The local Laplace transform of an elementary irregular meromorphic connection

The local Laplace transform of

an elementary irregular meromorphic connection

MARCOHIEN(*) - CLAUDESABBAH(**)

ABSTRACT- We give a definition of the topological local Laplace transformation for a Stokes- filtered 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.

MATHEMATICSSUBJECTCLASSIFICATION(2010). 14D07, 34M40

KEYWORDS. Laplace transformation, meromorphic connection, Stokes matrix, Stokes fil- tration.

1. Introduction

1.1 ±Riemann-Hilbert correspondence and Laplace transformation

Let M be a holonomic C‰tŠh@_ti-module and let be its Laplace transform, which is a holonomic C[t⁰]h@t⁰i-module through the correspondence t⁰ˆ@t and tˆ @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@_ti-moduleM corresponds to a Stokes-perverse sheaf on the projective lineP¹_t with affine coordinatet, which is a Stokes-filtered local system in the neighbourhood oftˆ 1. Similarly, corresponds to a Stokes-perverse sheaf onP¹_t0, which is a Stokes-filtered local system in the neighbourhood oft⁰ˆ 1. In the following, we will denote by t the coordinate ofP¹_t0 centered at c1such that

(*) Indirizzo dell'A.: Lehrstuhl fuÈr Algebra und Zahlentheorie, UniversitaÈtsstraûe 14, 86159 Augsburg, Deutschland.

E-mail: marco.hien@math.uni-augsburg.de

This research was supported by the grant DFG-HI 1475/2-1 of the Deutsche For- schungsgemeinschaft.

(**) Indirizzo dell'A.: UMR 7640 du CNRS, Centre de MatheÂmatiques Laurent Schwartz, EÂcole polytechnique, F-91128 Palaiseau cedex, France.

E-mail: Claude.Sabbah@polytechnique.edu

This research was supported by the grants ANR-08-BLAN-0317-01 and ANR-13-IS01- 0001-01 of the Agence nationale de la recherche.

tˆ1=t⁰ on P¹_t0nnf0;c1g, and we continue to set c1 ˆ ftˆ0g. The topological 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 set- ting of enhanced ind-sheaves considered in [DK13], which corresponds, through the Riemann-Hilbert correspondence of loc. cit., to the Laplace transformation of ho- lonomicD-modules (see also [KS14]).

1.2 ±Riemann-Hilbert correspondence and local Laplace transformation In this article, we will consider the local Laplace transformationF^(0;1)from the category of finite dimensionalC…ftg†-vector spaces with connection to that of finite dimensionalC…ftg†-vector spaces with connection. It is defined as follows. A finite dimensionalC…ftg†-vector space with connectionMcan be extended in a unique way as a holonomicC‰tŠh@ti-moduleMwith a regular singularity at infinity and no other singularity at finite distance thantˆ0. ThenF^(0;1)(M) is by definition the

germ of at c1:ˆ ft⁰ˆ 1g. We will regard F^(0;1)(M) as a C…ftg†-vector

space with connection. It is well-known that has at most a regular singularity at t⁰ˆ0 and no other singularity at finite distance. Therefore, giving F^(0;1)(M) is equivalent to giving the localized moduleC[t⁰;t^{0 1}]C[t⁰] .

The Deligne-Riemann-Hilbert correspondence associates to M a Stokes-fil- tered local system (L;L) onS¹_tˆ0(the circle with coordinate argt). Similarly, we will denote by F^(0;1)_top (L;L) the Stokes-filtered local system on S¹_tˆ0 associated withF^(0;1)(M). In this setting, we will address the following questions:

To make explicit the topological local Laplace transformation functorF^(0;1)_top . To use this topological definition to compute, in some examples, the Stokes

structure ofF^(0;1)(M) in terms of that ofM.

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:u7!tˆu^p is a ramification of orderpof the variable t,'is a polynomial inu ¹ without constant term, andR is a finite dimensional C…fug†-vector space with a regular connection. Setting E ^'ˆ(C…fug†;d d'), we define

M ˆEl(r; ';R):ˆr_‡(E ^' R):

If we extendRas a freeC[u;u ¹]-moduleRof finite rank with a connection having a regular singularity at uˆ0 and uˆ 1 and no other singularity, and set

E ^'ˆ(C[u;u ¹];d d'), the extensionMofMconsidered above is nothing but the freeC[t;t ¹]-module with connection

MˆEl(r; ';R):ˆr_‡(E ^'R):

Letqbe the pole order of'. We also encodeRas a pair (V;T) of a vector space with an automorphism (the formal monodromy). We call'theexponential factorofr^‡M and set ExpˆExp(r^‡M)ˆ f'g.

ASSUMPTION1.3.1. We will assume in this article thatp;qare coprime.

By the stationary phase formula of [Fan09, Sab08] the formal Levelt-Turrittin type of the local Laplace transform F^(0;1)(M) is that of El(br; ';b R), whereb

the ramificationbr:h7!thas orderbp:ˆp‡q, i.e.,br2h^p‡q(c(r; ')‡hCfhg) for some nonzero constant c(r; '); writing r and ' in the h variable, it is expressed asbr(h)ˆ r⁰(h)='⁰(h);

the exponential factor'(h)b ˆ'(h) '⁰(h)r(h)=r⁰(h) has pole orderq;

the regular partRb corresponds to the pair (V;b T)b ˆ(V;( 1)^qT).

