Non-abelian unipotent periods

Monodromy of iterated integrals

Zdzis law Wojtkowiak

§0. Introduction.

0.1. Let X be a smooth, algebraic variety defined over a number field k. Let σ : k → C

be an inclusion. We set XC = X ×k

C. Let X(C) be the set of C-points of XC with its

complex topology. There is the canonical isomorphism

pcomp : HnB(X(C))⊗ C

≈−→ HnDR(X)⊗

kC

between Betti (singular) cohomology and algebraic De Rham cohomology. The period

matrix (pij) is defined by equations ωi =∑pjiσj where ωi and σi are bases of Hn

DR(X)

and HnB(X(C)) or pji =

∫σ∗

jωi where σ∗

j is the dual base of Hn(X(C)).

Let us assume that X is an abelian variety. Let G be the largest subgroup of

GL(H1B(X(C))⊗Q)×Gm which fixes all tensors clB(Z), where Z is an algebraic cycle on

some Xn (see [D2]). Let P be the functor of k-algebras, such that any element of P (A) is

an isomorphism p : H1B(X(C))⊗A→ H1

DR(X)⊗A mapping clB(Z)⊗1 to clDR(Z)⊗1 for

any algebraic cycle Z on any Xn. The isomorphism pcomp belongs to P (C), the functor

P is represented by an algebraic variety over k, which is a Gk-torsor under the natural

action. It is a subtorsor of the GL(H1B(X(C)⊗ k)-torsor Iso(H1

B(X(C))⊗ k,H1DR(X)).

Let T be a smallest subtorsor defined over k of the torsor Iso(H1B(X(C))⊗k,H1

DR(X)),

which contains pcomp as a C-point and let G be the corresponding subgroup (defined over

k) of GL(H1B(X(C))⊗ k). Let Z(pcomp) be the Zariski closure of pcomp in Iso(H1

B(X(C)⊗k,H1

DR(X)) i.e. the smallest Zariski closed subset defined over k, which contains pcomp as

a C-point.

Then we have

Z(pcomp) ⊂ T ⊂ P

1

and

G ⊂ G.

In order to calculate Z(pcomp) and to show that Z(pcomp) = P one need to show that

certain number are transcendental. On the other hand to calculate T and G seems to be

an easier task. The requirement that T is a subtorsor of Iso(H1B(X(C))⊗ k,H1

DR(X)) is

very strong and usually relatively weak informations about numbers pij are necessary

to calculate T and G. We give an obvious example. If X = P 1Q then in order to show that

Z(pcomp) ≈ Gm we must know that 2πi is transcendental. But already the fact that 2πi

is not a kth-root of a rational number for any k ∈ N implies that T ≈ Gm and G = Gm.

In this note we shall discuss periods for fundamental groups. We shall concentrate on

analogues of T and G for fundamental groups. On the other hand we have no analogue of

P and G.

The plan of the paper.

0. Introduction.

1. Torsors.

2. Torsors associated to non-abelian unipotent periods.

3. Canonical connection with logarithmic singularities.

4. The Gauss-Manin connection associated with the morphism X∆[1] → X∂∆[1] of cosim-

plicial schemes.

5. Torsors associated to the canonical unipotent connection with logarithmic singulari-

ties.

6. Partial informations about GDR(P 1(C) \ 0, 1,∞).7. Homotopy relative tangential base points on P 1(C) \ a1, . . . , an+1.8. Generators of π1(P 1(C) \ a1, . . . , an+1, x).

9. Monodromy of iterated integrals on P 1(C) \ a1, . . . , an+1.10. Calculations.

11. Configuration spaces.

12. The Drinfeld-Ihara Z/5-cycle relation.

2

13. Functional equations of iterated integrals.

14. Subgroups of Aut(π2).

A.1. Malcev completion.

The dependence of sections

0 3∣∣ ∣∣1 7∣∣ ∣∣∣∣ 4 8

2∣∣∣∣ 9

14 5∣∣∣∣ 10∣∣ ∣∣

6 11∣∣12

0.2. Below we shall briefly discuss the contents of the paper. Let us assume that X is

a smooth, quasi-projective, algebraic variety defined over a number field k. Let x be a k-

point of X. In [W1] we defined affine, connected, pro-unipotent group schemes over k and

Q respectively; πDR1 (X, x) — the algebraic De Rham fundamental group and πB

1 (X(C), x)

— the Betti fundamental group. We have also the inclusion (of Q-points into C-points)

Φx : πB1 (X(C), x)(Q)→ πDR

1 (X, x)(C)

such that the induced homomorphism on C-points

ϕx : πB1 (X(C), x)(C)→ πDR

1 (X, x)(C)

is an isomorphism.

The affine, pro-algebraic scheme over k

Iso := Iso(πB1 (X(C), x)× k, πDR

1 (X, x))

3

is an Aut(πDR1 (X(C), x))-torsor. Let Z(ϕx) be the Zariski closure of ϕx in Iso. Let T (ϕx)

be the smallest subtorsor of Iso, defined over k which contains ϕx as a C-point. Let

GDR(ϕx) ⊂ Aut(πDR1 (X, x)) be the corresponding subgroup. One can hope that

Z(ϕx) = T (ϕx)(0.1.)

We shall denote by GDR(X) the image of GDR(ϕx) in Out(πDR1 (X, x)).

The calculation of the homomorphism ϕx is equivalent to the calculation of the mon-

odromy of all iterated integrals on X. We shall see that the monodromy representa-

tion of iterated integrals have a lot of properties similar to the action of Gal(k/k) on

fundamental groups. For example if S is a loop on a curve around a missing point,

then “σ(S) ∼ Sχ(σ)” in the l-adic case and “θ(S) ∼ S−2πi” for iterated integrals. Let

ϕ : Gk := Gal(k/k) → Out(π1(X ×kk, x)(l)) be the natural homomorphism. Very opti-

mistically we can state the following conjecture:

Lie(ϕ(Gk))⊗ k ≈ Lie(GDR(X))⊗Ql.(0.2)

We also point out that (0.1) and (0.2) will imply that values of the Riemann zeta function

at odd integers are all transcendental over Q(−2πi).

0.3. The calculations of non-abelian unipotent periods are in fact the calculations of

monodromy of iterated integrals. This causes that the paper contains in fact two dif-

ferent papers. In one part (sections 3,7,8,9,10,11,12,13) we are studying monodromy of

iterated integrals. In Section 3 are established some general properties of the canonical

unipotent connection. In Section 7 we present a “naive” approach to the Deligne tangen-

tial base point, which is sufficient for our applications. In Sections 9 and 10 we describe

the monodromy of iterated integrals on P 1(C) \ a1, . . . , an+1 and in more details on

P 1(C)\0, 1,∞. In Sections 11–13 we study monodromy of iterated integrals on configu-

ration spaces. We give a proof of Drinfeld-Ihara Z/5-cycle relation which is different from

the proof in [Dr]. The proof should be (is) analogous to the proof in [I2]. We point out

that the main point in our proof is the functoriality property of the universal unipotent

4

connection. This property can be shortly written as f∗ω = f∗ω and it is also fundamental

in our results on functional equations of polylogarithms and iterated integrals (see [W4]).

In “the second paper” contained in this paper we discuss torsors and corresponding

groups associated to non-abelian unipotent periods. In Section 1 we give some general

results and definitions. In Section 2 we define a torsor and a corresponding group asso-

ciated to non-abelian unipotent periods and we state some conjectures related to Galois

representations on fundamental groups. In Section 5 we define a torsor associated to the

monodromy of iterated integrals. Using results from Section 4 we show that this torsor

and the corresponding group coincides with the ones from Section 2. In Section 6 we cal-

culate some part of the group GDR(X) for X = P1(C) \ 0, 1,∞. This part corresponds to

the Galois representation on π1(X,−→01)/[π′, π′] where π′ := [π1(X,

−→01), π1(X,

−→01)] (see [I3]).

About this part of GDR(X), let us call it G, we have the following result.

The group G contains a group H = ft|ft(X) = t ·X, ft(Y ) = t ·Y |t ∈ C∗ if and only

if all numbers ζ(2k+ 1) are irrational. This leads to a definition of a new group associated

to unipotent periods, which should be relatively easily calculated. This new group is not

considered in this paper, see however Corollary 6.4 and Theorem 6.7 iii).

Acknowledgment. We would like to thank Professor Deligne, who once showed the

one-form considered in Section 3 in the case of C \ 0, 1. Thanks are due to Professor

Y. Ihara for showing us his proof of 5-cycle relation, which help us to find an analogous

one for unipotent periods. We would like to express our thanks to Professor Hubbuck for

his invitation to Aberdeen, where Section 12 was written and where in May 1993 we had

a possibility to give seminar talks on 5-cycle relation for unipotent periods.

We would like to thank very much Professor Y. Ihara for his invitation to Kyoto.

We would like to thank Professors Oda, Matsumoto, Tamagawa for useful discussions and

comments during my seminar talks.

Finally thanks are due to Professor L. Lewin, who once invited us to write a chapter

in the book on polylogarithms and suggested to include also results about monodromy of

iterated integrals. This encourage us very much to continue to work on this subject (see

preprints [W2] and [W3] which some parts are included in the present paper).

5

§1. Torsors.

Let G1 and G2 be two groups. We say that a set T is a (G1, G2)-bitorsor if T is equipped

with a free, transitive, left action of G1 and with a free, transitive right action of G2 and

if the actions of G1 and G2 commute.

We say that a subset S ⊂ T is a subtorsor of T if there exist subgroups H1 ⊂ G1 and

H2 ⊂ G2 such that S is a (H1, H2)-bitorsor under the natural actions of H1 and H2.

We say that a subset S ⊂ T is a left (resp. right) subtorsor of T if there is a subgroup

H1 ⊂ G1 (resp. H2 ⊂ G2 ) such that the natural action of H1 (resp. H2) on S is free and

transitive.

Main example. Let G1 and G2 be two groups. Assume that G1 and G2 are isomorphic.

Then the set of isomorphisms from G1 to G2, which we denote by Iso(G1, G2), is an

(Aut(G1),Aut(G2))-bitorsor.

For any non-empty subset S ⊂ Iso(G1, G2), the intersection of all subtorsors (resp.

right subtorsors, resp. left subtorsors) of Iso(G1, G2), which contain S, is a subtorsor (resp.

right subtorsor, resp. left subtorsor) of Iso(G1, G2), which we denote by T (S) (resp. Tr(S),

resp. Tl(S)).

1.1. Unipotent, affine, algebraic groups and torsors.

Let k be a field of characteristic zero. We say that X is an algebraic variety (de-

fined) over k if X is an algebraic scheme over Spec k. If A is a k-algebra, we set XA :=

X ×Spec k

SpecA. The set of A-points of X we denote by X(A). We say that G is an affine,

algebraic group (defined) over k, if G is an affine, algebraic group scheme over Spec k.

1.1.1. Let G be an affine, unipotent, connected, algebraic group over k. Then there is an

affine, algebraic group Aut(G) over k such that for any k-algebra A we have

Aut(G)(A) = Aut(GA).

6

Let G1 and G2 be two connected, affine, unipotent, algebraic groups over k. Then

there is a smooth, affine, algebraic variety Iso(G1, G2) over k, such that for any k-algebra

A we have

Iso(G1, G2)(A) := Iso(G1A, G2A).

Proof of 1.1.1. Let G be a Lie algebra of G. The exponential map exp: G → G is

an isomorphism of affine, algebraic groups if we equip G with a group law given by the

Baker-Campbell-Hausdorff formula. The automorphisms of the group G coincides with the

automorphisms of the Lie algebra G. One can easily give an ideal defining AutLie(G) in

k[GL(G)]. In the similar way one constructs Iso(G1, G2).

1.1.2. In [S] page 149 there is a definition of a (left) G-torsor (a principal homogeneous

space of G) if G is a linear, algebraic group over k. This definition extends immediately

for bitorsors of algebraic groups.

We say that an affine, algebraic variety T over k is a (G1, G2)-bitorsor, if there are

morphisms G1× T → T and T ×G2 → T over Spec k, which define free, transitive actions

of G1 and G2 on T and if these actions commute.

Observe that if T has a k-point then G1 and G2 are isomorphic.

The definitions of a subtorsor, a right subtorsor and a left subtorsor we left to a reader,

as well as a proof of the following lemma.

Lemma 1.1.2. Let T1 be an (H1, H2)-subtorsor of T and let T2 be an (H ′1, H

′2)-subtorsor

of T . Assume that T1 ∩ T2 6= ∅. Then the intersection T1 ∩ T2 is an (H1 ∩H2, H′1 ∩ H ′

2)-

subtorsor of T . The similar statements hold for right and left subtorsors of T .

1.1.3. Main example. Let G1 and G2 be two connected, unipotent, affine, algebraic

groups over k. Assume that there is an isomorphism G1k → G2k. Then the algebraic

variety Iso(G1, G2) is an (Aut(G1),Aut(G2))-bitorsor, if we equip Iso(G1, G2) with the

obvious actions of Aut(G1) and Aut(G2).

Let k ⊂ C be a subfield of C. Let Θ : G1(C) → G2(C) be an isomorphism. Then

Θ is a C-point of Iso(G1, G2). We denote by Z(Θ) the Zariski closure of Θ in Iso(G1, G2)

7

i.e. the smallest algebraic subset of Iso(G1, G2) (defined) over k, which contains Θ as a

C-point.

The connected, unipotent, affine, algebraic group Gi is isomorphic as an algebraic

variety over k to the affine space Amk , hence Θ can be view as a C-point (Θij)1≤i,j≤n of

Am2

k . Let k(Θ) be a subfield of C generated over k by all Θi,j .

Lemma 1.1.3. The field k(Θ) does not depend on the choice of isomorphisms Gi ≈ Amk

and the transcendental degree of the field k(Θ) over k is equal to the dimension of Z(Θ).

Proof. This follows (is) Lemma 1.7 in [D2].

Definition-Proposition 1.1.4. Let T (Θ) (resp. Tr(Θ), resp. Tl(Θ)) be the intersection

of all subtorsors T (resp. right-subtorsors Tr, resp. left subtorsors Tl) defined over k of

Iso(G1, G2), which contain Θ as a C-point. Then T (Θ) (resp. Tr(Θ) resp. Tl(Θ) ) is a

(Gbir (Θ), Gbi

l (Θ))-subtorsor (resp. right Gr(Θ)-subtorsor, resp. left Gl(Θ)-subtorsor) of

Iso(G1, G2) for some Gbir (Θ) ⊂ Aut(G2) and Gbi

l (Θ) ⊂ Aut(G1) (resp. Gr(Θ) ⊂ Aut(G2),

resp. Gl(Θ) ⊂ Aut(G1)).

Proof. The intersection of a family of algebraic varieties coincides with an intersection of

a finite number of them. Hence it follows from Lemma 1.1.2 that T (Θ), Tr(Θ) and Tl(Θ)

exist and are unique. The groups are also unique because they are intersections of the

corresponding subgroups of Aut(Gi).

If T (Θ) has a k-point f , then Gbil (Θ) = f−1Gbi

r (Θ)f. If Tl(Θ) (resp. Tr(θ)) has a k-

point, then T (Θ) = Tl(Θ) and Gbil (Θ) = Gl(Θ) (resp. T (Θ) = Tr(Θ) and Gbi

r (Θ) = Gr(Θ).

Lemma 1.1.5. Let G be a unipotent, connected, affine, algebraic group over k. Then

Aut(G) is an extension of an algebraic subgroup of GL(Gab) by a connected, unipotent,

affine, algebraic group. Hence the groups Gbir (Θ), Gbi

l (Θ), Gr(Θ), Gl(Θ) are extensions of

algebraic subgroups of GL(Gab) by connected, unipotent, affine, algebraic groups.

Proof. Let G be the Lie algebra of G and let (G(i))i be a filtration of G by the lower central

series. Any automorphism of the Lie algebra G preserves the filtration and the induced

automorphism of G(i)/G(i+1) is determined by the automorphism of Gab = G(1)/G(2). Hence

8

AutLie(G) is an extension of a closed subgroup of GL(Gab) be a unipotent group. The

lemma follows from the identification of Aut(G) with AutLie(G) by the exponential map

exp : G ≈→ G.

Lemma 1.1.6. Assume thatGr(Θ) is an extension ofGm (orG such thatH1(Gal( k/k), G)=

0) by a connected, unipotent, affine, algebraic group N . Then Tr(Θ) has a k-point.

Proof. It follows from [S] Proposition 4.1 that H1(Gal( k/k), N) = 0. It follow from

[S] Proposition 2.2 and the assumption of the lemma that H1(Gal( k/k), Gr(Θ)) = 0.

Proposition 1.1 from [S] implies that Tr(Θ)(k) 6= ∅.

