The twisted Alexander polynomial for finite abelian covers
over three manifolds with boundary
JÉRÔMEDUBOIS
YOSHIKAZU YAMAGUCHI
We provide the twisted Alexander polynomials of finite abelian covers over three-dimensional manifolds whose boundary is a finite union of tori. This is a generaliza-tion of a well-known formula for the usual Alexander polynomial of knots in finite cyclic branched covers over the three-dimensional sphere.
57M25; 57M27
1 Introduction
The classical Alexander polynomial is defined for null–homologous knots in rational homology spheres, where null–homologous means that the homology class of a knot is trivial in the first homology group with Z–coefficients of the ambient space.
If the pair .M; yK/ of a rational homology sphereM and a null-homologous knot y K in M is given by a finite cyclic branched cover over S3 branched along a knot K , where yK is the lift of K , then we can compute the Alexander polynomial of yK by using the well–known formula
Ky.t/ D
Y
2fx2C j xkD1g
K.t/ up to a factor ˙ta.a 2 Z/
where k is the order of the covering transformation group, runs all over the k th roots of unity and K.t/ is the Alexander polynomial of K. Such formulas have been
investigated from the viewpoint of Reidemeister torsion for a long time. In particular, V Turaev gave a formula for the Alexander polynomial of yK in a finite cyclic branched cover over S3, and a generalization in the case of links in general three–dimensional manifolds (we refer to Turaev[8, Theorems 1.9.2 and 1.9.3]).
The purpose of this paper is to provide the generalization of the above formula giving the Alexander polynomial of a knot in a finite cyclic branched cover over S3 to a formula for the twisted Alexander polynomial of finite abelian covers, which is a special kind of Reidemeister torsion. Especially, we also consider the twisted Alexander polynomial
for a link in a three–dimensional manifold from the viewpoint of Reidemeister torsion in the same way as V Turaev. But to deal with finite abelian covers beyond finite cyclic covers, we adopt the approach of J Porti in his work[5]. Porti gave a new proof of Mayberry–Murasugi’s formula, which gives the order of the first homology group of finite abelian branched covers over S3 branched along links, by using Reidemeister torsion theory. We call the twisted Alexander polynomial the polynomial torsion regarded as a kind of Reidemeister torsion.
In this paper, we are interested in the Reidemeister torsion for a finite sheeted abelian covering. We are mainly intested in link exteriors in homology three–spheres and their abelian covers. Our main theorem (see Theorem 4.1) is stated for an abelian cover
M ! M between two three–dimensional manifolds whose boundary is a finite union of tori as follows '˝ yy M .t/ D Y 2 yG .'˝/˝ M .t/ where'˝ yy M .t/ and .'˝/˝
M .t/ are the signed twisted Alexander polynomials, yG is
the set of homomorphisms from the covering transformation group G to the non–zero complex numbers and is a sign determined by the homology orientations of M and M .
To be more precise, we need two homomorphisms of the fundamental group to define the twisted Alexander polynomial of a manifold. The symbol' denotes a surjective ho-momorphism from1.M / to a multiplicative group Znand denotes a representation
of 1.M /, ie, a homomorphism from 1.M / to a linear automorphism group Aut.V /
of some vector space V (seeSection 3for the definition of the polynomial torsion). In the definition of the twisted Alexander polynomial of M , we use the pull–backs
y
' and y of ' and to 1.M/. The homomorphisms ' and determine variables in the twisted Alexander polynomial of M . In our main theorem, we assume that the composition of with the quotient homomorphism 1.M / ! 1.M /=1.M/ ' G factors through homomorphism ' (seeSection 4).
When we choose M ! M as a finite cyclic cover of a knot exterior EK of K in S3,'
as the abelianization homomorphism1.EK/!1.EK/=Œ1.EK/; 1.EK/'Z and
as the one–dimensional trivial representation,Theorem 4.1reduces to the classical formula for the Alexander polynomial of yK , where yK is the lift of the knot in the finite cyclic branched cover over S3
Ky.t/ D
Y
2fx2C j xkD1g
up to a factor ˙ta .a 2 Z/, where k is the order of 1.M /=1.M/. Our formula also provides the Alexander polynomial of a link in finite abelian branched covers over S3 branched along the link.