Our goal is then to compute explicitly the Stokes structure of the non-ramified meromorphic connectionbr^‡F^(0;1)(M), which is usually non trivial. It is of pure level q. Stokes structures of pure levelq 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 br^‡F^(0;1)(M), which consist of the family Expd :ˆExp(br^‡F^(0;1)(M))ˆ f'b (h)j 2m_p‡qg (m_n is the group of n-th roots of unity and 'b ˆ'(b h(1‡o(h))) and the pair ( _2m

p‡qV; _2m

p‡q( 1)^qT), we add:

(1.3.2) the choice of a generic argument#b_o2S¹_hˆ0, giving rise to a total ordering of the set Exp, hence an order preserving numberingd f0;. . .;p‡q 1g 'm_p‡q, (1.3.3) a set ofunramified linear Stokes data (L<sub>`</sub>);(S<sup>`‡1</sup>_{`</sub> );(FL<sub>`})

`ˆ0;...;2q 1of pure

level q, which consists of

a family (L_{`</sub>)<sub>`ˆ0;...;2q 1} consisting of 2qC-vector spaces, isomorphismsS<sup>`‡1</sup><sub>`</sub> :L_{`</sub> !L<sub>`‡1},

for each m2 f0;. . .;q 1g, a finite exhaustive (¹) increasing filtration FL2m (resp. decreasing filtration FL2m‡1) indexed by f0;. . .;p‡q 1g, with the property that the filtrations are mutually opposite with respect

(¹) We say that an increasing filtrationFLindexed by a totally ordered finite setOis exhaustiveifFmaxOLˆL. We then setF<minOLˆ0. For a decreasing filtration, we have F^minOLˆLand we setF^>maxOLˆ0.

to the isomorphismsS<sup>`‡1</sup><sub>`</sub> , i.e., for any m2 f0;. . .;q 1g:

L2mˆ^p‡qM¹

kˆ0

FkL2m\S^2m_{2m 1}…F^kL_{2m 1†}

; (1.3.3)

L2m‡1ˆ^p‡qM¹

kˆ0

F^kL2m‡1\S^2m‡1_2m …FkL2m†

; with the convention that L 1ˆL2q 1.

Note that the opposite filtrations induce uniquely determined splittings, that is, filtered isomorphisms

t_2m:L_2m !gr^FL_2mˆ:L_{p‡q 1}

kˆ0 gr^F_kL_2m; t2m‡1:L2m‡1 ! gr_FL2m‡1 ˆ:L_{p‡q 1}

kˆ0 gr^k_FL2m‡1; (

(1:3:4)

where the right hand side carries its natural increasing (resp. decreasing) filtration.

The resulting isomorphismsS^{`‡1</sup><sub>`</sub> :ˆt<sub>`‡1</sub>S<sup>`‡1}_{`</sub> t<sub>`}¹:gr_FL_{`</sub>!gr<sub>F</sub>L<sub>`‡1}are upper/

lower block triangular - depending on the parity of`- and their blocks (S<sup>`‡1</sup>_` )_j;jon the diagonal are isomorphisms. These matrices are commonly called the Stokes matrices or Stokes multipliers. We call the family

S br^‡F^(0;1)(M)

:ˆ (L`);(S<sup>`‡1</sup><sub>`</sub> );(FL`)

`ˆ0;...;2q 1

…1:3:5†

the linear part of the Stokes data ofbr^‡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 byS^std(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 br^‡F^(0;1)(M)

, for M ˆEl(r; ';(V;T)), are isomorphic to the standard linear Stokes dataS^std(V;T;p;q).

REMARK1.3.7. (1) In order to descend from this result to the linear Stokes data of F^(0;1)(M), we would need to identify the m_p‡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 topo- logical monodromy ofF^(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.

1.4 ±The standard linear Stokes data

We will now define the setS^std(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 onm_p‡q.

DEFINITION1.4.1. For ; ⁰2m_p‡q, we set

(1:4:1) <odd 0 if Re…e ^{"i… p 0p}††<0 for some"such that 0<"1 and <_ev ⁰, ⁰<_odd . We enumeratem_p‡qaccording to the even ordering, i.e., (1:4:1) m_p‡qˆ f _k jkˆ0;. . .;p‡q 1gwith ₀<ev <ev p‡q 1:

Since (p;q)ˆ1, we also have (p‡q;q)ˆ1 and there exist a;b2Z with apˆ1‡b(p‡q). Setj:ˆexp

2pi p‡q

2m_p‡q. Then the even orderingon m_p‡q is expressed by

1ˆj⁰<_evj^a<_evj ^a<_ev <_evj^ka<_evj ^ka<_ev <_ev^ev _max; (1:4:2)

withk2h 1;^p‡q₂

, with

ev max ˆ exp

api p‡q

if p‡q is odd andais even (‡) or odd ( );

exp (api)ˆ 1 if p‡q is even:

8<

:

Theodd orderingis the reverse ordering.

