Moduli of Stokes torsors and singularities of differential equations

Let M be a meromorphic connection with poles along a smooth divisor D in a smooth algebraic variety. Let Sol M be the solution complex of M. We prove that the good formal decomposition locus of M coincides with the locus where the restrictions to D of Sol M and Sol End M are local systems. By contrast to the very different natures of these loci (the first one is defined via algebra, the second one is defined via analysis), the proof of their coincidence is geometric. It relies on the moduli of Stokes torsors.

The problematic of this paper is to understand how the geometry of the Stokes phenomenon in any dimension sheds light on the interplay between the singularities of a differential equation and the singularities of its solutions.
Consider an algebraic linear system M of differential equations with n variables where Ω i is a square matrix of size r with coefficients into the ring Crx 1 , . . ., x n srx ´1 n s of Laurent polynomials with poles along the hyperplane D in C n given by x n " 0. At a point away from D, the holomorphic solutions of the system M are fully understood by means of Cauchy's theorem.At a point of D, the situation is much more complicated.It is still the source of challenging unsolved problems.We call D the singular locus of M. Two distinguished open subsets of D where the singularities of M are mild can be defined.First, the set GoodpMq of good formal decomposition points of M is the subset of D consisting of points P at the formal neighbourhood of which M admits a good decomposition.For P being the origin, and modulo ramification issues that will be neglected in this introduction, this means roughly that there exists a base change with coefficients in C x 1 , . . ., x n rx ´1 n s splitting M as a direct sum of well-understood systems easier to work with.
Good formal decomposition can always be achieved in the one variable case [Sv00].It is desirable in general because it provides a concrete description of the system, at least formally at a point.In the higher variable case however, it was observed in [Sab00] that M may not have good formal decomposition at every point of D. Thus, the set GoodpMq is a non trivial invariant of M. As proved by André [And07], the set GoodpMq is the complement in D of a Zariski closed subset F of D either purely of codimension 1 in D or empty.Traditionally, F is called the Turning point locus of M, by reference to the way the Stokes directions of M move along a small circle in D going around a turning point.In a sense, the good formal decomposition locus of M is the open subset of D where the singularities of the system M are as simple as possible.
To define the second distinguished subset of D associated to M, let us view M as a D-module, that is a module over the Weyl algebra of differential operators.Let us denote by Sol M the solution complex of the analytification of M. Concretely, H 0 Sol M encodes the holomorphic solutions of our differential system while the higher cohomologies of Sol M keep track of higher Ext groups in the category of D-modules.As proved by Kashiwara [Kas75], the complex Sol M is perverse.From a theorem of Mebkhout [Meb90], the restriction of Sol M to D, that is, the irregularity complex of M along D, denoted by Irr D M in this paper, is also perverse.In particular, pSol Mq |D is a local system on D away from a closed analytic subset of D. The smooth locus of pSol Mq |D denotes the biggest open in D on which pSol Mq |D is a local system.In a sense, the smooth locus of pSol Mq |D is the open subset of D where the singularities of the (derived) solutions of M are as simple as possible.
As observed in [Tey13], the open set GoodpMq is included in the smooth locus of pSol Mq |D and pSol End Mq |D , and the reverse inclusion was conjectured in [Tey13,15.0.5].Coincidence of GoodpMq with the smooth locus of pSol Mq |D and pSol End Mq |D seems surprising at first sight, since goodness is an algebraic notion whereas Sol M is transcendental.The main goal of this paper is to prove via geometric means the following Theorem 1. -The good formal decomposition locus of an algebraic meromorphic connection M with poles along a smooth divisor D in a smooth algebraic variety is exactly the locus of D where pSol Mq |D and pSol End Mq |D are local systems.
Other criteria detecting good points of meromorphic connections are available in the literature.Let us mention André's criterion [And07,3.4.1] in terms of specialisations of Newton polygons.Let us also mention Kedlaya's criterion [Ked10,4.4.2] in terms of the variation of spectral norms under varying Gauss norms on rings of formal power series.This criterion is numerical in nature.By contrast, the new criterion given by Theorem 1 is transcendental.