Organization The outline of the paper is as follows. Section 2 deals with some reviews on the sign–determined Reidemeister torsion for a manifold. InSection 3, we give the definition of the polynomial torsion (the twisted Alexander polynomial) for a manifold whose boundary is a finite union of tori. InSection 4, we consider the polynomial torsion of finite abelian covering spaces (seeTheorem 4.1).
Acknowledgments The authors gratefully acknowledge the many helpful ideas and suggestions of J Porti during the preparation of the paper. They also wish to express their thanks to C Blanchet and S Friedl for several helpful comments and for their encouragement. Our thanks also go to T Morifuji, K Murasugi and V Maillot for giving us helpful informations and comments regarding this work. The first author (JD) is partially supported by the French ANR project ANR-08-JCJC-0114-01. The second author (YY) is partially supported by the GCOE program at Graduate School of Mathematical Sciences, University of Tokyo and Research Fellowships of Japan Society for the promotion of Science for Young Scientists. The paper was conceived when JD visited the CRM. He thanks the CRM for hospitality. YY visited CRM and IMJ while writing the paper. He thanks CRM and IMJ for their hospitality. The authors also would like to thank the referee for his comments and helpful remarks.
2 Preliminaries
2.1 The Reidemeister torsion
We review the basic notions and results about the sign–determined Reidemeister torsion introduced by V Turaev which are needed in this paper. Details can be found in Milnor’s survey[3]and in Turaev’s monograph[10].
Torsion of a chain complex Let CD .0 ! Cn dn
! Cn 1 dn 1
! d! C1 0! 0/ be
a chain complex of finite dimensional vector spaces over a field F . Choose a basis ci of Ci and a basis hi of the i th homology group Hi.C/. The torsion of C with
respect to these choices of bases is defined as follows.
For each i , let bi be a set of vectors in Ci such that di.bi/ is a basis of Bi 1 D
im.diW Ci ! Ci 1/ and let zhi denote a lift of hi in Zi D ker.diW Ci ! Ci 1/. The
the determinant of the transition matrix between those bases (the entries of this matrix are coordinates of vectors in diC1.biC1/zhibi with respect to ci). The sign-determined
Reidemeister torsionof C (with respect to the bases c and h) is the following alternating product (see Turaev’s book[9, Definition 3.1]):
(2-1) Tor.C; c; h/ D . 1/jCj n Y iD0 ŒdiC1.biC1/zhibi=ci. 1/ i C1 2 F: Here jCj D X k>0 ˛k.C/ˇk.C/;
where˛i.C/ D PikD0dim Ck andˇi.C/ D PikD0dim Hk.C/.
The torsion Tor.C; c; h/ does not depend on the choices of bi nor on the lifts zhi.
Note that if C is acyclic (ie if HiD 0 for all i ), then jCj D 0.
Torsion of a CW-complex Let W be a finite CW-complex and .V; / be a pair of a vector space with an inner product over F and a homomorphism of 1.W / into
Aut.V /. The vector space V turns into a right ZŒ1.W /–module denoted V by
using the right action of 1.W / on V given by v D . / 1.v/, for v 2 V and
2 1.W /. The complex of the universal cover with integer coefficients C.W I Z/ also inherits a left ZŒ1.W /–module structure via the action of 1.W / onW as the covering group. We define the V–twisted chain complex of W to be
C.W I V/ D V˝ZŒ1.W /C.W I Z/:
The complex C.W I V/ computes the V–twisted homologyof W which is denoted by H.W I V/.
Let ˚ei1; : : : ; enii be the set of i –dimensional cells of W . We lift them to the universal
cover and we choose an arbitrary order and an arbitrary orientation for the cells ˚zei
1; : : : ; ze i
ni . If we choose an orthonormal basis fv1; : : : ; vmg of V , then we consider
the corresponding basis
ciD˚ v1˝ ze1i; : : : ; vm˝ ze1i; ; v1˝ zenii; : : : ; vm˝ ze
i ni
of Ci.W I V/ D V˝ZŒ1.W /C.W I Z/. We call the basis c D ˚ici a geometric basisof C.W I V/. Now choosing for each i a basis hi of the V–twisted homology Hi.W I V/, we can compute the torsion
We mainly consider the torsion of acyclic chain complexes C.W I V/, ie, the
homol-ogy group H.W I V/ D 0 . For acyclic chain complex C.W I V/, this definition
only depends on the combinatorial class of W , the conjugacy class of , the choices of c. The basis c for C.W I V/ depends on the following choices:
(1) an order of cells fejig and an orientation of each fe i jg;
(2) a lift ezji of eji and;
(3) an orthonormal basis of the vector space V .