Fork2Zwe set

evin(k):ˆ qk p‡q‡1

2

;

evout(k):ˆk ^evin(k)ˆ pk p‡q

1 2

: 8>

><

>>

: (1:4:3)

Both are increasing functions of k. If k2 f0;. . .;p‡q 1g, we have

evin(k)2 f0;. . .;qgand^evout(k)2 f0;. . .;pg. Furthermore, let min^ev_in(k):ˆmin k⁰5 p‡q

2q j^evin(j)ˆ^evin(k) for allj2[k⁰;k]

(1:4:4) and

(1:4:5) max^ev_out(k):ˆmax k⁰⁰<p‡q‡ p‡q

2p j^evout(j)ˆ^evout(k) for allj2[k;k⁰⁰]

(see Remark 6.1.3(3) for an explanation). In a similar way we set fork2Z:

oddin(k):ˆ qk p‡q

;

oddout(k):ˆk 1 ^oddin(k)ˆ pk p‡q 1

2 f 1;. . .;p 1g;

8>

><

>>

(1:4:6) :

and if k2 f0;. . .;p‡q 1g we have ^oddin(k)2 f0;. . .;q 1g and ^oddout(k)2 f 1;. . .;p 1g. We define min^odd_in (k) and max^odd_out(k) literally as in (1.4.4) and (1.4.5) replacing ``ev'' by ``odd''. Note that

h qk p‡q

1 2; qk

p‡q

i\Nˆ[()^evin(k)ˆ^oddin(k)‡1:

Let V be a finite dimensional complex vector space together with an auto- morphism T2Aut(V). Let us denote by1_k the complex vector space of rank one with chosen basis element 1_k. Let us consider direct sum V^p‡qofp‡qcopies of V and let us keep track of the indices by writing

V^p‡q ˆ^p‡qM¹

kˆ0

V_C1_k: (1:4:7)

CONVENTION1.4.8. For anyj2Z, we will write

v1_j:ˆT ^s(v)1_j‡s(p‡q) fors2Zsuch thatj‡s(p‡q)2[0;p‡q 1]:

DEFINITION1.4.9. Assumep‡q53. Then:

(1) The isomorphisms^odd_ev :L_{p‡q 1}

kˆ0 V1_k!L_{p‡q 1}

kˆ0 V1_k is defined as (1:4:9)

s^odd_ev …v1k†:ˆ

v1_k; if h

pqk‡q 1 2;_p‡^qk_qi

\Nˆ[ v1_min^odd

in (k) 1

v1_maxodd out(min^odd_in (k))

‡v1_maxodd

out(k)‡1; if h

p‡qkq 1 2;_p^qk_‡qi

\N6ˆ[ 8>

>>

>>

><

>>

>>

>>

: (2) The isomorphisms^ev_odd:L_{p‡q 1}

kˆ0 V1k!L_{p‡q 1}

kˆ0 V1k is defined as (1:4:9)

s^ev_odd…v1_k†:ˆ

v1_k; if h

p‡qqk 1 2;_p‡^qk_qi

\N6ˆ[ v1_min^ev_{in(k) 1}

v1_max^ev_out(min^ev_in(k))

‡v1_max^ev_out(k)‡1; if h

p‡qqk 1 2;_p‡^qk_qi

\Nˆ[ 8>

>>

>>

>>

<

>>

>>

>>

>:

Ifpˆqˆ1, the isomorphisms^odd_ev is defined as

(1:4:10) s^odd_ev (v10):ˆ v10‡(Id‡T ¹)v11 and s^odd_ev (v11):ˆv11; ands^ev_oddis defined as

(1:4:10) s^ev_odd…v1₀†:ˆv1₀ and s^ev_odd…v1₁†:ˆ …Id‡T ¹†v1₀ v1₁: We are now ready to define the following particular set of Stokes data.

DEFINITION1.4.11. The standard linear Stokes dataS^std(V;T;p;q) are given by (1) the vector spaces L<sub>`</sub>:ˆV<sup>p‡q</sup> for`ˆ0;. . .;2q 1,

(2) for eachmˆ0;. . .;q 1, the isomorphisms

S^2m_{2m 1}:ˆs^ev_odd:L_{2m 1} !L_2m (cf:(1:4:9));

S^2m‡1_2m :ˆs^odd_ev :L2m !L2m‡1 (cf:(1:4:9)):

(3) the isomorphismS^2q_{2q 1}:ˆdiag(T;. . .;T)s^ev_odd:L_{2q 1} ! L₀. (4) the filtrationsF_kL_2m:ˆL

k⁰4kV1_k⁰andF^kL_2m‡1:ˆL

k⁰5kV1_k⁰. Note that the opposedness property (1.3.3) for the above data is not ob- vious. 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 fol- lowing explicit computation of the topological monodromy of br^‡F^(0;1)(M), where we set

in(k):ˆ^evin(k)ˆ qk p‡q‡1

2

; maxout(k):ˆmax^ev_out(k):

(1:4:12)

PROPOSITION 1.4.13. The topological monodromy of br^‡F^(0;1)(M) is con- jugate to the automorphism of the vector spaceV^p‡qˆL_{p‡q 1}

kˆ0 VC1k (using the notation(1.4.7)and Convention1.4.8) given by

Tbtop(v1_k)ˆ v1_k‡1 if in(k‡1)ˆin(k);

v(1_k 1_{maxout (k)}‡1_{maxout (k)‡1}) if in(k‡1)ˆin(k)‡1:

(

REMARK1.4.14. The topological monodromy can of course be deduced from the Stokes data by the formula

T_top:ˆS⁰_{2q 1}S^2q_2q ¹₂ S¹₀2Aut(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.

