REND. SEM. MAT. UNIV. PADOVA, Vol. 134 (2015)
DOI 10.4171/RSMUP/134-4
The local Laplace transform of
an elementary irregular meromorphic connection
MARCO HIEN (*) - CLAUDE SABBAH (**)
ABSTRACT - We give a definition of the topological local Laplace transformation for a Stokesfiltered local system on the complex affine line and we compute in a topological way the
Stokes data of the Laplace transform of a differential system of elementary type.
MATHEMATICS SUBJECT CLASSIFICATION (2010). 14D07, 34M40
KEYWORDS. Laplace transformation, meromorphic connection, Stokes matrix, Stokes filtration.
1. Introduction
1.1 ± Riemann-Hilbert correspondence and Laplace transformation
Let M be a holonomic C‰tŠh@t i-module and let
be its Laplace transform,
which is a holonomic C[t0 ]h@t0 i-module through the correspondence t0 ˆ @t and
t ˆ @t0 , that is, given by the kernel exp ( tt0 ). According to the Riemann-Hilbert
correspondence as stated by Deligne (see [Del07], [Mal91] and [Sab13a, Chap. 5]),
the holonomic C‰tŠh@t i-module M corresponds to a Stokes-perverse sheaf on the
projective line P1t with affine coordinate t, which is a Stokes-filtered local system in
the neighbourhood of t ˆ 1. Similarly,
corresponds to a Stokes-perverse sheaf
on P1t0 , which is a Stokes-filtered local system in the neighbourhood of t0 ˆ 1. In
c such that
the following, we will denote by t the coordinate of P1t0 centered at 1
(*) Indirizzo dell'A.: Lehrstuhl fu
Èr Algebra und Zahlentheorie, UniversitaÈtsstraûe 14,
86159 Augsburg, Deutschland.
E-mail: [email protected]
This research was supported by the grant DFG-HI 1475/2-1 of the Deutsche Forschungsgemeinschaft.
(**) Indirizzo dell'A.: UMR 7640 du CNRS, Centre de MatheÂmatiques Laurent
 cole polytechnique, F-91128 Palaiseau cedex, France.
Schwartz, E
E-mail: [email protected]
This research was supported by the grants ANR-08-BLAN-0317-01 and ANR-13-IS010001-01 of the Agence nationale de la recherche.
134
Marco Hien - Claude Sabbah
cg, and we continue to set 1
c ˆ ft ˆ 0g. The topological
t ˆ 1=t0 on P1t0 n f0; 1
Laplace transformation is the corresponding transformation at the level of the
category of Stokes-perverse sheaves. We will use results of [Moc14] to make clear
its definition. A topological Laplace transformation can also be defined in the setting of enhanced ind-sheaves considered in [DK13], which corresponds, through the
Riemann-Hilbert correspondence of loc. cit., to the Laplace transformation of holonomic D -modules (see also [KS14]).
1.2 ± Riemann-Hilbert correspondence and local Laplace transformation
In this article, we will consider the local Laplace transformation F (0;1) from the
category of finite dimensional C…ftg†-vector spaces with connection to that of finite
dimensional C…ftg†-vector spaces with connection. It is defined as follows. A finite
dimensional C…ftg†-vector space with connection M can be extended in a unique
way as a holonomic C‰tŠh@t i-module M with a regular singularity at infinity and no
other singularity at finite distance than t ˆ 0. Then F (0;1) (M) is by definition the
c :ˆ ft0 ˆ 1g. We will regard F (0;1) (M) as a C…ftg†-vector
germ
of
at 1
space with connection. It is well-known that has at most a regular singularity at
t0 ˆ 0 and no other singularity at finite distance. Therefore, giving F (0;1) (M) is
equivalent to giving the localized module C[t0 ; t0 1 ] C[t0 ] .
The Deligne-Riemann-Hilbert correspondence associates to M a Stokes-filtered local system (L ; L ) on S1tˆ0 (the circle with coordinate arg t). Similarly, we
1
will denote by F (0;1)
top (L ; L ) the Stokes-filtered local system on Stˆ0 associated
with F (0;1) (M). In this setting, we will address the following questions:
To make explicit the topological local Laplace transformation functor F (0;1)
top .
To use this topological definition to compute, in some examples, the Stokes
structure of F (0;1) (M) in terms of that of M.
1.3 ± Statement of the results
The first goal will be achieved in §3. As for the second one, we will restrict to
the family of examples consisting of elementary meromorphic connections,
denoted by El(r; '; R) in [Sab08]. Namely, r : u 7! t ˆ up is a ramification of
order p of the variable t, ' is a polynomial in u 1 without constant term, and R
is a finite dimensional C…fug†-vector space with a regular connection. Setting
E ' ˆ (C…fug†; d d'), we define
M ˆ El(r; '; R) :ˆ r‡ (E
'
R):
If we extend R as a free C[u; u 1 ]-module R of finite rank with a connection having a
regular singularity at u ˆ 0 and u ˆ 1 and no other singularity, and set
The local Laplace transform of an elementary irregular etc.
135
E ' ˆ (C[u; u 1 ]; d d'), the extension M of M considered above is nothing but
the free C[t; t 1 ]-module with connection
M ˆ El(r; '; R) :ˆ r‡ (E
'
R):
Let q be the pole order of '. We also encode R as a pair (V; T) of a vector space with
an automorphism (the formal monodromy). We call ' the exponential factor of r‡ M
and set Exp ˆ Exp (r‡ M) ˆ f'g.
ASSUMPTION 1.3.1. We will assume in this article that p; q are coprime.
By the stationary phase formula of [Fan09, Sab08] the formal Levelt-Turrittin
b where
b R),
r; ';
type of the local Laplace transform F (0;1) (M) is that of El (b
the ramification b
r : h 7! t has order b
p :ˆ p ‡ q, i.e., b
r 2 hp‡q (c(r; ') ‡ hCfhg)
for some nonzero constant c(r; '); writing r and ' in the h variable, it is
expressed as b
r (h) ˆ r0 (h)='0 (h);
b ˆ '(h) '0 (h)r(h)=r0 (h) has pole order q;
the exponential factor '(h)
b corresponds to the pair (V;
b T)
b ˆ (V; ( 1)q T).
the regular part R
Our goal is then to compute explicitly the Stokes structure of the non-ramified
meromorphic connection b
r ‡ F (0;1) (M), which is usually non trivial. It is of pure
level q. Stokes structures of pure level q can be represented in various ways by
generalized monodromy data. We make use of the following description (see Section
2.2 for details).
To the formal data attached to b
r ‡ F (0;1) (M), which consist of the family
d :ˆ Exp (b
b (h) j 2 m g (m is the group of n-th roots of
Exp
r ‡ F (0;1) (M)) ˆ f'
p‡q
b h(1 ‡ o(h))) and the pair ( b ˆ '(
unity and '
n
2mp‡q
V; 2mp‡q
( 1)q T), we add:
(1.3.2) the choice of a generic argument #bo 2 S1hˆ0 , giving rise to a total ordering
d hence an order preserving numbering f0; . . . ; p ‡ q 1g ' m ,
of the set Exp,
p‡q
`‡1
(1.3.3) a set of unramified linear Stokes data (L` ); (S` ); (FL` ) `ˆ0;...;2q 1 of pure
level q, which consists of
a family (L` )`ˆ0;...;2q
isomorphisms
S`‡1
`
1
consisting of 2q C-vector spaces,
: L` ! L`‡1 ,
for each m 2 f0; . . . ; q 1g, a finite exhaustive (1) increasing filtration
F L2m (resp. decreasing filtration F L2m‡1 ) indexed by f0; . . . ; p ‡ q 1g,
with the property that the filtrations are mutually opposite with respect
(1) We say that an increasing filtration F L indexed by a totally ordered finite set O is
exhaustive if Fmax O L ˆ L. We then set F<min O L ˆ 0. For a decreasing filtration, we have
F min O L ˆ L and we set F>max O L ˆ 0.
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Marco Hien - Claude Sabbah
to the isomorphisms S`‡1
` , i.e., for any m 2 f0; . . . ; q
L2m ˆ
p‡q
M1
k
Fk L2m \ S2m
2m 1 …F L2m
kˆ0
(1.3.3)
L2m‡1 ˆ
p‡q
M1
kˆ0
with the convention that L
1
1†
1g:
;
Fk L2m‡1 \ S2m‡1
2m …Fk L2m † ;
ˆ L2q 1 .
Note that the opposite filtrations induce uniquely determined splittings, that is,
filtered isomorphisms
(
L
1
t2m : L2m ! grF L2m ˆ: p‡q
grFk L2m ;
kˆ0
(1:3:4)
L
1
t2m‡1 : L2m‡1 ! grF L2m‡1 ˆ: p‡q
grkF L2m‡1 ;
kˆ0
where the right hand side carries its natural increasing (resp. decreasing) filtration.
:ˆ t`‡1 S`‡1
t` 1 : grF L` ! grF L`‡1 are upper/
The resulting isomorphisms S `‡1
`
`
lower block triangular - depending on the parity of ` - and their blocks (S`‡1
` )j; j on
the diagonal are isomorphisms. These matrices are commonly called the Stokes
matrices or Stokes multipliers. We call the family
…1:3:5†
S b
r ‡ F (0;1) (M) :ˆ (L` ); (S`‡1
` ); (FL` ) `ˆ0;...;2q 1
the linear part of the Stokes data of b
r ‡ F (0;1) (M). The notion of morphism between
such families is obvious.
On the other hand, we will define in § 1.4 standard linear Stokes data that we
denote by Sstd (V; T; p; q).
The main result regarding the explicit determination of the Stokes data of the
local Laplace transform of El(r; '; R) can be stated as follows.
THEOREM 1.3.6. With the previous notation and assumptions, for a suitable
choice of #bo , the linear Stokes data S b
r ‡ F (0;1) (M) , for M ˆ El(r; '; (V; T)),
are isomorphic to the standard linear Stokes data Sstd (V; T; p; q).
REMARK 1.3.7. (1) In order to descend from this result to the linear Stokes data
of F (0;1) (M), we would need to identify the mp‡q -action on the standard
linear Stokes data. This will not be achieved in this article.
(2) The arguments in the proof also lead to an explicit computation of the topological monodromy of F (0;1) (M) up to conjugation (see Proposition 1.4.13).
(3) Similar results were obtained by T. Mochizuki in [Moc10, § 3], by using
explicit bases of homology cycles, while we use here cohomological methods.
Also, the results of loc. cit. do not make explicit a standard model for the
linear Stokes data, although it should be in principle possible to obtain such a
model from these results.
The local Laplace transform of an elementary irregular etc.
137
1.4 ± The standard linear Stokes data
We will now define the set Sstd (V; T; p; q) attached to a finite dimensional vector
space V equipped with an automorphism T, and to a pair of coprime integers
(p; q) 2 (N )2 . We refer to Section 6 for a geometric motivation which leads to the
definitions below. We start by defining two orderings on mp‡q .
DEFINITION 1.4.1. For ;
and
0
<odd
(1:4:1)
<ev
0
,
if Re…e
0<
odd
(1:4:1) mp‡q ˆ f
k
0
"i
2 mp‡q , we set
…
p
0p
†† < 0 for some " such that 0 <" 1
. We enumerate mp‡q according to the even ordering, i.e.,
j k ˆ 0; . . . ; p ‡ q
1g with
0
<ev <ev
p‡q 1 :
Since (p; q) ˆ 1, we also have (p ‡ q; q) ˆ 1 and there exist a; b 2 Z with
2pi
ap ˆ 1 ‡ b( p ‡ q). Set j :ˆ exp p‡q 2 mp‡q . Then the even ordering on mp‡q is
expressed by
1 ˆ j 0 <ev j a <ev j
(1:4:2)
a
<ev <ev j ka <ev j
ka
<ev <ev ev
max ;
h
p‡q
with k 2 1; 2 , with
ev
max
ˆ
8
< exp api
p‡q
:
exp (api) ˆ
if p ‡ q is odd and a is even ( ‡ ) or odd (
);
1 if p ‡ q is even:
The odd ordering is the reverse ordering.
For k 2 Z we set
8
qk
1
ev
>
>
in(k)
:ˆ
‡
;
<
p‡q 2
(1:4:3)
>
pk
>
ev
ev
: out (k) :ˆ k
in(k) ˆ
p‡q
1
:
2
Both are increasing functions of k. If k 2 f0; . . . ; p ‡ q 1g, we have
ev
in(k) 2 f0; . . . ; qg and evout (k) 2 f0; . . . ; pg. Furthermore, let
p ‡ q ev
ev
0
ev
0
minin (k) :ˆ min k 5
(1:4:4)
j in( j) ˆ in(k) for all j 2 [k ; k]
2q
and
(1:4:5)
maxev
out (k)
p ‡ q ev
00
ev
00
:ˆ max k < p ‡ q ‡
j out( j) ˆ out (k) for all j 2 [k; k ]
2p
138
Marco Hien - Claude Sabbah
(see Remark 6.1.3(3) for an explanation). In a similar way we set for k 2 Z:
8
qk
>
odd
>
;
in(k)
:ˆ
<
p‡q
(1:4:6)
pk
>
>
: oddout(k) :ˆ k 1 odd in(k) ˆ
1 2 f 1; . . . ; p 1g;
p‡q
and if k 2 f0; . . . ; p ‡ q 1g we have odd in(k) 2 f0; . . . ; q 1g and oddout(k) 2
odd
f 1; . . . ; p 1g. We define minodd
in (k) and maxout (k) literally as in (1.4.4) and (1.4.5)
replacing ``ev'' by ``odd''. Note that
h qk
p‡q
1 qk i
;
\ N ˆ [ () ev in(k) ˆ odd in(k) ‡ 1:
2 p‡q
Let V be a finite dimensional complex vector space together with an automorphism T 2 Aut(V). Let us denote by 1k the complex vector space of rank one
with chosen basis element 1k . Let us consider direct sum Vp‡q of p ‡ q copies of V
and let us keep track of the indices by writing
Vp‡q ˆ
(1:4:7)
p‡q
M1
V C 1k :
kˆ0
CONVENTION 1.4.8. For any j 2 Z, we will write
v 1j :ˆ T s (v) 1j‡s( p‡q) for s 2 Z such that j ‡ s(p ‡ q) 2 [0; p ‡ q
DEFINITION 1.4.9. Assume p ‡ q53. Then:
Lp‡q 1
L
(1) The isomorphism sodd
V 1k ! p‡q
ev :
kˆ0
kˆ0
(1:4:9)
sodd
ev …v 1k † :ˆ
8
>
v 1k ;
>
>
>
>
>
< v 1minodd (k)
in
>
>
>
>
>
>
:
(2) The isomorphism sev
odd :
(1:4:9)
sev
odd …v 1k † :ˆ
V 1k is defined as
if
in
out
‡ v 1maxodd (k)‡1 ;
if
out
Lp‡q
kˆ0
1
h
qk
p‡q
1 qk
;
2 p‡q
qk
p‡q
1 qk
;
2 p‡q
i
\Nˆ[
1
v 1maxodd (minodd (k))
V 1k !
8
>
v 1k ;
>
>
>
>
>
>
< v 1minevin (k)
>
>
>
>
>
>
>
:
1
1]:
Lp‡q
kˆ0
if
h
1
h
i
\ N 6ˆ [
V 1k is defined as
qk
p‡q
1 qk
;
2 p‡q
qk
p‡q
1 qk
;
2 p‡q
i
\ N 6ˆ [
1
v 1maxevout (minevin (k))
‡ v 1maxevout (k)‡1 ; if
h
i
\Nˆ[
The local Laplace transform of an elementary irregular etc.
139
If p ˆ q ˆ 1, the isomorphism sodd
ev is defined as
(1:4:10) sodd
ev (v 10 ) :ˆ
v 10 ‡ (Id ‡ T 1 )v 11 and sodd
ev (v 11 ) :ˆ v 11 ;
and sev
odd is defined as
ev
1
(1:4:10) sev
odd …v 10 † :ˆ v 10 and s odd …v 11 † :ˆ …Id ‡ T †v 10
v 11 :
We are now ready to define the following particular set of Stokes data.
DEFINITION 1.4.11. The standard linear Stokes data Sstd (V; T; p; q) are given by
(1) the vector spaces L` :ˆ Vp‡q for ` ˆ 0; . . . ; 2q
(2) for each m ˆ 0; . . . ; q
S2m
2m
1
1, the isomorphisms
:ˆ sev
odd : L2m
1
! L2m
:ˆ sodd
S2m‡1
ev : L2m ! L2m‡1
2m
(3) the isomorphism S2q
2q
1,
(cf: (1:4:9 ));
(cf: (1:4:9 )):
:ˆ diag(T; . . . ; T) sev
odd : L2q 1 ! L0 .
L
L
k
(4) the filtrations Fk L2m :ˆ k0 4k V 1k0 and F L2m‡1 :ˆ k0 5k V 1k0 .
1
Note that the opposedness property (1.3.3) for the above data is not obvious. We will not give a direct proof since it follows a posteriori from our main
result by which these data are the Stokes data attached to some Stokes
structure. The arguments in the proof of Theorem 1.3.6 also lead to the following explicit computation of the topological monodromy of b
r ‡ F (0;1) (M),
where we set
qk
1
ev
in(k) :ˆ in(k) ˆ
(1:4:12)
‡ ; maxout (k) :ˆ maxev
out (k):
p‡q 2
PROPOSITION 1.4.13. The topological monodromy of b
r ‡ F (0;1) (M) is conLp‡q 1
p‡q
jugate to the automorphism of the vector space V
ˆ kˆ0 V C 1k (using
the notation (1.4.7) and Convention 1.4.8) given by
(
v 1k‡1
if in(k ‡ 1) ˆ in(k);
b top (v 1k ) ˆ
T
v (1k 1maxout (k) ‡ 1maxout (k)‡1 ) if in(k ‡ 1) ˆ in(k) ‡ 1:
REMARK 1.4.14. The topological monodromy can of course be deduced from
the Stokes data by the formula
Ttop :ˆ S02q
1
S2q
2q
1
2
S10 2 Aut(L0 ):
However, we will give a more direct computation in § 6.3. This approach is less
involved than the detour over the Stokes data and could perhaps be applied in
more general situations without having to understand the full information of the
Stokes data.
140
Marco Hien - Claude Sabbah
The article is organized as follows. Section 2 recalls the notion of Stokes-filtered
local system and explains the correspondence with that of linear Stokes data generalized monodromy data - describing Stokes structures of pure level q. In
Section 3 the construction of the topological Laplace transformation is discussed in
general, and the main tool (Th. 3.3.1) is [Moc14, Cor. 4.7.5]. We then concentrate on
the case of an elementary meromorphic connection. A description of the Stokes
data for b
r ‡ F (0;1) (M) in cohomological terms is proved in the subsequent section Theorem 4.3.3.