We summarize the effect of changing these choices to Tor.C.W I V/; c; ∅/ in the
following three remarks.
Remark 2.1 We have the same Tor.C.W I V/; c; ∅/ for all orthonormal bases of V since the effect of change of orthonormal bases in V is given by multiplying the determinant of the bases change matrix with power of .W /. If the Euler characteristic .W / is zero, then we have the same torsion for any basis of V .
Remark 2.2 The torsion Tor.C.W I V/; c; ∅/ depends on the choice of the lifts z
eji under the action of 1.W / by . The effect of different lift of a cell is expressed
as the determinant of . / for some in 1.W /. To avoid this problem, we often
use representations into SL.V /.
Remark 2.3 To define the Reidemeister torsion, we order the cells ˚eji and choose an orientation of each eji, if we choose a different order and different orientations of cells, we could change the torsion sign. To remove this sign ambiguity, that only occurs when m is odd, we use the fact that the sign of the torsions Tor.C.W I R/; cR; hR/
and Tor.C.W I R/; c; h/ change in the same way.
Therefore we usually consider the torsion Tor.C.W I V/; c; ∅/ up to the above
indeterminacy, namely up to a factor ˙ det . / for some in 1.W /.
We can construct the additional sign term referred to inRemark 2.3as follows. The cells ˚zei
j
ˇ
ˇ0 6 i 6 dim W; 1 6 j 6 ni are in one–to–one correspondence with the cells of W , their order and orientation are induced an order and an orientation for the cells ˚ei
j
ˇ
ˇ0 6 i 6 dim W; 1 6 j 6 ni . Again, corresponding to these choices, we get a basis ciR over R of Ci.W I R/.
Choose a homology orientation of W , which is an orientation of the real vector space H.W I R/ D Li>0Hi.W I R/. Let o denote this chosen orientation. Provide each
vector space Hi.W I R/ with a reference basis hiR such that the basis˚h
0
R; : : : ; h
dim W
R
of H.W I R/ is positively oriented with respect to o. Compute the sign–determined
Reidemeister torsion Tor.C.W I R/; cR; hR/ 2 Rof the resulting based and homology
based chain complex and consider its sign
0D sgn Tor.C.W I R/; cR; h
R/ 2 f˙1g:
We define the sign–refined twisted Reidemeister torsion of W (with respect to o) to be (2-2) 0m Tor.C.W I V/; c; ∅/ 2 F
where mD dimFV . This sign refinement also works for the twisted chain complex
C.W I V/ with non–trivial homology group. When the dimension of V is even, we do not need the sign refinement, ie, the torsion Tor.C.W I V/; c; ∅/ is determined
up to det. / for some in 1.W /.
One can prove that the sign–refined Reidemeister torsion is invariant under cellular subdivision, homeomorphism and simple homotopy equivalences. In fact, it is precisely the sign . 1/jCj in Equation (2-1) which ensures all these important invariance
properties to hold (see Turaev’s monograph[10]).
3 Definition of the polynomial torsion
In this section we define the polynomial torsion. This gives a point of view from the Reidemeister torsion to polynomial invariants of topological spaces.
Hereafter M denotes a compact and connected three–dimensional manifold such that its boundary @M is empty or a disjoint union of b two–dimensional tori:
@M D T2
1 [ : : : [ T 2 b:
In the sequel, we denote by V a vector space over C and by a representation of 1.M / into Aut.V /, and such that det . / D 1 for all 2 1.M /.
Next we introduce a twisted chain complex with some variables. It will be done by using a ZŒ1.M /–module with variables to define a new twisted chain complex. We
regard Zn as the multiplicative group generated by n variables t1; : : : ; tn, ie,
ZnD˝t1; : : : ; tnj titj D tjti.8i; j /˛
and consider a surjective homomorphism 'W 1.M / ! Zn. We often abbreviate the n
variables.t1; : : : ; tn/ to t and the rational function field C.t1; : : : ; tn/ to C.t/.
