D

r+1

is eitheremptyortheunionofsomeelements

D

r+1

l

in

D

r+1

. Then,to provethat allpossibleinterse tionsofstrataof

E

r+1

_{∪ D}

r+1

isTate,itisenoughtoprovethat theinterse tion ofsome

D

r+1

i

withanyinterse tion

E

r+1

1

∩ · · · E

k

r+1

isTate. If two of the

E

r+1

i

are ex eptional divisors of

Bl

D

r

60

(Bl

D

6r

X) → Bl

D

6r−1

X

then the interse tion is empty be ausethe orresponding strata

D

r

i

and

D

r

j

have anemptyinterse tion(they havebeenseparatedatapreviousstage).

Hen e at most oneof the

E

r+1

i

is an ex eptionaldivisor omingfrom the last blow-up,andwe ansupposethatthestrata

D

r+1

i

, E

r+1

1

, . . . , E

r+1

k−1

omefromstrata at thepreviousstage

D

r

i

, E

1

r

, . . . , E

k−1

r

.

•

Supposethat

E

r+1

k

isthepropertransformofanex eptionaldivisor

E

r

k

in

E

r

. Thesubvariety

Y = D

r

i

∩E

1

r

∩· · · E

k

r

isTatebytheindu tionhypothesis anditspropertransformis

Bl

D

r

j

∩Y ;rank(D

j

)=r

Y

whi hisaTatevariety(

D

r

j

∩ Y

iseitheremptyorTateand

Y

isTate). On theothersidethepropertransformof

Y

istheinterse tion

D

r+1

i

∩ E

r+1

1

∩