The remaining sections provide the proof that these data are isomorphic to the
standard linear Stokes data of Definition 1.4.11. Section 5 identifies the filtered
vector spaces (L` ; FL` )` obtained by the cohomological computation of Theorem
4.3.3 with the filtered vector spaces entering in the standard linear Stokes data.
This is done by using Morse theory. However, the Morse-theoretic description of
the linear Stokes data expressed through Theorem 4.3.3 does not seem suitable to
compute the Stokes matrices S`‡1
` . This is why, in Section 6, we construct a simple
Leray covering giving rise to a basis of the cohomology space L` for each `, which
allows us to identify the matrices S`‡1
given by Theorem 4.3.3 to those given by the
`
standard linear Stokes data.
It remains to compare the Stokes filtration obtained in Theorem 4.3.3 with that
obtained from the standard linear Stokes data. The problem here is that the simple
Leray covering constructed in Section 6 may not be adapted to the Morse-theoretic
computation. In Section 7 we change the construction of the Leray covering in
order both to keep the same combinatorics to compute the matrices S`‡1
and to
`
identify the filtration of Theorem 4.3.3 with the standard Stokes filtration 1.4.11(4).
This is the contents of Corollary 7.2.5, which concludes the proof of Theorem 1.3.6.
2. Stokes-filtered local systems and Stokes data
2.1 ± Reminder on Stokes-filtered local systems
We refer to [Del07], [Mal91] and [Sab13a, Chap. 2] for more details.
f
f
Let $ : A1t ! A1t be the real blowing-up of the origin in A1t . Then A1t ˆ
f
A1t [ S1tˆ0 is homeomorphic to a semi-closed annulus, with S1tˆ0 :ˆ $ 1 (0) A1t .
f
Similarly, we consider A1u , and r extends as the p-fold covering : S1uˆ0 ! S1tˆ0 .
For any # 2 S1uˆ0 , the order on u 1 C[u 1 ] at # is the additive order defined by
(2:1:1) ' 4# 0 ()
exp (W(u)) has moderate growth in some open sector centered at #:
We set
(2:1:2)
' <# 0 () ' 4# 0 and ' 6ˆ 0:
The local Laplace transform of an elementary irregular etc.
141
This is equivalent to exp ('(u)) having rapid decay in some open sector centered
at #.
DEFINITION 2.1.3 (see e.g. [Sab13a, Chap. 2]). A Stokes-filtered local system
with ramification r consists of a local system L on S1tˆ0 such that 1 L is equipped
1 L indexed by a finite subset
with a family of subsheaves L 4' 1
1
Exp u C[u ] with the following properties:
(1) For each # 2 S1uˆ0 , the germs L 4';# (' 2 Exp) form an increasing filtration
of ( 1 L )# ˆ L (#) .
P
(2) Set L < ';# ˆ c <# ' L 4c;# , where the sum is taken in L (#) (and the sum
indexed by the empty set is zero). Then L <';# is the germ at # of a subsheaf
L <' of L 4' .
S
T
1 L and
(3) The filtration is exhaustive, i.e., '2Exp L 4' ˆ
'2Exp L <' ˆ 0.
(4) Each quotient gr' L :ˆ L 4' =L <' is a local system.
P
(5)
'2Exp rk gr' L ˆ rk L .
(6) For each z 2 mp and ' 2 Exp, setting 'z (u) :ˆ '(zu), we have L 4'z '
ez 1 L 4' through the equivariant isomorphism
1L ' e
z 1 1 L , where
1
1
ez : S
uˆ0 ! Suˆ0 is induced by the multiplication by z.
REMARK 2.1.4. It is equivalent to start with a family of subsheaves L <' such that L <';# L <c;# if ' 4# c, and to define L 4' by the formula
T
L 4';# ˆ c; c <# ' L <c;# with the convention that the intersection indexed by the
empty set is L # . The family L <' will be simpler to obtain in our computation of
Laplace transform.
1L
REMARK 2.1.5 (Extension of the index set). It is useful to extend the indexing
1 L for any
set of the filtration and to define the subsheaves L 4c 1
1
1
c 2 u C[u ]. For such a c and for any ' 2 Exp, we denote by S'4c the open
subset of S1uˆ0 defined by
# 2 S1'4c () ' 4# c:
We shall abbreviate by ( b'4c )! the functor on sheaves composed of the restriction to
this open set and the extension by zero from this open set to S1uˆ0 . Then L 4c is
defined by the formula
X
(2:1:5)
…b '4c †! L 4' :
L 4c ˆ
'2Exp
The germ of L 4c at any # 2
(2:1:5)
S1uˆ0
can also be written as
X
L 4c;# ˆ
L 4';# ;
' 2 Exp
'4# c
142
Marco Hien - Claude Sabbah
with the convention that the sum indexed by the empty set is zero. Note also that,
if c is not ramified, i.e. c 2 t 1 C[t 1 ], then L 4c is a subsheaf of L . A similar
definition holds for L <c , but one checks that L <c ˆ L 4c if c 2
= Exp. Of particular
interest will be L 40 L .
2.2 ± Stokes data of pure level q
In this section, we will define the notion of Stokes data attached to a given
Stokes structure (L ; L ), which - after various choices to be fixed a priori describes the Stokes filtration in terms of opposite filtrations on a generic stalk.
After choosing appropriate basis, the passage from one filtration to the other
could be expressed by matrices. These matrices are usually referred to as the
Stokes matrices or Stokes multipliers in different approaches to Stokes structures (see e.g. [BJL79]). In level q ˆ 1, the following description has already
been given in [HS11], section 2.b.
We will restrict to the case of an unramified Stokes-filtered local system, i.e., we
assume that, in Definition 2.1.3, r is equal to Id, and we will work with the
variable u. We will set S1 ˆ S1uˆ0 . We identify S1 ˆ R=2pZ and will call an element
# 2 S1 an argument.
The description of these data by combinatorial means will highly depend on the
various pole orders of the factors ' 2 Exp and their differences ' c as well. The
situation we have in mind as an application allows to get rid of many of these
difficulties, since we will be able to restrict to a single pole order:
DEFINITION 2.2.1. We say that a finite subset E u 1 C[u 1 ] is of pure level q if
the pole order of each ' 2 E as well as
the pole order of each difference '
c for '; c 2 E, ' 6ˆ c,
equals q. We say that an unramified Stokes structure (L ; L ) is pure of level q if the
associated set Exp of exponential factors is pure of level q.
Let E u 1 C[u 1 ] be a finite subset of pure level q. In particular, there are
exactly 2q Stokes directions for any pair ' 6ˆ c in E, where a Stokes direction is an
argument # 2 S1 at which the two factors '; c do not satisfy one of the two inequalities '4# c or c4# '. More precisely, if
(c
')(u) ˆ u
q
g(u)
with a polynomial g(u) with g(0) 6ˆ 0, the Stokes directions are
n
o
p
3p
St('; c) :ˆ # 2 S1 j q# ‡ arg (g(0)) or
mod 2p :
2
2
At these directions, the asymptotic behaviour of exp (' c) changes from rapid
decay to rapid growth and conversely. Let StDir(E) denote the set of all Stokes
The local Laplace transform of an elementary irregular etc.
directions StDir(E) :ˆ
with respect to E if
(2:2:2)
S
'6ˆc2E
n
#o ‡
143
St('; c). An argument #o 2 S1 is said to be generic
o
`p
j ` 2 Z \ StDir(E) ˆ [:
q
Let us fix a generic #o and let us set #` :ˆ #o ‡ `p=q (` ˆ 0; . . . ; 2q 1). Then each of
the open intervals (#` ; #`‡1 ) contains exactly one Stokes direction for each pair
' 6ˆ c in E. For " > 0 small enough, the same holds for the intervals
S 2q 1
I` :ˆ (#` "; #`‡1 ‡ ") with which we cover the circle S1 ˆ `ˆ0
I` .
The category of unramified linear Stokes data of pure level q indexed by the
finite set f1; . . . ; rg over the field C is defined by (1.3.3). A morphism of two linear
Stokes data of this type is given by C-linear maps between the vector spaces which
commute with all the S`‡1
and respect the filtrations. We denote this category by
`
Stdata (q; r). A similar definition can be given over any field k.
We now fix a finite set E u 1 C[u 1 ] of pure level q. Let Ststruct (E) be the
category of unramified Stokes structures (L ; L ) on S1 with Exp ˆ E. After
choosing an appropriate argument #o as above, we construct a functor
(2:2:3)
FE;#o : Ststruct (E) ! Stdata (q; #E)
as follows.
We can arrange the elements in E according to the ordering (2.1.1) with respect
to the generic direction #o , i.e., we write
E ˆ f'1 <#o '2 <#o '3 <#o <#o '#E g:
We then have the same ordering at any #2m with even index and the reverse order
for #2m‡1 . This follows from the fact that each inequality 'i <# 'j becomes reversed
whenever the argument # passes a Stokes direction for the pair 'i , 'j , and there is
exactly one Stokes direction for each pair inside (#` ; #`‡1 ).
We invoke the fundamental result going back to Balser, Jurkat and Lutz:
PROPOSITION 2.2.4. For each open interval I 4 S1 of width p=q ‡ 2" for " > 0
small enough, there is a unique splitting
M
gr' L jI
t : L jI '
'2E
compatible with the filtrations.
PROOF. See [Mal83, Lem. 5.1] or [BJL79, Th. A].
p
L
In particular, writing grL :ˆ '2E gr' L for the associated graded sheaf
endowed with the Stokes filtration naturally induced by using the order (2.1.1), we
144
Marco Hien - Claude Sabbah
have unique filtered isomorphisms
(2:2:5)
t` : (L ; L )jI` ! (grL ; (grL ) )jI`
over the intervals I` chosen above.
For each ` ˆ 0; . . . ; 2q 1, we let L(`) :ˆ G(I` ; L ) be the sections of the local
system L over I` . We have canonical isomorphisms
a` : L(`) ! L #`
and
b` : L(`) ! L #`‡1
between L(`) and the corresponding stalks of L . Writing L` :ˆ L #` for the stalk
at #` , we define
:ˆ b` 1 a` : L` ! L`‡1 :
S`‡1
`
We call this isomorphism the clockwise analytic continuation. Additionally, we define, for m ˆ 0; . . . ; q 1,
(
Fj L2m :ˆ (L 4'j )#2m L2m ;
F j L2m‡1 :ˆ (L 4'j )#2m‡1 L2m‡1 :
Then the data
(L` )` ; (S`‡1
` )` ; (FL` )`
define a set of linear Stokes data as in (1.3.3).
PROPOSITION 2.2.6. Given a finite set E of pure level q and a generic #o 2 S1 ,
the construction above defines an equivalence of categories
FE;#o : Ststruct (E) ! Stdata (q; #E)
between the Stokes structures with exponential factors E and the Stokes data of
pure level q indexed by f1; . . . ; #Eg.
PROOF. In the case q ˆ 1, a similar statement is proved in [HS11] or [Sab13b].
The same proof holds in our situation as well.
p
2.3 ± Reminder on the local Riemann-Hilbert correspondence.
Let M be a finite dimensional C…ftg†-vector space with connection, and let
r : u 7! up ˆ t be a ramification such that r‡ M has a formal Levelt-Turrittin decomposition
M
C……u†† C…fug† r‡ M '
(E '(u) R' );
'2Exp (M)
with Exp :ˆ Exp (M) u 1 C[u 1 ] (we use an opposite sign
'(u) with respect to
The local Laplace transform of an elementary irregular etc.
145
the usual notation as it will be more convenient for the Stokes filtration). Moreover,
by the uniqueness of the Levelt-Turrittin decomposition, we have, for each z 2 mp
(p-th root of the unity)
(2:3:1)
R'z ˆ R' ;
with 'z (h) :ˆ '(zh):
It may happen that 'z ˆ ' for some ' 2 Exp and z 2 mp .
The Riemann-Hilbert correspondence M 7! (L ; L ) goes as follows (see
[Del07], [BV89], [Mal91], [Sab13a, Chap. 5]). Let us denote by F the local system
attached to M on a punctured neighbourhood of ft ˆ 0g, that we regard as well as a
f
local system on the open annulus (A1t ) or on the semi-closed annulus A1t . Similarly,
f
f
extending r as a covering map (A1u ) ! (A1t ) or A1u ! A1t , we define r 1 F . We also
use the notation
1 L :ˆ F
L :ˆ F jS1 ;
:
jS1
tˆ0
uˆ0
One can attach to M the family of subsheaves L 4' :ˆ H 0 DRmod 0 (r‡ M E ' )jS1
uˆ0
1 L (' 2 Exp), where DRmod 0 denotes the de Rham complex with coefficients
of
f
in the sheaf of holomorphic functions on A1u n S1uˆ0 having moderate growth along
1
Suˆ0 . The equivariance property 2.1.3(6) is obtained through the isomorphisms
z ‡ r‡ M ˆ r‡ M. Similarly, we can consider the family of subsheaves L <' :ˆ
1 L (' 2 Exp), where DRrd 0 denotes the de Rham
H 0 DRrd 0 (r‡ M E ' ) of
f
complex with coefficients in the sheaf of holomorphic functions on A1u n S1uˆ0 having
rapid decay along S1uˆ0 .
REMARK 2.3.2. The equivalence of categories M !M considered in § 1.2 reads
as follows, by introducing the notion of Stokes-filtered sheaves. A ramified Stokesf
filtered sheaf on A1t is defined (in the present setting) as the pair formed by a local
f
system F on A1t and a Stokes-filtration L of its restriction L :ˆ F jS1 . Since
tˆ0
f
S1tˆ0 is a deformation retract of A1t , giving F is equivalent to giving L , hence the
equivalence.
With this notion of Stokes-filtered sheaf, we can define a subsheaf F 40 of F ,
which coincides with F away from S1tˆ0 , and whose restriction to S1tˆ0 is L 40 (see
e 1 of P1
Remark 2.1.5). In the following, we will also consider the real blow-up space P
t
t
at t ˆ 0 and t ˆ 1, which is homeomorphic to a closed annulus, and we will still denote
f
e 1.
by F (resp. F 40 ) the push-forward of F (resp. F 40 ) by the open inclusion A1 ,! P
t
t
3. The topological Laplace transformation F (0;1)
top
3.1 ± Reminder on the local Laplace transformation F (0;1)
We keep the notation as in the introduction, and we will focus on the coordinate t,
so that the Laplace kernel is now exp ( t=t). Moreover, since we only want to
146
Marco Hien - Claude Sabbah
deal with F (0;1) (M), we only consider the localized Laplace transform
C[t0 ; t0 1 ] C[t0 ] , that we will denote by
from now on. We consider the following diagram:
The localized Laplace transform
is defined by the formula
ˆb
p‡ (p‡ M E
t=t
):
We can regard the C[t; t; t 1 ]h@t ; @t i-module p‡ M E t=t both as a holonomic
D P1 A1 -module and as a meromorphic bundle with connection on P1t A1t with
t
t
poles along the divisor D :ˆ D0 [ D1 [ D in P1t A1t (with D0 ˆ f0g A1t ,
cg). It has irregular singularities along D. As
D1 ˆ f1g A1t and D ˆ P1t f 1
indicated in §1.2, the germ of at t ˆ 0 only depends on the germ M of M at t ˆ 0,
and it is denoted by F (0;1) (M).
r‡
Let b
r : h 7! h ˆ t be a ramification such that b
that is,
M
C……h†† C…fhg† b
r ‡ F (0;1) (M) '
2
d(
with Exp
…
is non-ramified at infinity,
(h)
(E
b );
R
†
) h 1 C[h 1 ]. Then b
r ‡ F (0;1) (M) is obtained through the diagram
by the formula
b
r ‡ F (0;1) (M) ˆ C…fhg† C[h;h
1]
b
p‡ (p‡ M E
t= (h)
):
Moreover, by the uniqueness of the Levelt-Turrittin decomposition, we have, for
p-th root of the unity)
each 2 m ( b
b
R
b ;
ˆR
with
(h) :ˆ (
1
h):
The stationary phase formula of [Fan09, Sab08] makes explicit the correspondence
d (R
b )
p; Exp;
).
(p; Exp; (R' )' 2 Exp ) 7! ( b
2
The local Laplace transform of an elementary irregular etc.
147
3.2 ± The monodromy of F (0;1) M
c the local system attached to F (0;1) M on S1 . The local
Let us denote by L
tˆ0
c can equivalently be regarded as a local system F
c on Gan . Let us
system L
m;t
btop the
denote by Ttop the topological monodromy of M around the origin, and by T
(0;1)
topological monodromy of F
M around t ˆ 0 (with the notation as in § 1.2).
b 1 is the topological monodromy of
Since
has no singularity except at 0; 1, T
top
0
around t ˆ 0. Therefore, there are various ways of calculating the topological
b top :
monodromy T
either by comparing it with the monodromy at t ˆ 1 of M, which is nothing,
up to changing orientation, but the topological monodromy of M at t ˆ 0,
since M has no singular point on (A1;an
t ) , and its singular point at t ˆ 1 is
regular; this will be done in Proposition 3.2.1;
or by a direct topological computation at t ˆ 1; this is done in § 6.3;
lastly, by obtaining it from the Stokes data at t ˆ 1, once we have computed
them; this is however not the most economical way.
Given an automorphism T, we will denote by Tl (l 2 C ) the component of T
with eigenvalue l. Since
has a regular singularity at t ˆ 1, with monodromy
equal to Ttop1 , classical results give:
b top;l ˆ Ttop;l . Moreover, if M is a
PROPOSITION 3.2.1. For l 6ˆ 1, we have T
successive extension of germs of rank-one meromorphic connections, the Jordan
b top;1 are in one-to-one correspondence with the Jordan
blocks of size k 5 2 of T
blocks of size k 1 of Ttop;1 .
SKETCH OF PROOF. Only for this proof, we denote by
the non localized Laplace
0
transform of M. It has a regular singularity at t ˆ 0. The monodromy Ttop1 of M at
infinity is known to be equal to the monodromy of the vanishing cycles of
at
0
1
0
b
t ˆ 0, while T top is the monodromy of the nearby cycles of
at t ˆ 0 (cf. e.g.
[Sab06, Prop. 4.1(iv)] with z ˆ 1). The first assertion follows. For the second assertion, note that, if M is a rank-one meromorphic connection, then M is irreducible as a C‰tŠh@t i-module, and thus
is also irreducible, so is a minimal extension at t0 ˆ 0, which implies the second assertion for M. In general, note that in a
exact sequence 0 ! 0 !