When we consider the right action of 1.M / on V .t/ D C.t/ ˝ V by the tensor
representation
' ˝ 1
we have the associated twisted chain C.M I V.t// given by
C.M I V.t// D V.t/ ˝ZŒ1.M /C.M I Z/
where f ˝ v ˝ is identified with f '. / ˝ . / 1.v/ ˝ for any 2 1.M /,
2 C.M I Z/, v 2 V and f 2 C.t/. We call this complex the V .t/–twisted chain complexof M .
Definition 3.1 Fix a homology orientation on M . If C.M I V.t// is acyclic, then the sign–refined Reidemeister torsion of C.M I V.t//
'˝
M .t1; : : : ; tn/ D
m
0 Tor.C.M I V.t//; c; ∅/ 2 C.t1; : : : ; tn/ n f0g
is called the polynomial torsion of M .
Observe that the sign–refined Reidemeister torsion'˝M .t1; : : : ; tn/ is determined up
to a factor tm1
1 : : : t mn
n like the classical Alexander polynomial is.
Example 3.2 (J Milnor [4], P Kirk & C Livingston [2]) Suppose that M is the knot exterior EK D S3n N .K/ of a knot K in S3 where N.K/ is an open tubular
neighbourhood of K .
If the representation 2 Hom.1.EK/I Q/ is the trivial homomorphism and ' is the
abelianization of 1.EK/, ie, 'W 1.EK/ ! H1.EKI Z/ ' hti, then the twisted chain
complex C.EKI Q.t// is acyclic and the Reidemeister torsion '˝EK .t/ is expressed
as a rational function which is the Alexander polynomial K.t/ divided by .t 1/
(see Turaev’s book[9]and monograph[10]).
Example 3.3 Suppose now that M is the link exterior EL D S3n N .L/ of a link
L in S3. We suppose that L has n components, where n > 2. We denote by i the meridian of the i th component. Consider the abelianization 'W 1.EL/ !
Zn defined by '.i/ D ti. Let W 1.EL/ ! GL.1I C/ D C n f0g be the one–
dimensional representation such that .i/ D i. Then the twisted chain complex
C.ELI C.t// is acyclic and the Reidemeister torsion '˝EL .t1; : : : ; tn/ is given by
(up to˙.11t1/k1: : : .n1tn/kn, ki2 Z)
'˝
EL .t1; : : : ; tn/ D L.
1
1 t1; : : : ; n1tn/
4 Torsion for finite sheeted abelian coverings
4.1 Statement of the resultLet M be a finite sheeted abelian covering of M , where M denotes a compact and connected three–dimensional manifold such that its boundary@M is empty or a disjoint union of b two–dimensional tori:
@M D T2
1 [ : : : [ T 2 b:
We denote by p the induced homomorphism from 1.M/ to 1.M / by the covering
map M ! M . The associated deck transformation group is a finite abelian group G of order jGj. We endow the manifolds M and M with some arbitrary homology orientations.
We have the following exact sequence: (4-1) 1 //1.M/
p //
1.M / //G //1:
When we consider the polynomial torsion for M , we use the pull–back of homo-morphisms of 1.M / as homomorphisms of 1.M/. We denote by ' a surjective homomorphism from1.M / to Zn and by ' the pull-back by p. We also supposey
that factors through ' . Our situation is summarized as follows: 1.M/ p // y ' '' 1.M / ' // G Zn 99s s s s
Similarly we use the symbol for the pull–back of W y 1.M / ! Aut.V / by p, where
V is a vector space. For homomorphisms of the quotient group G ' 1.M /=1.M/, we use the Pontrjagin dual of G which is the set of all representationsW G ! CD Cn f0g from G to non–zero complex numbers. Let yG denote this space.
We give the statement of the polynomial torsion for abelian coverings via that of the based manifold.
Theorem 4.1 With the above notation, we suppose that the twisted chain complex C.M I V.t// is acyclic. Then the twisted chain complex C.M I V y.t// is also acyclic and the polynomial torsion is expressed as
(4-2) '˝ yy M .t/ D Y 2 yG .'˝/˝ M .t/
where is a sign equal to 0.M/m 0.M /mjGj and mD dim V .
Remark 4.2 As we already observed, the sign term in Equation(4-2)is not relevant when m is even.