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 levelq. 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 forbr^‡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 matricesS<sup>`‡1</sup><sub>`</sub> . 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 matricesS<sup>`‡1</sup><sub>`</sub> 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 matricesS<sup>`‡1</sup><sub>`</sub> 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.

Let $:Af¹_t !A¹_t be the real blowing-up of the origin in A¹_t. Then Af¹_tˆ A¹_t[S¹_tˆ0 is homeomorphic to a semi-closed annulus, with S¹_tˆ0:ˆ$ ¹(0)Af¹_t. Similarly, we consider Af¹_u, andrextends as thep-fold covering :S¹_uˆ0!S¹_tˆ0. For any#2S¹_uˆ0, the order onu ¹C[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

' <#0()'4#0 and '6ˆ0:

(2:1:2)

This is equivalent to exp('(u)) having rapid decay in some open sector centered at#.

DEFINITION2.1.3 (see e.g. [Sab13a, Chap. 2]). A Stokes-filtered local system with ramificationrconsists of a local systemL onS¹_tˆ0such that ¹L is equipped with a family of subsheaves L_4' ¹L indexed by a finite subset Expu ¹C[u ¹] with the following properties:

(1) For each#2S¹_uˆ0, the germsL_4';#('2Exp) form an increasing filtration of ( ¹L)_# ˆL _(#).

(2) SetL_{< ';#}ˆP

c<#'L_4c;#, where the sum is taken inL _(#) (and the sum indexed by the empty set is zero). ThenL<';#is the germ at#of a subsheaf

L_<'ofL_4'.

(3) The filtration is exhaustive, i.e.,S

'2ExpL_4'ˆ ¹L andT

'2ExpL_<'ˆ0.

(4) Each quotient gr_'L :ˆL_4'=L_<'is a local system.

(5) P

'2Exprk gr_'L ˆrkL.

(6) For each z2m_p and '2Exp, setting '_z(u):ˆ'(zu), we have L4'_z ' ez ¹L4' through the equivariant isomorphism ¹L 'ez ^{1 1}L, where ez:S¹_uˆ0 !S¹_uˆ0 is induced by the multiplication by z.

REMARK2.1.4. It is equivalent to start with a family of subsheaves L<'

1L such that L<';#L<c;# if '4#c, and to define L4' by the formula L4';#ˆT

c;c<#'L<c;#with the convention that the intersection indexed by the

empty set isL_#. The familyL_<'will be simpler to obtain in our computation of Laplace transform.

REMARK2.1.5 (Extension of the index set). It is useful to extend the indexing set of the filtration and to define the subsheaves L_4c ¹L for any c2u ¹C[u ¹]. For such acand for any'2Exp, we denote byS¹_'4c the open subset ofS¹_uˆ0defined by

#2S¹_'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 toS¹_uˆ0. ThenL4c is defined by the formula

L_4cˆ X

'2Exp

…b_'4c†_!L_4': (2:1:5)

The germ ofL_4cat any#2S¹_uˆ0can also be written as L4c;#ˆ X

'2Exp

'4#c

L4';#; (2:1:5)

with the convention that the sum indexed by the empty set is zero. Note also that, ifc is not ramified, i.e.c2t ¹C[t ¹], then L_4c is a subsheaf of L. A similar definition holds forL_<c, but one checks thatL_<c ˆL_4cifc2=Exp. Of particular interest will beL₄₀L.

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 orStokes multipliers in different approaches to Stokes struc- tures (see e.g. [BJL79]). In level qˆ1, 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 variableu. We will setS¹ˆS¹_uˆ0. We identifyS¹ˆR=2pZand will call an element

#2S¹an argument.

The description of these data by combinatorial means will highly depend on the various pole orders of the factors'2Exp and their differences' cas 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:

DEFINITION2.2.1. We say that a finite subsetEu ¹C[u ¹] is of pure level qif the pole order of each'2Eas well as

the pole order of each difference' cfor';c2E,'6ˆc,

equalsq. We say that an unramified Stokes structure (L;L) is pure of levelqif the associated set Exp of exponential factors is pure of levelq.