Let Γi(G) be a filtration of a group G by the lower central series. Let us set G(i) :=

G/Γi+1G. The isomorphism Θ : G1(C) → G2(C) induces isomorphisms Θ(i) : G

(i)1 (C) →

G(i)2 (C). Let k < i. The projections G

(i)j → G

(k)j for j = 1, 2 induce

ρik : Iso(G

(i)1 , G

(i)2 )→ Iso(G

(k)1 , G

(k)2 ),

ρ(j)ik : Aut(G

(i)j )→ Aut(G

(k)j ) for j = 1, 2.

Lemma 1.1.8. We have

i) ρik(Z(Θ(i))) = Z(Θ(k)), ii) ρi

k(T (Θ(i))) = T (Θ(k)),

iii) ρik(Tl(Θ

(i))) = Tl(Θ(k)), ρi

k(Tr(Θ(i))) = Tr(Θ(k)),

iv) ρ(1)ik(Gl(Θ

(i))) = Gl(Θ(k)),

ρ(1)ik(Gbi

l (Θ(i))) = Gbil (Θ(k)),

ρ(2)ik(G

(resp.bi)r (Θ(i))) = G

(resp.bi)r (Θ(k)).

Proof. In the point i) ( ) means the Zariski closure and we skipped its proof because we

do not need this fact later.

Let us set p = ρik and p′ = ρ(2)i

k. Observe that the image p′(Gr(Θ(i))) of the group

Gr(Θ(i)) by the morphism p′ is a closed subgroup of Aut(G(k)2 ) defined over k. This implies

that p(Tr(Θ(i))) is a closed subvariety of Iso(G(k)1 , G

(k)2 ) and a p′(Gr(Θ(i)))-torsor defined

over k. This torsor contains Θ(k) as a C-point, so we have

Tr(Θ(k)) ⊂ p′(Tr(Θ(i))) and Gr(Θ(k)) ⊂ p′(Gr(Θ(i))).

9

Let P (resp. P ′) be the projection p (resp. p′) restricted to Tr(Θ(i)) (resp. Gr(Θ(i))). Then

P−1(Tr(Θ(k))) is P ′−1(Gr(Θ(k)))-torsor defined over k, which contain Θ(i) as a C-point.

Hence we get P−1(Tr(Θ(k))) = Tr(Θ(i)) and P ′−1(Gr(Θ(k))) = Gr(Θ(i)) This implies that

p(Tr(Θ(i))) = Tr(Θ(k)) and p′(Gr(Θ(i))) = Gr(Θ(k)). All other statements are proved in

the same way.

1.2. Affine, pro-algebraic, pro-unipotent groups and torsors.

1.2.1 We assume that G = lim←−i

G(i) where the groups G(i) are affine, connected, unipotent,

algebraic groups over k. We assume further that G(i) = G/Γi+1G. Finally we assume that

the Lie algebra G of G is finitely presented i.e. for i big enough the number of relations,

defining G/Γi+1G, of degree less than i+ 1 does not depend on i.

1.2.2. The condition that G is finitely presented implies that for i big enough the mor-

phisms

Aut(G(i+1))→ Aut(G(i))

are surjective. We set

Aut(G) := lim←−i

Aut(G(i)).

Similarly, if G1 and G2 satisfy 1.2.1 and if there is an isomorphism G1,L → G2,L for some

extension L of k, then the morphisms

Iso(G(i+1)1 , G

(i+1)2 )→ Iso(G

(i)1 , G

(i)2 )

are surjective for i big enough. We set

Iso(G1, G2) := lim←−i

Iso(G(i)1 , G

(i)2 ).

Observe that Iso(G1, G2) is an (Aut(G1),Aut(G2))-bitorsor defined over k.

1.2.3. Examples of groups satisfying 1.2.1.

10

1) Let G be a finitely presented group, for example G = π1(X, x) where X is a complex

algebraic variety. Let Lie(G) :=∞⊕

n=1(ΓnG

/Γn+1G) ⊗ Q, let L(G) := lim←−

i

Lie(G)/ΓiLie (G)

be a completed Lie algebra with respect to the lower central series and let π(G) be L(G)

equipped with the multiplication given by the Baker-Campbell-Hausdorff formula. Then

there is an affine group scheme Π(G) over Q satisfying 1.2.1 such that Π(G)(Q) = π(G).

2) G = Spec(H0DR((V, x)•)) where V is a smooth, algebraic variety over k.

Lemma 1.2.4. Let G be as in 1.2.1. The affine group scheme Aut(G) is an extension of

a closed subgroup of GL(Gab) by an affine, connected, pro-unipotent, pro-algebraic group.

The lemma follows from Lemma 1.1.5 and the definition of Aut(G).

1.2.5 Let G1 and G2 satisfy 1.2.1. Let Θ : G1(C) → G2(C) be an isomorphism. Then

Θ = lim←−i

Θ(i) where Θ(i) : G(i)1 (C)→ G

(i)2 (C) are isomorphisms.

Let us set

Z(Θ) := lim←−i

Z(Θ(i)),

T (Θ) := lim←−i

T (Θ(i)), Gbir (Θ) := lim←−

i

Gbir (Θ(i)),

Gbil (Θ) := lim←−

i

Gbil (Θ(i)),

Tr(Θ) := lim←−i

Tr(Θ(i)), Gr(Θ) := lim←−i

Gr(Θ(i));

Tl(Θ) := lim←−i

Tl(Θ(i)), Gl(Θ) := lim←−

i

Gl(Θ(i)).

Then T (Θ) is a (Gbil (Θ), Gbi

r (Θ))-bitorsor, Tr(Θ) is a right Gr(Θ)-torsor and Tl(Θ) is a

left Gl(Θ)-torsor.

Lemma 1.2.6. If for each i Tr(Θ(i))(k) 6= ∅ (resp. Tl(Θ(i)) 6= ∅, resp. T (Θ(i)) 6= ∅) then

Tr(Θ)(k) 6= ∅ (resp. Tl(Θ) 6= ∅, resp. T (Θ) 6= ∅).

The lemma follows from the fact that the morphisms Tr(Θ(i+1))(k)→ Tr(Θ(i))(k) are

surjective for i big enough by 1.2.2.

11

§2. Torsors associated to non-abelian unipotent periods

Let X be a smooth, quasi-projective algebraic variety defined over the field k of

characteristic zero. Assume that X has a k-point x. Let us fix an embedding k → C. In

[W1] we have constructed connected, affine, pro-algebraic, pro-unipotent, finitely presented

group schemes πB1 (X(C), x) and πDR

1 (X, x) over Spec Q and Spec k respectively.

We set

πB1 (X(C), x)n := πB

1 (X(C), x)/Γn+1πB1 (X(C), x);

πDR1 (X, x)n := πDR

1 (X, x)/Γn+1πDR1 (X, x).

Then we have:

πB1 (X(C), x) = lim←−

n

πB1 (X(C), x)n, πDR

1 (X, x) = lim←−n

πDR1 (X, x)n.

In [W1] Proposition 7.5 we have also constructed a homomorphism (called B α in [W1])

Cx : πB1 (X(C), x)(Q)→ πDR

1 (X, x)(C)

such that the induced map on C-points, also denoted by Cx,

Cx : πB1 (X(C), x)(C)→ πDR

1 (X, x)(C)

is an isomorphism.

We have Cx = lim←−n

Cnx , where Cn

x : πB1 (X(C), x)n(C) → πDR

1 (X, x)n(C) is induced by

Cx.

For each n we have an(Aut(πB

1 (X(C), x)n × k); Aut(πDR1 (X, x)n)

)-bitorsor Ison :=

Iso(πB

1 (X(C), x)n × k;πDR1 (X, x)n

). We have also a (left) Aut(πB

1 (X(C), x)n × k)-torsor

Ison and a (right) Aut(πDR1 (X, x)n)-torsor Ison.

We shall concentrate mainly on the (right) Aut(πDR1 (X, x)n)-torsor Ison.

Applying the construction from Section 1 to the isomorphism Cnx we get the Gr(Cn

x )-

torsor Tr(Cnx ) over Spec k. The group Gr(Cn

x ) is a subgroup of Aut(πDR1 (X, x)n).

12

We set

Gr(Cx) := lim←−n

Gr(Cnx )

and

Tr(Cx) := lim←−n

Tr(Cnx ).

We have projections

pn : Aut(πDR1 (X, x)n)→ Out(πDR

1 (X, x)n) := Aut(· · ·)/Inn(· · ·)

and

p : Aut(πDR

1 (X, x))→ Out

(πDR

1 (X, x)).

Definition 2.1. i) The group G(X, x)(n) is the image of Gr(Cnx ) in Out(πDR

1 (X, x)n).

ii) The group G(X, x) is the image of Gr(Cx) in Out(πDR1 (X, x)n).

Let x and y be two k-points of X. Let γ be a path in X(C) from x to y. Then γ

induces an isomorphism cγ : πDR1 (XC, x) → πDR

1 (XC, y). The induced isomorphisms of

outer automorphisms groups

(cγ)∗ : Out(πDR

1 (XC, x))→ Out

(πDR

1 (XC, y))

does not depend on the choice of γ and gives the canonical identification. We need this

identification over Spec k.

Let us consider the morphism p• : X∆[1] → X∂∆[1] of cosimplicial spaces. Let us

set (X;x, y)• := p•−1(x, y). Then Spec(H0

DR

((X;x, y)•

))is a left πDR

1 (X, x)-torsor and

a right πDR1 (X, y)-torsor (see [W1] Section 3). It follows from [S] Proposition 4.1 that

any torsor over πDR1 (X, x)n is trivial, hence any torsor over πDR

1 (X, x) is trivial (in-

verse limit of surjective maps of sets (k-points) is always non empty). Any k-point η

of Spec(H0

DR

((X;x, y)•

))determines on isomorphism

cη : πDR1 (X, x)→ πDR

1 (X, y).

13

The isomorphism is unique up to conjugation by elements of πDR1 (X, x). Hence the induced

isomorphism

(cη)∗ : Out(πDR

1 (X, x))→ Out

(πDR

1 (X, y))

is canonical (does not depend on the choice of a k-point of Spec(H0

DR

((X;x, y)•

))).

The monodromy representations Cx and Cy are related by the following commutative

diagram

πB1 (X(C), x)(C)

Cx−→ πDR1 (X, x)(C)ycγ

ycγ

πB1 (X(C), y)(C)

Cy−→ πDR1 (X, x)(C)

where cγ and cγ are induced by the path γ. Observe that cγ = cη conj(g), where η

is a k-point of Spec(H0

DR

((X;x, y)

))and conj(g) is a conjugation by an element g ∈

πDR1 (X, x)(C). This implies the following result.

Proposition 2.2. The groups G(X, x)n and G(X, y)(n) coincides under the canonical iso-

morphism (cη)∗. The groups G(X, x) and G(X, y) coincides under the canonical isomor-

phism (cη)∗. We shall denote these groups by GDR(X)(n) and GDR(X) respectively.

Some calculations and results considers in the sequel seems to lead to the following

conjectures.

Let Ck : Gk := Gal(k/k) → Out(π1(X ×

kk, x)l

)be the natural representation of the

Galois groups. Let Cnk : Gk → Out

(π1(X ×

kk, x)l/Γ

n+1(· · ·))

be induced by Ck.

If G is a group (algebraic or p-adic analytic) we denote by Lie(G) its Lie algebra.

Conjecture 2.3. For each n there exists a Lie algebra G(n) over Q such that

i) Q⊗ Lie(Cn

k (Gk))≈ G(n) ⊗Ql,

ii) Q⊗ Lie(GDR(X)(n)

)≈ G(n) ⊗ k.

The Lie algebras G(n) form an inverse system. Let us set G = lim←−n

Gn. Then

i) Q⊗ Lie(Ck(Gk)

)≈ G ⊗Ql,

ii) Q⊗ Lie(GDR(X)

)≈ G ⊗ k.

14

§3. Canonical connection with logarithmic singularities.

Let X be a smooth, projective scheme of finite type over a field k of characteristic zero.

Let D be a divisor with normal crossings in X and let V = X \D. Let

A∗(V ) := Γ(X,Ω∗X〈logD〉)

be a differential algebra of global sections of the algebraic De Rham complex on X with

logarithmic singularities along D.

3.1. It follows from [D1] Corollaire 3.2.14 that each element of A∗(V ) is closed and the

natural map A∗(V )→ H∗DR(V ) is injective.

We shall denote by ∧2(A1(V )) the exterior product of the vector space A1(V ) with

itself and by A1(V ) ∧A1(V ) the image of ∧2(A1(V )) in A2(V ).

Let H(V ) := (A1(V ))∗ and R(V ) := (A1(V ) ∧ A1(V ))∗ be dual vector spaces. The

map ∧2(A1(V )) A1(V ) ∧ A1(V ) induces a map R(V )→ Λ2(H(V )).

Let Lie (H(V )) be a free Lie algebra over k on H(V ). Observe that R(V ) is contained

in degree 2 terms of Lie (H(V )). Let (R(V )) be a Lie ideal generated by R(V ). We set

Lie (V ) := Lie (H(V ))/(R(V ))

and

L(V ) := lim←−n

(Lie (V )

/ΓnLie (V )

).

The Lie algebra L(V ) we equip with the multiplication given by the Baker-Campbell-

Hausdorff formula and the obtained group we shall denote by π(V ). Its Lie algebra can be

identified with L(V ). We define a one form ωV on V with values in the Lie algebra L(V )

in the following way. The form ωV corresponds to the identity homomorphism id A1(V )

under the natural isomorphism

A1(V )⊗H(V ) = A1(V )⊗ (A1(V ))∗ ≈ Hom (A1(V ), A1(V ))).

15

Lemma 3.2. The one-form ωV is integrable.

Proof. It is sufficient to show that dωV + 12[ωV , ωV ] = 0. It follows from 3.1 that dωV = 0.

We have exact sequences

0→ K ∧2(A1(V )) A1(V ) ∧ A1(V )→ 0

and

0← K∗ ∧2(H(V )) R(V )← 0.

The two-form [ωV , ωV ] ∈ A1(V ) ∧ A1(V ) ⊗ ∧2H(V )/R(V ) ≈

(∧2(A1(V ))

/K)⊗ K∗ is

represented by a map K → ∧2(A1(V ))→ A1(V ) ∧ A1(V ), hence it is zero.

Let T [H(V )] be a tensor algebra over k on H(V ) and let (R(V )) be an ideal of

T [H(V )] generated by R(V ). Let Q(V ) := T [H(V )]/R(V ) be the quotient algebra and let

Q(V ) be its completion with repect to the augmentation ideal I := ker (Q(V ) → k), i.e.

Q(V ) := lim←−n

(Q(V )

/In)

. Let P (V ) be the group of invertible elements in Q(V ), whose

constant terms are equal 1.

The elements of L(V ) we identify with Lie elements (can be of infinite length) in

P (V ). The exponential series defines an injective homomorphism

exp : π(V )→ P (V ).

The inverse of exp is defined on the subgroup exp(π(V )) of P (V ) and it is given by the

formula

logz = (z − 1)− 1/2(z − 1)2 + 1/3(z − 1)3 − 1/4(z − 1)4 + . . . .

Let us assume that k is the field of complex numbers C. Then V is a complex variety

with the standard complex topology.

Let x, z ∈ V be two points in V and let γ be a smooth path in V from x to z.

Let ΛV (z;x, γ) (resp. LV (z;x, γ)) be a horizontal section along γ of the principal

P (V ) (resp. π(V ))-bundle

V × P (V )→ V (resp. V × π(V )→ V )

16

equipped with the connection given by ωV and such that ΛV (x;x, γ) = 1 (resp. LV (x;x, γ) =

0).

Definition 3.3. Let x ∈ V and let α ∈ π1(V, x) be a loop. We shall define a homomor-

phism

θx,V : π1(V, x)→ P (V ) (resp. θx,V : π1(V, x)→ π(V ))

by the formula

θx,V (α) := ΛV (α(1);x, α) (resp. θx,V (α) := LV (α(1);x, α))

and we call it the monodromy homomorphism of the form ωV (at the point x).

Proposition 3.4. Let x1, x2 ∈ V . Then the monodromy homomorphisms θx1,V and θx2,V

of the form ωV are conjugated.

Proof. It is a property of a connection on a principal fiber bundle.

Proposition 3.5. Assume that

A1(V )→ H1DR(V )

is an isomorphism. Then the Lie algebra Lie (V ) is isomorphic to the Lie algebra of the

fundamental group of V .

Proof. Let Ω∗(V ) be a differential algebra of complex valued, global, smooth, differential

forms on V . The inclusion A∗(V ) → Ω∗(V ) induces an isomorphism of the “stage one”

minimal models. Moreover the “stage one” minimal model of A∗(V ) is formal as d1 =

d2 = 0. This implies the statement of the proposition.