The main tool at stake in the proof of Theorem 1 is geometric, via moduli of Stokes torsors [Tey17].For a detailed explanation of the line of thoughts that brought them into the picture, let us refer to 2.1.In this introduction, we explain how these moduli are used by giving the main ingredients of the proof of Theorem 1 in dimension 2. In that case, we have to show the goodness of a point 0 P D provided we know that pSol Mq |D and pSol End Mq |D are local systems in a neighbourhood of 0. The main problem is to extend the good formal decomposition of M across 0. This decomposition can be seen as a system of linear differential equations N defined in a neighbourhood of a small disc ∆ ˚of D punctured at 0.
To show that N extends across 0, we first construct via Stokes torsors a moduli space X parametrizing very roughly systems defined in a neighbourhood of ∆ and formally isomorphic to M along ∆.A distinguished point of X is given by M itself.Similarly, we construct a moduli space Y parametrizing roughly systems defined in a neighbourhood of ∆ ˚and formally isomorphic to M |∆ ˚along ∆ ˚.Two distinguished points of Y are M |∆ ˚and N .Restriction from ∆ to ∆ ˚provides a morphism of algebraic varieties res : X ÝÑ Y.The problem of extending N is then the problem of proving that res hits N .The moduli X and Y have the wonderful property that the tangent map T M res of res at M is exactly the map Γp∆, H 1 Sol End Mq ÝÑ Γp∆ ˚, H 1 Sol End Mq associating to s P Γp∆, H 1 Sol End Mq the restriction of s to ∆ ˚.In this geometric picture, the smoothness of pH 1 Sol End Mq |D around 0 thus translates into the fact that T M res is an isomorphism of vector spaces.Since X and Y are smooth, we deduce that res is étale at the point M. Thus, the image of res in Y contains a non empty open set.We prove furthermore that res is proper, so its image is closed in Y. Since Y is irreducible, we conclude that res is surjective, which proves the existence of the sought-after extension of N .
As a by-product of the tools developed to prove Theorem 1, we show furthermore the following rigidity result refining [Tey17, Th 3].In a sense, it says that at a singular point of a divisor, the existence of a non trivial Stokes structure is an exceptional phenomenon Theorem 2. -Let N be a good unramified split meromorphic flat bundle in a neighbourhood of the origin in C n .If the pole locus of N has at least two components, and if N is very general, then N itself is the only germ of good meromorphic flat bundle formally isomorphic to N at 0.
In this statement, very general means roughly that the residues of each regular constituent contributing to N lie away from a countable union of strict Zariski closed subsets in an affine space.The main idea to prove Theorem 2 is to show that under the genericity assumption, the moduli of Stokes torsors of N has dimension 0 and is connected.It is thus reduced to a point.
A last application of the tools developed to prove Theorem 1 deals with degenerations of irregular singularities.Let X be a smooth algebraic variety and let D be a germ of smooth divisor at 0 P X.Let M be a germ of meromorphic connection defined in a neighbourhood of D in X and with poles along D. Motivated by Dubrovin's conjecture and the study of Frobenius manifolds, Cotti, Dubrovin and Guzzetti [CDG17] studied how much information on the Stokes data of M can be retrieved from the restriction of M to a smooth curve C transverse to D and passing through 0.
Under the assumption that M p D splits as a direct sum of regular connections twisted by meromorphic functions a 1 , . . ., a n P O X p˚Dq with simple poles along D, they proved that the Stokes data of the restriction M |C determine in a bijective way the Stokes data of M in a small neighbourhood of 0 in D. This is striking, since the numerators of the a i ´aj may vanish at 0, thus inducing a discontinuity at 0 in the configuration of the Stokes directions.Using different methods, this was reproved by Sabbah in [Sab17b,Th 1.4].If X is a surface, we give a short conceptual proof of a stronger version of Cotti, Dubrovin and Guzzetti's injectivity theorem, which generalizes it in several directions: we don't make any assumption on the shape of M p D , nor suppose that D is smooth, nor assume that C is transverse to D. The price to pay for this generality is the use of resolution of turning points in dimension 2, as proved in the fundamental work of Kedlaya [Ked10] and Mochizuki [Moc09].The intuition that the techniques developed in this paper could be applied to the questions considered by Cotti, Dubrovin and Guzzetti is due to C. Sabbah.