! 00 ! 0, if the extreme terms are minimal ex0
tensions at t ˆ 0, then so is the middle term. This concludes the proof of the second
assertion.
p
e 1 ! P1 of P1
As in Remark 2.3.2, let us consider the real blowing-up map $ : P
t
t
t
1
e
at t ˆ 0 and t ˆ 1 (so that Pt is a closed annulus) and let us denote by
e
e 1 Gan ! Gan the projection. Then we have
b:P
t
(3:2:2)
m;t
m;t
e
c ' R1 b DRmod (D0 [D1 ) (p‡ M E
F
t=t
)
148
Marco Hien - Claude Sabbah
and all other Rk e
b vanish. We note that, for t 6ˆ 0, twisting with E t=t has no effect
near t ˆ 0. Similarly, since M is regular at 1, E t=t is the only important factor
along t ˆ 1.
Computation. Let us use the notation of Remark 2.3.2. Let us fix to 2 C and let us
ct . Recall that P
e 1 is a closed annulus with boundary components
compute the fibre F
o
t
S1tˆ0 and S1tˆ1 . Let F 40 be the sheaf H 0 DRmod 0 M on the semi-closed annulus
e 1 )4 1 the closed annulus P
e 1 with the closed interval
e 1 n S1 . We denote by (P
P
t
tˆ1
t
t
to
arg t arg to 2 [ p=2; p=2] mod 2p in S1tˆ1 deleted. This is the domain where the
function exp (t=to ) has rapid decay. We consider the inclusions
a1
b1
to
to
e 1 )4 1 ,!
e 1:
e 1 n S1 ,!
(
P
P
P
t
tˆ1
t
t
to
We then have (similarly to [Sab13a, §7.3]):
ct is equal to H1 (P
e 1 ; b1 a1 F 40 ) and the
PROPOSITION 3.2.3. The fibre F
o
t
to ;! to ;
other Hk vanish. It is isomorphic to the vanishing cycle space at t ˆ 0 of the
c with respect to t is obtained by
perverse sheaf $ F 40 on A1t . The monodromy of F
1
rotating the interval (p=2; 3p=2) ‡ arg to Stˆ1 in the counterclockwise direction
with respect to to .
p
b
REMARK 3.2.4. We can similarly compute the fibre of e
r
replacing arg to with b
p arg ho in the formula above.
c
1F
at any ho by
3.3 ± The Stokes structure of F (0;1) M
c the Stokes filtration of L
c , regarded as a family of
Let us denote by L
e
c ). Due to the ramification b
subsheaves of b
r 1(L
r, we now work with the variable h.
1
1
1
1
Let Pg
t Ah be the real oriented blowing-up of Pt Ah along the components
1
1
D0 ; D1 ; D of D regarded now in Pt Ah . We have a similar diagram (see e.g.
[Sab13a, Chap. 8]):
1
1
e 1 f1
e1
We note that Pg
t Ah is the product Pt A h of the closed annulus Pt by the semif
closed annulus A1 h . One defines on such a real blown-up space the sheaf of holomorphic functions with rapid decay near the boundary (i.e., the inverse image by the
real blow-up map of the divisor D). There is a corresponding de Rham complex that
we denote by DRmod D (see loc. cit.).
The local Laplace transform of an elementary irregular etc.
THEOREM 3.3.1 (Mochizuki, [Moc14, Cor. 4.7.5]). For each
natural morphism
e
DRmod b
r ‡ F (0;1) (M) E (h) ! R
b DRmod D (p‡ M E
149
2 h 1 C[h 1 ] the
(h) t= (h)
)[1]
is a quasi-isomorphism and the natural morphism
e
R1 b DRmod D (p‡ M E
(h) t= (h)
e
b DRmod (D0 [D1 ) (p‡ M E
)!b
| R1 t= (h)
)
is injective.
REMARKS 3.3.2 (1) The result of [Moc14, Cor. 4.7.5] refers to the real blow-up
f
P1t A1 h along D , not along all the components of D. The above statement
f1
1
1
1
is obtained by using that, with respect to $ : Pg
t Ah ! Pt A h , we have
R$ DRmod D (p‡ M E
t= (h)
) ˆ DRmod D (p‡ M E
t= (h)
mod D
This follows from the identification R$ A mod D ˆ A 1
Pt as in [Sab13a, Prop. 8.9].
(2)
):
( D0 [ D1 ),
Set b
r (h) ˆ ch (1 ‡ o(h)) and let us choose a b
p-th root c1= (1 ‡ o(h)) of b
r (h)=h .
Then the b
p-th roots of b
r (h) are written b
r
Now, for
2 h 1 C[h 1 ], we set
1=
(h) ˆ h c1= (1 ‡ o(h)) ( 2 m ).
1=
b
r (h) .
(h) :ˆ
The m -equivariance is then induced by the isomorphism
‡
E
where we still denote by
(h) t= (h)
ˆE
;
1=
the map h 7! b
r
(h).
e
The theorem expresses therefore the Stokes filtration as the push-forward by b
d The questions stated in the introduction
of a family of complexes indexed by Exp.
reduce now to
d the complex DRmod D (p‡ M E
expressing, for any 2 Exp,
gether with the isomorphisms
1
DRmod D (p‡ M E
(h) t= (h)
) ' DRmod D (p‡ M E
(h) t= (h)
), to-
);
in terms of the Stokes-filtered local system attached to M,
e
computing in a topological way the push-forward by b once this complex is well
understood.
While H 0 DRmod D is easy to compute from the data (F ; L ), it happens in
general that the complex DRmod D has higher cohomology, which is not easy to
express in terms of (F ; L ). The main reason is that the meromorphic connection
150
Marco Hien - Claude Sabbah
c ), due to inp‡ M E (h) t= (h) may not be good (in the sense of [Sab00]) at (0; 1
determinacy 0=0 of the functions
v'; (u; h) :ˆ (h)
(3:3:3)
r(u)=b
r (h)
'(u)
d
2 Exp).
for u ! 0 and h ! 0 (' 2 Exp,
PROPOSITION 3.3.4. There exists a projective modification e : Z ! P1t A1h
c ), such that for each
which consists of a succession of point blowing-ups above (0; 1
d the pull-back p‡ M E (h) t= (h) is good along the normal crossing
2 Exp,
divisor DZ :ˆ e 1 (D).
p
This result is a particular case of a general result due to Kedlaya [Ked10]
and Mochizuki [Moc09a] (see also [Moc09b]), but can be proved in a much easier
way in the present setting, since it essentially reduces to resolving the
indeterminacy 0=0 of the rational functions v'; (u; h) for ' 2 Exp (M) and
2 Exp ( ), and to using [Sab00, Lem. III.1.3.3]. A particular case will be
made precise in §3.4. The results below do not depend on the choice of Z
satisfying the conclusion of Theorem 3.3.5. We decompose DZ as the union of
the strict transforms of D0 ; D1 ; D that we denote by the same letters, and the
exceptional divisor E.
The main tool is then the higher dimensional Hukuhara-Turrittin theorem
(originally due to Majima [Maj84], see also [Sab93, Proof of Th. 7.2], [Moc11,
e
Chap. 20], [Sab13a, Th. 12.5 & Cor. 12.7]). We consider the real blow-up space Z
1
e ! Pg
of Z along the components of DZ and the natural lift e : Z
A1 of
t
e : Z ! P1t A1h .
h
THEOREM 3.3.5. If Z is as in Proposition 3.3.4, then the complex
DRmod DZ e‡ (p‡ M E
(h) t= (h)
)
has cohomology in degree zero only.
p
We can then use a particular case [Sab13a, Prop. 8.9] of [Moc14, Cor. 4.7.5],
which is easier to prove because e is a projective modification, to get:
Re DRmod DZ e‡ (p‡ M E
(h) t= (h)
) ˆ DRmod D (p‡ M E
(h) t= (h)
):
We will apply Theorem 3.3.1 through its corollary below.
COROLLARY 3.3.6 (of Theorem 3.3.1). Let Z be as in Proposition 3.3.4. Then,
for each 2 h 1 C[h 1 ] the natural morphism
DRmod
‡
F (0;1) (M) E
(h)
e
b e ) H 0 DRmod DZ e‡ (p‡ M E
! R1 ( (h) t= (h)
†
The local Laplace transform of an elementary irregular etc.
151
is a quasi-isomorphism and the natural morphism
e
b e) H 0 DRmod DZ e‡ (p‡ M E
R1 ( (h) t= (h)
b
)!e
r
1
Fb
is injective.
p
The pull-back (b
p e) 1 (fh ˆ 0g) ˆ e 1 (D ) is the union D [ E. Its pull-back
e
e denoted by D
e [ Ee , is equal to (
b
b e) 1 (S1hˆ0 ). We will denote by e
| the open
in Z,
e [ Ee ) ,! Z.
e
e n (D
inclusion Z
‡
Let r : u 7! up ˆ t be a ramification
such
that r M is non-ramified at t ˆ 0.
1
1
e as a topological space:
Z
Consider the fibre product Ze0 :ˆ Pg
u Ah e 0 :ˆ e
e Z ) and eb
e0 n e
e [ Ee ) ,! Z
e 0 . Note that the restrics 1 (D
Let us write D
s 1 (D
|0 : Z
Z
e 0 is a homeomorphism.
e0 n D
tion of e 0 to Z
Z
e
Let $ : Z ! Z be the oriented real blow-up map. Below we use the convention
e 0 we set
of Remark 2.3.2, and for ez0 2 Z
(
e 2 S1uˆ0 ;
Exp if u
1
0
0
0
e
e ˆ (e
u
p e ) ( ez ) 2 Pu ; Exp ˆ
e 2 P1u n S1uˆ0 :
0
if u
e 0 over z :ˆ $ e
and, as usual, for a point ez 0 2 Z
s(ez 0 ) 2 Z and a function g(u; h):
e0 '4 e0 g () Re e0 ' e0 g < 0 in some neighb: of ez0
or e0 '
e0 g is bounded in some neighb: of ($ e
s ) 1 (z):
Recall that the Stokes filtration (F 0 ; F
1
the equivariance condition z F
0
4'
0
4
ˆF
) on the local system F 0 :ˆ e
r
0
4'z
DEFINITION 3.3.7. (1) The sheaf G 0 on
the subsheaf of (pe0 ee0 ) 1 F 0 on
e 0 n (e
e [ Ee )) is defined by
ez0 2 Z
s 1 (D
X
G 0 :ˆ
1
0
1
F satisfies
0
inside z F ˆ F for each z 2 mp .
e 0 is defined as e
b
Z
| 0 G 0 , where G 0 is
0n
1 e
e
e
Z (e
s (D [ E )) whose germ at
F
0
4';
F 0:
0
(2) The sheaf G 4
is the subsheaf of G 0 which coincides with the latter away
e 0 is defined by the formula
e Z0 and whose germ G 0
at each ez0 2 Z
from D
4 ;
152
Marco Hien - Claude Sabbah
(same convention as in (2.1.5))
0
G4
;
:ˆ
X
0
(We note that the inclusion G 4
G
F
0
0
4';
F 0:
e
is obvious away from e
s 1 (D
[ Ee )
since c plays no role there, and therefore it is also clear on this set, by the definition of G 0 there.)
0
LEMMA AND DEFINITION 3.3.8 (Sheaves G and G 4 ). The sheaves G 0 and G 4
e
descend with respect to e
s to subsheaves G and G
of (e
p e) 1 F on Z.
4
PROOF. Since r(u) ˆ up , the group mp acts transitively on the fibres of e
s. This action
e Z . By definition, for any z 2 m ,
is free on the fibres over ez 2 D
p
e0 '4z~z0 e0 ( (h)
r(u)=b
r (h)) () e0 'z 4~z0 e0 ( (h)
r(u)=b
r (h))
since the right-hand side is invariant with respect to the mp -action. By the equivariance property of F 04 , we deduce the equality of subspaces of F ~t :
X
X
0
0
F 04';z~u ˆ
F 04' ;~u ˆ G 4
G4
;z~z0 ˆ
; ;
z
where we recall that ' 2 Exp , 'z 2 Exp for all z 2 mp . Therefore, we can define
0
the subsheaf G 4 (e
p e) 1 F by requiring its stalk at ez to be G 4 ; :ˆ G 4
;
independent on the choice of the lift ez0 of ez. The same works for G 0 descending to a
subsheaf G of (e
p e ) 1 F . Obviously, G 4 G .
p
0
REMARK 3.3.9. The indexing condition in the sum defining G 4
;
can be
e1
P
h
h2
be the image
written as 04 e0 v'; , where v'; is defined by (3.3.3). Let e
e
b e 0 . We will be mainly interested in the case where e
h 2 S1hˆ0 , in which
of ez 0 by b Then, when ' is fixed, the condition e0 v 4 e0 v0 is
case we denote it by #.
4
0
. Therefore,
0
';
';
4
implies G 4 ; G 4 0 ; G . The
1 e e
e
b ) G 4 ) is included in (R1 (
b e) G 4 0 )
theorem below implies then that (R (
e
and defines a filtration of (R1 (
b e) G ) .
equivalent to
THEOREM AND DEFINITION 3.3.10. Let (F ; F ) be a Stokes-filtered sheaf
e 1 , with trivial Stokes filtration away from S1 and a Stokes filtration inon P
t
tˆ0
d and b
r according to the
dexed by Exp on S1tˆ0 having ramification r. Define Exp
stationary phase formula, and G ; G 4 as in Definition 3.3.8.
e
e
d the complexes R(
Then, for each 2 Exp,
b e) G and R(
b e) G
have
4
The local Laplace transform of an elementary irregular etc.
153
cohomology in degree one at most, and R1 (e
b e) G 4 is a subsheaf of
1 e e
1
e
c. This family of subsheaves, together with the natural
b ) G ˆ b
r F
R (
d defines a Stokes
b
isomorphisms induced by e
r 1 G 4 ' G 4 for 2 Exp,
c.
filtration of F
The Stokes-filtered sheaf obtained by the procedure of the theorem is called
the localized (0; 1)-Laplace transform of the Stokes-filtered local system
c c
(L ; L ), and denoted by F (0;1)
top (L ; L ), or simply by (L ; L ).
e
e
Note that the inclusion R1 (
b e) G 4 R1 (
b e) G is far from obvious from
the sheaf-theoretic point of view.
THEOREM 3.3.11. If (L ; L ) is the Stokes-filtered local system attached to M,
c; L
c ) is the Stokes-filtered local system attached to F (0;1) M.
then (L
PROOF OF THEOREMS 3.3.10 AND 3.3.11. Given a Stokes-filtered local system
(L ; L ) we can choose a finite dimensional C…ftg†-vector space with connection M
such that (L ; L ) corresponds to M by the Riemann-Hilbert correspondence
recalled in §2.1. Both results are proved simultaneously, in order to apply Corollary
d a
3.3.6, which reduces the problem to showing that there is, for each 2 Exp,
natural isomorphism
(3:3:12)
G4
! H 0 DRmod DZ e‡ (p‡ M E
(h) t= (h)
):
Naturality means here the following. Let us denote by e
| the open inclusion
1
e
e 1 satisfies
n
e) F of the local system F on P
Z DZ ,! Z. The pull-back (e
t
1
1
1
‡
‡
(h)
t=
(h)
| e
e) F . On Z n DZ , DRe (p M E
) is naturally
(e
e) F ˆ e
| (e
e) 1 F , and both sheaves considered in the isomorphism
identified with e
| 1 (e
e) 1 F . The isomorphism
above are naturally regarded as subsheaves of e
| e
| 1 (e
above is nothing but the equality as such.
1
e and thus on Pg
The question is therefore local on Z,
A1 , and we can assume
h
t
that p‡ M is decomposed with respect to its Hukuhara-Turrittin decomposition in
the neighbourhood of ez. Then we are reduced to the case where Exp is reduced to
one element, and then the assertion is easy to check.
p
CONCLUSION 3.3.13. Computing F (0;1) (L ; L ) reduces therefore to the following steps:
d obtained through
(1) Given the subset Exp and its ``Laplace transform'' Exp
the stationary phase formula, to compute a sequence of blowing-ups
e : Z ! P1t A1h such that each e‡ (p‡ M E
(h) t= (h)
) is good.
1
e) F .
(2) To compute the family of subsheaves G 4 of (e
154
Marco Hien - Claude Sabbah
e
b
(3) To compute the push-forwards R1 (
b e ) G 4 as subsheaves of e
r
1
Fb .
(4) To make explicit, in the formula thus obtained, the m -action induced by the
isomorphisms 1 G 4 ' G 4 .
REMARK 3.3.14 (Moderate growth versus rapid decay). One can also determine
c . The
c; L
c ) by computing the subsheaves L
the Stokes filtered local system (L
<
use of the moderate growth condition is mainly dictated by (3.2.2), in order to avoid
mixed moderate growth/rapid decay in Theorem 3.3.1. On the other hand, assume
that M is purely irregular. Then
DRmod (D0 [D1 ) (p‡ M E
t=t
) ˆ DRrd (D0 [D1 ) (p‡ M E
t=t
):
As a consequence, with such an assumption, we can replace everywhere in §3.3 the
moderate growth property with the rapid decay property, provided we change G 4
with G < and the order 4 with the strict order < .
3.4 ± The Stokes structure on F (0;1) M under a simplifying assumption
The main example of this article satisfies the following assumptions.
ASSUMPTION 3.4.1. (1) M is purely irregular,
(2) there exists a ramification r : u 7! up ˆ t of order p and a C…fug†-module N
with connection such that M ˆ r‡ N (equivalently, M ˆ r‡ N) and for which
the Levelt-Turrittin decomposition of N is not ramified:
M
C……u†† N ˆ
(Eb ' Rb' );
'2Exp
where Exp is a finite subset in u 1 C[u 1 ].
Assumption 3.4.1(2) means that no Stokes phenomenon for M is produced
by the ramification. With this assumption, b
r ‡ F (0;1) (M) is obtained through the
diagram
by the formula
b
r‡
ˆb
p‡ (p‡ N E
r(u)= (h)
):
In other words, we can work directly with the u coordinate, as if M were non-ramified, and the only difference with the case where M is non-ramified is the twist
The local Laplace transform of an elementary irregular etc.
155
by E r(u)= (h) , not by E u= (h) . For that reason, we will use the same notations for the
maps in the u-variable, and we will now denote by (L ; L ) the non-ramified Stokesfiltered local system attached to N . We regard N as a holonomic D P1 -module which
u
is localized at u ˆ 1. Considering now the diagram
we have
b
p‡ (p‡ N E
r ‡ F (0;1) (M) ˆ b
r(u)= (h)
):
d h 1 C[h 1 ] be the set of exponential factors of C……h†† b
Let Exp
r ‡ F (0;1) (M) as
above.
DEFINITION 3.4.2. A proper modification e : Z ! P1u A1h is said to be suitable
for M if
c ) 2 P1u A1h ,
(1) it is a succession of point blowing-ups above (0; 1
c ) of each function v'; (u; h) (' 2 Exp,
(2) the indeterminacy at (0; 1
d
2 Exp),
cf. (3.3.3), is mostly resolved on Z, that is, the components of the divisors of
zeros of v'; (u; h) which have multiplicity > 2 do not meet the divisor of poles
c ), and those having multiplicity 42 meet it at most
of v'; e along e 1 (0; 1
at smooth points.