Proposition 3.6. If A1(V ) → H1DR(V ) is an isomorphism then the monodromy homo-

morphism θx,V : π1(V, x)→ π(V ) induces an isomorphism of the Malcev C-completion of

π1(V, x) into π(V ).

Proof. The Sullivan theory of minimal models recovers the Malcev Q-completion π1(V, x)0(Q)

of π1(V, x). (See §A.1 of Appendix for the notation π1(V, x)0.) The formality of the “state

one” minimal model of V implies that the Malcev C-completion of π1(V, x) is (isomorphic

17

to) π(V ). Hence the groups π1(V, x)0(C) and π(V ) are isomorphic. By the universal prop-

erty of the Malcev C-completion there is an isomorphism θx,V : π1(V, x)0(C)≈−→ π(V ) of

affine, pro-nilpotent groups such that the diagram

π1(V, x)

rC θx,V

π1(V, x)0(C)≈

−−−−−→θx,V

π(V )

commutes.

Let Xi (for i = 1, 2) be smooth, projective schemes of finite type over k. Let Di be

divisors with normal crossings in Xi (for i = 1, 2). Let Vi = Xi \Di (for i = 1, 2) and let

f : X1 → X2 be a morphism such that f−1(D2) = D1. Then f induces f∗ : A1(V2) →A1(V1).

Let f∗ : H(V1)→ H(V2) be the dual map. This map induces group homomorphisms

f∗ : P (V1)→ P (V2)

and

f∗ : π(V1)→ π(V2)

Lemma 3.7. We have

f∗(ωV1) = f∗(ωV2

).

Corollary 3.8. We have

f∗(ΛV1(z;x, γ)) = ΛV2

(f(z); f(x), f(γ)).

and

f∗(LV1(z;x, γ)) = LV2

(f(z); f(x), f(γ)).

The lemma follows from the definition of ωVias id A1(Vi).

Let ω1, . . . , ωn be a base of A1(V ). Let X1, . . . , Xn be a dual base of H(V ). Then

P (V ) is a multiplicative group of the algebra of formal power series in non-commuting

variables X1, . . . , Xn divided by the ideal generated by R(V ).

If α(X1, . . . , Xn) is a formal power series in non-commuting variables X1, . . . , Xn we

shall denote by ′α(X1, . . . , Xn) its image in P (V ).

18

Proposition 3.9. We have

i) (z,ΛV (z;x, γ)) = (z, ′1 +∑

((−1)k

∫ z

x,γ

ωi1 , . . . , ωik)Xik

· . . . ·Xi1) ∈ V × P (V );

(the summation is over all non-commutative monomials in variables X1, . . . , Xn, the inter-

ated integrals are calculated along the path γ)

ii) LV (z;x, γ) = log(ΛV (z;x, γ))

Proof. The principal bundle V × CX1, . . . , Xn∗ → V equipped with the connection

given byn∑

i=1ωi ⊗Xi has horizontal sections given by

z → (z, 1 +∑

((−1)k

∫ z

x,γ

ωi1 , . . . , ωik)Xik

· . . . ·Xi1).

Hence the point i) follows. Observe that exp : π(V ) → P (V ) identifies ωV ∈ A1(V ) ⊗Lie (π(V )) with ωV ∈ A1(V )⊗ Lie (P (V )). Hence the point ii) follows.

§4. The Gauss-Manin connection associated with the morphism X∆[1] → X∂∆[1]

of cosimplicial schemes.

4.0. Let V be a smooth, quasi-projective scheme over a field k of characteristic zero. The

inclusion of simplicial sets

∂∆[1] → ∆[1]

induces a morphism of cosimplicial schemes

ρ• : V ∆[1] → V ∂∆[1].

Imitating the construction of Katz and Oda (see [KO]) we equipped the sheaves

Hi(tRρ•

∗Ω∗

(V ∆[1])/(V ∂∆[1])

)with the integrable connection dk (see [W1]). We recall that

(V, x)• = ρ• −1(x, x).

4.1. Let X be a smooth, projective variety defined over C. Let D be a divisor with

normal crossings in X and let V = X \ D. Let Ω1〈D〉(X) be global, meromorphic one

forms on X with logarithmic singularities along D. We shall assume that Ω1〈D〉(X) ≈

19

H1DR(V ). We shall calculate the monodromy representation of the connection dC on the

sheaf H0(tRρ•∗Ω∗

(V ∆[1])/(V ∂∆[1])). Let

L(V )∗ := lim−→n

(Lie (V )

/ΓnLie (V )

)∗.

Then the algebra of polynomial functions on π(V ) coincides with the symmetric algebra

on L(V )∗ i.e.

Alg (π(V )) = S(L(V )∗).

The group π(V ) acts on Alg (π(V )) on the left by the formula

a) g(f)(x) := f(x · g) b) g(f)(x) := f(g−1x)

We form the associated vector bundle

V × Alg (π(V )) ≈ V × π(V ) ×π(V )

Alg (π(V ))→ V

and we equipped it with the connection induced by ωV . This connection we shall denote

by ω′V in the case a) and by ′ωV in the case b).

Lemma 4.1. The monodromy representation of the associated vector bundle is given by

π1(V, x) 3 γ → (f → θx(γ)(f)) ∈ Aut (Alg (π(V ))).

Proof. This follows from the definition of an associated vector bundle.

4.2. Let us set A(V ) := Ω1〈D〉(X) and A2(V ) := A(V )∧A(V ) ⊂ Ω2

〈D〉(X). The inclusion

of complexes of differential graded algebras

(C0−→A(V ))

0−→A2(V )) → Ω∗(V ).

induces an isomorphism

H0 := H0(Bar(C

0−→A(V )0−→A2(V ))

)≈ H0

DR((V, x)•).

Let T [A(V )∗] be a tensor algebra on the dual vector space A(V )∗. Let ∧2(A(V )) be

an exterior product of complex vector spaces. The surjective map ∧2(A(V )) A2(V )

20

induces an inclusion (A2(V ))∗ → ∧2(A(V )∗). Let us set R(V ) := (A2(V ))∗. Let (R(V ))

be an ideal in T [A(V )∗] generated by R(V ) and let Q(V ) := T [A(V )∗]/(R(V )) be the

quotient algebra. Let I be its augmentation ideal and let Q(V )∗ := lim−→n

(Q(V )

/In)∗

be

the direct limit of dual vector spaces.

Let us fixed a base ω1, . . . , ωn of A(V ). Let X1, . . . , Xn be a dual base of A(V )∗.

Then T [A(V )∗] = CX1, . . . , Xn is a polynomial algebra on non-commuting variables

X1, . . . , Xn. Let a(Xi1 , . . . , Xik) denotes a linear form on T [A(V )∗] which to f associates its

coefficient at Xi1 . . .Xik. Let T ∗ be a vector space generated by all a(Xi1 . . . , Xk) including

a(φ). Observe that Q(V )∗ is a subspace of T ∗. We equipped Q(V )∗ with a shuffle multipli-

cation and a Hopf algebra structure given by a(Xi1 . . . , Xik)→

k∑l=0

a(Xi1 . . . , Xil)a(Xil+1

, . . . , Xik).

Lemma 4.3. There is an isomorphism of Hopf algebras

P : H0 ≈−→Q(V )∗

induced by ωi1 ⊗ . . .⊗ ωik→ a(Xi1 , . . . , Xik

).

The symmetric algebra on the vector space Q(V )∗ divided by an ideal (a(φ)−1) is the

algebra of polynomial functions on P (V ), which we denonte by Alg (P (V )). Let Sh(V ) be

an ideal in Alg (P (V )) generated by shuffle product relations. Then the morphism

4.4. Q : Alg (P (V ))/Sh(V )

≈−→Q(V )∗

is an isomorphism of Hopf algebras.

It follows from a theorem of R. Ree (see [R] Theorem 2.5) that a formal power series

f ∈ P (V ) is in the image of exp : π(V )→ P (V ) if and only if its coefficients (i.e. coefficients

of some lifting to CX1, . . . , Xn) satisfy shuffle product relations. This implies that the

injective homomorphism

exp : π(V )→ P (V )

induces an isomorphism of Hopf algebras

4.5. R : Q(V )∗≈−→ Alg (π(V )).

21

Let H0x (resp. xH0) be a restriction of the bundle H0(tRρ•

∗Ω∗

(V ∆[1])/(V ∂∆[1])) to X×x

(resp. x ×X.) Both bundles are equipped with the induced connection, which we also

denote by dC.

Our main result in this section is the following theorem.

Theorem 4.6. Let V be such as in 4.1. Then the vector bundles equipped with connec-

tions

(H0x, dC) and (V × Alg (π(V ))→ V, ω′

V )

are isomorphic in the category of algebraic vector bundles equipped with algebraic inte-

grable connections.

((xH0, dC) and (V ×Alg (π(V ))→ V, ω′V ) are also isomorphic.)

Proof. We shall calculate horizontal sections of the connection dC on

H := H0(tRρ•∗Ω∗(V ∆[1])/(V ∂∆[1])

). We recall the construction of dC. We start with the

filtration F i of Ω∗V ∆[1] . The filtrations F i on Ω∗

V n+2 are defined by

F iΩ∗V n+2 := image (Ω∗−i

V n+2 ⊗(ρn)∗ ΩiV ×V → Ω∗

V n+2).

These filtrations give a filtration F i of Ω∗V ∆[1] . The connection dC is defined as the boundary

homomorphism of the short exact sequence of complexes on V ∆[1]

0→ F 1/F 2 → F 0/

F 2 → F 0/F 1 → 0.

On V n+2 we have

i) F 0/F 1 ≈ Ω∗

V n+2/V × V

ii) F 0/F 2 ≈ Ω∗

V n+2/Ω∗−2

V n+2

⊗ Ω2V ×V .

Observe that OV ⊗ Ω∗V n ⊗OV ⊂ F 0/

F 2.

Let us assume that ω = ω1 ⊗ . . . ⊗ ωn ∈ A(V )⊗n is closed in the total complex of

Ω∗((V, x)•). Its class [ω] belongs to H0DR((V, x)•) — the fiber of H over (x, x) ∈ V × V .

We face two problems. We must extend [ω] to a continous section of H and we must

show that this section is horizontal. All computations are carried in the total complex

22

T := Tot (n → OV ⊗ Ω∗(V n) ⊗ OV ). Then ω1 ⊗ . . . ωn can be interpreted as a global

section of T . One verifies that the class of the element

∑

0≤k,l≤nk+l≤n

(∫

x

ω1, . . . , ωk

)⊗ ωk+1 ⊗ . . .⊗ ωn−l ⊗ (−1)l

(∫

x

ωn, . . . , ωn−l+1

)

∈∑OV ⊗ Ωn−1−k(V n−1−k)⊗OV

is a horizontal section of H. Hence we get.

4.7. Let (α, β) ∈ π1(V × V, (x, x)). Then the representation of π1(V × V, (x, x)) is given

by

(α, β) : ω1⊗ . . .⊗ωn →∑

0≤k,l≤nk+l≤n

(∫

α

ω1, . . . , ωk

)ωk+1⊗ . . .⊗ωn−l

(∫

β−1

ωn−l+1, . . . , ωn

).

By 4.5. the Hopf algebra Q(V )∗ is isomorphic to Alg (π(V )). Hence we shall consider

ω′V on the associated bundle V × Q(V )∗ → V . It follows from Proposition 3.9.i) that the

monodromy representation is given by

π1(V, x) 3 α : a(Xi1 , . . . , Xik)→

∑

l

a(Xi1 , . . . , Xil)

(∫

α−1

ωil+1, . . . , ωik

)

for the connection ω′V , and by

π1(V, x) 3 α : a(Xi1 , . . . , Xik)→

∑

l

(∫

α

ωi1 , . . . , ωil

)a(Xil+1

, . . . , Xik)

for the connection ′ωV .

The isomorphism H0 ∼→Q(V )∗ defined by ωi1 ⊗ . . .⊗ ωik→ a(Xi1 , . . . , Xik

) gives an

isomorphism of monodromy representations of bundles (HX , dC) and (V ×Q(V )∗ → V, ω′V )

(resp. (xH, dC) and (V ×Q(V )∗ → V,′ ωV )) at the point x ∈ V .

Observe that both connections are regular. The theorem follows from [D3] Theoreme 5.9.

23

§5. Torsors associated to the canonical unipotent connection with logarithmic

singularities.

5.0. Let V be as in section 3. We shall assume that A1(V )→ H1DR(V ) is an isomorphism.

We denote by V (C) the set of complex points of V . In 3.1 we defined the group π(V ).

It is easy to see that π(V ) is a group of k-points of a connected, affine, pro-unipotent,

pro-algebraic group scheme over k. In this section we denote by π(V ) the corresponding

group scheme, and by π(V )(R) the group of R-points of π(V ) for a k-algebra R. Simillar

notation is used for P (V ).

Let us define a homomorphism

θ−1x : π1(V (C), x)→ π(V )(C)

by the formula (θ−1x )(α) := (θx(α))−1, where θx is the monodromy homomorphism of the

form ωV .

Let us set π0 := π1(V (C), x)0-the Malcev Q-completion of π1(V (C), x). By the univer-

sal property of rC : π1(V (C), x)→ π0(C) (see A.1) there exists a unique homomorphism

(θ−1x )0 : π0(C)→ π(V )(C)

such that (θ−1x ) = (θ−1

x )0 rC. It follows from the Sullivan theory of minimal models that

the groups (affine, pro-algebraic group schemes over C) π0(C) and π(V )(C) are isomorphic.

The morphism (θ−1x )0 is an isomorphism mod Γ2, hence it is an isomorphism.

Applying the construction from section 1.2 to the isomorphism (θ−1x )0 and to the

Aut(π(V ))-torsor Iso(π0 × Spec k, π(V )) we get a torsor over k Tr((θ−1x )0), and an affine,

pro-algebraic group scheme over k Gr((θ−1x )0). If we work mod Γn+1, the map (θ−1

x )0

induces (θ−1x )n

0 : π0(C)/Γn+1··· → π(V )(C)/Γn+1··· and we get a subtorsor Tr((θ−1x )0)n of

Iso((π0/Γn+1π0) × Spec k, π(V )/Γn+1π(V )) and a subgroup Gr((θ−1

x )0)n. By the definition

we have Gr((θ−1x )0) = lim←−

n

Gr((θ−1x )0)n and Tr((θ−1

x )0) = lim←−n

Tr((θ−1x )0)n.

5.1. Let π1(V (C), x) = 〈A1, . . . , Ak|rj(A1, . . . , Ak) = 0, j = 1, . . . , l〉 be a presentation

of π1(V (C), x) in terms of generators and relations. We shall denote this presentation by

P (Ai)). For simplicity let us set P = P ((Ai)).

24

Notation. 〈w1, . . . , wn〉 is a vector subspace generated by vectors w1, . . . , wn; x denotes

the class of x mod Γ2.

We define a functor IP ( ) on k-algebras in the following way. For a k-algebra R we

set:

IP (R) := xi ∈ π(V )(R), i = 1, . . . , k∣∣∣rj(x1, . . . , xk) = 0, j = 1, . . . , l;

〈x1, . . . , xk〉 = π(V )(R)ab.

For each n we define functors InP ( ). We set

I(n)P (R) := xi ∈ π(V )(R)/Γn+1π(V )(R), i = 1, . . . , k

∣∣∣ . . ..

Proposition 5.1. The functor InP ( ) (resp.IP ( )) on k-algebras is representable by

an affine, algebraic (resp. pro-algebraic) scheme over k, which we denote by InP (resp.

IP ). The scheme InP (resp. IP ) is an Aut(π(V )/Γn+1π(V ))-torsor (resp. Aut(π(V ))-torsor)

isomorphic to the Aut(π(V )/Γn+1π(V ))-torsor (resp. Aut(π(V ))-torsor) Iso(π0/Γn+1π0×

Spec k, π(V )/Γn+1π(V )) (resp. Iso(π0 × Spec k, π(V ))). Hence we have IP = lim←−n

InP .

Proof. Observe that any f ∈ IP (R) determines a unique homomorphism f ′ : π1(V (C), x)→π(V )(R). By the universal property of the Malcev completion there exists a unique R-

homomorphism (which is an isomorphism) f : π0(R) → π(V )(R) such that rR f = f ′.

Moreover any R-isomorphism ϕ : π0(R)→ π(V )(R) defines an element of IP (R); the im-

age of generators A1 . . . , Ak by ϕ rR. Hence we have a natural isomorphism (∗) IP (R) ≈IsoR(π0(R), π(V )(R)). But the functor R → IsoR(π0(R), π(V )(R)) is representable by an

affine, pro-algebraic scheme Iso(π0 × Spec k, π(V )). Finally observe that the isomorphism

(∗) of functors is compatible with action of AutR(π(V )(R)) = (Aut(π(V ))(R)).