Let Eu ¹C[u ¹] be a finite subset of pure levelq. In particular, there are exactly 2qStokes directions for any pair'6ˆcinE, where a Stokes direction is an argument #2S¹ at which the two factors ';c do not satisfy one of the two in- equalities'4_#corc4_#'. More precisely, if

(c ')(u)ˆu ^qg(u)

with a polynomialg(u) withg(0)6ˆ0, the Stokes directions are St(';c):ˆn

#2S¹j q#‡arg (g(0))p 2 or 3p

2 mod 2po :

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

directions StDir(E):ˆS

'6ˆc2ESt(';c). An argument#o2S¹is said to begeneric

with respect to Eif

n#o‡`p

q j`2Zo

\StDir(E)ˆ[:

(2:2:2)

Let us fix a generic#_oand let us set#_{`</sub> :ˆ#<sub>o</sub>‡`p=q(`ˆ0;. . .;2q 1). Then each of the open intervals (#<sub>`}; #<sub>`‡1</sub>) contains exactly one Stokes direction for each pair '6ˆc in E. For " >0 small enough, the same holds for the intervals I`:ˆ(#` "; #`‡1‡") with which we cover the circleS¹ˆS_{2q 1}

`ˆ0 I`.

The category of unramified linear Stokes data of pure level q indexed by the finite setf1;. . .;rgover the fieldCis defined by (1.3.3). A morphism of two linear Stokes data of this type is given byC-linear maps between the vector spaces which commute with all theS<sup>`‡1</sup><sub>`</sub> and respect the filtrations. We denote this category by Stdata(q;r). A similar definition can be given over any fieldk.

We now fix a finite set Eu ¹C[u ¹] of pure level q. L et Ststruct(E) be the category of unramified Stokes structures (L;L) on S¹ with ExpˆE. After choosing an appropriate argument#_o as above, we construct a functor

F_E;#o :St_struct(E)!St_data(q;#E) (2:2:3)

as follows.

We can arrange the elements inEaccording to the ordering (2.1.1) with respect to the generic direction#_o, i.e., we write

Eˆ f'₁<#o '₂<#o'₃<#o <#o '_#Eg:

We then have the same ordering at any#2mwith even index and the reverse order for#_2m‡1. This follows from the fact that each inequality'_i<_# '_jbecomes reversed whenever the argument#passes a Stokes direction for the pair'_i,'_j, and there is exactly one Stokes direction for each pair inside (#_{`</sub>; #<sub>`‡1}).

We invoke the fundamental result going back to Balser, Jurkat and Lutz:

PROPOSITION2.2.4. For each open interval I₄S¹ of width p=q‡2"for " >0 small enough, there is a unique splitting

t:Lj_I'M

'2E

gr_'Lj_I compatible with the filtrations.

PROOF. See [Mal83, Lem. 5.1] or [BJL79, Th. A]. p

In particular, writing grL :ˆL

'2Egr_'L for the associated graded sheaf endowed with the Stokes filtration naturally induced by using the order (2.1.1), we

have unique filtered isomorphisms

t_{`</sub>:(L;L)j<sub>I`} ! (grL;(grL))j_I` (2:2:5)

over the intervalsI`chosen above.

For each `ˆ0;. . .;2q 1, we let L<sup>(`)</sup> :ˆG(I`;L) be the sections of the local systemL overI`. We have canonical isomorphisms

a`:L<sup>(`)</sup> !L_{#`</sub> and b`:L<sup>(`)</sup> ! L<sub>#`‡1}

betweenL<sup>(`)</sup> and the corresponding stalks ofL. Writing L<sub>`</sub>:ˆL_{#`</sub> for the stalk at#<sub>`}, we define

S<sup>`‡1</sup><sub>`</sub> :ˆb<sub>`</sub><sup>1</sup>a`:L` !L`‡1:

We call this isomorphism the clockwise analytic continuation. Additionally, we de- fine, formˆ0;. . .;q 1,

F_jL_2m:ˆ(L_4'j)_#2mL_2m; F^jL2m‡1:ˆ(L4'j)_#2m‡1 L2m‡1: (

Then the data

(L`)<sub>`</sub>;(S<sup>`‡1</sup><sub>`</sub> )<sub>`</sub>;(FL`)_` define a set of linear Stokes data as in (1.3.3).

PROPOSITION2.2.6. Given a finite set E of pure level q and a generic#o2S¹, the construction above defines an equivalence of categories

F_E;#o :St_struct(E)!St_data(q;#E)

between the Stokes structures with exponential factors E and the Stokes data of pure level q indexed byf1;. . .;#Eg.

PROOF. In the caseqˆ1, 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:u7!u^pˆtbe a ramification such thatr^‡Mhas a formal Levelt-Turrittin de- composition

C……u†† _C…fug†r^‡M ' M

'2Exp (M)

(E ^'(u) R');

with Exp:ˆExp(M)u ¹C[u ¹] (we use an opposite sign '(u) with respect to

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 eachz2m_p (p-th root of the unity)

R_'z ˆ R_'; with'_z(h):ˆ'(zh):

(2:3:1)

It may happen that'_zˆ'for some'2Exp andz2m_p.