It is known that such a suitable modification always exists. The results on
b
r ‡ F (0;1) M obtained in the previous subsection can be adapted in a straightforward way to the present setting, even for a suitable modification, provided that
(1) we denote by (F ; F ) the Stokes-filtered sheaf attached to N (in the uvariable),
e in the formulas.
(2) we replace t with r(u) and et with u
With Assumption 3.4.1(2), we do not have to take care of the ramification of M.
With Assumption 3.4.1(1), we can work in the rapid decay setting. The explicit
description of the family of subsheaves G < is then simpler, due to the following
lemma.
LEMMA 3.4.3. If Z is suitable for M, then it satisfies the conclusion of Theorem
3.3.5 for N in the rapid decay setting.
PROOF. Clearly the goodness condition holds away from the intersection of the zero
set of the functions v'; with the divisor DZ . We can then argue as in [Sab13b,
Lem. A.1] for the zeros of multiplicity 42.
p
156
Marco Hien - Claude Sabbah
Note that we use the convention that the inequality 0 < e v'; is not satisfied
at a point ez whose image in Z is an indeterminacy point of e v'; . We also use
[Sab13b, Lem. A.1] to prove (3.3.12) at these points.
3.5 ± The case of an elementary meromorphic connection
We now restrict to the case where M ˆ El(r; '; R), with 0 6ˆ ' 2 u 1 C[u 1 ],
r : u 7! up ˆ t and R is a regular connection. We thus have M ˆ r‡ N with
N ˆ E ' R. Assumption 3.4.1 is thus satisfied. Since R is a successive extension
of rank-one meromorphic connections, the monodromy of F (0;1) M is given by
Proposition 3.2.1, where Ttop is the monodromy corresponding to r‡ R, that is, such
p
that Ttop
ˆ T, if T is the monodromy of R.
We set ' ˆ 'q u q (1 ‡ o(u)) with 'q 2 C n f0g. According to [Fan09, Sab08], we
b ( 1)q R), with
have C……t†† F (0;1) M ' El(b
r; ';
r0
p p‡q
ˆ
h (1 ‡ o(h));
'0 q'q
b
r (h) ˆ
b ˆ '(h) ‡
'(h)
(3:5:1)
r(h) p ‡ q
'q h q (1 ‡ o(h)):
ˆ
b
p
r (h)
We thus have
C……h†† b
r ‡ F (0;1) M '
b (h) ˆ '
b
with '
M
C……h†† (E
(h)
( 1)q R);
2mp‡q
h(1 ‡ o(h) . As a consequence,
Exp ˆ f'g
and
d ˆ f'
b (h) j
Exp
2 mp‡q g:
The blowing-up e : Z ! P1u A1h (see §3.4) is obtained by resolving the family of
rational functions
(3:5:2)
b (h)
v (u; h) :ˆ '
'(u)
r(u)=b
r (h);
2 mp‡q :
c ). Let us denote by " the single blow-up of the point (0; 1
c ). In the
Blow-up of (0; 1
c ), the divisor D reduces to D0 [ D , with D0 ˆ fu ˆ 0g and
chart centered at (0; 1
D ˆ fh ˆ 0g (cf. §3.1, by using the ramified coordinates u; h). We denote by the
same name their strict transforms by the blow-up map ", and by E the exceptional
c ).
divisor " 1 (0; 1
Figure 1. The blow-up map " : Z ! P1u A1h .
The local Laplace transform of an elementary irregular etc.
157
The chart (u; h) is covered by two charts with respective systems of coordinates
(x; y) and (w; z), such that "(x; y) ˆ (xy; y) and "(w; z) ˆ (w; wz). The strict transform of D0 , defined by x ˆ 0, is located in the first chart, and that of D , defined by
z ˆ 0, is located in the second chart. Moreover, (x:z) forms a system of projective
coordinates on E ' P1 .
In the coordinates (x; y) we find
'q q q p‡q
(3:5:3)
( p ‡ q) q xq ‡ p ‡ o(xy) :
x y qx
" v (x; y) ˆ
p
In these coordinates, the indeterminacy locus of v (x; y) consists of the points with
coordinate x on the divisor E ˆ fy ˆ 0g which are roots of the polynomial
f (X) :ˆ f (X= ) with
f (X) ˆ qX p‡q
(3:5:4)
( p ‡ q)X q ‡ p:
LEMMA 3.5.5. Under Assumption 1.3.1, the polynomial has a single double
root at X ˆ 1 and ( p ‡ q) 2 simple roots which are all nonzero and whose absolute value is > 1 for p 1 of them and < 1 for the remaining q 1.
p
c ) is suitable for M in
As a consequence, the blow-up " : Z ! P1u A1h of (0; 1
the sense of Definition 3.4.2.
4. The case of an elementary connection - cohomological description
Following Conclusion 3.3.13 together with Remark 3.3.14, we are now going to describe the family of subsheaves G < of G (e
p "e) 1 F and their push-forwards
e
c ).
b "e) G , where " : Z ! P1 A1 is the blow-up of (0; 1
R1 ( <
4.1 ± The sheaves G and G <
u
h
e
on the fibres of b
e
e Let us first describe the map e!P
e1 .
4.1.a - The real blow-up space Z.
b "e : Z
h
1 1
e
It will be enough for us to understand the description of (
b "e) (Shˆ0 ). This space
lies over D [ E.
e with polar
Over the coordinate chart (x; y) of Z (see Figure 1) is the chart of Z
coordinates (rx ; #x ; ry ; #y ). In this chart, the pull-back Ee of E is defined by ry ˆ 0
e
and the map b "e is given by (rx ; #x ; ry ; #y ) 7! (ry ; #y ). Therefore, in this chart,
1
1
e
e
(
b ") (Shˆ0 ) is the product of the annulus [0; 1) S1xˆ0 with the circle
e "e takes values in S1uˆ0 and is given by
S1yˆ0 ˆ S1hˆ0 . The restriction to Ee of (rx ; #x ; #y ) 7! # ˆ #x ‡ #y . Therefore, it will be more convenient for us to use the
description Ee ˆ [0; 1) S1uˆ0 S1hˆ0 , with the coordinate #b ˆ #y on S1hˆ0 ˆ S1yˆ0 .
e with polar coordinates
Over the coordinate chart (w; z) of Z is the chart of Z
e of E [ D is defined by
(rw ; #w ; rz ; #z ). In this chart, the pull-back Ee [ D
158
Marco Hien - Claude Sabbah
e
rw rz ˆ 0 and the map b "e is given by (rw ; #w ; rz ; #z ) 7! (rw rz ; #w ‡ #z ), while the
map e "e is given by (rw ; #w ; rz ; #z ) 7! (ru ; #) ˆ (rw ; #w ). We can therefore identify
e
e
e 1 S1 so that the restriction of the map D with P
e "e (resp. b "e) is identified
u
hˆ0
with the first (resp. the second) projection.
e [ Ee is obtained by gluing
As a consequence, D
(4:1:1)
e
D
b j ru 2 [0; 1]; # 2 S1 ; #b 2 S1 g
ˆ f(ru ; #; #)
uˆ0
hˆ0
with
b j rx 2 [0; 1]; # 2 S1 ; #b 2 S1 g
Ee ˆ f(rx ; #; #)
uˆ0
hˆ0
along ru ˆ 0 (resp. rx ˆ 1) by the identity on S1uˆ0 S1hˆ0 .
e . We denote by b : B ,! Ee [ D
e the inclusion of the
4.1.b - The sheaf G on Ee [ D
open subset consisting of the union of
e n fru ˆ 1g,
Ee n frx ˆ 0g, D
3
b
frx ˆ 0g,
(0; #; #) j q# 2 arg ('q )
;
2 2
3
b
b
fru ˆ 1g.
(1; #; #) j p# ( p ‡ q) # 2 arg ('q ) ‡ 2 ; 2
e
e e the projection Ee [ D
LEMMA 4.1.2. Let us still denote by 1
Gj [
ˆ b ! (e
e) F .
e
Figure 2. The annulus D
;
e 1 . Then
!P
u
e .
[ E
f1
e!P
e1 A
PROOF. By analyzing the map e : Z
h in the neighbourhood of
u
e
e
E [D .
p
e . Assume that
4.1.c - The sheaves G < on Ee [ D
b . According to Definition
ˆ'
3.3.8 (in the rapid decay setting) and recalling that Exp ˆ f'g, we obtain:
e
LEMMA 4.1.3. The restriction to Ee [ D
given by
(4:1:3 )
G<
j
[
ˆ (b v
of the sheaf G <
>0 )! G j
[
is the subsheaf of G
The local Laplace transform of an elementary irregular etc.
where b v
(4:1:3 )
>0
159
denotes the inclusion of the open subset
e j 0< " v (u; h) ,! Ee [ D
e :
Bv >0 :ˆ ez 2 Ee [ D
PROOF. Let us make precise the reasoning at the points ez whose projection on
E [ Z is an indeterminacy point for v . On D , the factor " v written in the coordinates (w; z) reads as
1
" v (w; z) ˆ q p‡q l (w; z)
w z
for some holomorphic function l such that l (w; 0) ˆ
q'q =p 6ˆ 0, a value which is
independent of 2 mp‡q . Hence there is no indeterminacy point of v on D . We
conclude from Lemma 3.5.5 that these points lie on E n (D0 [ D ) and that, in a
suitable local coordinate v on E vanishing at such a point, " v can be written as
b the
va =yq with a ˆ 1 or a ˆ 2. In any punctured neighbourhood of a point (v ˆ 0; #),
argument of " v takes all possible values, therefore such a point cannot be inside
Bv >0 (nor inside Bv <0 ). It thus lies on the boundary of Bv >0 and, according to a
computation similar to [Sab13b, Lem. A.1], Formula (4.1.3) also holds at such a
point.
p
In other words, we will forget the indeterminacy points in the remaining part of
this article.
4.1.d - The sheaf G <
(4:1:4)
Bv
e
on D
e
>0 \ D
ˆ
;
. We have
b j p#
(ru ; #; #)
(p ‡ q) #b 2
arg ('q ) ‡
3
;
:
2 2
This is the pull-back under the radial projection [0; 1] 3 ru 7! 1 of the set
B \ fru ˆ 1g considered before Lemma 4.1.2 and is independent of . As a cone , G < jfr ˆ1g is equal to G jfr ˆ1g and G < jfr <1g G jfr <1g .
sequence, on D
u
u
u
u
4.1.e - The sheaf G <
on Ee . From (3.5.2) and (3.5.3), the condition 0 < " v for
b 2 Ee lying over (x; 0) 2 E n f 1 (0) reads
the point ez ˆ (rx ; #; 0; #)
3
1
x)) 2 arg ('q ) ‡
q# arg ( f (
;
…mod 2p†:
2 2
Therefore
b j q# arg ( f ( 1 x)) 2 arg (' ) ‡ ; 3 mod 2p :
(4:1:5) Bv >0 \ Ee ˆ (rx ; #; #)
q
2 2
From (4.1.3 ) it clearly follows:
b the fibres (Bv
(4.1.6) For each fixed #,
2 mp‡q
\ Ee ) are increasing with respect to
b
b ) at #.
ordered according to the order of the family ('
>0
160
Marco Hien - Claude Sabbah
Let us observe that for jxj 1, we have q#x
the limit jxj ! 1, we have
(4:1:7) Bv
>0
\ Ee \ frx ˆ 1g
b j p#
ˆ (1; #; #)
arg ( f (
(p ‡ q) #b 2
1
x)) p#x , so that in
arg ('q ) ‡
3
;
;
2 2
e \ fru ˆ 0g (see
which is independent of 2 mp‡q and coincides with Bv >0 \ D
(4.1.4)).
Similarly, for jxj ! 0, we see that q#x ‡ arg ( f ( 1 x)) q#x and hence
Bv
(4:1:8)
e
>0 \ E \ frx ˆ 0g ˆ
b j q# 2 arg (' ) ‡
(0; #; #)
q
3
;
2 2
;
is independent on 2 mp‡q , and coincides with B \ frx ˆ 0g.
As a consequence, G < j is obviously contained in G j , and coincides with G
on frx ˆ 0g.
e
4.2 ± The Stokes-filtered sheaf (b
1
c; L
c )
L
From Theorems 3.3.10 and 3.3.11 in the rapid decay case and after the ramification e
b, we obtain:
c; L
c ) on S1 is given by:
b 1 L
COROLLARY 4.2.1. The Stokes-filtered sheaf ( e
hˆ0
e
1
1
c
e
b L ˆ R (
b "e) G ,
for each
c
2 mp‡q , L
<
e
ˆ R1 ( b "e) G < .
In particular, the latter is contained in the former.
p
We will simplify the topological situation in order to ease the explicit compue . We notice
tation of these objects. We continue to restrict the sheaves to Ee [ D
that Bv
>0
e
\D
e [
defined by (4.1.4) is independent of . We thus set B :ˆ (B \ E)
e ) (for any ). We have, for any , the inclusions Bv >0 B B. The
\D
e , while the second one is so on fru ˆ 1g. We
first inclusion is an equality on D
e and we set G :ˆ (b )! (e
e) 1 F . We
denote by b the inclusion B ,! Ee \ D
(Bv
>0
thus have a sequence of inclusions G <
,! G ,! G .
LEMMA 4.2.2. (1) The inclusion G ,! G induces an isomorphism
e
b "e ) G
R1 ( e
! R1 ( b "e ) G :
The local Laplace transform of an elementary irregular etc.
161
(2) The natural restriction morphisms
e
e
R1 ( b "e) G ! R1 (
b "e ) G
j
and
e
R1 ( b "e ) G <
e
! R1 ( b "e ) G<
are isomorphisms.
j
p
e ; (Bv >0 )
; B)
We can thus replace, in Corollary 4.2.1, the triple (Ee [ D
2mp‡q
e
e
e
; B \ E) and the pair ((G < )
; G ) with the pair
with (E ; (Bv >0 \ E )
((G <
j
)
2mp‡q
;G
j
).
2mp‡q
2mp‡q
4.3 ± Stokes data for b ‡ F (0;1) M - abstract version
The computation of the Stokes data can be done with the simplified model above.
Namely, we denote by A the annulus [0; 1] S1uˆ0 with coordinates (r; #), where r
now abbreviates rx . The open annulus A is defined as (0; 1) S1uˆ0 and the
boundary components are denoted by @ in A (r ˆ 0) and @ out A (r ˆ 1). We denote by
f the local system on A S1 ' Ee pull-back of F by e e and we consider the
F
hˆ0
subsets Bv >0 and B, which now denote the intersection of Bv >0 (resp. B ) with
f the restriction of F
f to the fibre A f #g
f
b (so F
A S1 . We will denote by F
hˆ0
is canonically equal to F ), and similarly for Bv >0 and B, as well as for bv
Let us choose an argument #bo 2 S1 such that the set
>0
and b.
hˆ0
f #b` :ˆ #bo ‡ `p=q j ` 2 Zg
b
does not contain a Stokes direction for any difference of two exponential factors '
f )
b for ; 0 2 m . Let I` :ˆ [ #b` ; #b`‡1 ]. We then define L` :ˆ H1 (A; (b )! F
and '
0
p‡q
for ` ˆ 0; . . . ; 2q 1 and S`‡1
: L` ! L`‡1 by the diagram of isomorphisms induced
`
by the restriction isomorphisms
…4:3:1†
d through
We arrange the elements of mp‡q , identified with Exp
b
the ordering (2.1.2) at #o :
mp‡q ˆ f
0
<
o
<
o
b , according to
7! '
p‡q 1 g:
This ordering repeats itself at ` for each even ` ˆ 2m and is completely opposite for
each odd ` ˆ 2m ‡ 1. From Corollary 4.2.1 in this simplified setting, we conclude
162
Marco Hien - Claude Sabbah
that the family
H1 A; (b v
(4:3:2)
k
)
>0 !
f
F
kˆ0;...;p‡q 1
f ) for ` even
forms a increasing (resp. decreasing) filtration of H1 (A; ( b )! F
(resp. ` odd) in f0; . . . ; 2q 1g. We denote by FL` such an increasing/decreasing
filtration. We can conclude:
THEOREM 4.3.3. The Stokes structure of the de-ramified local Fourier transform b ‡ F (0;1) M of the elementary meromorphic connection M ˆ El(r; '; R)
can be represented by the following linear Stokes data:
f ` -see (4.3.1)- for ` ˆ 0; . . . ; 2q 1,
the vector spaces L` ˆ H1 A; (b ` )! F
the isomorphisms S`‡1
from (4.3.1),
`
the filtrations FL` given by (4.3.2).
p
5. Topology of the support of the sheaves G <
on the annulus
In this section, we will describe the shape of the support Bv
in order to obtain Corollary 5.1.12 below.
>0
inside the annulus A
5.1 ± Morse theory on the closed annulus
In this section, it will be more convenient to use the argument arg x ˆ #x , related to # as follows:
b
# ˆ #x ‡ #:
(5:1:1)
We consider the function (cf. (3.5.4))
G
A n fx j f (x) ˆ 0g ! S1 ;
x 7 ! u ˆ arg ( f (x)=xq ) ˆ
with f (x) given by (3.5.4). For
G (x) :ˆ G(
1
x)
2 mp‡q and #b 2 S1hˆ0 , we set
q( #b ‡ arg ) ‡ arg ('q ) ˆ
We have
(5:1:2)
Bv
q#x ‡ arg f (x)
>0
ˆ (G )
1
3
;
2 2
q# ‡ arg ('q ) ‡ arg f (
1
x):
:
Let us denote by b : A0 ! A the real oriented blow-up of the roots of the
polynomial f (x) (where G is not defined) and let us set A0 ˆ An f 1 (0). We will use
The local Laplace transform of an elementary irregular etc.
163
classical Morse-theoretic arguments for the extension of G to the manifold with
boundary A0 (cf. [HL73, §3]).
LEMMA 5.1.3. The function G extends to a differentiable function on A0 . Its
restriction to @A0 has no critical point. The set of critical points in A0 , which are all
of Morse type of index one, is equal to mp‡q f1g embedded in fx j jxj ˆ 1g. To
each critical point 2 mp‡q f1g corresponds the critical value arg ( p 1) ˆ
p
kp
p 3p ‡ p‡q mod 2p for some k ˆ 1; . . . ; p ‡ q 1, which belongs to 2 ; 2 . The critical
2
p
kp
2qkp value 2 ‡ p‡q mod 2p comes from exactly one critical point, which is exp p‡q for
k 2 f1; . . . ; p ‡ q 1g such that k ak mod (p ‡ q), with a as in (1.4.2).
PROOF. The assertion on the critical points and their values is obtained by noticing
that G is obtained as the composition of the function x 7! f (x)=xq from A0 to C with
the projection to S1 .