To state our result, let us recall that a M-marked connection is the data of a couple pM, isoq where M is a germ of meromorphic connection with poles along D defined in a neighbourhood of D in X, and where iso : Theorem 3. -Let X be a smooth algebraic surface, let 0 P X and let D be a divisor defined in a neighbourhood of 0. Let M be a germ of meromorphic connection at 0 and with poles along D. Let C be a smooth curve passing through 0 and not contained in any of the irreducible components of D. If pM 1 , iso 1 q and pM 2 , iso 2 q are M-marked connections such that pM 1 , iso 1 q |C » pM 2 , iso 2 q |C then pM 1 , iso 1 q and pM 2 , iso 2 q are isomorphic in a neighbourhood of 0.
Let us give an outline of the paper.In section 1, we recall the Level filtration for the Stokes sheaf in any dimension.We then apply it to prove Theorem 2. In section 2, we introduce the global variant of the moduli of Stokes torsors constructed in [Tey17] suited for the proof of Theorem 1.We then prove Theorem 3. In section 3, we show how to reduce the proof of Theorem 1 to the dimension 2 case.We then show in dimension 2 that Theorem 1 reduces to extending the good formal model of M across the point 0 under study.In the last section, we show that the sought-after extension exists provided the moduli of Stokes torsors associated to a resolution of the turning point 0 for M satisfies suitable geometric conditions.Finally, we show that these geometric conditions are always satisfied when the hypothesis of Theorem 1 are satisfied, thus concluding the proof of Theorem 1.
for Mathematics, Bonn.We thank the Hausdorff Institute for providing outstanding working conditions.

Level filtration and application
We first introduce some notations and recall some definitions.A reference for good meromorphic flat bundles is Part I, Chapter 2 from [Moc11b].For basics concerning Stokes torsors in any dimension, we refer to [Tey17].
1.1.Irregular values and truncation.-Let D be the germ of normal crossing divisor at 0 P C n given by z 1 ¨¨¨z m " 0. We endow Z m with the order given by m ď m 1 if and only if m i ď m 1 i for every i " 1, . . .m.For a P O C n p˚Dq{O C n , we write a " ř mPZ m a m z m and denote by ord a the minimum of tm P Z m ď0 such that a m ‰ 0u when it exists.
Let I be a good set of irregular values with poles contained in D. By definition, I is a subset of O C n p˚Dq{O C n such that -For every non zero a P I, ord a exists and a ord a is invertible in a neighbourhood of 0.
-For every distinct a, b P I, ord a ´b exists and pa ´bq ord a´b is invertible in a neighbourhood of 0.
-The set ΦpIq :" tord a ´b, a, b P I distinctu is totally ordered.
1.2.Real blow-up.-Let p : r X ÝÑ C n be the fiber product of the real blow-ups of C n along the z i " 0, i " 1, . . ., m.We have r X » pr0, `8rˆS 1 q m ˆCn´m and p reads ppr k , θ k q k , yq ÝÑ ppr k e iθ k q k , yq In particular, T :" p ´1p0q is a torus.Let π : R m ÝÑ T be the canonical projection.
1.3.Good unramified split bundle.-For every a P I, set E a " pO C n ,0 p˚Dq, dd aq.We fix once for all a germ of split unramified good meromorphic flat bundle of rank r with poles along D N :" where the R a are regular.Let i a : E a b R a ÝÑ N be the canonical inclusion and p a : N ÝÑ E a b R a the canonical projection.For i " 0, . . ., L `1, we set Ipiq :" ξ mpiq pIq and N piq :" The levels of N piq belong to tmp0q, . . ., mpi ´1qu.For α P Ipiq, we set The levels of N α belong to tmpiq, . . ., mpL `1qu.
1.4.The Stokes sheaf.-Let St N be the Stokes sheaf of N .This is a sheaf of complex unipotent algebraic groups over T. By definition, the germs of St N at θ P T are the automorphisms of N defined on small sectors emanating from 0 containing the direction θ and asymptotic to id at 0 along the direction θ. s ab has rapid decay.Since F a and F b have moderate growth at 0, we deduce that the constant matrix ιpsq ab has rapid decay.Hence, ιpsq ab " 0. Thus St i N Ă ϕpSt N piq q.On the other hand, let s P ϕpSt N piq q and let c with ord c ą mpi ´1q.We have to show that for every a, b P I with a ‰ b, e c s ab " e c`a´b F a ϕpsq ab F ´1 b has rapid decay.We can suppose ξ mpiq paq ă S ξ mpiq pbq.In particular a ă S b.Since the leading term of c `a ´b is the leading term of a ´b, the exponential e c`a´b has rapid decay on S. Thus, so does e c s ab .Hence, s P St i N and we deduce St i N " ϕpSt N piq q.