For the given presentation P = ((Ai)) of π1(V (C), x), the sequence (θ−1x (Ai))i ∈ IP (C)

and the sequence ((θ−1x )n(Ai))i ∈ I(n)

P (C),where (θ−1x )n : π1(V (C), x)→ π(V )(C)/Γn+1π(V )(C)

is induced by θ−1x . We denote by Tr((θ−1

x )(Ai))n the smallest subtorsor of I(n)

P defined over

k, which contains ((θ−1x )n(Ai))i as a C-point. The corresponding subgroup of Aut(π(V )/Γn+2π(V ))

we shall denote by Gr((θ−1x )(Ai))

n.

The following result follows immediately from Proposition 5.1.

25

Corollary 5.2. The isomorphism of Aut(π(V )/Γn+1π(V ))-torsors from Proposition 5.1

identifies Gr((θ−1x )(Ai))

n-torsor Tr((θ−1x )(Ai))

n with Gr((θ−1x )0)n-torsor Tr((θ−1

x )0)n.

5.3. We recall from section 2 that we have an isomorphism

Cx : πB1 (V (C), x)(C)→ πDR

1 (V,x)(C)

and the associated Gr(Cnx )-torsor Tr(Cn

x ). We shall relate this torsor to the torsors consid-

ered in this section.

Proposition 5.3. The group schemes Gr((θ−1x )0)n and Gr(Cn

x ) are isomorphic and the

corresponding torsors Tr((θ−1x )0)n and Tr(Cn

x ) are isomorphic torsors.

Proof. It follows from 4.3, 4.4 and 4.5 that we have an isomorphism

v : π(V )→ πDR1 (V, x)

induced by isomorphisms P and R of Hopf algebras. The homomorphism

b : π1(V (C), x)→ πB1 (V (C), x)(Q)

is given by evaluating iterated integrals on loops in π1(V (C), x). There is a unique homo-

morphism

b0 : π1(V (C), x)(Q)→ πB1 (V (C), x)(Q),

which is an isomorphism by the Sullivan theory of minimal models, such that b0 rQ = b.

We shall show that the diagram

π1(V (C), x)0(C)(θ−1

x )0−→ π(V )(C)yb0

yν

πB1 (V (C), x)(C)

Cx−→ πDR1 (V, x)(C)

commutes. The morphism

π1(V (C), x)→ πB1 π1(V (C), x)

Cx−→πDR1 (V, x)(C)

26

is induced by evaluating iterated integrals on elements of π1(V (C), x). Let exp : π(V )(C)→P (V )(C) be the exponential map. Then if follows from Proposition 3.9 and the formula∫

αω1, . . . , ωn = (−1)n

∫α−1 ωn, . . . , ω1 (see [Ch]) that exp(θ−1

x (α)) = 1+∑

(∫

αωi1 , . . . , ωik

)

Xi1 . . . , Xik.The map u∗ : Alg(P (V ))→ H0

DR((V, x)•), which induces u : πDR1 (V, x)

ν−1

−→π(V )exp−→P (V ), associates the coefficient at Xi1 . . . , Xik

(Xi = (ωi)∗) to the class of [ωi1 ⊗

. . . ,⊗ ωik] in H0

DR((V, x)•). This implies that the diagram commutes. The commutativity

of the diagram implies the proposition.

§6. Partial calculations of GDR(P 1 \ 0, 1,∞).

6.1. Let X = P1Q \ 0, 1,∞. We recall that Lie(X) is a free Lie algebra over Q on

X =(

dzz

)∗and Y =

(dz

z−1

)∗. We set Z := −X − Y . Let Fk

(Lie(X)

)be a Lie subalgebra

of Lie(X) generated by commutators which contain X at least k-times. We recall that

L(X) is a completion of Lie(X) with respect to the filtration induced by the lower central

series. The closure of Fk

(Lie(X)

)in L(X) we denote by Lk(X) and the corresponding

closed subgroup of π(X) by πk. Observe that πk∞k=0 is a filtration of π(X) by normal

subgroups. We set πk := π(X)/πk.

For any Q-algebra R, we set

Aut∗(π(X)

)(R) :=

f ∈ Aut

(π(X)R

)∣∣ ∃α ∈ R∗, f(X) = αX,

f(Y ) ≈ αY, f(Z) ∼ αZ.

Here ≈ means a conjugation by an element of [π(X)(R), π(X)(R)], ∼ means a conjugation

by an element of π(X)(R).

Similarly we shall define

Aut∗(πk)(R) :=f ∈ Aut(πk

R)∣∣∃α ∈ R∗, f(X) = αX,

f(Y ) ≈ αY, f(Z) ∼ αZ.

These functors on Q-algebras are represented by affine, pro-algebraic group schemes over

Q, which we denote by Aut∗(π(X)

)and Aut∗(πk) respectively.

27

6.1.2. The projection map pk : π(X)→ πk induces

(pk)∗ : Aut∗(π(X)

)−→ Aut∗(πk).

It follows from the fact that any f ∈ Aut∗(π(X)

)maps πk into itself.

6.1.3. Observe also that the map

Aut∗(π(X)

)−→ Out

(π(X)

):= Aut

(π(X)

)/Inn(π(X)

)

is injective. This follows from the fact that

Aut∗(π(X)

)∩ Inn

(π(X)

)= Id.

6.1.4. We recall from section 10 that the monodromy homomorphism

θ := θ−→01

: π1(X(C),−→01) −→ π(X)(C)

is given by the formula

θ(S0) = (−2πi)X, θ(S1) = α−1 · (−2πi)Y · α, θ(S∞) = α′−1 · (−2πi)Z · α′,

where α = α−→10−→01

(X,Y ) ∈[π(X);π(X)

]. Let us set Φ(α) :=

(θ(α)

)−1. Let π1(kX) ⊂

π1

(X(C);

−→01)

be a subgroup generated by commutators in S0 and S1 which contain S0

at least k-times. Let πk1 := π1

(X(C);

−→01)/π1(kX). The monodromy homomorphism Φ

induces

Φ(k) : πk1 −→ πk(C)

such that the induced map

Φ(k)0 : (πk1 )0(C) −→ πk(C)

is an isomorphism.

28

Proposition 6.1. We have

i) Gr(Φ0) ⊂ Aut∗(π(X)

)

ii) Gr

(Φ(k)0

)⊂ Aut∗(πk).

Proof. The presentation P = S0, S1, S∞ | S0 · S1 · S∞ of π1(X(C),−→01) defines an

Aut(π(X)

)-torsor IP over Spec Q (see section 5). The triple τ =

((2πi)X,α−1 ·

((2πi)Y

)·

α, α′−1 ·((2πi)Z

)· α′)∈ IP (C). Let

T =βX, b−1 · (βY ) · b, b′−1 · (βZ) · b′

∣∣ β′ ∈ C∗, b ∈ [π(X), π(X)],

b′ ∈ π(X), (βX) ·(b−1 · (βY ) · b

)·(b′−1 · (βZ) · b′

)= 0.

T is an affine scheme over Spec Q and T is Aut∗(π(X)

)-torsor. Observe that τ ∈ T (C).

Hence Corollary 5.2 implies that G(Φ0) ⊂ Aut∗(π(X)

).

For each k, the presentation P defines an Aut(πk)-torsor IkP . For each n, the pre-

sentation P defines an Aut(πk/Γn+2πk)-torsor Ik,nP . Let τk (resp. τk,n) be the image of τ

in πk(C) (resp. πk(C)/Γn+2πk(C)). The smallest subtorsor of Ik,nP defined over Q, which

contains τk,n as a complex point, we shall denote by T (τk,n). The corresponding group we

shall denote by G(τk,n). By definition we have

T (τk) := lim←−n

T (τk,n), G(τk) := lim←−n

G(τk,n).

Let

Tk =βX, b−1 · (βY ) · b, b′−1 · (βZ) · b′

∣∣ β ∈ C∗, b ∈ [πk(C), πk(C)],

b′ ∈ πk(C), (βX) ·(b−1 · (βY ) · b

)·(b′−1 · (βZ) · b′

)= 0.

Tk is an affine scheme over Q and Tk is an Aut∗(πk)-torsor. Observe that τk ∈ Tk(C). One

must notice that analogues of Proposition 5.1 and Corollary 5.2 hold for the groups πk

and πk1 , the presentation P and the isomorphism Φ(k)0. Then the analogue of Corollary

5.2 implies Gr

(Φ(k)0

)⊂ Aut∗(πk).

6.1.5. The projection pk : π(X)→ πk induces (pk)∗ : Aut∗(π(X)

)→ Aut∗(πk) (see 6.1.2)

and (pk)′∗ : T → Tk. We have

(pk)∗(Gr(Φ0)

)= Gr

(Φ(k)0

)and (pk)′∗

(T (τ)

)= T (τk).

29

The proof is analogous to the proof of Lemma 1.1.8.

6.1.6. Below we shall calculate the torsor T (τ2) and the corresponding group Gr

(Φ(2)0

).

By 6.1.5 we get some information about Gr(Φ0) and hence by 6.1.3 about GDR(X).

6.2. The elements θ(S0) = (−2πi)X and θ(S1) = (−2πi)Y +∞∑

k=2

(2πi)ζ(k)((YX)Y k−1

)

are images in π2(C) of the generators S0 and S1 by the monodromy homomorphism θ (see

section 10). Let Θn(S0) and Θn(S1) be images of θ(S0)−1 and θ(S1)−1 in π2(C)/Γn+2π2(C).

We shall calculate subtorsors Tn := T(Θn(S0),Θn(S1)

)of I2,n

P and the corresponding sub-

groups Gn := G(Θn(S0),Θn(S1)

)of Aut(π2/Γn+2π2). Observe that by Lemma 1.1.8 we

have

Pn+1n (Tn+1) = Tn and Pn+1

n (Gn+1) = Gn,(∗)

where Pn+1n is induced by the projection π2/Γn+3π2 → π2/Γn+2π2.

0. Calculations of T0 and G0.

We have Θ0(S0) = (2πi)X and Θ0(S1) = (2πi)Y . Observe that (2πi) is not a k-th

root of a rational number for any k = 1, 2, 3 . . . . This implies that T0 = αX,αY | α ∈ C∗and G0 = fα | fα(X) = αX, fα(Y ) = αY | α ∈ C∗.

1. Calculations of T1 and G1.

We have Θ1(S0) = (2πi)X and Θ1(S1) = (2πi)Y . The property (∗) implies that

T1 = αX,αY | α ∈ C∗ and G1 = fα | fα(X) = αX, fα(Y ) = αY | α ∈ C∗.

2. Calculations of T2 and G2.

We have Θ2(S0) = (2πi)X, Θ2(S1) = (2πi)Y + (−2πi)ζ(2)((Y X)Y

). Observe that

(−2πi)ζ(2) = 124(2πi)3. This equality and the property (∗) (or the fact that (−2πi) is not

a square root of a rational number) implies that

T2 =αX,αY +

1

24α3((YX)Y

) ∣∣ α ∈ C∗

and

G2 =fα

∣∣ fα(X) = αX, fα(Y ) = αY∣∣ α ∈ C∗

.

3. Calculations of T3 and G3.

30

We have Θ3(S0) = (2πi)X, Θ3(S1) = (2πi)Y + (−2πi)ζ(2)((Y X)Y

)+

+(−2πi)ζ(3)((Y X)Y 2

). Assume that dimG3 = dimT3 = 1. Then it follows from (∗) and

Corollary 14.2 that G3 = G(0, 0, c3) where c3 ∈ Q. As T3 is a one-dimensional variety we

have T3 =αX, αY + 1

24α3((Y X)Y

)+ β3

((Y X)Y 2

) ∣∣ α ∈ C∗, β3 ∈ C, p(α, β3) = 0

for

some Laurent polynomial p(x, y) ∈ Q[x, 1x , y]. Observe that to each value of α corresponds

exactly one value β3(α), because T3 is a G3-torsor. Hence β3(α) = p(α) where p(x) ∈Q[x, 1

x]. If

(αX,αY + 1

24α3((YX)Y ) + p(α)(YX)Y 2

)∈ T3(C), then

((αt)X, (αt)Y +

1

24(αt)3((YX)Y ) +

(p(α)t4 + c3(t− t4)α

)((Y X)Y 2)

)∈ T3(C).

Hence p(α · t) = p(α) · t4 + c3 · (t − t4) · α. Therefore p(x) = a · x4 + c3 · x. The equality

(−2πi)ζ(3) = a(2πi)4 + c3(2πi) implies a = 0 and c3 = −ζ(3). Therefore we get

dimG3 = 1 if and only if ζ(3) ∈ Q.

n. Calculations of Tn and Gn.

We have Θn(S0) = (2πi)X, Θn(S1) = (2πi)Y +n−1∑k=2

(−2πi)ζ(k)((Y X)Y k−1

)+

+(−2πi)ζ(n)((Y X)Y n−1

). Assume that Gn−1 = G(0, 0, c3, 0, c5, 0, c7, . . . | 0, 0, ε3, 0, ε5, 0,

ε7, . . .) (see Corollary 14.4), where c1 = c2 = c4 = · · · = c2k = · · · = 0, c2k+1 = −ζ(2k+ 1)

and ε2k+1 = 0 if ζ(2k + 1) ∈ Q, and c2k+1 = 0 and ε2k+1 = 1 if ζ(2k + 1) /∈ Q.

Then we have that the torsor Tn−1 corresponding to the group Gn−1 is given by

Tn−1 =αX,αY +

n−1∑

k=2,k-pair

rk · αk+1((Y X)Y k−1

)+

n−1∑

k=3,k-impair

((1− εk)ck · α+

+ εkβk

)((Y X)Y k−1

) ∣∣ α ∈ C∗, ∀k βk ∈ C

.

Assume that n = 2p. Then ζ(n)(−2πi) = rn · (2πi)n+1, rn ∈ Q. This equality, the fact

that (2πi)n /∈ Q and the property (∗) imply that

Gn = G(0, 0, c3, 0, c5, . . . , cn−1, 0∣∣ 0, 0, ε3, 0, . . . , εn−1, 0)

and

31

Tn =αX,αY +

n∑

k=2,k-pair

rk · αk+1((Y X)Y k−1

)+

n∑

k=3,k-impair

((1− εk)ck · α+

+ εk · βk)((Y X)Y k−1

) ∣∣ α ∈ C∗, ∀k, βk ∈ C

.

Assume that n = 2p+ 1. Assume that dimGn = dimGn−1. Then

Gn = G(0, 0, c3, . . . , 0, cn∣∣ 0, 0, ε3, 0, ε5, . . . , 0, 0) (by Corollary 14.4)

and

Tn = αX,αY +

n−1∑

k=2,k-pair

rkαk+1((Y X)Y k−1

)+

n−1∑

k=3,k-impair

((1− εk) · ck · α+

+ εk · βk

)((Y X)Y k−1

)+ βn(Y nX)

∣∣ α ∈ C∗; ∀k, βk ∈ C; p(α, ε3 ·β3, ε5 ·β5, . . . , βn) = 0

for some polynomial p(x, y3, y5, . . . , yn) ∈ Q[x,1

x, y3, y5, . . . , yn].

Assume that for some α0 there are two different βn. Then there is g in Gn such

that g(X) = X, g(Y ) = Y + · · ·+ bn((Y X)Y n−1

)with bn 6= 0. Then it follows from the

proof of Lemma 14.5 that Gn = G(0, 0, c3, 0, . . . , 0, 0 | 0, 0, ε3, 0, . . . , 0, 1) and dimGn =

dimGn−1 +1. Hence for any α ∈ C∗ there is a unique βn corresponding to that α. Observe

that βn is an algebraic function of α, β3, β5, . . . , which depends only on α. Hence βn = p(α)

for some p(x) ∈ Q[x, 1

x

].

Choose any f ∈ Gn. Then f(X) = tX, f(Y ) = tY +n−1∑

k=3,k-impair

((1−εk)·ck ·(t−tk+1)+

εk · bk)(

(YX)Y k−1)

+ cn · (t − tn+1)((YX)Y n−1

)for some t ∈ C∗, b3, b5, . . . , bn−2 ∈ C.

Acting by f on a chosen element of Tn we get

((α · t)X, (α · t)Y +

n−1∑

k=2,k-pair

rk(α · t)k+1((YX)Y k−1

)+

+

n−1∑

k=3,k-impair

(α · (1− εk) · ck · (t− tk+1) + α · εk · bk + (1− εk) · ck · α · tk+1+

+ εk · βk · tk+1)(

(Y X)Y k−1)

+(α · cn · (t− tn+1) + βn · tn+1

)((Y X)Y n−1

).

Therefore we get

p(α · t) = cn · α · (t− tn+1) + p(α) · tn+1.