Let us consider the local behaviour of G around a root xo 2 A of f (x). Let S1xo
denote the boundary circle over xo and assume first that xo is a simple root. The
function G then reads
G(x) ˆ arg (f (x)=xq ) ˆ arg ((x
xo ) g(x)=xq )
in a neighborhood S1xo A0 for some polynomial g(x) which does not vanish at xo .
This shows that G extends differentiably to S1xo and its restriction S1xo ! S1 has
maximal rank everywhere. Locally around the double zero x ˆ 1, the situation
amounts to
G(x) ˆ arg (x 1)2 g(x)=xq
with g(x) non-vanishing at x ˆ 1. The same arguments as above apply regarding the
extension of G to S11 . Lastly, the behaviour of G on @A has been given in §4.1. p
As a consequence, we can regard G as a Morse function on the pair (A0 ; @A0 ), and
also on the pair (A n f 1 (0); @A) (cf. [HL73, Th. 3.1.6]).
PROPOSITION 5.1.4. For each u 2 S1 , each connected component of the subset
p 3p
B :ˆ G 1 u ‡ ( 2 ; 2 ) A n f 1 (0) is homeomorphic to the union of an open disc
with holes (i.e., interior closed discs deleted) and nonempty disjoint open intervals
(at least one) in its boundary. It is embedded in A in such a way that the
nonempty open intervals in the boundary are contained in @A.
u
PROOF. We will start with the case u ˆ p.
p p LEMMA 5.1.5. The subset G 1 ( 2 ; 2 ) A n f 1 (0) is homeomorphic to a
disjoint union of (p ‡ q) semi-closed discs, each of them being a closed disc with a
closed interval in its boundary deleted, embedded in A in such a way that the
remaining nonempty open interval in the boundary is contained in @A.
164
Marco Hien - Claude Sabbah
Consider the annulus A ˆ [0; 1] S1 with coordinate (r; #x ). We will use the
notation
8
2mp
p 2mp
p
in
:ˆ fr ˆ 0g q
; q ‡ 2q ;
< evIm
2q
(5:1:6)
m ˆ 0; . . . ; q 1
(2m‡1)p
p (2m‡1)p
p
: odd in
Im :ˆ fr ˆ 0g ;
‡
;
q
2q
q
2q
8
< evInout :ˆ fr ˆ 1g (5:1:7)
: odd
Inout :ˆ fr ˆ 1g 2np
p
p 2np
;
2p p
(2n‡1)p
p
p
‡ 2p ;
p (2n‡1)p
; p
2p
p
‡ 2p ;
n ˆ 0; . . . ; p
1:
Figure 4. The topological space G 1 (0).
Each of the red circles stands for a
(deleted) root of f(x). The number of
curves starting at S1rˆ0 is q, the number
of curves ending in S1rˆ1 is p.
in odd in
Figure 3. The intervals ev Im
,
Im ,
ev out odd out
In , In in the chosen numbering.
We then have
@ in G
1
(
p p ; )
2 2
ˆ
q[1
mˆ0
ev in
Im ;
@ out G
1
(
p p ; )
2 2
ˆ
p[1
nˆ0
ev out
In :
p p PROOF OF LEMMA 5.1.5. By Morse theory (cf. loc. cit.), G 1 ( 2 ; 2 ) is diffeop p
morphic to the product G 1 (0) ( 2 ; 2 ). It is therefore enough to check that G 1 (0)
is the disjoint union of ( p ‡ q) semi-closed intervals with closed end points on @A.
Since the closure G 1 (0) cuts the boundary @A0 transversally, we conclude that
in the neighbourhood of
(m ˆ 0; . . . ; q 1),
ev in
Im ,
G 1 (0) is a smooth curve abutting to (0;
2mp
)
q
in the neighbourhood of evInout , G 1 (0) is a smooth curve abutting to (1;
(n ˆ 0; . . . ; p 1),
2np
)
p
in the neighbourhood of a simple root of f , G 1 (0) is a smooth curve with one
end in this puncture,
in the neighbourhood of the double root of f , G 1 (0) is a smooth curve with
two ends in this puncture.
The local Laplace transform of an elementary irregular etc.
The lemma is then a consequence of the properties claimed below.
165
p
CLAIMS (See Figure 4). (1) The connected component of G 1 (0) abutting to
2mp
(0; q ) (m ˆ 1; . . . ; q 1) is a semi-closed interval with its open end at some
simple root of f with absolute value < 1.
2np
(2) The connected component of G 1 (0) abutting to (1; p ) (n ˆ 1; . . . ; p 1) is a
semi-closed interval with its open end at some simple root of f with absolute
value > 1.
(3) The connected components of G 1 (0) abutting to (0; 0) and (1; 0) are semiclosed intervals with open end at x ˆ 1.
PROOF OF THE CLAIMS. Writing x ˆ rei#x with r 2 (0; 1), one has
(5:1:8)
G(x) 2 f0; pg ()
f (x)
2 R n f0g () qrp‡q sin (p#x )
xq
p sin (q#x ) ˆ 0:
We consider a fixed argument #x . If sin (q#x ) ˆ 0 ˆ sin ( p#x ) - which is the case for
#x ˆ 0 or #x ˆ p only, since we assume p; q to be co-prime - then all r > 0 are
solutions to the last equation. Since p or q is odd, f (x)=xq ˆ qxp (p ‡ q) ‡ px q is
negative for x < 0, so (0; 1) f#x ˆ pg is contained in G 1 (p). On the other hand,
f (x)=xq is > 0 on [(0; 1) [ (1; 1)] f#x ˆ 0g, hence the third claim.
In case sin ( p#x ) 6ˆ 0, the equation reads
(5:1:9)
r p‡q ˆ
p sin (q#x )
;
q sin (p#x )
and has exactly one positive solution r > 0 whenever sin ( p#x ) and sin (q#x ) have the
same sign, and no solution otherwise.
Now, for fixed r, we see that for small r 1, Equation (5.1.8) has exactly q
solutions #x contributing to G 1 (0), given in the limit r & 0 by the
2mp
arguments f q j m ˆ 0; . . . q 1g. For large r 1, it has exactly p solutions #x
2np
contributing to G 1 (0), given in the limit r % 1 by f p j n ˆ 0; . . . ; p 1g.
Since 0 and p are the only common zeroes of sin (q#x ) or sin (p#x ), i.e., at each
other zero of one of these functions, the corresponding function changes its sign
whereas the other one doesn't. This means that locally at these arguments there
is an open sector (with the argument as a boundary of it) where equation (5.1.8)
does not admit any solution r > 0. As a consequence, between each of the
arguments
n 2mp
o n 2np
o
j m ˆ 0; . . . ; q 1 [
j n ˆ 0; . . . ; p 1
q
p
there is an open sector such that equation (5.1.8) has no solution r > 0 for any argument #x inside this sector.
166
Marco Hien - Claude Sabbah
Lastly, for jxj ˆ 1, we get G(x) ˆ 0 if and only if p sin (q#x ) ˆ q sin ( p#x ) and
q cos (p#x ) ‡ p cos (q#x ) (p ‡ q) > 0, which cannot occur. Therefore
(5:1:10)
fx ˆ ei#x j #x 6ˆ 0g \ G 1 (0) ˆ [;
keeping in mind that x ˆ 1 is a root of f (x).
Therefore, a connected component of G 1 (0) meeting fr ˆ 0g (resp. fr ˆ 1g)
at a nonzero argument cannot end at another similar point, either because it cannot
cross a forbidden sector, so cannot end at fr ˆ 0g (resp. fr ˆ 1g), or because it
cannot cross the circle r ˆ 1, so cannot end at fr ˆ 1g (resp. fr ˆ 0g). Such a
component must then have an end at a simple root of f . This gives the first two
claims, according to the last part of Lemma 3.5.5.
p
END OF THE PROOF OF PROPOSITION 5.1.4. The proof will be done
by induction on the
3 number of critical points contained in Bu :ˆ G 1 u ‡ 2 ; 2 , and Lemma 5.1.5
provides the initial step of the induction. We now consider the subset Bu for u
varying from p to 2p. The case from p to 0 is very similar. When u varies, no critical
p
3p
point can go out from G 1 (u ‡ 2 ), but a critical point may enter in G 1 (u ‡ 2 ). This
occurs successively for
kp
; k ˆ 1; . . . ; p ‡ q 1;
u ˆ uk ˆ p ‡
p‡q
with corresponding critical point given by Lemma 5.1.3. Attaching a 1-cell at such
a u consists in connecting a connected component of Bu " intersecting fr < 1g with
one intersecting fr > 1g.
An example with p ˆ 4 and q ˆ 5.
Figure 5. The subset Bp for the starting
value p of u. The green circles are the
zeroes of f(x). The small yellow circles
are the roots of unity mp‡q , i.e., the critical
points of the Morse function.
Figure 6. The subset Bu when the first
critical point enters the picture
p
(u1 ˆ p ‡ p‡q). We see the gluing of the
corresponding discs in the upper half of
the picture.
The behaviour of Bu when u varies between uk " and uk ‡ " (" > 0 small) is
pictured in Figure 7, where the dotted lines are the part of the boundary which do not
belong to Bu . Indeed, for u < uk , the boundary of Bu is contained in @A by induction,
The local Laplace transform of an elementary irregular etc.
167
hence far from the critical point and, by Morse theoretic arguments, the properties
of Bu remain constant away from a neighbourhood of the corresponding critical
point. The assertion of the proposition is thus also satisfied for u 2 [uk ; uk ‡ "). p
2kpi
Figure 7. Local view at the critical point ˆ exp( p‡q ) of the set Bu at u ˆ uk
u ˆ uk ‡ ", with dotted lines deleted.
" and
COROLLARY 5.1.11. Let K be a local system on A. Then, for each u 2 S1 ,
Hj (A; b u! K ) ˆ 0 for j 6ˆ 1 and
M
H1 (A; b u! K ) '
K :
2Bu \mp‡q
PROOF. For the first assertion, we note that the topological boundary of each
connected component of Bu intersects the component but is not contained in it,
according to Proposition 5.1.4. Therefore, b u! K has no global section, nor has its
dual sheaf, hence the conclusion by using PoincareÂ-Verdier duality.
The second assertion is proved by induction as in Proposition 5.1.4 on the
number of critical points contained in Bu . The case u ˆ p follows from Lemma 5.1.5
and [Sab13a, Lem. 7.13]. When u is in the neighbourhood of a critical value uk , we
decompose Bu in two closed subsets Bu1 [ Bu2 , so that Bu2k " resp. Bu2k ‡" are as in
Figure 7 and Bu1k " is diffeomorphic to Bu1k ‡" by a Morse-theoretic argument. We
then have an exact sequence, according to the first part and with obvious notation,
0 ! H1 (A; bu! K ) ! H1 (A; bu1;! K ) H1 (A; b u2;! K ) ! H1 (A; b u12;! K ) ! 0
and H1 (A; bu1;! K ) remains constant, as well as H1 (A; bu12;! K ), when u varies
around uk . The only changes come from H1 (A; b u2;! K ), which is zero for
u 2 (uk "; uk ] and has dimension rk K for u 2 (uk ; uk ‡ ").
p
Going back to the sets Bv
>0 ,
we conclude:
COROLLARY 5.1.12. Let K be a local system on the annulus A. Then, for each
#b 2 S1hˆ0 and 2 mp‡q , we have
M
K 0;
H1 A; (b v >0 )! K '
0 2m
p‡q n f1g
0 2B
v >0
where we regard mp‡q as embedded in A ˆ [0; 1] S1 as f(1; #) j #p‡q ˆ 1g.
p
168
Marco Hien - Claude Sabbah
5.2 ± The family (Bv` >0 ) for
2 mp‡q and ` ˆ 0; . . . ; 2q
1
We now make precise the choice of an argument #bo 2 S1hˆ0 , and so the set
`p
f #b` :ˆ #bo ‡ q j ` ˆ 0; . . . ; 2q 1g. Due to the previous results, the choice of a base
point #b 0 2 S1 such that q #b 0 arg (' ) ˆ p seems appropriate. However, since #b 0
o
o
hˆ0
o
q
is a Stokes direction for some pair in (b
' )
2mp‡q
, we modify it and set #bo :ˆ #bo0 ‡ "=q
for " > 0 small enough in such a way that #bo is not a Stokes direction. We thus have
q #bo arg ('q ) ˆ p ‡ ".
b ) at #b` is defined by
We now fix ` 2 f0; . . . ; 2q 1g. The order on the family ('
b
b 0 ) < 0 in some neighbourhood of #b` , which amounts to
b 4 '
b 0 () Re('
'
'
`
0p
Re e "i ( p
) < 0 for ` odd, and the reverse order for ` even, according to
Formula (3.5.1) and to the relation q #b` arg ('q ) ˆ (` ‡ 1)p ‡ ". Therefore it
corresponds to the even (resp. odd) order of Definition 1.4.1 on mp‡q when ` is even
(resp. odd). Recall that, with respect to this order, the family indexed by :
n
3 o
;
Bv` >0 ˆ x 2 A j G( 1 x) 2 q arg ‡ q #b` arg ('q ) ‡
2 2
is increasing. One can also check directly with the formula given in Lemma 5.1.3
that the order on mp‡q at #b` coincides with the order by the number of critical points
contained in Bv` >0 . We thus have
8
p p
ev
1
>
B
:ˆ
G
‡
"
‡
;
if ` is even,
q
arg
>
v
>0
<
2 2
(5:2:1)
Bv` >0 ˆ
>
> oddBv >0 :ˆ G 1 q arg ‡ " ‡ p ; 3p
:
if ` is odd.
2 2
We will set
(5:2:2)
ev in
Im (")
odd in
Im (")
Then we have
(5:2:3) @ in Bv` >0 ˆ
in
:ˆ evIm
"=q;
in
:ˆ oddIm
"=q;
8S
1 ev in
< qmˆ0
Im (")
: Sq
1 odd in
Im (")
mˆ0
ev out
In (")
odd out
In (")
(` even)
(` odd)
:ˆ evInout ‡ "=p;
:ˆ oddInout ‡ "=p:
Sp
1 ev out
nˆ0 In (")
Sp
9
=
1 odd out
In (") ;
nˆ0
ˆ @ out Bv` >0 :
In the following, " > 0 will be chosen strictly smaller of any constant quantity
like p=( p ‡ q), etc.
6. Topological model for the standard linear Stokes data
In this section we introduce a ``linear model'' and we develop a computation of the
Stokes data in this model. That this model suffices for the computation of the desired Stokes data will be shown in Section 7.
The local Laplace transform of an elementary irregular etc.
169
6.1 ± Standard filtration on a local system on the annulus
We are given two coprime integers p; q 2 N. Recall that we defined two opposite orderings on the set of ( p ‡ q)th roots of unity in Definition 1.4.1. We define
the following two subsets related to those constructed in §4.1 (notation as in (5.1.6)
and (5.1.7)):
pG1
qG1
ev
ev in
ev out
ev
B :ˆ A [
Im t
In
, ! A;
mˆ0
(6:1:1)
odd
B :ˆ A [
qG1
mˆ0
nˆ0
odd in
Im
t
pG1
nˆ0
odd out
In
odd
, ! A:
Both spaces are obtained from the closed annulus A by deleting p closed intervals in
the outer boundary and q closed intervals in the inner boundary. On the other hand,
we embed mp‡q into the circle with r ˆ 1.
6.1.a - The even case. The combinatorics which will play a role in the following is
governed by the following map (cf. (1.4.3) for the notation):
(6:1:2)
f0; . . . ; p ‡ q
1g ! f0; . . . ; qg f0; . . . ; pg
k 7! ev in(k); evout(k) :
We will also consider the composition evc with the projection to Z=qZ Z=pZ, and
we will denote by m (resp. n) the representative of ev in(k) mod q in f0; . . . ; q 1g
(resp. of evout(k) mod p in f0; . . . ; p 1g), so we also regard evc : k 7! (m; n) as a map
f0; . . . ; p ‡ q 1g ! f0; . . . ; q 1g f0; . . . ; p 1g.
REMARKS 6.1.3. (1) The map (6.1.2) clearly is injective. Moreover, if p and q are
6ˆ 1, evc is also injective. Indeed, let k1 ; k2 be such that (m1 ; n1 ) ˆ (m2 ; n2 ).
If m1 ˆ m2 ˆ
6 0 and n1 ˆ n2 ˆ
6 0, then ev in(k1 ) ˆ ev in(k2 ), evout(k1 ) ˆ evout(k2 ),
ev
ev
and out(k1 ) ˆ k1
in(k1 ), evout(k2 ) ˆ k2 ev in(k2 ), so k1 ˆ k2 .
If m1 ˆ m2 ˆ 0 and n1 ˆ n2 ˆ
6 0, then evout(k1 ) ˆ evout(k2 ) ˆ: n 2
ev
f1; . . . ; p 1g and in(k1 ) ˆ k1 n, ev in(k2 ) ˆ k2 n. It is thus enough to
consider the case ev in(k1 ) ˆ 0 and ev in(k2 ) ˆ q, which is then equivalent to
n
1
n‡q
1
k1 ˆ n, k2 ˆ n ‡ q, p‡q < 2q and p‡q 51 2q . This implies p ‡ q > pq,
which can occur only if q ˆ 1 (since p ˆ
6 1 with our starting assumption).
The case m1 ˆ m2 6ˆ 0 and n1 ˆ n2 ˆ 0 is treated similarly.
Assume m1 ˆ m2 ˆ 0 and n1 ˆ n2 ˆ 0.
- If ev in(k1 ) ˆ ev in(k2 ) ˆ 0, the only case to consider is evout(k1 ) ˆ 0
and evout(k2 ) ˆ p, which implies p ‡ q > 2pq and cannot occur.
- If ev in(k1 ) ˆ ev in(k2 ) ˆ q, we have k1 ˆ evout(k1 ) ‡ q and
k2 ˆ evout(k2 ) ‡ q with ni ˆ 0 or p, but ni ˆ p is not possible since
ki < p ‡ q, so evout(k1 ) ˆ evout(k2 ) ˆ 0, and therefore k1 ˆ k2 .
170
Marco Hien - Claude Sabbah
- If ev in(k1 ) ˆ 0 and ev in(k2 ) ˆ q, then k1 ˆ evout(k1 ), k2 ˆ evout(k2 ) ‡ q,
so the only problematic cases are k1 ˆ 0; p, evout(k2 ) ˆ 0 and k2 ˆ q.
But evout(k2 ) ˆ 0 and k2 ˆ q imply qp=(p‡q) < 1=2, which cannot occur.
If q ˆ 1 and p is even (resp. odd) the pairs k1 ˆ p=2; k2 ˆ (p ‡ 2)=2 (resp.
k1 ˆ (p 1)=2; k2 ˆ ( p‡1)=2) give rise to the same pair (m; n) ˆ (0; p=2)
(resp. (m; n) ˆ (0; (p 1)=2)). The case p ˆ 1 is similar.