1.6.Quotients of the level filtration.-Lemma 1.6.1.-There is a split exact sequence of sheaves of algebraic groups Proof.-Let us define s ÝÑ ˆps ab q ξ mpiq paq"α From the local description of St N piq given in the proof of 1.5.1, we see that the only a priori non obvious thing to prove is the fact that ψ is a group homomorphism.
Let S be an open of T, let s, t P ΓpS, St N q, let α P Ipiq and let a, b P I such that ξ mpiq paq " ξ mpiq pbq " α.Let us denote by ψ α the component of ψ associated to α.Then If c ďm ‰ α, the leading coefficient of c ´a is that of ξ mpiq pcq ´ξmpiq paq " ξ mpiq pcq ´α.Hence, a ď S c if and only if α ă S ξ mpiq pcq.Similarly, c ď S b if and only if ξ mpiq pcq ă S α.Hence, for ξ mpiq pcq ‰ α, the condition a ď S c ď S b is empty.Thus pψ σ pstqq ab " ÿ cPI ξ mpiq pcq"α s ac t cb " pψ α psqψ α ptqq ab 1.7.Action of the fundamental group in the one level case.-We consider in this paragraph the case where N has a unique level m and we fix a smooth curve C passing through 0 as in 1.2.We denote by S 1 C Ă T the circle of directions in C emanating from 0. For an hyperplane H of R n and for an interval I of S 1 C , set TpH, Iq :" πpH `π´1 pIqq.For m P Z m ď0 , we set Tpm, Iq :" Tp For every x P T, the translation t x by x provides an isomorphism π 1 pTpm, 0q, 0q ÝÑ π 1 px `Tpm, 0q, xq.Hence, π 1 pTpm, 0q, 0q acts on St N C » pSt M q |S 1 C via the parallel transport.We deduce that π 1 pTpm, 0q, 0q acts on H 1 pS 1 C , St N C q.To simplify notations, we will denote by H 1 pS 1 C , St N C q π1 the set of invariants of the action of π 1 pTpm, 0q, 0q on For a connected open S Ă T, the path γ acts on ΓpS, LpN qq via a linear map ρpγq.The induced action on ΓpS, LpEnd N qq is the conjugation by ρpγq.
Lemma 1.7.1.-Let C be a smooth curve passing through the origin.For every cover I " pI i q iPZ{N Z of S 1 C adapted to N C , the morphisms in the commutative triangle are isomorphisms.In particular, H 1 pT, St N q is an affine space.
For a single level connection in dimension 1, an adapted cover is a cover by consecutive intervals with empty triple intersections such that every I P I and every pair of irregular values a, b, the interval I contains exactly one Stokes direction associated to a ´b.
Proof.-Since N has only one level, the same holds for Since St N has no non trivial global section on Tpm, I i q, the section t i extends uniquely into a section of T on Tpm, I i q.Hence, the cocycle corresponding to T S 1 C via (1.7.2) extends uniquely into a cocycle for T relative to the cover Tpm, Iq.In particular, the upper horizontal arrow of the diagram a connected open is determined by its germ at a point.Hence, the left vertical arrow of (1.7.3) is injective.We deduce that the upper horizontal arrow of (1.7.3) is a bijection and that res C is injective.Tautologically, the image of the left vertical arrow is exactly formed by those collections of g P Z 1 pI, St N C q extending to Tpm, Iq.These are exactly the invariants under the action of π 1 pTpm, 0q, 0q constructed in 1.7.
To conclude, we observe that St N being a sheaf of unipotent algebraic groups, the scheme ΓpU, St N q is an affine space for every open subset U Ă T. Since Z 1 pTpm, Iq, St N q is a product of such schemes, it is also an affine space.