32

This implies p(x) = a · xn+1 + cn · x. The equality ζ(n)(−2πi) = a · (2πi)n+1 + cn · (2πi)implies p(x) = cn · x and cn = −ζ(n). Therefore we get

dimGn = dimGn−1 if and only if ζ(n) ∈ Q.

The final result is the following.

Proposition 6.2. Let G = lim←−n

Gn. Then

G = G(0, 0, c3, 0, c5, 0, c7, . . . , 0, c2k+1, 0 . . . | 0, 0, ε3, 0, ε5, 0 . . . , 0, ε2k+1, 0 . . .),

where ε2k+1 = 0 if and only if ζ(2k + 1) ∈ Q, and then c2k+1 = ζ(2k + 1).

Corollary 6.3. The group G contains the group H := fα | fα(X) = αX, fα(Y ) = αY |α ∈ C∗ if and only if all numbers ζ(2k + 1) are irrational.

Proof. It follows from Proposition 6.2 that all ζ(2k + 1) are irrational if and only if G =

G(0, 0, 0, . . . , 0, . . . | 0, 0, 1, 0, 1, 0, . . .0, 1, 0, 1, 0, . . .) (all ci = 0, ε1 = 0, all ε2k = 0 and all

ε2k+1 = 1 for k = 1, . . .), but then H ⊂ G. Let H ⊂ G = G(0, 0, c3, 0, c5, . . . , 0, c2k+1, 0 . . . |0, 0, ε3, 0, ε5, 0, . . . , 0, ε2k+1, 0, . . .). Then there is f ∈ G such that f(X) = αX, f(Y ) =

αY+∞∑

k=1

x2k+1((YX)Y 2k), where all x2k+1 6= 0. Corollary 14.5 implies thatG = G(0, 0 . . . , 0, . . . |0, 0, 1, 0, 1, . . .0, 1, 0, . . .) (all ci = 0, ε1 = ε2k = 0, ε2k+1 = 1 for k = 1, 2, . . .).

Corollary 6.4. Let G be the smallest subgroup of Aut(π2) defined over Q, which contains

G and H. Then G = G(0, 0 . . . , 0, . . . | 0, 0, 1, 0, 1, 0 . . .) (all ci = 0, ε1 = ε2k = 0, ε2k+1 = 1

for k = 1, 2, . . .).

Proof. The group G contains f such that f(X) = X, f(Y ) = Y +∞∑

k=1

x2k+1((YX)Y 2k)

where all x2k+1 6= 0. Lemma 14.5 implies that the corollary.

Corollary 6.5. Image of GDR(X) in Aut(π2) is the group G from Proposition 6.2.

6.6. We recall from section 10 that the monodromy homomorphism

θ := θ−→01

: π1(X,−→01) −→ π2(X)

33

is given by

S0 → (−2πi)X, S1 → (−2πi)Y +

∞∑

i=0,j=0

(2πi)αi+1,j+1

((YX)XiY j+1

)

(see section 10, formula (7)). We shall describe a torsor and a corresponding group asso-

ciated to θ. It is an observation of Drinfeld that the numbers αi,j satisfy the equation

−1 +∑

n≥0,m≥1

αn+1,m · un+1 · vm = −exp(−

∞∑

k=2

∑

i+j=ki≥1,j≥1

(k − 1)!

i! j!· α1,k−1 · ui · vj

)

(see [Dr]). We have also α1,k−1 = αk−1,1 = ζ(k). Now we can describe the monodromy

homomorphism θ in the following way

S0 → (−2πi)X, S1 →((−2πi)Y, exp

(−

∞∑

k=2

∑

i+j=ki≥1,j≥1

(k − 1)!

i! j!· α1,k−1 ·Xi · Y j

))

(see section 14 for the notation(. . . , exp(. . .)

)). Repeating arguments from the proof of

6.2–6.4 and using Propositions 14.1 and 14.3 instead of Corollaries 14.2 and 14.4 we get

the following result.

Theorem 6.7. i) The torsor T (θ) associated to θ (i.e. to the map obtained from θ by

taking θ−1, passing to suitable quotient and C-completion) is equal

αX, (αY, exp

( ∞∑

k=1

∑

i+j=2ki≥1,j≥1

(k − 1)!

i! j!r2kα

2k+

∞∑

k=3k odd

∑

i+j=ki≥1,j≥1

(k − 1)!

i! j!

((1− ε(k)) · ck · (1− αk) + ε(k) · b1,k−1

)XiY j

)) ∣∣∣

α ∈ C∗, −ζ(2k) = r2k · (2πi)2k, ∀kb1,k−1 ∈ C, ck = ζ(k)

and ε(k) = 0 if ζ(k) ∈ Q, ε(k) = 1 if ζ(k) /∈ Q.

ii) The corresponding group

G(θ) =f ∈ Aut∗(π2(X))

∣∣ f(X) = tX, f(Y ) =

(tY, exp

(∑

k=3k-odd

∑

i+j=ki≥1,j≥1

(k − 1)!

i! j!

(1− ε(k)

)· ck · (1− tk) + ε(k)β1,k−1

)XiY j

) ∣∣∣

t ∈ C∗, β1,k−1 ∈ C.

34

iii) Let G ⊂ Aut∗(π2(X)

)be the smallest group which contains G(θ) and G0 := ft ∈

Aut∗(π2(X)

)| ft(X) = tX, ft(Y ) = tY | t ∈ C∗. Then

G =f ∈ Aut∗

(π2(X)

)| f(X) = tX, f(Y ) =

(tY, exp

( ∞∑

k=3k odd

∑

i+j=ki≥1,j≥1

(k − 1)!

i! j!β1,k−1X

iY j)) ∣∣∣ t ∈ C∗, β1,k−1 ∈ C

.

§7. Homotopy relative tangential base points on P 1(C)\a1, · · · , an+1.

7.1. Let X = P 1(C)\a1, · · · , an+1. Let Tx(P 1(C)) be the tangent space to P 1(C) in x.

Let us set X = X ∪n+1⋃i=1

(Tai(P 1(C))\0).

Let J ′(X) be the set of all continous maps from the closed unit interval [0; 1] to P 1(C)

such that

i) ϕ((0, 1)) ⊂ X;

ii) if ϕ(0) = ai then ϕ is smooth near ai and ϕ(0) 6= 0, and if ϕ(1) = ak then ϕ is smooth

near ak and ϕ(1) 6= 0.

In the sequel we shall identify ϕ(0) (resp. ϕ(1)) with a tangent vector to ϕ in

Tai(P 1(C)) (resp. Tak

(P 1(C)).This tangent vector we shall denote also by ϕ(0) ∈ Tai(P 1(C))

(resp. ϕ(1) ∈ Tak(P 1(C)).

If ϕ(0) = x ∈ X and ϕ(1) = y ∈ X then we say that ϕ is a path from x to y.

If ϕ(0) = ai (resp. ϕ(1) = ak) then we say that ϕ is a path from ϕ(0) ∈ Tai(P 1(C))

(resp. to −ϕ(1) ∈ Tak(P 1(C)) and we shall write ϕ(0) = ϕ(0) (resp. ϕ(1) = −ϕ(1)).

To J ′(X) we joint all constant maps from [0,1] to X and the resulting set we denote

by J(X).

We shall define a relation of homotopy in the set J(X). Let ϕ, ψ ∈ J(X). If ϕ(0) =

ψ(0) = x ∈ X and ϕ(1) = ψ(1) = y ∈ X then we say that ϕ and ψ are homotopic if they

are homotopic maps in the space map([0, 1], 0, 1;X, x, y).

If ϕ(0) = ψ(0) = v ∈ Tai(P 1(C)) and ϕ(1) = ψ(1) = y ∈ X then we say that ϕ and ψ

are homotopic if there is a homotopy

Hs ∈ map([0, 1], 0, 1;X ∪ ai, ai, y) such that

i) Hs ∈ J(X) and Hs(0) = v for all s ∈ [0, 1];

35

ii) Hs((0, 1)) ⊂ X for all s ∈ [0, 1];

iii) H0 = ϕ and H1 = ψ.

We left to the reader the cases when ϕ(0) = ψ(0) = x ∈ X,ϕ(1) = ψ(1) = w ∈Tak

(P 1(C)) and ϕ(0) = ψ(0) = v ∈ Tai(P 1(C)), ϕ(1) = ψ(1) = w ∈ Tak

(P 1(C)).

Let ϕ ∈ J(X) be such that ϕ(0) = ϕ(1) = v ∈ Tai(P 1(C)) and let ψ ∈ J(X) be a

constant map equal to v. We say that ϕ and ψ are homotopic if there is a homotopy

Hs ∈ map([0, 1], 0, 1;X ∪ ai, ai, ai)

such that

i) Hs ∈ J(X) and Hs(0) = Hs(1) = v for all s ∈ [0, 1];

ii) Hs((0, 1)) ⊂ X for all s ∈ [0, 1];

iii) H0 = ϕ and H1(t) = ai for t ∈ [0, 1].

Observe that Gt := H1−t defines a homotopy between ψ and ϕ.

With the definition given above paths α, β ∈ J( C\0) are not homotopic.

We shall write ϕ ∼ ψ if ϕ and ψ are homotopic. The relation ∼ is an equivalence

relation on the set J(X). Let ι(X) := J(X)/ ∼ be the set of equivalence classes.

We define a partial composition in ι(X) in the following way. Let φ,Ψ ∈ ι(X) and let

ϕ, ψ ∈ J(X) be its representatives.

If ϕ(1) = ψ(0) = y ∈ X then we set Ψ φ := [ψ ϕ], the class of ψ ϕ in π(X).

If ϕ(1) = ψ(0) = v ∈ TaiP 1(C) then we can assume that ϕ and ψ coinside near ai

and we define Ψ φ := [ψε ·ϕη], where ϕ1−ε := ϕ|[0,1−ε], ψη := ψ|[η,1] and ϕ(1− ε) = ψ(η).

The map pr:J(X) → X × X/pr(ϕ) = (ϕ(0), ϕ(1)) which associates to a path its

beginning (ϕ(0)) and its end (ϕ(1)) agrees with the relation ∼ and it defines p : ι(X) →X × X. The partial composition o makes p : ι(X)→ X × X into a groupoid over X × X.

36

Let x ∈ X. We set π1(X, x) := p−1(x, x). This is a fundamental group with a base

point in x ∈ X.

7.2. We shall construct a family of horizontal sections of ωx, where a base point x is

replace by a tangent vector.

Let us set V = C\a1, · · · , an. Let x0 ∈ C and let δ : [0, 1] 3 t → ai + t.(x0 − ai) be

an interval joining ai and x0. Let γ be a path from ai to z ∈ V (not passing through any

ak, k = 1, · · ·n) tangent to δ in ai. We assume that in a small neighbourhood of ai the

path γ coincides with δ.

Observe that v = x0 − ai can be canonically identified with a tangent vector to C in ai.

Let ω1 = dzz−a1

, · · · , ωn = dzz−an

. We set

Λai,v(α1, · · · , αk)(z) :=

∫ z

ai,γ

ωα1, · · · , ωαk

if α1 6= i.

Let ε ∈ im(δ) be near ai. Let γε be a part of γ from ε to z, and let δε be a part of δ

from ε to x0. We set

Λai,v(i, · · · , i, αk+1 · · · , αl)(z) :=

limε→ai

∫ z

ε,γε

(∫ z

x0,γε+(δε)−1

dz

x− ai, · · · , dz

z − ai

)ωαk+1

, · · · , ωαl

if αk+1 6= i,

and

Λai,v(i, · · · , i)(z) =

∫ z

x0,γε+δ−1ε

dz

z − ai, · · · , dz

z − ai.

Lemma 7.2.1. The integrals Λai,v(α1, · · · , αk)(z) exist and they are analytic, multivalued

functions on V .

Proof. Assume that αt 6= i for t ≤ l and αl+1 = i. The function g(z) :=∫ z

aiωα1

, · · · , ωαl

is analytic, multivalued on V ∪ ai and vanishes in ai. Hence the integral g1(z) :=∫ z

aig(z) dz

z−aiexists, the function g1(z) is analytic, multivalued on V ∪ ai and vanishes

in ai. Hence by induction we get that Λai,v(α1, · · · , αn)(z) exists, and it is analytic,

multivalued on V ∪ ai.

37

Assume now that αt = i for t ≤ l and αl+1 6= i. Without loss of generality we can

assume that ai = 0 and x0 = 1.

Observe that limε→0

∫ z

x0,γε(δε)−1 zn(logz)mdz = zn+1(

m∑i=0

βi(logz)m−i) where βi are ra-

tional numbers. The function zg(logz)p for g and p positive integers, is analytic, multvalued

on V , continous on any small cone with a vertex in ai (0 in this case) and it vanishes in

ai. The function 1z−aj

for j 6= i is bounded on any sufficiently small neighbourhood of ai.

Hence the integral

limε→ai

∫ z

ε,γε

(∫

x0,γε(δε)−1

dz

z − ai, · · · , dz

z − ai

)dz

z − aj, · · · , dz

z − aik

is an analytic, multivalued function on V , continous, univalued on any small cone with a

vertex in ai and it vanishes in ai.

Let us set

ΛV (z; v, γ) = 1 +∑

(−1)kΛai,v(α1, · · · , αk)(z)Xαk· · ·Xα1

.

We recall that X1, · · · , Xn are duals of dzz−a1

, · · · , dzz−an

.

Lemma 7.2.2. The map

V 3 z → (z,ΛV (z; v, γ)) ∈ V × P (V )

is horizontal with respect to ωV .

It rests to define functions ΛX(z; v, γ) if ai =∞ or all ai are different from ∞.Let f : Y = C\b1, · · · , bn → X = P 1(C)\a1, · · · , an+1 be a regular map of the

form az+bcz+d

with det

(a bc d

)6= 0. It follows from corollary 3.8 that f∗(ΛY (z; y, γ)) =

ΛX(f(x); f(y), f(γ)) if y ∈ Y . We shall use this fact to define ΛX(z; v, γ) where v is a

tangent vector to P 1(C) in ai and γ is a path from ai to z, which is tangent to v and

f−1(γ) coincides with f−1(v) near f−1(ai).

We set

ΛX(z; v, γ) := f∗(ΛY (f−1(z); f−1∗ (v), f−1(γ))).

It is clear that ΛX(z; v, γ) does not depend on the choice of f .

38

Lemma 7.2.3. The map

X 3 z → (z,ΛX(z; v, γ)) ∈ X × P (X)

is horizontal with respect to ωX .

We set LX(z; v, γ) := logΛX(z; v, γ). If we are dealing with only one space X we shall

usually omit subscript X and we shall write Λ(z; v, γ) and L(z; v, γ), or even Λv(z; γ) and

Lv(z; γ), or Λv(z) and Lv(z).

We summarize the constractions from 7.1 and 7.2 in the following proposition.

Proposition 7.2.4. The functions ΛX(z; v, γ) and LX(z; v, γ) depend only on the homo-

topy class of γ in ι(X) = J(X)/ ∼.

§8. Generators of π1(P 1(C)\a1, . . . , an+1, x).

Let X = P 1(C)\a1, . . . , an+1 and let x ∈ X. We shall describe how to choose generators

of π1(X, x). Let vi be a tangent vector in ai. Then the loop around ai at the base point vi

is the following element Sviof π1(X, vi) (see picture).

For each i let us choose a tangent vector vi ∈ Tai(P 1(C)). Let us choose a family of

paths Γ = γin+1i=1 in J(X) from x to each vi such that any two paths do not intersect

and no path self intersects. The indices are chosen in such a way that when we make a

small circle around x (around ak if x is a tangent vector at ak) in the opposite clokwise

direction, starting from γ1 we meet γ2, γ3, . . . , γn+1. If γ1 is a constant path equal v1 we

meet γ2, . . . , γn+1. To the path γi we associate the following element Si in π1(X, x). We

39

move along γi, we make the loop Sviaround ai and we veturn along γ−1

i to x. If γ1 is a

constant path equal v1 then S1 is Sv1.

The following lemma is obvious.

Lemma 8.1. The elements S1, S2, . . . , Sn+1 are generators of π1(X, x). We have Sn+1 ·

. . . · S2 · S1 = 1.

Definition 8.2. The ordered sequence (S1, . . . , Sn+1) of elements of π1(X, x) obtained

from the family of paths Γ = γin+1i=1 we shall call a sequence of geometric generators of

π1(X, x) associated to Γ = γin+1i=1 .

§9. Monodromy of iterated integrals on P 1(C)\a1, · · · , an+1.

Let X = P 1(C)\a1, · · · , an+1. Let x1, x2, x3 ∈ X and let z0 ∈ X. Let γi for i = 1, 2, 3

be a path belonging to J(X) from xi to z0. Let us set γij := γ−1j γi.