(2) The map evc arises in the following way: m is the unique element in
f0; . . . ; q 1g such that the rotation of evI0in by j k has non-empty intersection
in
with evIm
(") as defined by (5.2.2):
2kp
in
‡ evI0in \ evIm
6ˆ [ 8" > 0 small enough:
p‡q
Equivalently, this condition decomposes as
(
6ˆ [; or
2kp
ev in
ev in
‡ I0 \ Im
2kp
p
2mp
p‡q
ˆ [ and p‡q ‡ 2q ˆ q
p
2q
mod 2p;
and the latter case can occur only if p ‡ q is even. Similarly, n is the unique
element of f0; . . . ; p 1g such that
2kp
‡ evI0out \ evInout 6ˆ [ 8" > 0 small enough:
p‡q
(3) It will be suggestive to write k 7! (m; n) as
ev in
Im
? k ? evInout :
We will construct a family of curves evgk in Proposition 6.1.5 below such that
ev
in
gk connects evIm
and evInout passing through j k . If q52 we can characterize
ev
in
the index minin (k) in (1.4.4) as the index k0 such that evgk0 starts from evIm
but
ev
0
ev
not gk0 1 (where k is understood mod ( p ‡ q)) - and similarly for maxout (k).
(4) In the remaining part of this section, we will assume that p > q. Otherwise,
the same arguments hold by interchanging the roles of the inner and outer
circle of the annulus A ˆ [0; 1] S1 with coordinates (r; #) in the following
constructions.
The following lemma is straightforward and is illustrated by Figure 8.
in
LEMMA 6.1.4. Assuming that evIm
? k ? evIout
n one of the following holds:
i)
ii)
iii)
ev in
Im 1 ? (k 1) ? evIout
n
ev in
(k 1)
ev out
Im ? ? In 1
ev in
(k 1)
Im ? ? evIout
n 1
and
ev in
Im
and
ev in
Im ? (k‡1) ? evIout
n‡1
ev in
(k‡1)
ev out
Im‡1 ? ? In :
and
? (k‡1) ? evIout n‡1
The local Laplace transform of an elementary irregular etc.
Moreover,
i) holds
()
k
p‡q
2
ii) holds
()
k
p‡q
2
iii) holds
()
k
p‡q
2
m
q
1 m
;
2q q
q
1
2q
q
‡ 2q
m
m
1
2q
1
1 ‡ p‡q ;
m
1
‡ p‡q ; q ‡ 2q
1
1
m
;
p‡q q
In particular, #(evcout ) 1 (n)42 for each n 2 f0; . . . ; p
Figure 8. The three possible situations for evc(k
In the following we use the convention that (p ‡ q
1
‡ 2q :
171
1 ;
p‡q
1g.
p
1); evc(k); evc(k ‡ 1).
1) ‡ 1 ˆ 0.
PROPOSITION 6.1.5. There exists a closed covering (evSk )kˆ0;...;p‡q
satisfying the following properties:
1
of A
(1) Each evSk is the homeomorphic image of the unit square with edges denoted
successively by a; b; c; d.
(2) The image of a (resp. c) is a closed interval in the inner (resp. outer)
boundary of A.
(3) For each k, there is exactly one of the intervals oddIin and oddIout which
in
intersects @(evSk ), either oddIm
or oddInout with (m; n) :ˆ evc(k), and it is entirely
ev
contained in @( Sk ).
(4) The image of b (resp. d) is a smooth curve evgk (resp. evgk‡1 ) which satisfies
(a)
ev
gk \ @A @(evgk ),
(b)
ev
gk \ mp‡q ˆ fj k g,
in
gk connects evIm
with evIout
n ,
8
[
>
<
(5) We have evSk \ evSk0 ˆ evgk
>
: ev
gk‡1
(c)
ev
where (m; n) :ˆ evc(k).
if k0 6ˆ k ‡ 1; k
if k0 ˆ k
0
1;
if k ˆ k ‡ 1:
1;
172
Marco Hien - Claude Sabbah
PROOF. We will represent A as obtained from [0; 1] [0; 2p] by identifying
[0; 1] f0g with [0; 1] f2pg through the identity map. We start by considering
2kp
the closed covering (evS0k ) cut out by the line segments evg0k ˆ [0; 1] fp‡qg. However, Properties (3) and (4c) are not satisfied in general by some line segments evg0k
for k ˆ
6 0. We then move the ends of the line segment evg0k , keeping j k fixed, so that the
new curve evgk fulfills (4c). Then, if q > 1, (3) is fulfilled, according to Lemma 6.1.4.
Figure 9. The closed covering (evS0k ) (example with p ˆ 4, q ˆ 3).
Figure 10. The closed covering (evSk ) (example with p ˆ 4, q ˆ 3).
The case q ˆ 1, which needs a special treatment since evc is not injective in this
case, is checked on Figure 11. In the case q ˆ 1 and p 1 mod 2, we use the
convention that the curve evgp‡1 is chosen as in Figure 11.
p
2
Figure 11. The closed coverings (evS0k ), (evSk ) (p ˆ 5, q ˆ 1).
COROLLARY 6.1.6. Let K be a local system on A. Then the covering
( Sk )kˆ0;...;p‡q 1 is a Leray covering for ev! K (ev as in (6.1.1)) and the identification H0 (evgk ; K ) ' Kjk induces an isomorphism
M
ev
…6:1:6†
: H1 (A; ev ! K ) !
K :
ev
2mp‡q
The local Laplace transform of an elementary irregular etc.
173
PROOF. The Leray property follows from Proposition 6.1.5(3) together with
[Sab13a, Lem. 7.13].
p
Let us fix the even ordering mp‡q ˆ f
Definition 1.4.1).
L
0
<ev
1
<ev <ev
p‡q 1 g
on mp‡q (cf.
DEFINITION 6.1.7. The even standard Stokes (increasing) filtration on
L
L
K is defined by the even ordering Fk (
K ) :ˆ j4k K .
2mp‡q
2mp‡q
j
6.1.b - The odd case. Similarly to (6.1.2), consider the map (cf. (1.4.6) for the notation):
(6:1:8)
f0; . . . ; p ‡ q
1g ! f0; . . . ; q
k 7!
odd
1g f 1; . . . ; p
in(k); oddout(k)
1g
and we also consider the composition odd c with the projection to Z=qZ Z=pZ, and
we still denote by m (resp. n) the representative of odd in(k) mod q in f0; . . . ; q 1g
(resp. of oddout(k) mod p in f0; . . . ; p 1g), so we regard odd c : k 7! (m; n) as a map
f0; . . . ; p ‡ q 1g ! f0; . . . ; q 1g f0; . . . ; p 1g. We thus have m ˆ odd in(k).
REMARK 6.1.9. As in Remark 6.1.3, (m; n) is characterized by the properties
8
2kp
p 2kp
p odd in
>
>
;
‡
\ Im (") 6ˆ [;
>
< p ‡ q 2q p ‡ q 2q
8" > 0 small enough:
2kp
>
>
p 2kp
p odd out
>
:
;
‡
\ In (") 6ˆ [;
p ‡ q 2q p ‡ q 2q
We thus have
(6:1:10)
odd in
Im
? k ? oddIout
k m
1
if
h m m 1
k
2
; ‡ mod 1:
p‡q
q q q
Let us denote by odd min ˆ ev max 2 mp‡q the minimal element with respect to
the odd ordering, which is the maximal one with respect to the even ordering.
There are two cases to be distinguished.
If p ‡ q is even, we have
( p ‡ q)=2.
odd
min
ˆ j a( p‡q)=2 ˆ
If p ‡ q is odd, we have odd min ˆ j a( p‡q
respondingly odd kmin :ˆ a(p ‡ q 1)=2.
1)=2
1 and we set
odd
kmin :ˆ
(cf. (1.4.2) for a). We set cor-
REMARK 6.1.11. Although we could proceed as in the even case to define the
Leray covering (odd Sk )kˆ0;...;p‡q 1 , it will be convenient for a future use to define it
by rotation from the even covering already constructed.
174
Marco Hien - Claude Sabbah
For k 2 f0; . . . ; p ‡ q
1g, let us set
1g 3 k0 :ˆ k ‡ odd kmin mod (p ‡ q):
f0; . . . ; p ‡ q
We now define
odd
(6:1:12)
Sk0 :ˆ odd
to be the rotation of evSk by odd
min
evSk
min .
PROPOSITION 6.1.13. If we assume that the paths evgk satisfy the following
supplementary properties when p ‡ q is odd:
8
p
ev
in
in
>
;
g \ evIm
2 evIm
>
< k
q(p ‡ q)
(6:1:13)
>
p
>
: evgk \ evInout 2 evInout ‡
;
q(p ‡ q)
then the odd analogue of Proposition 6.1.5 holds for (odd Sk )kˆ0;...;p‡q 1 (where we
replace evIin and evIout in 6.1.5(3) with the corresponding rotated intervals).
in
We note that since the length of evIm
(resp. evInout ) is p=q, (6.1.13) can be realized
ev
by moving the end points of the paths gk .
PROOF. We only need to check (4c). Assume first that p ‡ q is even (so that p and q
are odd, arg odd min ˆ p and odd kmin ˆ (p ‡ q)=2). We will show
odd
(6:1:14)
odd
min
min
This amounts to showing
2 h q(k0 (p ‡ q)=2)
q
p‡q
in
in
evIev
ˆ oddIodd
;
in(k)
in(k0 )
out
out
evIev
ˆ oddIodd
out(k)
‡
2 l p(k0 ( p ‡ q)=2)
p
p‡q
out(k0 )
:
1i
2 h qk0 i 1
‡
‡1
mod 2;
2
q p‡q
q
m 1
1m
2 l pk0
1 ‡
mod 2;
‡1
2
p p‡q
p
which is easily checked.
Assume now that p ‡ q is odd. Let us write aq ˆ c( p ‡ q)
(then c aq is odd), so that
arg odd
min
ˆ
ap ‡
ap
(c aq)p
ˆ
p‡q
p‡q
1 for some c 2 Z
p
q(p ‡ q)
and
odd
min
in
in
evIm
ˆ oddIm
0
p
;
q( p ‡ q)
with f0; . . . ; q
1g 3 m0 :ˆ m ‡ c
aq 1
2
mod q;
The local Laplace transform of an elementary irregular etc.
and therefore, since oddImin0 has length p=q, (odd
odd
min
evInout ˆoddInout
0 ‡
min
p
; with f0; . . . ; p
p( p ‡ q)
175
in
evIm
) \ oddImin0 6ˆ [. Similarly,
1g 3 n0 :ˆ n ‡
a c ap 1
2
mod p:
Let us check that m0 ˆ odd in(k) and n0 ˆ oddout(k), therefore proving the odd analogue of 6.1.5(4c). We have
qk0
qk
aq( p ‡ q 1)=2
aq( p ‡ q 1)=2 h 1 1
ˆ
2m
‡
; mod q
p‡q p‡q
p‡q
p‡q
2 2
h 1 1
1
1
ˆ m0 ‡
‡
; mod q
2 2( p ‡ q)
2 2
h
1
1
ˆ m0
; m0 ‡ 1
mod q;
2(p ‡ q)
2(p ‡ q)
which implies qk0 =( p ‡ q) 2 [m0 ; m0 ‡ 1), hence odd in(k0 ) ˆ m0 . Similarly,
pk0
p‡q
1ˆ
pk
p‡q
1
ap(p ‡ q 1)=2
p‡q
ap( p ‡ q 1)=2 1 1 i
‡
; mod p
p‡q
2 2
1 1i
1
1
‡
‡
; mod p
ˆ n0
2 2( p ‡ q)
2 2
i
1
1
ˆ n0 1 ‡
; n0 ‡
mod p;
2(p ‡ q)
2( p ‡ q)
2n
1
hence oddout(k0 ) ˆ n0 .
p
The obvious odd analogues of Lemma 6.1.4 and Corollary 6.1.6 hold, and we
have an isomorphism
M
odd
(6:1:15)
: H1 (A; odd ! K ) !
K :
2mp‡q
DEFINITION 6.1.16. The odd standard Stokes (decreasing) filtration on
K is defined by the odd ordering, i.e., if we number the elements of mp‡q
2mp‡q
L
by the even ordering on mp‡q ˆ fz 0 <ev z 1 <ev g we have Fk (
K ) :ˆ
2mp‡q
L
K
.
j5k
L
j
6.2 ± The topological model
We will now mimic the description of §4.3, from which we keep the notation, in
b on
the present simplified setting. However we choose the coordinates (r; #x ; #)
1
the product A Shˆ0 . This will make formulas simpler. We thus introduce the
176
Marco Hien - Claude Sabbah
isomorphism
(6:2:1)
[0; 1] S1xˆ0 S1hˆ0
[0; 1] S1uˆ0 S1hˆ0 !
b 7 ! (r; #x ; #)
b ˆ (r; #
(r; #; #)
b #):
b
#;
The subset b : B ,! A S1 contains A and has its inner (resp. outer) boundary
described by (4.1.8) (resp. (4.1.7)). If we fix #b` (` ˆ 0; . . . ; 2q 1) as in §5.2 but with
" ˆ 0 for simplicity, we obtain
8S
Sp 1 ev out 9
in
=
< q 1 evIm
if ` is even
mˆ0
nˆ0 In
out
`
@ in B ` ˆ S
Sp 1 odd out ; ˆ @ B :
: q 1 oddIin
if ` is odd
In
m
mˆ0
nˆ0
With this notation, and given a local system F on [0; 1] S1uˆ0 , we set
and K ` :ˆ jAf g . Then (4.3.1) reads
f
:ˆ r F
`
…6:2:2†
where ` (resp. b ` ) is ev or odd (resp. ev or odd) according to the parity of `.
Let (V; T) be the monodromy pair of the local system F on the annulus.
Then
can be described as the triple (V; T; Id), where T is the monodromy with
respect to S1xˆ0 and Id that with respect to S1hˆ0 .
CONVENTION 6.2.3. We fix an identification G (S1xˆ0 n f#x ˆ "g) S1hˆ0 ;
with V, and T is recovered as usual by going in the positive direction from
L
#x ˆ 2" to #x ˆ 0. This fixes, for each `, an identification
K`ˆ
2mp‡q
L
p‡q 1
V 1k , with the notation as in (1.4.7).
kˆ0
DEFINITION 6.2.4. The data (
L
2mp‡q
K ` )` ; (s`‡1
` )` ; F) - the last entry being
defined by 6.1.7 and 6.1.16 - will be called the topological model for the standard
Stokes structure associated to (V; T).
For each ` ˆ 0; . . . ; 2q 1, let us fix a diffeomorphism A I` ! A I` by
lifting the vector field @ to A S1hˆ0 in such a way that the lift is equal to @
away from a small neighbourhood of @A and such that the diffeomorphism in
duces BI` ! B `‡1 I` . It also induces a diffeomorphism A f #b` g ! A f #b`‡1 g
The local Laplace transform of an elementary irregular etc.
177
and the push-forward of K ` is isomorphic to K `‡1 . Moreover, (4.1.8) and (4.1.7)
show that
in
for ` even, `Im
is sent to
in
for ` odd, `Im
is sent to
`‡1 in
`‡1 out
Im 1 and `Iout
In ,
n to
`‡1 in
` out
`‡1 out
Im and I n to I n‡1 .
e k ) and the curves (evg )
The Leray covering (evSk )k is sent to a Leray covering (evS
k
k k
ev
ev e
are sent to curves ( ek )k . The Leray covering ( Sk )k induces the isomorphism
M
e ` : H1 (A; (b `‡1 )! K `‡1 ) !
K `‡1 ;
2mp‡q
which is transported by the previous isomorphism from ` , and the lower part of
(6.2.2) reads
…6:2:5†
for ` even. Due to Proposition 6.1.5, we know
6.2.a - Explicit description of s`‡1
`
that (regarding m resp. k m modulo q resp. p as indicated above)
(6:2:6)
ev
gk :
8
in
>
? k ? evIout
< evIm
k m
>
: evIin
? k ? evIout
k m
8
in
>
< oddIm
k
odd out
Ik m
1 ? ?
m‡1
and consequently
(6:2:7)
ev
ek :
1
>
: oddIin ? k ? oddIout
m
From h(6.1.10) we deduce
that
k
m
1 m
1
2
‡
;
‡
mod
1.
p‡q
q
2q q
q
k m 1
odd
k and
h
1
‡ 2q mod 1;
h
k
m
1 m
1
if p‡q 2 q ‡ 2q ; q ‡ q mod 1;
if
k
p‡q
2
h
m m
;
q q
1
‡ 2q mod 1;
h
k
m
1 m
1
if p‡q 2 q ‡ 2q ; q ‡ q mod 1:
if
k
p‡q
2
m m
;
q q
ev
ek show the same behaviour whenever
in
Fixing an m 2 f0; . . . ; q 1g and considering the interval oddIm
, we see that the
m( p‡q)
(m‡1)( p‡q)
odd
curves k with kmin (m) :ˆ d q e4k4d
1e
ˆ:
k
max (m) are exactly
q
(2m‡1)( p‡q)
odd in
e. We have
the ones starting at Im . Additionally, we define kmid (m) :ˆ d
2q
kmid (m) kmax (m 1)52 and kmax (m) kmid (m)51. Moreover, for fixed m, the
curves odd k and evek
connect the same intervals if kmid (m)4k4kmax (m) and
connect different intervals if kmin (m)4k < kmid (m).
178
Marco Hien - Claude Sabbah
in
in
Figure 12. The curves odd k starting at oddIm
and oddIm
1 for fixed m. Exactly for
ev
kmid (m)4k4kmax (m), the curves ek have the same behaviour as odd k (we denote k
instead of j k ).
Let us consider the situation between kmax (m
m 2 f0; . . . ; q 1g, as sketched on Figure 13.
1) and kmid (m) for a given
Figure 13. The sector S and the curves odd k and evek in the range [kmax (m
We set
s :ˆ kmid (m)
kmax (m
1); kmid (m)].
1)52:
This picture allows us to define the matrix ev S(m) 2 Mat(s‡1)(s‡1) (End(V)). Writing
1 ˆ IdV , we set
0
(6:2:8)
1
B0
B
B
B0
B
B
ev
S(m) :ˆ B 0
B.
B.
B.
B
@0
0
1
1
1
0
..
.
0
0
1
1
0
1
..
.
0
0
1
1
0
0
..
.
...
0
...
...
...
...
..
.
1
...
1
1
0
0
..
.
0
1
1
0
0C
C
C
0C
C
0C
C:
.. C
C
.C
C
0A
1
The local Laplace transform of an elementary irregular etc.
179
In case s takes its minimal value s ˆ 2, the block with diagonal zeros disappears and
we obtain
0
1
1 1 0
ev
S(m) ˆ @ 0
1 0 A:
0 1 1
REMARK 6.2.9. The heuristic rule is that, on Figure 13, the red segment going
through kmax (m 1) 1 ‡ j ( j ˆ 1; . . . ; s ‡ 1) oriented from in to out ( jth column
of ev S(m)) is written as a sum of black segments oriented from in to out with the
same origin and end horizontal segments. The vertical black segments are also
regarded to be red.