The Riemann-Hilbert correspondence M 7!(L;L) goes as follows (see [Del07], [BV89], [Mal91], [Sab13a, Chap. 5]). Let us denote byF the local system attached toMon a punctured neighbourhood offtˆ0g, that we regard as well as a local system on the open annulus (A¹_t)or on the semi-closed annulusAf¹_t. Similarly, extendingras a covering map (A¹_u) !(A¹_t)orAf¹_u!Af¹_t, we definer ¹F . We also use the notation

L :ˆF _jS1

tˆ0; ¹L :ˆF _jS1 uˆ0:

One can attach toMthe family of subsheavesL4':ˆH⁰DR^{mod 0}(r^‡M E^')_jS1

of ¹L('2Exp), where DR^{mod 0}denotes the de Rham complex with coefficientsuˆ0

in the sheaf of holomorphic functions onAf¹_uⁿⁿS¹_uˆ0 having moderate growth along S¹_uˆ0. The equivariance property 2.1.3(6) is obtained through the isomorphisms z^‡r^‡M ˆr^‡M. Similarly, we can consider the family of subsheaves L_<':ˆ

H⁰DR^{rd 0}(r^‡M E^') of ¹L ('2Exp), where DR^{rd 0} denotes the de Rham complex with coefficients in the sheaf of holomorphic functions onAf¹_uⁿⁿS¹_uˆ0having rapid decay alongS¹_uˆ0.

REMARK2.3.2. The equivalence of categoriesM !Mconsidered in § 1.2 reads as follows, by introducing the notion of Stokes-filtered sheaves. A ramified Stokes- filtered sheaf on fA¹_tis defined (in the present setting) as the pair formed by a local system F onAf¹_t and a Stokes-filtrationL of its restrictionL :ˆF _jS¹

tˆ0. Since S¹_tˆ0is a deformation retract ofAf¹_t, givingF is equivalent to givingL, hence the equivalence.

With this notion of Stokes-filtered sheaf, we can define a subsheafF₄₀ ofF , which coincides withF away fromS¹_tˆ0, and whose restriction toS¹_tˆ0isL40(see Remark 2.1.5). In the following, we will also consider the real blow-up spacePe¹_t ofP¹_t attˆ0 andtˆ 1, which is homeomorphic to a closed annulus, and we will still denote byF (resp.F₄₀) the push-forward ofF (resp.F₄₀) by the open inclusionAf¹_t,!Pe¹_t. 3. The topological Laplace transformationF^(0;1)_top

3.1 ±Reminder on the local Laplace transformationF^(0;1)

We keep the notation as in the introduction, and we will focus on the coordinatet, so that the Laplace kernel is now exp( t=t). Moreover, since we only want to

deal with F^(0;1)(M), we only consider the localized Laplace transform C[t⁰;t^{0 1}]_C[t⁰_] , that we will denote by from now on. We consider the fol- lowing diagram:

The localized Laplace transform is defined by the formula

ˆbp‡(p^‡ME ^t=t):

We can regard the C[t;t;t ¹]h@_t; @_ti-module p^‡ME ^t=t both as a holonomic D_P1

tA1t-module and as a meromorphic bundle with connection on P¹_t A¹_t with poles along the divisor D:ˆD0[D1[D in P¹_t A¹_t (with D0ˆ f0g A¹_t, D₁ˆ f1g A¹_tandD ˆP¹_t fc1g). It has irregular singularities alongD. As indicated in §1.2, the germ of attˆ0 only depends on the germMofMattˆ0, and it is denoted byF^(0;1)(M).

Letbr:h7!h ˆtbe a ramification such thatbr^‡ is non-ramified at infinity, that is,

C……h†† _C…fhg†br^‡F^(0;1)(M)' M

2 … †

(E ^(h)Rb );

withExp( )d h ¹C[h ¹]. Thenbr^‡F^(0;1)(M) is obtained through the diagram

by the formula

br^‡F^(0;1)(M)ˆC…fhg† _{C[h;h 1]}bp‡(p^‡ME ^{t= (h)}):

Moreover, by the uniqueness of the Levelt-Turrittin decomposition, we have, for each 2m (bp-th root of the unity)

Rb ˆRb ; with (h):ˆ ( ¹h):

The stationary phase formula of [Fan09, Sab08] makes explicit the correspondence (p;Exp;(R_')_'2Exp)7!(bp;Exp;d (Rb ) ₂ ).

3.2 ±The monodromy ofF^(0;1)M

Let us denote by Lc the local system attached toF^(0;1)MonS¹_tˆ0. The local system Lc can equivalently be regarded as a local system Fc onG^an_m;t. Let us denote byTtopthe topological monodromy ofMaround the origin, and byTbtopthe topological monodromy of F^(0;1)M around tˆ0 (with the notation as in § 1.2).

Since has no singularity except at 0;1,Tb_top¹is the topological monodromy of around t⁰ˆ0. Therefore, there are various ways of calculating the topological monodromyTb_top:

either by comparing it with the monodromy attˆ 1ofM, which is nothing, up to changing orientation, but the topological monodromy of M attˆ0, sinceMhas no singular point on (A^1;an_t ), and its singular point attˆ 1is regular; this will be done in Proposition 3.2.1;

or by a direct topological computation attˆ 1; this is done in § 6.3;

lastly, by obtaining it from the Stokes data attˆ 1, once we have computed them; this is however not the most economical way.

Given an automorphismT, we will denote byT_l (l2C) the component of T with eigenvaluel. Since has a regular singularity attˆ 1, with monodromy equal toT_top¹, classical results give:

PROPOSITION 3.2.1. For l6ˆ1, we have Tbtop;lˆTtop;l. Moreover, if M is a successive extension of germs of rank-one meromorphic connections, the Jordan blocks of size k52 of Tb_top;1 are in one-to-one correspondence with the Jordan blocks of size k 1of T_top;1.