Proposition 9.1. Let us prolongate each function Λxi(z) along γi to the point z0. There

exist elements axixj

(γij) ∈ P (X) such that

Λxi(z) · axi

xj(γij) = Λxj

(z)

for all z in a small neibourhood of z0. The elements axixj

(γij) satisfy the following relations

axixi

(γii) = 1,

axixj

(γij) · axjxi

(γji) = 1,

axixj

(γij) · axjxk

(γjk) = axixk

(γik).

Proof. The existence of axixj

(γij) follows from the fact that Λxi(z)′s are horizontal sections.

The first two relations are obvious. The last relation follows from equalities Λxi(z) ·

axixj

(γij) = Λxj(z),Λxj

(z) · axjxk(γjk) = Λxk

(z) and Λxi(z) · axi

xk(γik) = Λxk

(z).

40

Proposition 9.2. Let vk ∈ TakP 1(C)\0. Let Sk be a loop around ak based at vk ∈

Tak(P 1(C))\0, (see picture)

The monodromy of Λvkalong Sk is given by

Sk : Λvk(z)→ Λvk

(z) · e−2πiXk

Proof. The monodromy of Λvk(km)(z) := Λak,vk

(k, k, · · · , k)(z) along Sk is given by

Sk : Λvk(km)(z) → Λvk

(km)(z) +m∑

l=1

Λvk(km−l)(z) (−2πi)l

l! . This implies that the mon-

odromy of Λvk(α1, · · · , αp, k

m)(z) along Sk is given by Sk : Λvk(α1, · · · , αp, k

m)(z) →Λvk

(α1, · · · , αp, km)(z) +

m∑l=1

Λvk(a1, · · · , ap, k

m−l)(z) (−2πi)l

l! . Hence it follows the formula

for the monodromy of Λvk(z) along Sk.

Let x ∈ X. Let us choose vi ∈ TaiP 1(C)\0 for i = 1, 2, · · · , n+ 1. Let (S1, · · · , Sn+1)

be a sequence of geometric generators of π1(X, x) associated to Γ = γ1n+1i=1 where Γ is a

family of paths in J ′(X) from x to vi for i = 1, 2, · · · , n+ 1.

Theorem 9.3. The monodromy of the function Λx(z) along the loop Sk is given by

Sk : Λx(z)→ Λx(z) · axvk

(γk) · e−2πiXk · (axvk

(γk))−1.

Proof. It follows from Proposition 9.1 that

(∗1) Λx(z) · axvk

(γk) = Λvk(z)

for z in the small neibourhood of some γ(z0). This equality is preserved after the mon-

odromy transformation along Sk, hence we have

(∗2) (Λx(z))Sk · axvk

(γk) = (Λvk(z))Sk ,

41

where ( )Sk denotes the function ( ) after the monodromy transformation along Sk. If

follows from Proposition 9.2 that

(∗3) (Λvk(z))Sk = Λvk

(z) · e−2πiXk .

If we substitute (∗3) in (∗2) and then substitute (∗1) for Λvk(z) we get the formula for

(Λx(z))Sk .

Corollary 9.4. The monodromy of the function Lx(z) along the loop Sk is given by

Sk : Lx(z)→ Lvk(z) · αx

vx(γk) · (−2πiXk) · αx

vk(γk)−1

where αxvk

(γk) = log(αxvk

(γk)).

Proof. The corollary follows immediately from Proposition 3.9. ii).

9.5 It follows from above, that the definition of the monodromy homomorphism (Definition

3.3) extends to any x ∈ X. Hence for any v ∈ X we have a monodromy homomorphism

θv,X : π1(X, v)→ P (X)

and if v, v′ ∈ X, then the homomorphisms θv,X and θv′,X are conjugated.

Proposition 9.6. Let f : X = P 1(C)\a1, . . . , an+1 → Y = P 1(C)\b1, . . . , bm+1 be a

regular map. Then for any v, w ∈ X, a path γ from v to z and a path δ from v to w we

have

f∗(ΛX(z; v, γ)) = ΛY (f(z); f(v), f(γ))

and

f∗(avw(δ)) = a

f(v)f(w)(f(δ)).

(notation: f(v) := f∗(v) if v is a tangent vector).

Proof. The proposition follows from the definition of ΛX(z; v, γ) and avw for tangent vec-

tors and from Corollary 3.8.

42

§10. Calculations

Let X = P1(C)\0, 1,∞. The forms dzz

and dzz−1

form a base of A1(X). Let X := ( dzz

)∗

and Y := ( dzz−1

)∗ be the dual base of (A1(X))∗. Let us set Z := −X − Y . The group P (X)

is the group of invertible power series with a constant term equal 1 in non-commuting

variables X and Y .

Let us fix a path γ1 = interval [0, 1] from−→01 to

−→10. It follows from Proposition 9.1

that

Λ−→10

(z) · a−→10−→01

(X,Y ) = Λ−→01

(z).(1)

Let f(z) = 1− z. It follows from Proposition 9.6 that

f∗(a1001(X,Y )) = a01

10(X,Y ).

Proposition 9.1 implies

a0110(X,Y ) = (a10

01(X,Y ))−1.

Observe that f∗(X) = Y and f∗(Y ) = X. Hence we get the Deligne formula

a1001(X,Y ) = (a10

01(Y,X))−1.(2)

(The proof of (2) given here repeats essentially the Deligne proof.)

Let us fix a path γ∞ = interval [∞,−ε]+ arc from −ε to ε passing by (−i) ·ε (ε > 0) +

interval from ε to 0) from−→∞0 to

−→01. Let S0 (around 0), S1 (around 1) and S∞ (around

∞) be geometric generators of π1(X,−→01) associated to the family γ0, γ1, γ∞, where γ0 is

the constant path equal−→01. Then we have S0 S1 S∞ = 1. The monodromy of Λ−→

01(z)

is given by the following formulas (see Theorem 9.3)

S0 : Λ−→01

(z)→ Λ−→01

(z) · e(−2πi)X ,

S1 : Λ−→01

(z)→ Λ−→01

(z) · (a0110(X,Y ))−1 · e(−2πi)Y · a01

10(X,Y ),(3)

S∞ : Λ−→01

(z)→ Λ−→01

(z) · e−πiX · (a0110(Z,X))−1 · e(−2πi)Z · (a01

10(Z,X)) · eπiX .

The monodromy along S∞ needs some explanations. By Theorem 9.3 it is given by the

formula S∞ : Λ−→01

(z) → Λ−→01

(z) · (a∞001 (X,Y ))−1 · e(−2πi)Z · a∞0

01 (X,Y ). By Proposition

43

9.1 a∞001 (X,Y ) = a∞0

0∞(X,Y ) · a0∞01 (X,Y ). One calculates that a0∞

01 (X,Y ) = eπi·X . Let

f(z) = z−1z

. Then it follows from Proposition 9.6 that a0110(Z,X) = a∞0

0∞(X,Y ). Hence we

get the formula describing the monodromy along S∞.

The Lie algebra L(X) is the completion of the free Lie algebra on two generators X

and Y . Let us set α(X,Y ) := α0110(X,Y ) := loga01

10(X,Y ). The monodromy of L−→01

(z) is

given by the following formulas (see Corollary 9.4).

S0 : L−→01

(z)→ L−→01

(z) · (−2πi)X,

S1 : L−→01

(z)→ L−→01

(z) · α(X,Y )−1 · (−2πi)Y · α(X,Y ),(4)

S∞ : L−→01

(z)→ L−→01

(z) · (−πi)X · α(Z,X)−1 · (−2πi)Z · α(Z,X) · (πi) ·X.

We shall calcualte coefficients of a0110(X,Y ) and α(X,Y ). If ω is a monomial in X and

Y, a(ω) is the coefficient at ω of a0110(X,Y ). Let X be the first basic Lie element and let Y

be the second basic Lie element. We shall choose a base of a free Lie algebra on X and Y

as in [MKS] pages 324-325. If ω is an element of this base, let α(ω) be the coefficient at ω

of α(X,Y ). It follows from the formula (1) that

a(XnY ) = (−1)nζ(n+ 1), a(Y nX) = (−1)n+1ζ(n+ 1),(5)

a(XiY j) =

∫ 1

0

(− dz

z − 1)j , (−dzz

)i, a(Y jXi) =

∫ 1

0

(−dzz

)i, (− dz

z − 1)j .(6)

(If ω is a one-form then ωi := ω, ω, · · · , ω i-times.) It follows from (2) that a(X iY j) +

a(Y iXj) = 0. It follows from [Ch] that a(X iY j) + (−1)i+ja(Y jXi) = 0. Hence we get

αi,j := α((YX)Xi−1Y j−1) = (−1)ia(XiY j) = (−1)j−1a(Y jXi)

and

α((YX)Xj−1Y i−1) = α((YX)Xi−1Y j−1).

Let us set π′ := [π(X), π(X)] and π′′ := [π′, π′]. It follows from (3) that the monodromy

homomorphism

θ−→01

: π1(X,−→01)→ π2(X) := π(X)/π′′

44

is given by

S0 →(−2πi)X,

S1 →(−2πi)Y + [−2πiY, α(X,Y )](7)

= (−2πi)Y +

∞∑

i=0,j=0

(2πi)αi+1,j+1((YX)XiY j+1).

The formula

∫ z

0

F (z)dz

z, (dz

z)n =

n∑

i=0

(−1)n−i

(n− i)!i! (∫ z

0

F (z)(logz)n−i dz

z)(logz)i

implies

αn+1,m =(−1)n(−1)m

n!m!

∫ 1

0

(log(1− z))m(logz)n dz

z.(8)

§11. The configuration spaces

Let X = P 1(C)\a1, · · · , an+1 and X ′ = P 1(C)\a′1, · · · , a′n+1. If the sequences (a, x) :=

(a1, · · · , an+1, x) and (a′, x′) := (a′1, · · · , a′n+1, x′) are close then the groups π1(X, x) and

π1(X ′, x′) are canonically isomorphic. We shall study how the monodromy homomor-

phisms θx,a; = θx,X and θx′,a′ := θx′,X′ from sections 3 and 9

θx,a : π1(X, x) → π(X)oo ‖

θx′,a′ : π1(X ′, x′) → π(X ′)

depend on a and a′.

Let Xn = (z1, · · · , zn) ∈ Cn|zi 6= zj if i 6= j. The space of global one-forms on Xn

with logarithmic singularities, A1(Xn) is spanned bydzi−dzj

zi−zjfor i, j ∈ 1, 2, · · · , n and

i < j. Let Xij = (dzi−dzj

zi−zj)∗ be their formal duals. We set Xji = Xij = 0. Dualizing the

map∧2

(A1(Xn))→ A1(Xn) ∧A1(Xn)

we get that R(Xn) is generated by

[Xij, Xik +Xjk] with i, j, k different

45

and

[Xij , Xkl] with i, j, k, l different.

Let x = (x1, · · · , xn, xn+1) ∈ Xn+1 be a base point. Let pi : Xn+1 → Xn

(i = 1, · · · , n+ 1) be a projection pi(z1, · · · , zn+1) = (z1, · · · , zi, · · · , zn+1), let x(i) :=

(x1, · · · , xi−1, xi+1, · · · , xn+1) and let X(i, x) := p−1i

(x(i)

)= C\x1, · · · , xi, · · · , xn+1.

(z means z is omitted). Let ki : X(i, x)→ Xn+1 be given by ki(z) = (x1, · · · , xi−1, z, xi+1,

· · ·xn+1). The inclusion ki induces

(ki)∗ : P (X(i, x))→ P (Xn+1)

and

(ki)∗ : π(X(i, x))→ π(Xn+1)

where (ki)∗(Xj) = Xij and Xj is the formal dual of dzz−aj

on X(i, x). The map (ki)∗ is

injective and its image, (ki)∗(π(X(i, x)) is a normal subgroup of π(Xn+1).

Let x = (x1, · · · , xn, xn+1) ∈ Xn+1 and x′ = (x′1, · · · , x′n, x′n+1) ∈ Xn+1. Let us set

X := X(n + 1, x) and X ′ = X(n + 1, x′). We choose a family of non-intersecting paths

γ1, · · · , γn, γn+1 in C from x1 to x′1, · · · , xn to x′n and xn+1 to x′n+1. We shall identify

π1(X, xn+1) and π1(X ′, x′n+1) in the following way. Observe that γ = (γ1, · · · , γn, γn+1)

is a path in Xn+1 from x to x′. The identification isomorphism γ. : π1(X, xn+1) →π1(X ′, x′n+1) is the unique isomorphism making the following diagram commutative

π1(X, xn+1)(kn+1)∗−−−−−−→ π1(Xn+1, x)

∨

...γ. ↓ γ#

π1(X ′, x′n+1)(kn+1)∗−−−−→ π1(Xn+1, x

′).

(γ# is induced by the path γ is a standard way).

Proposition 11.1. After the identification of the fundamental groups ofX = C\x1, · · · , xnand X ′ = C\x′1, · · · , x′n by γ. the monodromy homomorphisms

θxn+1,X : π1(X, xn+1)→ π(X) and θx′n+1

,X′ : π1(X ′, x′n+1)→ π(X ′)

46

(π(X) = π(X ′)) are conjugated by an element of the group π(Xn+1). (The group (kn+1)∗π(X)

is a normal subgroup of π(Xn+1) so π(Xn+1) acts on π(X) by conjugations.)

Proof. The corollary follows from the commutative diagram

π1(X, xn+1)θxn+1,X

−−−−−−→ π(X)

(kn+1)∗ (3) (kn+1)∗

π1(Xn+1, x)θx,Xn+1

−−−−→ π(Xn+1)↓ γ. (4) ↓ γ# (1) ↓ cLXn+1

(x′;x,γ)

π1(Xn+1, x′)

θx′,Xn+1

−−−−→ π(Xn+1)(kn+1)∗ (2) (kn+1)∗

π1(X ′, x′n+1)

θx′n+1

,X′

−−−−→ π(X ′) = π(X)

where cLXn+1(x′;x,γ) is a conjugation by the element LXn+1

(x′;x, γ). It follows from Propo-

sition 3.4 that the square (1) commutes. Corollary 3.8 implies that (2) and (3) commutes.

The square (4) commutes by the construction.

Corollary 11.2. Let x = (x1, · · · , xn+1) ∈ Xn+1. Let us set X(i) := X(i, x). Let aij

be the following element of π1(X(i), xi)- a geometric generator of π1(X(i), xi), which is

a loop around the point xj. Let Aij be its image in π1(Xn+1, x). Then θx,Xn+1(Aij) is

conjugated to (−2πi)Xij in the group π(Xn+1).

Proof. It follows from Proposition 9.2 that θxi,X(i)(aij) is conjugated to (−2πi)Xij in the

group π(X(i)). Hence the statement follows from Corollary 3.8.

Now we shall study the relation between the monodromy representation for the con-

figuration spaces (C\0, 1)n∗ and (C\0, 1)m

∗ . We shall use the Ihara result (see [I1] The

Injectivity Theorem (i)). Let Yn := (P 1(C))n∗ . The group PGL2(C) acts diagonally on

Yn and let Yn := Yn/PGL2(C). Let ψk : Xn−1 → Yn be the composition of the map

(x1, · · · , xk−1, xk+1, · · · , xn) → (x1, · · · , xk−1,∞, xk, · · · , xn) and the projection Yn → Yn.

The map ψk induces (ψk)∗ : H(Xn−1) → H(Yn). Let us set Xij = (ψk)∗(Xij) where

Xij = (dxi−dxj

xi−xj)∗ ∈ H(Xn−1). (We use the same notation for Xij ∈ H(Xn−1) and its

image in H(Yn). Notice also that Xij in H(Yn) does not depend on the choice of ψk.)

Let Aij ∈ π1(Xn−1, x) be such as in Corollary 11.2. The image of Aij in Yn we shall

also denote by Aij .

47

Corollary 11.3. The element θy,Yn(Aij) is conjugated to (−2πi)Xij in π(Yn).

Proof. It follows from Corollary 11.2 and the commutative diagram

θx,Xn−1: π1(Xn−1, x) → π(Xn−1)

↓ (ψk)∗ ↓ (ψk)∗θy,Yn

: π1(Yn, y = ψk(x)) → π(Yn).

Let Aut∗(π(Yn)) be a subgroup of AutC(π(Yn)) defined in the following way:

Aut∗(π(Yn)) = f ∈ AutC(π(Yn))|∃αf ∈ C∗, f(Xij) ∼ αf ·Xij.

(AutC( ) dentoes C-linear automorphisms and ∼ means conjugated.)

Let us set

Tn(C) = ϕ ∈ Hom(π1(Yn,y);π(Yn))|∃α ∈ C∗, ∀Aij , ϕ(Aij) ∼ αXij

(Aij ∈ π1(Yn, y) are as in Corollary 11.3.). Observe that T n(C) is an Aut∗(π(Yn))-torsor.