Remark 4.3 (Explanation of Formula(4-2)with variables) If we denote by x the composition ı x as in the following commutative diagram
1.M / // '
G //C Zn x
99
then Formula(4-2)can be written concretely as follows: '˝ yy M .t1; : : : ; tn/ D Y 2 yG '˝ M .t1x.t1/; : : : ; tnx.tn//
In the special case where nD 1, G D Z=qZ and M is the q –fold cyclic covering Mq
of M , then we have that x.t/ D e2k
p
1=q, for kD 0; : : : ; q 1. Hence we have the
following covering formula for the polynomial torsion.
Corollary 4.4 Suppose that'.1.M // D hti and y'.1.Mq// D hsi hti, where we
suppose that sD tq. We have '˝ yy Mq .s/ D y '˝ Mq .t q/ D q 1 Y kD0 '˝ M .e 2kp 1=qt/: The torsion '˝Mqy .t
q/ inCorollary 4.4can be regarded as a kind of the total twisted
Alexander polynomial introduced in Silver and Williams[7]. Hirasawa and Murasugi[1]
worked on the total twisted Alexander polynomial for abelian representations as in
Example 3.2 and they observed the similar formula as inCorollary 4.4in terms of the total Alexander polynomial and the Alexander polynomial of a knot in the cyclic branched coverings over S3.
4.2 Proof ofTheorem 4.1
We use the same notation as inRemark 4.3. First observe the following key facts:
the torsion'˝y
M is computed using the twisted complex
V.t1; : : : ; tn/ ˝ZŒ
1. M/C.M I Z/;
whereas the torsion'˝M is computed using
V.t1; : : : ; tn/ ˝ZŒ1.M /C.M I Z/:
Lemma 4.5 Let x2 V.t1; : : : ; tn/, c 2C.M I Z/. For 2 1.M /, x 1˝
1. M/ c
only depends on . / 2 G . For g 2 G , choose 2 1.M / such that . / D g and
set
(4-3) g? x ˝
1. M/c D x
1
˝1. M/ c: This defines a natural action of G on V.t1; : : : ; tn/ ˝
1. M/C.M I Z/.
Further observe that, since for any lift of g, is not contained in p.1.M//, we can not reduce the right hand side in Equation .4-3/.
Proof Take another lift 0 in 1.M / of g 2 G . Since 0 D y for some y 2
p.1.M//, we can see that x 0 1˝
1. M/
0c D x 1
˝1. M/ c: The proof ofTheorem 4.1is based on the following technical lemma. Lemma 4.6 The map
(4-4) ˆW V.t/ ˝ZŒ
1. M/C.M I Z/ ! .V .t/ ˝CCŒG/ ˝ZŒ1.M /C.M I Z/
given by
ˆ x ˝1. M/c D .x ˝ 1/ ˝ c
is an isomorphism of complexes of CŒG–modules where the action of G on the twisted complex .V.t/ ˝CCŒG/ ˝ZŒ1.M /C.M I Z/ is given by
g .x ˝ g0˝ c/ D x ˝ gg0˝ c:
and the right action of 2 1.M / on V.t/ ˝CCŒG is defined by
..f ˝ v/ ˝ g/ D f '. / ˝ 1. /.v/ ˝ . /g;
Proof ofLemma 4.6 We first observe thatˆ is a well–defined chain map of C–vector spaces since1.M/ is a normal subgroup of 1.M /. By the definition, we can see that ˆ.x ˝1. M/ c/ D ˆ.x ˝1. M/c/ for any in 1.M/. Hence ˆ is well–defined. From ˆ.@.x ˝
1. M/c// D ˆ.x ˝1. M/@c/ D .x ˝ 1/ ˝ @c D @..x ˝ 1/ ˝ c/ D
@.ˆ.x ˝ c// it follows that ˆ ı @ D @ ı ˆ. The G –equivariance of ˆ follows from ˆ.g ? .x ˝1. M/c// D ˆ.x 1˝1. M/ c/
D .x 1˝ 1/ ˝ c D .x ˝ g/ ˝ c D g ˆ.x ˝
1. M/c/:
We can prove that ˆ is an isomorphism by taking its inverse ‰ as ‰..x ˝ g/ ˝ c/ D g? .x ˝ c/:
We mention bases of the chain complex V.t/ ˝ZŒ
1. M/C.M I Z/ before the next
step. The following basis (4-5) ycD[ i>0 ˚xk˝ 1. M/ gee i j ˇ ˇ1 6 j 6 ni; g 2 G; 1 6 k 6 m
is the geometric basis used to compute the polynomial torsion'˝ yy