L
L
1
1
LEMMA 6.2.10. The isomorphism s`‡1
: p‡q
V 1k ! p‡q
V 1k does
`
kˆ0
kˆ0
not depend on ` if ` is even and is the linear map decomposing into diagonal blocks
as follows:
(1) Id[kmid (m);kmax (m)) (maybe empty), m ˆ 0; . . . ; q
(2)
ev
S(m)[kmax (m
1);kmid (m)]
1,
(of size at least 3), m ˆ 1; . . . ; q
(3) diag(1; T; T; . . . ; T) ev S(0)[kmax (q
1);kmid (0))
1,
1
1
diag(1; T ; T ; . . . ; T 1 ).
e k with odd Sk when kmid (m)4k < kmax (m) without
PROOF. We can first replace evS
ev
e ` , and thus identify ek with odd k when kmid (m)4k4kmax (m). It follows
changing that the restriction of the linear isomorphism s`‡1
to the subspace
`
kmax
(m) 1
M
kˆkmid (m)‡1
Kj`‡1
k
equals the identity. Note that this block may be empty if p < 3q.
Next, let us understand the situation when kmax (m 1) < k < kmid (m) (cf.
Figure 13). In such a case,
(6:2:11)
odd
in
k : oddIm
? k ? oddIout
k m
1
and
ev
in
ek : oddIm
1
? k ? oddIout
k m:
Consider the closed sector
S(m) :ˆ
kmid (m)
[
kˆkmax (m 1) 1
odd
Sk ˆ
kmid (m)
[
ev e
Sk
kˆkmax (m 1) 1
of the closed annulus between the curves evekmax (m 1) ˆ odd kmax (m 1) 52 and
ev
ek (m) ˆ odd k (m) , due to our previous identification. Our goal is to replace both
mid
mid
Leray coverings on S(m) with a single one, in order to compute the matrix of s`‡1
in
`
the corresponding basis. Let us consider the closed coverings A :ˆ (Aj )jˆ 1;...;s and
e k (m 1)‡j as in Figure 14.
E :ˆ (Ej )jˆ 1;...;s , with Aj ˆ odd Skmax (m 1)‡j and Ej ˆ evS
max
180
Marco Hien - Claude Sabbah
Figure 14. The coverings A and E of the sector S, their refinements B; D, themselves
being refinements of a common covering C.
We first pass from both coverings to individual refinements. Let us denote by
B :ˆ fB 1 ; B0 ; Bs g [ fB
j j j ˆ 1; . . . ; s
D :ˆ fD 1 ; Ds 1 ; Ds g [
fD
j
j j ˆ 0; . . . ; s
1g the refinement of A and
2g the refinement of E
as shown on Figure 14, so that B 1 ˆ D 1 ˆ odd Skmax (m 1) 1 and Bs ˆ
Ds ˆ odd Skmid (m) . From these refinements we again pass back to a common coarser
covering C :ˆ fCj j j ˆ 1; . . . ; sg - in the sense of a closed covering which refines to
both B and D. The construction of these can be read off Figure 14. The proof of
Lemma 6.2.10 consists in a computation of the corresponding base change.
p
REMARKS 6.2.12. (1) The above computations have been carried out in the case
q < p. In the case q > p, we can proceed in the same way (which amounts to
interchanging the roles of the inner and outer boundary components). The
case p ˆ q ˆ 1 will be studied in §6.2.c below.
(2) The isomorphism s`‡1
of Lemma 6.2.10 coincides with sodd
ev defined by (1.4.9).
`
Let us verify this statement. Recall that evek \ @ in A 2 oddIevinin(k) 1 . By definition
of kmin (m); kmid (m) and kmax (m), we deduce that
k 2 [kmid (m); kmax (m)] () ev in(k)
1 ˆ odd in(k):
The local Laplace transform of an elementary irregular etc.
181
`‡1
odd
Then s`‡1
` (v 1k ) ˆ v 1k ˆ s ev (v 1k ). In the other case, s` (v 1k ) is
ev
determined by the corresponding column of the block S(m) (different from the
first or last column). The rows with the non-vanishing entries 1; 1; 1 correspond
to the places
minodd
in (k);
odd
maxodd
out (minin (k))
maxodd
out (k)
and
respectively.
6.2.b - Explicit description for s`‡1
for ` odd. The result in this case is similar. In
`
analogy to (6.2.6) and (6.2.7), we now have
8
h
k
m
1 m
in
k
odd out
>
I k m if p‡q 2 q 2q ; q mod 1
< oddIm
1? ?
odd
(6:2:13)
k :
h
>
k
m m
1
: oddIin ? k ? oddIout
if p‡q 2 q ; q ‡ 2q mod 1
m
k m 1
and consequently
(6:2:14)
odd
egk :
8
in
>
< evIm
k
ev out
1 ? ? I k m‡1
>
in
: oddIm
? k ? oddIout
k m
if
k
p‡q
2
if
k
p‡q
2
h
h
m
q
1 m
;
2q q
mod 1
m m
;
q q
1
‡ 2q mod 1:
k
m
Hence evgk and oddegk show the same behaviour whenever p‡q 2
q
m
1
; q ‡ 2q mod 1.
Picking up the notation from §6.2.a, we see that for fixed m 2 f0; . . . ; q
in
curves evgk start at evIm
exactly for
(6:2:15)
kmid (m
1) ˆ
l (2m
l (2m ‡ 1)( p ‡ q)
1)( p ‡ q) m
4k4
2q
2q
1g, the
m
1 ˆ kmid (m)
1;
and that among these the curves evgk and oddegk
connect the same intervals if kmin (m)4k4kmid (m)
connect different intervals if kmid (m
1 and
1)4k < kmin (m).
in
in
Figure 15. The curves evgk starting at evIm
and evIm
1 for fixed m. Exactly for kmin (m)4
odde
k4kmid (m) 1, the curves gk have the same behaviour as evgk .
182
Marco Hien - Claude Sabbah
Consequently, the situation between kmid (m 1) 1 and kmin (m) for a given m
gives the exact same picture as before, cf. Figure 12 and Figure 15. We now can
proceed as in the proof of Lemma 6.2.10:
e k with evSk when kmin (m)4k < kmid (m) 1 and thus identifying
Replacing odd S
odd
ev
to the subspace
ek with gk for these k, we obtain that the restriction of s`‡1
`
kmid (m) 1
M
kˆkmin (m)
is the identity. For kmid (m
1)
K `‡1
k
j
1 < k < kmin (m), we have
ev
in
gk : evIm
? k ? evIout
k m
and
odd
in
ek : evIm
1
? k ? evIout
k m‡1 :
Comparing with (6.2.11), we see that the appropriate sector of the annulus reproduces the same situation as in the proof of Lemma 6.2.10, cf. Figure 16.
Defining
s :ˆ kmin (m)
kmid (m
1) ‡ 152
the resulting matrix is the matrix odd S(m) 2 Mat(s‡1)(s‡1) (End(V )) with exactly the
same design as the one defined in (6.2.8) before. Therefore, we finally obtain the
analogous result to Lemma 6.2.10:
L
L
1
1
LEMMA 6.2.16. The isomorphism s`‡1
: p‡q
V 1k ! p‡q
V 1k does
`
kˆ0
kˆ0
not depend on ` if ` is odd and is the linear map decomposing into diagonal blocks
as follows:
(1) Id[kmin (m);kmid (m)
(2)
odd
S(m)[kmid (m
1)
(maybe empty), m ˆ 0; . . . ; q
1) 1;kmin (m)]
(3) diag(1; . . . ; 1; T) odd
1,
(of size at least 3), m ˆ 1; . . . ; q
S(0)[kmid (q
1) 1;kmin (0)]
1,
1
diag(1; . . . ; 1; T ).
Figure 16. The sector S and the curves evgk and oddek in the range [kmid (m 1) 1; kmin (m)].
The local Laplace transform of an elementary irregular etc.
183
REMARK 6.2.17. The case q < p can again be obtained in an analogous way
interchanging the role of the inner and outer boundary (cf. Remark 6.2.12). With
the same arguments as in Remark 6.2.12 we verify that s`‡1
for ` odd coincides with
`
ev
sodd of (1.4.9 ).
6.2.c - The case p ˆ q ˆ 1. Then the Leray coverings evS and odd S are given as in
Figure 17. The curves evek and oddek obtained by the appropriate isotopy are also
given in the picture.
Figure 17. The Leray covering evS (black) and the curves oddek (red) on the left hand
side, the Leray covering odd S (black) and the curves evek (red) on the right hand side.
By Remark 6.2.9, the heuristic rule provides an expectation on the form of sodd
ev
and sev
odd (namely to be the one defined in 1.4.9). These expectations can be verified
by the analogous procedure used in the proof of Lemma 6.2.10. Let us denote by
A :ˆ (A0 ; A1 ) the closed covering induced by the curves evgk and by C :ˆ (C0 ; C1 )
the one induced by the curves oddek . Furthermore, let B be the canonical common
(ordered) refinement B ˆ (B0 ; B1 ; B ; B‡ ) (cf. Figure 17), such that A0 ˆ B0 [ B‡ ,
A1 ˆ B1 [ B , C0 ˆ B0 [ B and C1 ˆ B1 [ B‡ . This refinement allows to comÏ ech complex associated to A to the one
pute the change of the basis from the C
associated to C as we did in the proof of Lemma 6.2.10. The resulting presentation
of the base change gives the map sev
odd as in Definition 1.4.9. The same arguments
apply for sodd
using
the
picture
on
the
right hand side of Figure 17.
ev
6.2.d - Standard Stokes data. We have now obtained the following result:
PROPOSITION 6.2.18. Let p; q 2 N be two co-prime numbers and let (V; T) be a
vector space with an automorphism. The topological model of Definition 6.2.4 is
isomorphic as a set of linear Stokes data to the standard linear Stokes data of
Definition 1.4.11.
p
184
Marco Hien - Claude Sabbah
b top
6.3 ± Explicit computation of T
We now prove Proposition 1.4.13. In principle, the topological monodromy of the
c can be obtained once we determined its
Laplace transform b ‡ F (0;1) M around 1
Stokes data (see Remark 1.4.14). However, using the techniques developed above
in this section, it is possible to compute in a straightforward way the monodromy.
In particular, we can work on the space P1u A1h and do not have to use the blow-up
c) - see sections 3.4 and 3.5 for the notation.
of (0; 1
e 1 be the closed annulus obtained as the real blow up of 0; 1 in P1 . We set
Let P
u
u
# ˆ arg u as in §4.1. The fiber at to 6ˆ 0 of
(cf. §1.1) is obtained as
(6:3:1)
e 1 ; H 0 DRmod (0;1) (E
H1 P
u
'(u) up =to
R) ;
p
c t :ˆ H 0 DRmod (0;1) (E '(u) u =to R) is the extension j K of the
where F
o
sheaf K (corresponding to R, i.e., defined by the monodromy T) on the open annulus Cu to the open part of the boundary defined by
(6:3:2)in
q# ‡ arg 'q 2 (p=2; 3p=2) mod 2p;
(6:3:2)out
p#
arg to 2 (p=2; 3p=2) mod 2p;
and extended by zero to the remaining part of the boundary. The action of the
monodromy to 7! e2pi to consists only in moving each of the p open intervals (6:3:2)out
(of length p=p) of the outer boundary in the positive direction to the next one. The
in
intervals (6:3:2)in can be written as evIm
‡ (arg Wq )=q, m ˆ 0; . . . ; q 1, and it is
possible to choose to so that the intervals (6:3:2)out are the intervals evInout ‡
(arg Wq )=q, n ˆ 0; . . . ; p 1. For simplicity we assume below that arg Wq ˆ 0. We can
therefore use the topological model of the annulus A with the subset evB defined by
(6.1.1) and the sheaf K on A, and obtain the isomorphism
c t ) ' H1 (A; (ev)! K ) ˆ H1 (evB; K ):
e1 ; F
H1 ( P
o
u
c
It will be simpler here, since we do not have to control Stokes filtrations, to work
with Borel-Moore homology, for which we refer to [BM60], [BH61] and [Bre97]. We
odd
Â
have an isomorphism H1c (evB; K ) ' HBM
1 ( B; K ): On the one hand, by Poincare
_ _
1 ev
1 odd
duality, Hc ( B; K ) ' Hc ( B; K ) ; on the other hand, the cap product
(cf. [Bre97, Th. V.10.4] - note that the result in loc. cit. is stated for K being of
rank one but generalizes to local systems of arbitrary rank) induces an isomorphism
_ _
odd
1 odd
HBM
1 ( B; K ) ' Hc ( B; K ) .
odd
The elements of HBM
1 ( B; K ) are represented by Borel-Moore cycles of the
form
h loc:fin:
i
X
s ws
s21 (oddB)
The local Laplace transform of an elementary irregular etc.
185
where 1 (oddB) denotes the set of closed simplicial 1-chains, ws 2 H0 (s; K ), and the
sum being locally finite. Note that a curve g : (0; 1) ! evB such that limt!0 g (t);
odd
limt!1 g (t) 2 @(A) defines a Borel-Moore cycle g w 2 HBM
1 ( B; K ) if w 2
H0 (g; K ). In particular, there is a well-defined morphism
p‡q
M1
(6:3:3)
kˆ0
odd
Kj k ! HBM
1 ( B; K );
defined by w 7! evgk w for w 2 Kj k . The long exact sequence for the open
embedding oddB A ([BM60, Th. 3.8] or [Bre97, §V.5]) yields that this is a basis for
odd
HBM
1 ( B; K ). Let us denote by
odd
BM : HBM
1 ( B; K ) !
(6:3:4)
p‡q
M1
kˆ0
Kj k
its inverse. We can therefore use the curves evgk to compute this homology by way of
the isomorphism BM .
We fix an isotopy A [0; 1] ! A [0; 1] of the identity to a diffeomorphism c of
A to itself by lifting the vector field @t on [0; 1] in a way that the lift equals @t outside
a neighbourhood of the outer boundary of A and that the diffeomorphism reev
ek be the images of the
stricted to the outer boundary maps evInout to evIout
n‡1 . Let ev
curves gk by the diffeomorphism c. We obtain another isomorphism
e BM : HBM (oddB; K ) !
1
p‡q
M1
kˆ0
Kj k :
e BM ) 1 .
The topological monodromy is then represented by BM (
We see that evgk and evek connect the boundary intervals in the following way:
ev
in
gk : evIm
? k ? evIout
n
and
ev
in
? k ? evIout
ek : evIm
n‡1 ;
with (m; n) ˆ (ev in(k); evout(k)) as defined in (6.1.2). Fixing m 2 f0; . . . ; q
have (cf. (6.2.15))
ev
in(k) ˆ m () k 2 kmid (m 1); kmid (m) 1 ;
1g, we
(2m‡1)( p‡q)
where kmid (m) ˆ d
e. Therefore, evek and evgk‡1 connect the same bound2q
in
and evIout
ary intervals evIm
n‡1 for all
(6:3:5)
k 2 kmid (m 1); kmid (m) 2
(which is possibly empty). Note, that if p ‡ q53, there is at least one m for which
this interval is non-empty (cf. Lemma 6.1.4). The case p ˆ q ˆ 1 was completely
determined in §6.2.c. The situation is sketched in Figure 18 and 19.
186
Marco Hien - Claude Sabbah
The Borel-Moore cycles evgk (black) and evek (red).
Figure 18. The case q < p, the interval
(6.3.5) always containing at least one integer.
Figure 19. The case q > p, the interval
(6.3.5) containing at most one integer.
Here, it is non-empty only for the values
in
is marked with an ( ).
of m where evIm
Now, the assertion of Proposition 1.4.13 can directly be read off from these
figures, the index maxout (k) denoting the index k0 5k such that out(k0 ) ˆ out(k), but
out(k0 ‡ 1) 6ˆ out(k). Note, that in the case q < p, we have
(
k
if k 6ˆ kmid (m) 1
maxout (k) ˆ
kmid (m) if k ˆ kmid (m) 1
with m ˆ in(k) (cf. Figure 18).
In case q > p, we have maxout (k) ˆ k if and only if in(k ‡ 1) ˆ in(k) (where k
is the lower index of the two curves starting at the marked ( ) intervals in
Figure 19).
p
7. Stokes data for b ‡ F (0;1) M
We will now prove Theorem 1.3.6 asserting that the linear Stokes data attached to
b ‡ F (0;1) M are isomorphic to the standard model of §1.4. We are left to proving that
for each ` 2 f0; . . . ; 2q 1g, the isomorphism ` in (6.2.2) is compatible with the
filtrations of (4.3.2) and of Definitions 6.1.7 and 6.1.16. For that purpose, we will
modify the construction of the Leray coverings evS resp. odd S in such a way that the
new Leray coverings moreover induce Leray coverings on each evBv
resp.
>0
odd
Bv ( 2 mp‡q ).
>0
7.1 ± The even case
Throughout this subsection, we use the even order <ev on mp‡q (cf. Definition
1.4.1) and we simply denote it by <. The key observation is the following Lemma:
The local Laplace transform of an elementary irregular etc.
LEMMA 7.1.1. There exists a family fg j
the following properties:
187
2 mp‡q g of smooth curves in A with
(1) g passes through ,
(2) g evBv
(3) setting
>0
,
to the intervals
(4) for
0
2kpi
ˆ exp( p‡q ) with k 2 f0; . . . ; p ‡ q
ev in
Im ("); evInout (")
1g, the end points of g belong
with (m mod q; n mod p) ˆ evc(k),
6ˆ , we have g 0 \ g ˆ [.
PROOF. For ` even, we have Bv` <0 ˆ evBv >0 , according to (5.2.1). Lemma 5.1.5
p p
shows that there exist exactly two connected components of G 1 ( 2 ; 2 )
containing the point x ˆ 1 in their closure: these are the components containing
ev in
I0 and evI0out (see Figures 3 and 5). Let us denote by B the union of their
closures.
We will construct paths g of the form g ˆ for some connecting evI0in
and evI0out via x ˆ 1 (so that (1) will be fulfilled) inside the region B and contained in
p p
p
p
the fiber G 1 (u ) for some argument u 2 ( 2 ; 2 ) \ (q arg ( ) ‡ ( 2 ‡ "; 2 ‡ "))
(so that (2) will be fulfilled, according to (5.2.1); note that this intersection is not
empty). We will then have
g \ @ in A and
g \ @ out A evI0in \ @ in (evBv
>0 );
evI0out \ @ out (evBv
>0 );
so (3) will be fulfilled according to (5.2.3) and Remark 6.1.3(2).