SKETCH OF PROOF. Only for this proof, we denote by the non localized Laplace transform ofM. It has a regular singularity att⁰ˆ0. The monodromyT_top¹ ofMat infinity is known to be equal to the monodromy of the vanishing cycles of at t⁰ˆ0, while Tb_top¹ is the monodromy of the nearby cycles of att⁰ˆ0 (cf. e.g.

[Sab06, Prop. 4.1(iv)] withzˆ1). The first assertion follows. For the second as- sertion, note that, if M is a rank-one meromorphic connection, thenM is irre- ducible as aC‰tŠh@ti-module, and thus is also irreducible, so is a minimal exten- sion att⁰ˆ0, which implies the second assertion forM. In general, note that in a exact sequence 0! ⁰! ! ⁰⁰!0, if the extreme terms are minimal ex- tensions att⁰ˆ0, then so is the middle term. This concludes the proof of the second

assertion. p

As in Remark 2.3.2, let us consider the real blowing-up map$:Pe¹_t !P¹_t of P¹_t at tˆ0 and tˆ 1 (so that Pe¹_t is a closed annulus) and let us denote by eb

:Pe¹_t G^an_m;t!G^an_m;t the projection. Then we have

Fc'R¹ebDR^{mod (D0[D1)}(p^‡ME ^t=t) (3:2:2)

and all otherR^kebvanish. We note that, fort6ˆ0, twisting withE ^t=thas no effect near tˆ0. Similarly, sinceMis regular at1,E ^t=t is the only important factor alongtˆ 1.

Computation. Let us use the notation of Remark 2.3.2. Let us fixt_o2Cand let us compute the fibreFcto. Recall thatPe¹_t is a closed annulus with boundary components S¹_tˆ0 andS¹_tˆ1. L etF₄₀ be the sheaf H⁰DR^{mod 0}M on the semi-closed annulus Pe¹_tⁿⁿS¹_tˆ1. We denote by (Pe¹_t)_4to1 the closed annulusPe¹_t with the closed interval arg t argto2[ p=2;p=2] mod 2pin S¹_tˆ1deleted. This is the domain where the function exp(t=to) has rapid decay. We consider the inclusions

Pe¹_tⁿⁿS¹_tˆ1,!a¹_to

(Pe¹_t)_4to1,!b¹_to Pe¹_t: We then have (similarly to [Sab13a, §7.3]):

PROPOSITION 3.2.3. The fibre Fc_to is equal to H¹(Pe¹_t;b¹_to;!a¹_to;F₄₀) and the other H^k vanish. It is isomorphic to the vanishing cycle space at tˆ0 of the perverse sheaf$F40onA¹_t. The monodromy ofFcwith respect totis obtained by rotating the interval (p=2;3p=2)‡argt_oS¹_tˆ1in the counterclockwise direction

with respect to t_o. p

REMARK3.2.4. We can similarly compute the fibre of ebr ¹Fc at any h_o by replacing argto with bpargh_o in the formula above.

3.3 ±The Stokes structure ofF^(0;1)M

Let us denote by Lc the Stokes filtration of Lc, regarded as a family of subsheaves ofebr ¹(Lc). Due to the ramificationbr, we now work with the variableh.

Let P

g

¹_t A¹_h be the real oriented blowing-up of P¹_t A¹_h along the components D0;D1;D of Dregarded now inP¹_t A¹_h. We have a similar diagram (see e.g.

[Sab13a, Chap. 8]):

We note thatP

g

¹_t A¹_his the productPe¹_t Af¹hof the closed annulusPe¹_t by the semi- closed annulusAf¹h. One defines on such a real blown-up space the sheaf of holo- morphic functions with rapid decay near the boundary (i.e., the inverse image by the real blow-up map of the divisorD). There is a corresponding de Rham complex that we denote by DR^modD(see loc. cit.).

THEOREM 3.3.1 (Mochizuki, [Moc14, Cor. 4.7.5]). For each 2h ¹C[h ¹] the natural morphism

DR^mod br^‡F^(0;1)(M) E ^(h)

!RebDR^modD(p^‡ME ^{(h) t= (h)})[1]

is a quasi-isomorphism and the natural morphism

R¹ebDR^modD(p^‡ME ^{(h) t= (h)})!b|R¹ebDR^{mod (D0[D1)}(p^‡ME ^{t= (h)}) is injective.

REMARKS3.3.2 (1) The result of [Moc14, Cor. 4.7.5] refers to the real blow-up P¹_t Af¹halongD , not along all the components ofD. The above statement is obtained by using that, with respect to$:P

g

¹_t A¹_h!P¹_t Af¹h, we have

R$DR^modD(p^‡ME ^{t= (h)})ˆDR^modD (p^‡ME ^{t= (h)}):

This follows from the identificationR$A^modD ˆA^mod_P1 ^D

t (D0[D1),

as in [Sab13a, Prop. 8.9].

(2) Setbr(h)ˆch (1‡o(h)) and let us choose abp-th rootc¹⁼ (1‡o(h)) ofbr(h)=h . Then thebp-th roots ofbr(h) are writtenbr¹⁼ (h)ˆ hc¹⁼ (1‡o(h)) ( 2m ).