1.8.Proof of Theorem 2. -We are now in position to prove Theorem 2. This is a local statement, so we work in a neighbourhood of 0 in C n and use notations from section 1.We argue recursively on the number of levels of N .Suppose that N has only one level.From 1.7.1, we know that H 1 pT, St N q is an affine space.From [Tey17, Th 3], we know that H 1 pT, St N q has dimension 0. Hence, H 1 pT, St N q is a point, so Theorem 2 is proved in the single level case.Suppose that N has at least two levels.Let pmp0q, . . ., mpLq, mpL `1qq be an auxiliary sequence for I.Then, there is an index i such that N piq has only one level and such that the number of levels of St Nα is strictly less than number of levels of N for every α P Ipiq.Since the N α are direct summands of N , they are also very generic.By recursion hypothesis applied to the N α , we obtain that the right term of the exact sequence of pointed sets Since N is very generic, so is N piq.Since N piq has only one level, H 1 pT, St N piq q is a point.This finishes the proof of Theorem 2.
2. Moduli of Stokes torsors.Global aspects 2.1.Why moduli of Stokes torsors?-Let us explain in this subsection how the moduli of Stokes torsors were found to be relevant to the proof of Theorem 1.We use the notations from the introduction and work in dimension 2. We suppose that 0 P D lies in the smooth locus of pSol Mq |D and pSol End Mq |D , and we want to prove that 0 is a good formal decomposition point for M.
From a theorem of Kedlaya [Ked10][Ked11] and Mochizuki [Moc09][Moc11b], our connection M acquires good formal decomposition at any point after pulling-back by a suitable sequence of blow-ups above D. To test the validity of the conjecture [Tey13, 15.0.5], a natural case to consider was the case where only one blow-up is needed.Using results of André [And07], it was shown in [Tey14] that the conjecture reduces in this case to the following Question.-Given two good meromorphic connections M and N with poles along the coordinate axis in C 2 and formally isomorphic at 0, is it true that It turns out that each side of (2.1.1)appeared as dimensions of moduli spaces of Stokes torsors constructed by Babbitt-Varadarajan in [BV89].These moduli were associated with germs of meromorphic connections in dimension 1. Babbitt and Varadarajan proved that they are affine spaces.This suggested the existence of a moduli X with two points P, Q P X such that the left-hand side of (2.1.1)would be dim T P X and the right-hand side of (2.1.1)would be dim T Q X .The equality (2.1.1)would then follow from the smoothness and connectedness of the putative moduli.This is what led to [Tey17], but the question of smoothness and connectedness was left open.In the meantime, a positive answer to the above question was given by purely analytic means by C. Sabbah in [Sab17a].

Relation with [Tey17].
-In [Tey17], a moduli for local Stokes torsors was constructed in any dimension.This moduli suffers two drawbacks in view of the proof of Theorem 1. First, the Stokes sheaf used in [Tey17] only makes sense at a neigbourhood of a point, whereas our situation will be global as soon as we apply Kedlaya-Mochizuki's resolution of turning points.Second, the relation between Irregularity and the tangent spaces of the moduli from [Tey17] only holds in particular cases.To convert the hypothesis on Irregularity appearing in Theorem 1 into a geometric statement on a moduli of torsors, we need to replace the Stokes sheaf St M of a connection M by a subsheaf denoted by St ăD M .We will abuse terminology be also calling the torsors under St Note that the only global moduli of Stokes torsors needed in this paper come from the case where X is a surface.Hence, this case is of independent interest regarding the general theory and thus deserves a special treatment.To keep the level of technicality as low as possible, we will thus stick to the case of surfaces.The general case will appear in a subsequent work, along with applications of different nature than the one we aim at in the present paper.
2.3.Geometric setup.-In this section, D denotes a normal crossing divisor in a smooth algebraic surface X.Let D 1 , . . ., D n be the irreducible components of D. For every sheaf of O X -module F, we set Let D i be the complement in D i of the singular locus SingpDq of D. Let p : r X ÝÑ X be the real blow-up of X along D. For every subset A Ă X, we set BA :" p ´1pAq X p ´1pDq and denote by ι A : BA ÝÑ BD the canonical inclusion.We denote by A the sheaf of functions on BD admitting an asymptotic development along D [Sab00].We denote by A ăD Ă A the sheaf of functions on BD with rapid decay along D. Concretely, this means the following.Let px 1 , x 2 q be local coordinates such that D is defined locally by x 1 x i " 0 with i P t1, 2u.Then, the germ of A ăD at θ P B0 is given by those holomorphic functions u defined over the trace on XzD of a neighbourhood Ω of θ in r X, and such that for every compact K Ă Ω, for every N 1 , N i P N, there exists a constant C N1,Ni ą 0 such that 2.4.Definition of the moduli.-Let M be a good meromorphic connection defined in a neighbourhood of D with poles along D. We set We define St ăD M as the subsheaf of H 0 DR B End M of sections asymptotic to the Identity along D, that is of the form Id `f where f has coefficients in A ăD .