We are thus reduced to find (u )
in order to fulfill (4). In the following, we
2mp‡q
will identify the paths g and their images, and we will consider the set
arg (g) :ˆ farg (x) j x 2 gg:
According to the ordering at #bo (cf. §5.2), we will inductively find u with the slightly
stronger condition (preparing for a variant necessary later) that
p
p p
p p u 2
(7:1:2)
‡ ";
\ q arg ( ) ‡ " ‡
;
;
2
2
2 2 p‡q
assuming that q arg ( ) 6ˆ p (in which case the intersection is non-empty). The
odd
case q arg ( ) ˆ p appears exactly when p ‡ q is even and ˆ 1 ˆ:
max is the
maximal element in mp‡q with respect to the even ordering. Therefore, we can also
use Condition (7.1.2) in the inductive argument in the case p ‡ q 0 mod 2 if we
odd
apply the induction hypotheses to all <
max only (recall that < means <ev ),
and it is enough to construct an appropriate uodd
at the end of the induction
max
process.
Assuming we already have found g 0 of the above form for 0 2 f 0 ; . . . ; k 1 g, the
task then is to find an argument u ˆ u with ˆ k , giving rise to the path
188
Marco Hien - Claude Sabbah
B (whose image is then uniquely determined by ˆ G 1 (u) \ B) with the
properties:
p p
p ;
u 2 q arg ( ) ‡ " ‡
(7:1:3)(i)
; and
2 2 p‡q
p
[
p
n
‡ ";
(7:1:3)(ii)
arg ( 1 g 0 ):
u2
2
2
0 2f
g
0 ;...; k 1
Condition (7:1:3)(i) ensures that G 1 (u) 1
evBv
>0
(cf. (5.2.1)) and Condition
1
(7:1:3)(ii) that u is such that :ˆ G (u) \ B does not intersect any of the curves
1
g
0
ˆ
1
0
0
for
0
< , which forces g \ g
To this end, consider a
0 ˆ G 1 (u 0 ) \ B for some u
G(x) ˆ arg g(x) ˆ u
1 0
x)
REMARK 7.1.5. Since
0
0.
0
0 1 0
x)
ˆ arg g(
ˆ [ for all these
2 f 0 ; . . . ; k 1 g. Then by induction, we have
as
desired. In particular, each x 2 0 satisfies
0
p
p
‡ ";
(g(x) :ˆ f (x)=xq ; cf: (3:5:4)):
2
2
2
1
Consequently, each point in
(7:1:4) G(
0
0
0
1
is of the form
0p g(x)
ˆ q arg ( ) ‡ arg
< , we know that arg (
0 x for such an x and satisfies
p
0p
‡ (p ‡ q)(
)2(
p p
; ].
2 2
p
0p
) :
Using the
p
notation of Definition 1.4.1, the pairs (
) such that < and arg ( 0 p
)ˆ
0
p‡q
ka
ka
k
a
0
p=2 are the pairs (j ; j ) with k 2 [1; 2 ) \ N. Indeed, set ˆ j and ˆ j ka
0;
p‡q p‡q
0
0
p‡q p‡q
p
with k0 2 (
; 2 ) \ Z and k 2 [
; 2 ) \ Z. Then 0 p
ˆ jk j k,
2
2
which has argument p=2 if and only if the conditions above are fulfilled. We then
2p
have p arg 2 (p; 2p p‡q ].
Additionally, since x 2 0 , we know from (7:1:3) (i) that
p p
p p arg ( 0 ) ‡ u 0 2 " ‡
;
:
2 2 p‡q
Notice also that if arg (
p
0p
arg (
0p
In such a case, the set f
)2(
p
)2
0 p g(x)
h
p p
; ),
2 2
we have more accurately
p
p p
p i
‡
;
:
2 p‡q 2 p‡q
‡ (p ‡ q)(
0p
p
0p
p
) j x 2 0 g is contained in a
half-line starting at the point ( p ‡ q)(
) - with fixed argument inside
p
p
p
p
p p
p
[ 2 ‡ p‡q ; 2 p‡q ] - and having direction p arg ( 0 ) ‡ u 0 2 " ‡ ( 2 ; 2 p‡q ).
Let us set
iu 0
p
(7:1:6)
) j r 2 R50 g:
U 0 :ˆ farg 0 p re ‡ ( p ‡ q)( 0 p
The local Laplace transform of an elementary irregular etc.
189
We deduce that there exists "0 > 0 such that
p
8
p
0 p
0
p
>
"
‡
;
) 6ˆ p=2;
‡
"
"
if arg ( 0 p
>
<
2
2 p‡q
(7:1:7)
U 0 >
p
pi
>
p
:
) ˆ p=2:
‡ " ‡ "0 ;
if arg ( 0 p
2
2
S
Let us set U :ˆ
0 < U 0 . Now, there are two cases to distinguish. Let us consider the interval
p
p p
p p S :ˆ p arg ( ) ‡
‡ ";
\ "‡
;
:
2
2
2 2 p‡q
Then S 6ˆ [, and only two cases can occur for the relative position of the two intervals.
Case 1. Assume that
S ˆ " ‡ p arg ( )
p p
;
2 2
p :
p‡q
p
In this case we have p arg 2 (0; p p‡q ), hence Remark 7.1.5 implies that there is
p
) ˆ p=2, so that the first line of (7.1.7) applies for all
no 0 < such that arg ( 0 p
0
0 <
and thus there exists " > 0 such that
p
p
p
‡ "0 ;
"0 :
U "‡
2
2 p‡q
It follows that S 6 U and we can find a u 2 q arg ( ) ‡ S n U , as desired (cf.
(7.1.3)).
Case 2. Assume that
Sˆ
that is, p ‡ " < p arg
p
2
< 2p
p
"0 ; 2 ],
p
p
‡ "; p arg ( ) ‡ ;
2
2
p
p‡q
‡ ". From the second line of (7.1.7) we deduce
that U (
‡"‡
so that S 6 U , and we conclude as in Case 1.
It remains to consider the special case when p ‡ q is even and ˆ ev max ˆ 1.
By the first part of the proof, we can assume that u 0 satisfies (7:1:3) (i) and
(7:1:3)(ii) for each
0
< . The first line of (7.1.7) applies to U . Now, we do not need
the stronger condition (7.1.3) to hold for since we do not have to continue with the
induction. Therefore, we consider the interval
p p p p
p p (7:1:8)
;
\ "‡
;
ˆ
; ‡"
S 0 :ˆ p arg ( ) ‡
|‚‚‚‚‚{z‚‚‚‚‚}
2 2
2 2
2 2
ˆp
instead of S. We conclude again that S 0 6 U
q arg ( ) ‡ (S 0 n U).
and hence we will find u 2
p
190
Marco Hien - Claude Sabbah
REMARK 7.1.9. The case q ˆ 1 and p 1 mod 2 is covered by the special case
above. Due to (7.1.8), the resulting picture looks as the one from the topological
model in Figure 11 (if p 6ˆ 1) or Figure 17 (if p ˆ 1).
We will now set gk :ˆ gjk for k ˆ 0; . . . ; p ‡ q 1.
PROPOSITION 7.1.10. The collection of curves fgk j k ˆ 0; . . . ; p ‡ q 1g from
Lemma 7.1.1 decomposes A into p ‡ q pieces, each being homeomorphic to a closed
disc. This decomposition is a Leray covering by closed subsets for each of the
sheaves (b v` >0 )! K
`
with
2 mp‡q . It induces the isomorphism (6.1.6).
PROOF. According to Lemma 7.1.1, it is clear that the family of curves (gk )k decomposes A into p ‡ q pieces homeomorphic to closed sectors of A. Let S be one of
these. Then S is bounded on each side by gk and gk‡1 for some k, and additionally
bounded by an interval in the circle S1rˆ0 and one of the circle S1rˆ1 in
A ˆ f(r; #x ) j r 2 [0; 1]; #x 2 S1 g, see Figure 8.
Now, for any 2 mp‡q , we know the shape of evBv >0 due to Proposition 5.1.4:
namely, evBv
>0
emerged from discs with part of the boundary contained in it
in
and evInout (")) which have been glued according at various
(namely the intervals evIm
points
2 mp‡q (depending on the relation <). The shape of
ev
Bv
>0
\ S is
therefore easily determined once we know the shape of the boundary component
gk \ evBv >0 for j k 2 S.
If j k 4
o
, it is clear that gk evBvj k >0 evBv
>0 .
If on the other hand
we proceed as in Lemma 7.1.1. For x 2 k ˆ j k gk , we have
1 k
j x) ˆ q arg ‡ arg j pk g(x) ‡ ( p ‡ q)(j pk
G(
(7:1:11)
p
< jk,
) :
p p
The term g(x) ˆ: reiuk has fixed argument uk 2 ( 2 ; 2 ) (see the construction of k
in Lemma 7.1.1). From (5.2.1), we deduce that
p p
p
; () j k x 2 evBv >0 :
(7:1:12) arg j pk reiuk ‡ (p ‡ q)(j pk
) 2"‡
|‚‚‚‚‚‚{z‚‚‚‚‚‚} |‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚}
2 2
(i)
(ii)
p 3p
The summand (ii) in (7.1.12) is a point with argument in ( 2 ; 2 ) (since < j k ) and
therefore h :ˆ (i)+(ii) describes a half-line starting from (ii) with direction
(i) ˆ
2pkp
‡ uk :
p‡q
Note that r ! 1 both for jxj ! 0 or jxj ! 1 and since the starting and endpoint
of gk is contained in evBv >0 , we see that the latter direction is contained
p p
in " ‡ ( 2 ; 2 ). Let t 7! k (t), t 2 [0; 2] be a regular parametrization of k with
k (1) ˆ 1 and consider the resulting parametrization
t 7! h(t) :ˆ r(t) j pk eiuk ‡ ( p ‡ q)(j pk
of the half-line h with r(t) :ˆ jg( k (t))j.
p
)
The local Laplace transform of an elementary irregular etc.
191
CLAIM. The parametrization h of the half-line h satisfies
h
1
"‡
p p
;
ˆ 0; t0 [ t1 ; 2
2 2
for some 0 < t0 < 1 < t1 < 2.
If this claim is proved, we deduce by (7.1.12) that
j k k (t) 2 evBv >0 () t 2 0; t0 [ t1 ; 2 ;
and therefore gk \ evBv >0 is homeomorphic to the union I in [ I out of two halfin
closed intervals, I in starting at evIm
("), and I out starting at evInout (") with
(m mod q; n mod p) ˆ evc(k).
PROOF OF THE CLAIM. Due to the construction of k , we know that G( k (t)) ˆ
arg (g( k (t))) ˆ uk for all t with k (t) 6ˆ 1 (recall that G(x) is not defined for x ˆ 1). We
deduce that
Im g( k (t))e iuk ˆ 0 and Re g( k (t))e iuk 50
for all t. Consequently, r(t) ˆ jg( k (t))j ˆ jg( k (t))e
d
d
r(t) ˆ g( k (t))e
dt
dt
iuk
iuk
ˆ g0 ( k (t)) j ˆ g( k (t))e
d
(t)e
dt k
iuk
. Now
iuk
and since g0 (x) ˆ 0 if and only if x 2 mp‡q , the real function r(t) is strictly monotone
on (0; 1) and (1; 2) with limt!0 r(t) ˆ limt!2 r(t) ˆ 1 and limt!1 r(t) ˆ 0.
p
We can now finally understand the shape of
ev
Bv
>0
\ S. Let gk ; gk‡1 S
be the boundary curves. Due to Lemma 6.1.4, we know that, up to interchanging
the inner and outer boundaries, the sector S looks as in Figure 20.
Figure 20. The typical shape of S.
192
Marco Hien - Claude Sabbah
We have two cases for the boundaries: either gk evBv
I
ev
in
Bv
>0
or gk \ evBv
>0
ˆ
[ I out , and the same for gk‡1 . From our knowledge of the homotopy type of
ev
>0 , we deduce that Bv >0 \ S is homeomorphic to one of the three spaces
sketched in Figure 21.
Figure 21. Up to homeomorphism, the three possible shapes of evBv
We conclude that each connected component of evBv
>0
>0
\ S.
\ S is homeomorphic
to a closed disc with a closed interval deleted in its boundary, and each connected
component of an intersection evBv
>0
\ S \ S0 is homeomorphic to a closed or a
semi-closed interval. There are no triple intersections for this covering. It is thus
clear that this covering is Leray for ( bv` >0 )! K
`
(` even).
Finally, the properties of the curves gk proven in Lemma 7.1.1 correspond to the
behaviour of the standard Leray covering evS described in Proposition 6.1.5, and
we obtain the final assertion.
p
7.2 ± The odd case
We now have Bv` >0 ˆ oddBv
>0
(cf. (5.2.1)). In analogy to the procedure in the
standard case as in section 6.1.b, we deduce the odd case from the even one by
rotation by odd min as defined before (6.1.12). Note that
8
if p ‡ q 0 mod 2
>
<p
odd
(7:2:1)
q arg (
p
min ) ˆ >
if p ‡ q 1 mod 2:
:p
p‡q
The minimal subset to be considered is oddBvodd >0 which is contained in oddBv >0
min
for all 2 mp‡q :
8
p
p
odd
1
>
‡
";
‡
"
if p ‡ q 0 mod 2
G
>
<
min
2
2
odd
Bvodd >0 ˆ
p
p
p
>
min
>
‡
‡ "; ‡ "
if p ‡ q 1 mod 2:
: odd min G 1
p‡q
2
2
The local Laplace transform of an elementary irregular etc.
Since there is no critical value of G inside
G
1
p
p‡q
p
2
‡(
193
p
‡ "; 2 ‡ "), the subset
p
p
p
‡
‡ "; ‡ "
p‡q
2
2
is homeomorphic to Bvo <0 . Therefore, oddBvodd
1
min
>0
is either equal to evBv1 >0 rotated
by odd min (in the first case), or a small perturbation of it in the sense of applying the
flow of the gradient vector field of the Morse function considered for a time interval
without critical points.
Furthermore, considering arbitrary 2 mp‡q , we have
p
3p
‡ "; ‡ "
(7:2:2) oddBv >0 ˆ G 1 q arg
q arg (odd min ) ‡ q arg (odd min ) ‡
2
2
p
p
p
ˆ G 1 q arg ( (odd min ) 1 ) s
‡
‡ "; ‡ "
p‡q
2
2
with s ˆ 0 if p ‡ q 0 mod 2 and s ˆ 1 otherwise. This space is again either equal to
odd
ev
>0 or a perturbation of it in the same sense as above.
min Bv
1
(odd
min )
Let us define eg :ˆ odd
min
g
(odd
min )
1
oddBv
>0 ,
with g as in Lemma 7.1.1. If
p ‡ q is even, the whole situation to be considered is obtained from the one in the
even case by rotation by p. Therefore, the statements of Lemma 7.1.1 and Proposition 7.1.10 remain true for the curves eg and using oddB, oddIin , oddIout and odd c
instead of their even counterparts. We have to verify that this is still true for p ‡ q
being odd.
Properties (1) and (4) of Lemma 7.1.1 for the family ( eg ) are trivially satisfied.
Furthermore, we have
eg ˆ odd
min
#b (odd
min )
1
(odd min ) 1
ˆ #b Now, each z 2 eg is of the form z ˆ #b x for some x 2 from (7.1.2) that
G( #b
1
z) ˆ G(x) ˆ u
(odd min ) 1
2 q arg ( #b (odd
:
(odd min ) 1
(odd
min
1
min ) ) ‡ " ‡
) 1
and we deduce
p p
;
2 2
p ;
p‡q
and (7.2.2) yields z 2 oddBv > 0 . This proves property (2) of Lemma 7.1.1 for the
family eg .
Property (3) of Lemma 7.1.1 follows due to the fact that (4) holds and therefore
and
eg \ @ in A odd
min
in
evIm
\ @ in (oddBv
eg \ @ out A odd
min
evInout \ @ out (oddBv
with (m; n) ˆ evc(k) for #b ˆ j k together with (6.1.14).
>0 );
>0 );
194
Marco Hien - Claude Sabbah
We also have the analogous statement to Proposition 7.1.10:
PROPOSITION 7.2.3. The collection of curves feg j #b 2 mp‡q g decomposes A into
p ‡ q pieces, each being homeomorphic to a closed disc. This decomposition is a
Leray covering by closed subsets for each of the sheaves (b v` >0 )! K
`
with
2 mp‡q . It induces the isomorphism (6.1.15) (` odd).
PROOF. We only have to consider the case p ‡ q being odd. The proof goes along the
same lines as the one of Proposition 7.1.10 with the following changes. Again, it
suffices to understand the shape of egjk \ oddBv >0 for j k ; 2 mp‡q .
If j k 4 , then egjk oddBvjk >0 oddBv >0 . Otherwise, we proceed in the same
`
way. For j k x 2 j k jk (odd
G(
min
)
1
ˆ egjk , we have the equality (7.1.11):
1 k
j x) ˆ q arg ‡ arg j pk g(x) ‡ ( p ‡ q)(j pk
p
) :
The term g(x) ˆ: reiu has fixed argument u :ˆ arg j k (odd min ) 1 . From (7.2.2), we
deduce that
p p
p
p arg j pk reiuk ‡ (p ‡ q)(j pk
‡
;
) 2"
(7:2:4)
|‚‚‚‚‚‚{z‚‚‚‚‚‚} |‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚}
p‡q
2 2
(i)
(ii)
() j k x 2 evBv
>0 :
The sum (i)+(ii) again describes a half-line in the complex plane. The argument used
in the proof (Claim) of Proposition 7.1.10 shows that choosing a regular parametrization of the curve egjk , the half-line h inherits a parametrization [0; 2] ! h,
t 7! (ii) ‡ r(t)eiu with r(1) ˆ 0 such that r(t) is monotone on (0; 1) and (1; 2) and
satisfies limt!0 r(t) ˆ limt!2 r(t) ˆ 1. Since the half-line has to be inside the region
determined by (7.2.4) for r big enough (due to the fact that the endpoints of egj k lie in
odd
Bv >0 ) and since (ii) cannot be inside this region, we obtain the same conclusion as
in the proof of Proposition 7.1.10.
p
Since the Leray coverings of Proposition 7.1.10 (` even) and 7.2.3 (` odd)
respect the filtrations (cf. (4.3.2)) given by the subspaces evBv >0 (resp. odd), we finally obtain the following corollary concluding the proof of Theorem 1.3.6:
COROLLARY 7.2.5. The isomorphisms ` (6.2.2) are compatible with the filtration (4.3.2) and the filtration defined in 6.1.7 for ` even (resp. 6.1.16 for ` odd).
Acknowledgments. The first author thanks Giovanni Morando for useful discussions. The second author thanks Takuro Mochizuki for many discussions on the
subject. Both authors thank the referee for having carefully read the manuscript
and for having proposed various improvements.
The local Laplace transform of an elementary irregular etc.
195
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Manoscritto pervenuto in redazione il 22 maggio 2014.