Now, for 2h ¹C[h ¹], we set (h):ˆ br¹⁼ (h) . Them -equivariance is then induced by the isomorphism

‡E ^{(h) t= (h)}ˆE ; where we still denote by the maph7!br¹⁼ (h).

The theorem expresses therefore the Stokes filtration as the push-forward byeb of a family of complexes indexed byExp. The questions stated in the introductiond reduce now to

expressing, for any 2Exp, the complex DRd ^modD(p^‡ME ^{(h) t= (h)}), to- gether with the isomorphisms

1DR^modD(p^‡ME ^{(h) t= (h)})'DR^modD(p^‡ME );

in terms of the Stokes-filtered local system attached toM,

computing in a topological way the push-forward byebonce this complex is well understood.

While H⁰DR^modD is easy to compute from the data (F ;L), it happens in general that the complex DR^modD has higher cohomology, which is not easy to express in terms of (F ;L). The main reason is that the meromorphic connection

p^‡ME ^{(h) t= (h)}may not be good (in the sense of [Sab00]) at (0;c1), due to in- determinacy 0=0 of the functions

v_'; (u;h):ˆ (h) r(u)=br(h) '(u) (3:3:3)

foru!0 andh!0 ('2Exp, 2Exp).d

PROPOSITION 3.3.4. There exists a projective modification e:Z!P¹_t A¹_h which consists of a succession of point blowing-ups above(0;c1), such that for each 2Exp, the pull-backd p^‡ME ^{(h) t= (h)} is good along the normal crossing

divisor DZ:ˆe 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 '2Exp(M) and 2Exp( ), 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 D_Z as the union of the strict transforms ofD₀;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, Chap. 20], [Sab13a, Th. 12.5 & Cor. 12.7]). We consider the real blow-up space Ze of Z along the components of D_Z and the natural lift ee:Ze !P

g

¹_t A¹_h of e:Z!P¹_t A¹_h.

THEOREM3.3.5. If Z is as in Proposition3.3.4, then the complex DR^modDZe^‡(p^‡ME ^{(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 becauseeis a projective modification, to get:

ReeDR^modDZe^‡(p^‡ME ^{(h) t= (h)})ˆDR^modD(p^‡ME ^{(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 Proposition3.3.4.Then, for each 2h ¹C[h ¹]the natural morphism

DR^{mod ‡}F^(0;1)(M) E ^(h)

!R¹(ebee)H⁰DR^modDZe^‡(p^‡ME ^{(h) t= (h)}†

is a quasi-isomorphism and the natural morphism

R¹(ebee)H⁰DR^modDZe^‡(p^‡ME ^{(h) t= (h)})!ebr ¹Fb

is injective. p

The pull-back (bpe) ¹(fhˆ0g)ˆe ¹(D ) is the unionD [E. Its pull-back in Z, denoted bye De [E, is equal to (ebe ee) ¹(S¹_hˆ0). We will denote byeb|the open inclusionZeⁿⁿ(De [Ee),!Z.e

Let r:u7!u^pˆt be a ramification such thatr^‡M is non-ramified at tˆ0.

Consider the fibre productZe⁰:ˆP

g

¹_uA¹_h

Ze as a topological space:

Let us writeDe⁰_Z:ˆes ¹(De_Z) andeb|⁰:Ze^{0 nn}es ¹(De [E)e ,!Ze⁰. Note that the restric- tion ofee⁰toZe^{0 nn}De⁰_Zis a homeomorphism.

Let$:Ze !Zbe the oriented real blow-up map. Below we use the convention of Remark 2.3.2, and forez⁰2Ze⁰we set

euˆ(ep⁰ee⁰) (ez⁰)2Pe¹_u; Exp ˆ Exp if eu2S¹_uˆ0; 0 if eu2P¹_uⁿⁿS¹_uˆ0: (

and, as usual, for a pointez⁰2Ze⁰overz:ˆ$es(ez⁰)2Zand a functiong(u;h):

e⁰'4 e⁰g()Re e⁰' e⁰g

<0 in some neighb:ofez⁰

ore⁰' e⁰g is bounded in some neighb:of ($ es) ¹(z):

Recall that the Stokes filtration (F ⁰;F ⁰₄ ) on the local systemF ⁰:ˆer ¹F satisfies the equivariance conditionz ¹F ⁰_4'ˆF ⁰_4'z insidez ¹F ⁰ˆF ⁰for eachz2m_p.

DEFINITION3.3.7. (1) The sheafG⁰onZe⁰is defined aseb|⁰G⁰, whereG⁰ is the subsheaf of (pe⁰ee⁰) ¹F ⁰ on Ze^{0 nn}(es ¹(De [E)) whose germ ate ez⁰2Ze^{0 nn}(es ¹(De [E)) is defined bye

G⁰ :ˆ X

F ⁰_4'; F ⁰:

(2) The sheafG₄⁰ is the subsheaf ofG⁰ which coincides with the latter away fromDeZ⁰ and whose germG₄⁰ _; at eachez⁰2Ze⁰is defined by the formula