The subgroup of inner automorphisms Inn(π(Yn)) is a normal subgroup of Aut∗(π(Yn)).

Hence tn(C) := Tn(C)/Inn(π(Yn)) is a Out∗(π(Yn)) := Aut∗(π(Yn))/Inn(π(Yn))-torsor.

The following result is an analog of the Ihara Injectivity Theorem (see [I1] page 4).

Proposition 11.4. The canonical map Out∗(π(Yn)) → Out∗(π(Yn−1)) is injective for

n ≥ 5.

Proof. Let Out∗1(π(Yn)) := ker(Out∗(π(Yn))N→ C∗), where N(f) = αf . The Lie algebra

of Out∗(π(Yn)) is the Lie algebra of special derivations of L(π(Yn)) modullo inner deriva-

tions. The Lie version is proved in [I1] page 12. Because Out∗1(π(Yn)) is pro-nilpotent, the

Lie version implies the result for Out∗1(π(Yn)), and then also for Out∗(π(Yn)).

The surjective homomorphisms (pn+1)∗ : π1(Yn+1, y) → π1(Yn, y′) and (pn+1)∗ :

π(Yn+1)→ π(Yn) induce the morphism of torsors

tn+1(C)→ tn(C)

compatible with Out∗(π(Yn+1))→ Out∗(π(Yn)).

Lemma 11.5. The canonical morphism of torsors tn+1(C)→ tn(C) is injective for n ≥ 4.

This follows immediately from Proposition 11.4.

48

Corollary 11.6. The monodromy homomorphism θy,Yn: π1(Yn, y) → π(Yn) is deter-

mined (up to conjugacy by an element of π(Yn)) by the homomorphism θy′,Y4: π1(Y4, y

′)→π(Y4).

Proof. Observe that θy,Yn∈ tn(C) and θy′,Y4

is the image of θy,Ynunder the canonical

morphism tn(C)→ t4(C).

Let a := (a1, · · · , an, an+1) be a sequence of n + 1 different points in P 1(C) and let

Xa := P 1(C)\a1, · · · , an, an+1. The vector space H(Xa) is spanned by Xi := ( dzz−ai

−dz

z−an+1)∗ i = 1, · · · , n. Let us set Xn+1 := −∑n

i=1Xi. Let Ak denotes a geometric

generator of Xa, which is a loop around ak. Let us set

Ta(C) := f ∈ Hom(π1(Xa, x)→ π(Xa))|∃αf ∈ C∗, ∀Ak f(Ak) ∼ αfXk.

Assume that a = (ai)n+1i=1 is such that a1 = 0, a2 = 1, a3 =∞. The fibration

Xa

kn+2

−−−−→ Yn+2

pn+2

−−−−→ Yn+1 (Xa = (pn+2)−1(a1, · · · , an+1))

realizes π(Xa) as a normal subgroup of π(Yn+2) ((kn+2)∗(Xi) = Xi,n+2). Hence the group

π(Yn+2) acts on Ta(C) and let

ta(C) := Ta(C)/π(Yn+2).

Observe that any π(Yn+2)-conjugate of Xi,n+2 is in the image of π(Xa). Hence the re-

striction map

(kn+2)∗ : tn+2(C)→ ta(C)

given by f → f|π1(Xa,x) is defined. We set

τa(C) := im (tn+2(C)→ ta(C)).

Observe that the diagram

tn+2(C)(kn+2)

∗

−−−−→ τa(C)↓pr ↓ pr1t4(C)

(k4)∗−−−−→

≈τ0,1,∞(C)

commutes where the map pr1 is induced by the inclusion Xa → P 1(C)\0, 1,∞. The

map (k4)∗ is bijective because Y4 = P 1(C)\0, 1,∞. Lemma 11.5 implies that the map

pr is injective. Hence both maps, (kn+2)∗ and pr1 are injective. Hence we have proved the

following result.

49

Proposition 11.7. i) The π(Yn+2)-conjugacy class of the monodromy homomorphism

θx,Yn+2: π1(Yn+2, x)→ π(Yn+2) is determined by its restriction to π1(Xa, x

′).

ii) The π(Yn+2)-conjugacy class of the monodromy homomorphism θx,Xa: π1(Xa, x) →

π(Xa) is determined by the monodromy homomorphism

θx′,P 1(C)\0,1,∞ : π1(P 1(C)\0, 1,∞, x′)→ π(P 1(C)\0, 1,∞).

§12. The Drinfeld-Ihara Z/5-cycle relation

In this section we show that the element which describes the monodromy of all iterated

integrals on P 1(C) \ 0, 1,∞ satisfies the Drinfeld-Ihara relation.

12.1. Configuration spaces

If T is a topological space we set T n∗ = (t1, · · · , tn) ∈ Tn | ti 6= tj if i 6= j. The

group Σn acts on Tn∗ by permutations.

Let us set Yn =(P 1(C)

)n∗

and Yn =(P 1(C)\0, 1,∞

)n−3

∗. Let a, b, c ∈ P 1(C) be

three different points and let ϕa,b,c(z) = b− cb− a ·

z − az − c . The map Φ4,5 : Y5 → Y5 given by

Φ4,5(x1, x2, x3, x4, x5) =(ϕx1,x2,x3

(x4), ϕx1,x2,x3(x5)

)induces a bijection

ϕ4,5 : Y5/PGL2(C)→ Y5.

The action of Σ5 on Y5 induces an action of Σ5 on Y5. The map σ : Y5 → Y5, σ(s, t) =(t− 1t− s ,

1s

)corresponds to the permutation σ of Ys given by

σ(x1, x2, x3, x4, x5) = (x2, x3, x4, x5, x1).

Observe that the points A =(√

5− 12 ,

√5 + 12

)∈ Y5 and B =

(−√

5− 12 , −

√5 + 12

)∈

Y5 are fixed by σ.

The one-forms dss ,dss− 1 ,

dtt ,

dtt− 1 ,

ds− dts− t generate A1(Y5) and H1

DR(Y5). Let

S0, S1, T0, T1 and N be their formal duals. The subspace R(Y5) of H(Y5)⊗2 is generated

by

[Si, N ] + [Ti, N ] i = 0, 1;

[Si, Ti] + [Si, N ] i = 0, 1;

[Ti, Si] + [Ti, N ] i = 0, 1;

[S0, T1] and [S1, T0]

50

where [A,B] = A⊗B − B ⊗ A.

Let G := P (Y5) i.e. G is a multiplicative group of the algebra of formal power series

in non-commuting variables S0, S1, T0, T1 and N divided by the ideal generated by R(Y5).

The principal fibration

Y5 ×G→ Y5

we equipped with the integrable connection given by the one form

ωY5=( dt

t− 1− dt

t

)⊗ T1 +

(− dt

t

)⊗ T∞

+(ds− dts− t −

dt

t

)⊗N +

ds

s⊗ S0 +

ds

s− 1⊗ S1

where T∞ = −T0 − T1 −N . We shall write shortly ω instead of ωY5.

12.2. Integration of w

We recall that on P 1(C)\0, 1,∞ we have

Λ−→∞1

(z) · a−→∞1−→1∞

= Λ−→1∞

(z) (see Proposition 9.1).12.2.0

The monodromy of Λ−→∞1

(z) is given by:

(around ∞) : Λ−→∞1

(z)→ Λ−→∞1

(z) · e−2πiT∞ ;

(around 1) : Λ−→∞1

(z)→ Λ−→∞1

(z) · a∞11∞ · e−2πiT1 · (a∞1

1∞)−1,

(see Theorem 9.3). We have f∗(a0110(T0, T1)

)= a∞1

1∞(T∞, T1) where f∗(T0) = T∞, f∗(T1) =

T1 and f(z) = 1/z.

We have assymptotically at ∞

Λ−→∞1

(z) ∼z=∞

e

(∫z

1

dtt

)T∞

12.2.1

i.e. limz→∞z>1

(Λ−→∞1

(z) · e−( ∫

z

1

dtt

)T∞)

= 1.

Let Pε = (ε, 1 + ε) ∈ Y5 where ε > 0 and small. Let ΛPε(z; path) be a horizontal

section of ω such that ΛPε(Pε) = 1. Let γ be a path in Y5 from Pε to σ(Pε) = (ε, 1

/ε)

which is constant (= ε) on the first coordinate.

51

Assuming s = constant (= ε) we have

ΛPε(z) · a1+ε

−→1∞

= Λ−→1∞

(z).

Hence we have assymptotically for positive, small ε

a1+ε−→1∞

∼ε=0

e

(−∫

1+ε

∞

(dtt− 1

−dtt

)T1

).12.2.2

It follows from 12.2.0, 12.2.1 and 12.2.2 and Proposition 9.1 that for ε positive, near 0

ΛPε

(σ(Pε); γ

) ∼ε=0

e

( ∫ 1/ε1

dzz)T∞ · a∞1

1∞(T∞, T1) · e( ∫

1+ε

∞

dtt− 1

−dtt

)T1.12.2.3

Let p = γ + σ(γ) + σ2(γ) + σ3(γ) + σ4(γ). Then ΛPε(Pε; p) = 1 because the path p is

contractible in Y5. On the other hand

1 = ΛPε(Pε, p) = Λσ4(Pε)

(Pε;σ4(γ)

)· Λσ3(Pε)

(σ4(Pε);σ3(γ)

)

· Λσ2(Pε)

(σ3(Pε);σ2(γ)

)· Λσ(Pε)

(σ2(Pε);σ(γ)

)· ΛPε

(σ(Pε); γ

).

The formula

(σi)∗(ΛPε

(σ(Pε), γ))

= Λσi(Pε)

(σi+1(Pε), σi(γ)

)

(see Corollary 3.8) implies that

1 = σ4∗(L) · σ3

∗(L) · σ2∗(L) · σ∗(L) · L

where L = ΛPε(σ(Pε), γ). Let

L = e

(∫ 1/ε1

dzz)T∞ · a∞1

1∞(T∞, T1) · e( ∫ 1+ε

∞

dtt− 1

−dtt

)T1.

It follows from 12.2.3 that

1 ∼ε=0

σ4∗( L) · σ3

∗( L) · σ2∗( L) · σ∗( L) · L.

52

The factors e

∫ 1+ε

∞

(dtt− 1

−dtt

)T∞

(=σ2

∗(T1))

and e

(∫ 1/ε1

dzz)T∞ can be placed together

in the product σ4∗( L) · . . . · L because T∞ = σ2

∗(T1) commutes with σ∗(T1) = S0 and

σ∗(T∞) = S1. After the calculations we get∫ 1+ε

∞

(dt

t− 1− dt

t

)−∫ 1/ε

1

dt

t= −log(1 + ε).

Repeating the same argument for S1, S1 +T1 +N , T1 and S0 and passing to the limit with

ε we get

σ4∗(a) · σ3

∗(a) · σ2∗(a) · σ∗(a) · a = 1

where a = a∞11∞(T∞, T1). The last formula we can write in the form

a(S0, S1 + T1 +N) · a(T1, S1) · a(S1 + T1 +N, T∞) · a(S1, S0) · a(T∞, T1) = 1

because σ∗(S0) = T∞, σ∗(S1) = S1 + T1 + N , σ∗(T0) = N , σ∗(T1) = S0 and σ∗(N) =

−S0 − S1 −N .

Let ψ5 : C4∗ → Y5 be given by ψ5(z1, z2, z3, z4) = Φ4,5(z1, z2, z3, z4,∞). Let (Aij)i,j

be formal duals of(dzi − dzjzi − zj

)i,j

. Then we have

ψ5∗(A12) = −S0 − S1 − T0 − T1 −N,

ψ5∗(A13) = S1 + T1 +N,

ψ5∗(A14) = S0,

ψ5∗(A23) = S0 + T0 +N,

ψ5∗(A24) = S1,

ψ5∗(A34) = −S0 − S1 −N.

Using ψ1 : C4∗ → Y5 given by ψ1(z2, z3, z4, z5) = Φ4,5(∞, z2, z3, z4, z5) we get

ψ1∗(A23) = S0 + T0 +N,

ψ1∗(A24) = S1,

ψ1∗(A25) = T1,

ψ1∗(A34) = −S0 − S1 −N,

ψ1∗(A35) = −T0 − T1 −N,

ψ1∗(A45) = N.

53

We set Xij := ψε∗(Aij) ε = 1, 5. Then X15 = T0. Hence finally we get a formula

a(X14, X13) · a(X25, X24) · a(X13, X35) · a(X24, X14) · a(X35, X25) = 1.12.2.4

If we use Φ2,4 : X5∗ → Y5 given by Φ2,4(0, s, 1, t,∞) = (s, t) and repeat the calculations

in Y5 we get the same formula as before, but the Xij ’s names of S0, S1, . . . are now different

and the resulting formula is:

a(X12, X15) · a(X34, X23) · a(X15, X45) · a(X23, X12) · a(X45, X34) = 1.12.2.5

This is exactly the formula which appears in [12] page 106 if we replace a( ) by a( )−1.

Proposition 12.3. For any permutation σ of five letters we have

a(Xσ(14), Xσ(13)) · a(Xσ(25), Xσ(24)) · a(Xσ(13), Xσ(35))i)

· a(Xσ(24), Xσ(14)) · a(Xσ(35), Xσ(25)) = 1,

a(Xσ(12), Xσ(15)) · a(Xσ(34), Xσ(23)) · a(Xσ(15), Xσ(45))ii)

· a(Xσ(23), Xσ(12)) · a(Xσ(45), Xσ(34)) = 1,

where σ(ij) = σ(i)σ(j).

Proof. It follows from 12.2.4, 12.2.5 and Corollary 3.8.

Remark. The formulas of Proposition 12.3 are in the group P (Y5). If we apply log we

get formulas in the group π(Y5).

In the sequel we shall work in the group π(Y5).

We finish this section with a formula from which the Deligne Z/3-cycle relation can

be obtained. The proof is an imitation of the Deligne proof.

Proposition 12.4. Let α := loga. In the group π(Y5) we have

α(X25, X23)(−πiX23)α(X23, X35)(−πiX35)α(X35, X25)(−πiX25) = −πiX14i)

and

α(X25, X23)(πiX23)α(X23, X35)(πiX35)α(X35, X25)(πiX25) = πiX14.ii)

54

Proof. Let σ(x1, x2, x3, x4, x5) = (x1, x5, x2, x4, x3). Then the induced map σ : Y5 → Y5

is given by σ(s, t) =(

t−1t · s

s−1 ,t−1

t

)and σ2(s, t) =

(s

s−t ,1

1−t

). Let P− = (r, 1 − r) and

P+ = (r, 1 + r) where r is positive and small. Let Q− = (−r, 1− r) and Q+ = (−r, 1 + r).

Let γ be a path from P+ = (r, 1 + r) to σ2(Q−) = (r, 1/r), which is constant on the first

coordinate. Let γ′ be a path from Q+ to σ2(P−) passing through the point(

r2r−1 , 1 + r

)

which is piecewise constant, first on the second coordinate, next on the first coordinate.

Let S be a path [0, π] 3 ϕ → (r, 1 + rei(ϕ+π)) and let S′ be a path [0, π] 3 ϕ →(−r, 1 + rei(ϕ+π)). Let us consider the composition p = σ(γ ′) σ(S′) σ2(γ) σ2(S) γ′ S′ σ(γ) σ(S) σ2(γ′) σ2(S′) γ S. If we integrate the form ω along this path and

pass to the limit if r → 0 we get the square of the left hand side of the expression i).

Let α be a loop in the opposite clockwise direction around (0, 0) in the plane P =

(s, t) ∈ C2 | αs+ βt = 0. The integration of the form ω along α gives (−2πi)(S0 +N +

T0) = (−2πi)X23. In the model of Y 5∗ /PGL2(C) in which the subspace (x1, x2, x3, x4, x5) |

x1 = x4 of (P 1(C))5 degenerates to a point (for example for Φ2,5(0, s, 1,∞, t) = (s, t)), the

path p is homotopic to a loop around one of the points (0, 0), (1, 1) or (∞,∞) in the plane

passing through the corresponding point (0, 0), (1, 1) or (∞,∞) (the point (1, 1) in the

case of the model Φ2,5). Hence the left hand side of the expression i) is also (−2πi) ·X14.

The proof of the second equality is similar.

Corollary 12.5. For any permutation σ of five letters 1, 2, 3, 4, 5 we have formulas i ′) and

ii ′), which are obtained from formulas i) and ii) by replacing indices 1, 2, 3, 4, 5 by σ(1),

σ(2), σ(3), σ(4), σ(5).

Proof. One consider the map of Y5 given by (xi)i=1,...,5 → (xσ(i))i=1,...,5. The induced

map σ : Y5 → Y5 satisfies σ∗ω = σ∗ω. This implies formulas i′) and ii′).