M . When we consider
the bases change from the basis in Equation(4-5)to the basis in the next equation (4-6) ˚g? xk˝ 1. M/ee i j D xk g1˝1. M/ gee i j ˇ ˇ1 6 j 6 ni; g 2 G; 1 6 k 6 m ;
we can see that the action of g1 arises the change in '˝ yy
M by multiplying its
determinant powered the Euler characteristic of M . Since the Euler characteristic of M is zero, the polynomial torsion '˝ yy
M can also be computed using the basis
in Equation (4-6). Finally observe that ˆ maps the basis in Equation(4-6) to the geometric basis (4-7) cGD[ i>0 ˚ .xk˝ g/ ˝ee i j ˇ ˇ1 6 j 6 ni; g 2 G; 1 6 k 6 m ; thus '˝ yy M D 0.M/ m Tor..V.t/ ˝CCŒG/ ˝ZŒ1.M /C.M I Z/; c G; ∅/:
Now, we want to compute the torsion of .V.t/ ˝CCŒG/ ˝ZŒ1.M /C.M I Z/ in
terms of polynomial torsions of M . To this end we use the decomposition along orthogonal idempotents of the group ring CŒG, see Serre[6]for details. Associated to
2 yG , we define:
fDjGj1
X
g2G
.g 1/g 2 CŒG:
The properties of f are the following f2 D f; ff0D 0 . if ¤ 0/; X 2 yG fD 1 and g fD .g/f; for all g 2 G:
We have the following CŒG–modules decomposition of the group ring as a direct sum according to its representations:
(4-8) CŒG DM
2 yG
CŒf:
Here each factor is the 1–dimensional C –vector space which is isomorphic to the CŒG–module associated to W G ! C.
Following Porti[5, Section 3], corresponding to the decomposition in Equation(4-8)
we have a decomposition of complexes of CŒG–modules: .V.t/˝CCŒG/˝ZŒ1.M /C.M I Z/ D
M
2 yG
.V.t/˝CCŒf/˝ZŒ1.M /C.M I Z/:
Remark 4.7 This decomposition implies that .V.t/˝CCŒG/˝ZŒ1.M /C.M I Z/
is acyclic, since one can see that each chain complex .V.t/ ˝CCŒf/ ˝ZŒ1.M /
C.M I Z/ is acyclic from our assumptions and a change of variables.
The geometric basis in Equation(4-7)induces a basis compatible with the decomposition in Equation(4-8)by replacing fg j g 2 Gg by ffj 2 yGg. The change of bases cancels
when we compute the torsion because Euler characteristic is zero, see[5, Lemma 5.2]. And thus decomposition in Equation(4-8)implies that (in the natural geometric bases):
(4-9) '˝ yy M D 0.M/ m Tor..V.t/ ˝CCŒG/ ˝ZŒ1.M /C.M I Z/; c G; ∅/ D 0.M/ m Y 2 yG Tor..V.t/ ˝CCŒf/ ˝ZŒ1.M /C.M I Z/; c ; ∅/:
Each factor in the right hand side is related to the polynomial torsion of M and its relation is given by the following claim.
Lemma 4.8 We have: .'˝/˝ M D 0.M / m Tor..V.t/ ˝CCŒf/ ˝ZŒ1.M /C.M I Z/; c ; ∅/:
Proof ofLemma 4.8 One can observe that, as a ZŒ1.M /–module, V.t/˝CCŒf
is isomorphic to V.t/ simply by replacing the action ' ˝ by .' ˝ / ˝ . This proves the equality of torsions.
Proof ofTheorem 4.1 Combining Equation .4-9/ andLemma 4.8, we obtain '˝ yy M D 0.M/ m 0.M /mjGj Y 2 yG .'˝/˝ M
which achieves the proof of Formula(4-2).
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Institut de Mathématiques de Jussieu, Université Paris Diderot–Paris 7, UFR de Mathématiques Case 7012, Bâtiment Chevaleret, 75205 Paris Cedex 13, France
Department of Mathematics, Tokyo Institute of Technology 2-12-1 Ookayama Meguro-ku, Tokyo 152-8551, Japan
dubois@math.jussieu.fr, shouji@math.titech.ac.jp Received: 3 September 2011 Revised: 20 December 2011