Remark. We have X23 + X25 + X35 = X14 in the Lie algebra Lie(Y5). If we set X14 = 0

then the formulas i) and ii) reduce to the Deligne formula.

55

§13. Functional equations of iterated integrals.

In this section we give necessary and sufficient conditions to have a functional equation

of iterated integrals in terms of exotic analogues of the Bloch group. We shall work only

on a pointed projective line.

Let Lie(C(z)∗) be a free Lie algebra on the abelian group C(z)∗. Let I ⊂ Lie(C(z)∗)

be a Lie ideal consisting of all brackets [. . . [f1, f2] . . . [. . . , fk] . . .] such that at least one fi

is in C∗. Observe that L(C(z)∗) is a free Lie algebra on the set (z − a) | a ∈ C.

Let X = P1(C)\a1, . . . , an,∞ and let Y = P1(C)\b1, . . . , bm,∞. We set

B(Y) := ⊕f∈C(z)\b1,...,bm

Z ,

the free abelian group on the set C(z)\b1, . . . , bm. The generator corresponding to f we

shall denote by [f ]. For any homogenous element e ∈ Lie(H(Y)

),

e =∑

i

αi[. . . [Bi1 . . . [. . . , Bik] . . .] ,

where Bj =(

dzz−bj

)∗we define a map

bY (e) : B(Y)→ L(C(z)∗)

by the formula

bY (e)([f ]) =∑

i

αi[. . . [f − bi1 , . . . [. . . , f − bik] . . .].

Let us fix an ordering B1 =(

dzz−b1

)∗, B2, . . . , Bm of the base of H(Y). Then there is a

canonical base B = eii∈I of Lie(H(Y)

)given by basic Lie elements (see [MKS]). Let

e∗i i∈I be the dual linear forms.

56

Theorem13.1. Let e1, . . . , en ∈ B be basic Lie elements of degree n. Let f1, . . . , fn : X→Y be regular maps. Let n1, . . . , nN be integers. Let γ be a path in X from x to z. The

following conditions are equivalent.

N∑

i=1

ni · e∗i(LY(fi(z); fi(x), fi(γ)

)= 0 ;i)

N∑

i=1

ni · e∗i (fi)∗ = 0 in Hom(

Γnπ(X)/Γn+1π(X); C

);ii)

N∑

i=1

ni · bY(ei)([fi]) = 0 .iii)

Proof. It follows from Corollary 2.2 in [W4] that conditions i) and ii) are equivalent.

Hence we must show that ii) and iii) are equivalent. Let e = [. . . [Bi1 , . . . [. . . , Bik] . . .] be

a basic Lie element of degree n and let e∗ be its dual. Let f : X → Y be a regular map.

The map f and e∗ induce

Lie(H(X)

) f∗−→ Lie(H(Y)

) e∗−→ C.

Let us set C∨ := Hom(C,C). Passing to dual objects we get a map

Lie(A1(X)

) f∗

←− Lie(A1(Y)

) (e∗)∨←− C∨.

Observe that the condition ii) is equivalent to the condition

N∑

i=1

ni · (fi)∗ (e∗i )∨ = 0 in Hom

(C∨,Lie

(A1(X)

)).ii′)

Observe that f∗(

dzz−bi

)= f ′(z)

f(z)−bidz and (e∗)∨(idC) =

[. . .[

dzz−bi1

. . .[. . . , dz

z−bik

]. . .].

Let us define maps

iX : Lie(A1(X)

)→ L

(C(z)∗

)and iY : Lie

(A1(Y)

)→ L

(C(z)∗

)

by the formula iX

(dz

z−ai

)= (z − ai). Observe that the diagram

Lie(A1(X)

) (f∗)←− Lie(A1(Y)

) (e∗)∨←−C∨yiX

yiY e := iY (e∗)∨

L(C(z)∗

) (f])←− L(C(z)∗

)

57

commutes, where f ](z − a) = f(z) − a and e(idC) = [. . . [z − bi1 , . . . [. . . , z − bik] . . .].

The maps iX and iY are inclusions, hence the condition ii’) is equivalent to the condi-

tion∑N

i=1 ni · (fi)] ei = 0. This last condition is equivalent to the condition

∑Ni=1 ni ·

(fi)](ei(idC)

)= 0. Observe that (fi)

](ei(idC)

)= bY(ei)[fi]. Hence we get that ii) and iii)

are equivalent.

§14. Subgroups of Aut(π2(X)).

Let us set L′ := L(X), L′′ := [L′, L′] and L2(X) := L(X)/L′′ . Then the Lie algebra L2(X)

equipped with the multiplication given by the Baker-Campbell-Hausdorff formula is the

group π2(X) from section 10.

Let p : π2(X) → π2(X)/Γ3π2(X) be the natural projection. The map p induces

p∗ : Aut(π2(X)) → Aut(π2(X)/Γ3π2(X)). Let Aut∗(π2(X)) := f ∈ Aut(π2(X))∣∣∣∃ α ∈

C∗, f(X) = αX, f(Y ) ≈ αY, f(Z) ∼ αZ(Z = −X − Y )(≈ is a conjugation by an el-

emt of [π2(X), π2(X)] and ∼ is a conjugation by an element of π2(X)). Let C : Gm →Aut(π2(X)/Γ3π2(X)) be given by Ct(X) = tX, Ct(Y ) = tY.We shall investigate liftings of C to

Aut∗(π2(X)). We recall that elements X,Y and Y X iY j−1 := (. . . (Y,X)X) . . .X)Y ) . . . Y )

i = 1, . . . , j = 1, . . . form a topological base of L2(X).

Notation. If exp(∑fi,jX

iY j) = 1 +∑

n=1,m=1FnmX

nY m, then (ω, exp(. . .)) := ω +∑

n=1,m=1Fn,m(. . . (ω,X) . . .X)Y ) . . . Y ).

We shall investigate pro-algebraic subgroups of Aut∗(π2(X)) and Aut∗(π2)) and alge-

braic subgroups of Aut∗(π2/Γn+2π2).

Proposition 14.1.

i) Let Φ : Gm → Aut∗(π2(X)) be given by

Φt(X) = tX

Φt(Y ) = (tY, exp(∞∑

n=2

∑

i+j=ni≥1,j≥1

ci,j(1− tn)XiY j)), where ci,j ∈ C.

Then Φ is a homomorphism.

58

ii) All homomorphisms Φ of Gm into Aut∗(π2(X)) such that p∗ Φ = C are of this form.

iii) All one-dimensional subgroups G of Aut∗(π2(X)), which the map p∗ projects onto

C(Gm) are of this form.

iv) The group G is defined over a subfield k of C if and only if all cij are in k.

Proof. The point i) is a straightforward verification. To show point ii) we can assume that

Φ has a form Φt(Y ) = (tY, exp(∑

i≥1,j≥1

fi,j(t)XiY j)), where fi,j(t) ∈ C[t, t−1] are Laurent

polynomials. Assume that Φ has a form Φt(Y ) = (tY, exp(N−1∑n=2

∑i+j=n

ci,j(1 − tn)XiY j +

∑i+j=N

fi,j(t)XiY j +∑

i+j>N

Fi,j(t)XiY j)). Then one gets fi,j(t)+ tNfi,j(s) = fi,j(s · t). This

implies fi,j(t) = ci,j(1− tN ). Hence the point ii) is proved. If a subgroup of Aut∗(π2(X))

is one dimensional then the coefficients fi,j are algebraic functions of t. They cannot be

multivalued functions because then the dimension of the subgroup would be greater than

1. Hence fi,j are Laurent polynomials of t. Now the point iii) follows from the proof of ii).

The last point is obvious because char k = 0.

In section 6 we need also results about the subgroups of Aut(π2/Γn+1π2) and Aut(π2).

Let C : Gm → Aut(π2/Γ3π2) be given by Ct(X) = tX, Ct(Y ) = tY. Let (pn)∗ : Aut(π2/Γn+2π2)→Aut(π2/Γ3π2) and p∗ : Aut(π2) → Aut(π2/Γ3π2) be induced by projections of π2/Γn+2π2

and π2 onto π2/Γ3π2 .

Corollary 14.2. All one dimensional subgroups G of Aut(π2/Γn+2π2) (resp. Aut(π2))

which the map (pn)∗ (resp. p∗) projects onto C(Gm) are of the form G = ft ∈

Aut(π2/Γn+2π2) (resp. Aut(π2)

∣∣∣ft(X) = tX, ft(Y ) = tY+n(resp.∞)∑

k=1

ck(t−tk+2)((YX)Y k)∣∣∣t ∈

C∗. The group G is defined over a subfield k of C if and only if all ci’s are in k.

The subgroup G of Aut(π2/Γn+2π2) considered in the corollary we shall denote by

G(c1, c2, . . . , cn). The subgroup G of Aut(π2) considered in the corollary we shall denote

by G((ci)∞i=1) = G(c1, c2, . . . , cn, . . .).

Let N be the set of natural numbers and let c : N × N → C be a function into

complex numbers. For each n let ln1 , . . . , lnkn

be linear forms with complex coefficients in

variables xi,j , where i + j = n, i ≥ 1, j ≥ 1. Let L := (ln1 , . . . , lnkn)n∈N. Let Cn−1 =

59

∑i+j=n

i≥i,j≥1

C ((a1,n−1, a2,n−2)...) ∈ Cn−1) and let Vn be a set of common zeroes of ln1 , . . . , lnkn.

Observe that for each n, the sequence (cij)i+j∈n ∈ Cn−1.

Proposition 14.3.

i) Let G(c, L) := f ∈ Aut∗(π2(X))∣∣∣f(X) = tX, f(Y ) = (tY, exp(

∞∑n=2

∑i+j=n

(cij(1− tn) +

βij)XiY j)∣∣∣t ∈ C∗, βij ∈ C, ∀

n∀

1≤i≤kn

lni (β1,n−1, β2,n−2, . . . , βn−1,1) = 0.Then G(c, L) is a subgroup of Aut∗(π2(X)).

ii) Any subgroup G of Aut∗(π2(X)), whose projection onto Aut∗(π2(X)/Γ3π2(X)) is C(Gm)

is of this form.

iii) Two subgroups G(c, L) and G(c′, L′) coincides if and only if for each n Vn = V ′n and

(cij)i+j=n + Vn = (c′ij)i+j=n + Vn.

iv) The group G(c, L) is defined over the subfield k of C if and only if for each n the

vector space Vn and the affine space (cij)i+j=n + Vn are defined over k.

Proof. The point i) is a standard checking. Let G be as in the point i). Let G1 :=

ker(p∗ : G → Aut(π2(X)/Γ3π2(X)). Any element f ∈ G1 is of the form f(X) = X, f(Y ) =

(Y, exp(∞∑

n=2

∑i+j=n

fi,jXiY j)). Observe that f → ((fi,j)i+j=n)n=2,3,...,N defines a homo-

morphism G1 →N∑

n=2

∑i+j=n

Ga. Hence there are only linear relations between various fij ’s.

The group G is an extension of Gm by the pro-unipotent group G1, hence there is a lifting

φ : Gm → G of C. If we calculate fi,j coefficients of φt f φ−1t we get that it can be

linear relations only between (fi,n−i)i=1,...,n−1. Hence for each n we have a finite number

of linear forms ln1 , . . . , lnkn

in variables x1,n−1, x2,n−2, . . . , xn−1,1 such that

G1 = f |f(X) = X, f(Y ) = (Y, exp(∞∑

n=2

∑

i+j=n

βi,jXiY j)

∣∣∣

βi,j ∈ C, ∀n

∀1≤i≤kn

lni (β1,n−1, β2,n−2, . . .) = 0.

Let L = (ln1 , . . . , lnkn)n∈N. The group Φ(Gm) is as in Proposition 14.1.i) for some function

c : N × N → C. Observe that G1 Φ(Gm) = G(c, L). Hence by dimensional reasons

G = G(c, L). The points iii) and iv) are evident.

60

Let (εi)∞i=1 be a sequence such that εi ∈ 0, 1 for each i. Let (ci)

∞i=1 be a se-

quence of complex numbers. We set G((ci)∞i=1

∣∣∣(εi)∞i=1) = G(c1, c2, . . . |ε1, ε2, . . .) := f =

ft,(εiβi)∞i=1∈ Aut(π2)

∣∣∣f(X) = t·X, f(Y ) = t·Y+∞∑

i=1

((εiβi)+(1−εi)ci(t−ti+2))((YX)Y i)∣∣∣t ∈

C∗, βi ∈ C for i = 1, 2, . . .. If we replace in the above definition ∞ by n we get a sub-

group, denoted by G(c1, . . . , cn|ε1, . . . , εn) of Aut(π2/Γn+2π2). Let δ((εi)ni=1) be a number

of εi equal to 1. Then dimG(c1, . . . , cn|ε1, . . . , εn) = δ((εi)ni=1) + 1.

Corollary 14.4. All subgroups of Aut∗(π2/Γn+2π2) (resp. Aut∗(π2)),which the map (pn)∗

(resp. p∗) projects onto C(Gm) are of the formG(c1, . . . , cn|ε1, . . . , εn) (resp. G((ci)∞i=1|(εi)

∞i=1)).

The group is defined over a subfield k of C if and only if all numbers (1− εi)ci are in k.

The subgroup ft ∈ Aut(π2/Γn+2π2

∣∣∣ft(X) = tX1ft(Y ) = tY∣∣∣t ∈ C∗ of Aut(π2/Γn+2π2)

we shall denote by G0. Let δ : C→ 0, 1 be a map defined by δ(0) = 0, δ(z) = 1 if z 6= 0.

Lemma 14.5. Let f0 ∈ Aut(π2/Γn+2π2) be such that f0(X) = α0X, f0(Y ) = α0Y +n−1∑i=1

β0i ((Y X)Y i). LetG ⊂ Aut(π2/Γn+2π2) be the smallest algebraic subgroup of Aut(π2/Γn+2π2)

such that G0 ⊂ G and f0 ∈ G. Then G = G(0, 0, . . . , 0∣∣∣δ(β0

1), . . . , δ(β0n−1)).

Proof. Let Ct(X) = tX and Ct(Y ) = tY and let f1 = f0 C−1α0. Then f1(X) = X and

f1(Y ) = Y +n∑

i=1β0

i α−10 ((YX)Y i−1). Let G1 ⊂ G be a subgroup consisting of h such that

h(X) = X, h(Y ) = Y +n∑

i=1βi((YX)Y i−1). Assume that for some i, β0

i 6= 0. Then G1 6= Idbecause f1 ∈ G1. The subgroup G1 is contained in (Ga)n, hence it is given by a finite num-

ber of linear forms. Let k = Ct h C−1t . Then k(X) = X, k(Y ) = Y +

n∑i=1

tiβi((YX)Y i−1).

Let l(X1, . . . , Xn) be one of linear forms defining G1. Then l(tβ1, t2β2, . . . , t

nβn) = 0 im-

plies l ≡ 0. Hence the β1, β2, . . . , βn, if they are non-zero, they are linearly independent.

Hence G = G(0, 0, . . . , 0∣∣∣δ(β0

1), . . . , δ(β0n−1)).

Corollary 14.6. Let G ⊂ Aut(π2/Γn+2π2) be the smallest subgroup such that the sub-

group G0 := ft ∈ Aut(π2/Γn+2π2)∣∣∣ft(X) = tX, ft(Y ) = tY

∣∣∣t ∈ C∗ is contained in G and

the subgroup G(c1, . . . , cn) is contained in G. Then G = G(0, . . . , 0∣∣∣δ(c1), δ(c2), . . . , δ(cn)).

Proof. One takes any element f of G(c1, . . . , cn−1) such that f(X) = αX and α 6= 1.

61

§A.1. Malcev completion

Let π be a nilpotent group. Then there exists a connected, affine, nilpotent group scheme

π0 over Spec Q, together with a homomorphism rQ : π → π0(Q), functorial with respect

to morphisms π → π′ such that

i) rQ : π → π0(Q) is a rationalization of π,

ii) for any field K of characteristic zero, the morhpism rK : πrQ→ π0(Q) → π0(K) is the

Malcev K-completion of π i.e. for any homomorphism f : π → G of π into a nilpotent

K-group G there is a unique homomorphism of K-group f ′ : π0(K) → G such that

f = f ′ rK .

If π is any group then we set

π0 := lim←−i

(π/Γiπ)0.

The group π0(K) together with the morhpism rK : π → π0(K) has a universal prop-

erty with respect to maps of π into pro-nilpotent K-groups.

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Universite de Nice-Sophia Antipolis Research InstituteDepartment de Mathematiques for Mathematical SciencesLaboratoire Jean Alexandre Dieudonne Kyoto UniversityU.R.A. au C.N.R.S., No 168 Kitashirakawa, Sakyo-ku,Parc Valrose - B.P.N 71 Kyoto 606, Japan06108 Nice Cedex 2, France

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