The sheaf St ăD M is a sheaf of complex unipotent algebraic groups.In particular, St ăD M pRq is defined as a sheaf of groups on BD for every R P C-alg.For every subset A Ă D, we denote by H 1 pBA, St ăD M q the functor 2.5.Representability by a scheme.-The purpose of this subsection is to prove that H 1 pBD, St ăD M q is representable by an affine scheme of finite type over C. To do this, the idea is to analyse separately the contributions coming from each stratum of D. On the smooth locus of D, representability will essentially be a consequence of Babbitt-Varadarajan's works [BV89].At a singular point P of D, representability will be achieved by comparison with the situation on a well-chosen component passing through P .
Let P P SingpDq and let D i be a component of D containing P .Then, there exists a disc ∆ Di,P Ă D i centred at P such that any ι ´1 P St ăD M -torsor extends above ∆ Di,P .
Set ∆ Di,P " ∆ Di,P ztP u.Let j i,P : B∆ Di,P ÝÑ BD be the canonical inclusion.Hence, there is a canonical morphism of functors (2.5.1) On the other hand, restriction of torsors provides a morphism of functors (2.5.2) The collection of morphisms (2.5.1) and (2.5.2) defines a finite diagram of functors.Since Stokes torsors have no non trivial automorphisms [Tey17, 1.8.1], the limit of this diagram is H 1 pBD, St ăD M q.In particular, to understand H 1 pBD, St ăD M q amounts to understand what happens at a singular point of D and what happens on the smooth locus.
Lemma 2.5.3.-For every i " 1, . . ., n, the functor H 1 pBD i , St ăD M q is a scheme of finite type over C. The restriction morphism (2.5.2) is a closed immersion.
Proof.-Let R P C-alg.Since Stokes torsors have no non trivial automorphisms, the relative non abelian cohomology functor M pRqq is a sheaf of sets on D. From [Sab02, II 6.1] (see also [Mal83a] for the one level case), the restriction of R 1 p ˚St ăD M pRq to D i is a local system on D i whose stack at P P D i is H 1 pBP, St ăD M pRqq.Hence, for a ball B in D i , for every connected open set U Ă B and every P P U , restriction induces an identification From works of Babbit-Varadarajan [BV89], the functor H 1 pBP, St ăD M q is an affine space.Hence, the restriction of R 1 p ˚St ăD M to D i is a local system of schemes in the sense of [Sim94].Let P P D i .Since M q identifies with the invariants of the action of π 1 pD i , P q on H 1 pBP, St ăD M q.Since this action is functorial in R, Yoneda lemma implies that π 1 pD i , P q acts on the scheme H 1 pBP, St ăD M q via algebraic maps.Hence, its invariants form a closed sub-scheme in H 1 pBP, St ăD M q.In particular, H 1 pBD i , St ăD M q is a scheme.The fact that (2.5.2) is a closed immersion is an immediate consequence of the fact that H 1 pBD i , St ăD M q / / H 1 pBP, St ăD M q is a closed immersion.
Proposition 2.5.4.-For every P P SingpDq, the functor H 1 pBP, St ăD M q is a scheme of finite type over C.
Proof.-If M has only one irregular value at P , then H 1 pBP, St ăD M q is the trivial scheme so there is nothing to do.Suppose that M has at least two irregular values at P .Goodness implies that there is a component of D passing through P such that the difference of any two irregular values of M at P has poles along this component.Let D i be the other component of D passing through P .We take local coordinates px, yq such that D i is given by x " 0. For every T P H 1 pB∆ Di,P , St ăD M q, the sheaf ι ´1 P j i,P ˚T is a ι ´1 P j i,P ˚St ăD M -torsor on BP .So if we prove that the adjunction morphism (2.5.5) is an isomorphism, then ι ´1 P j i,P ˚will provide us with an inverse for (2.5.1), and 2.5.4 will be a consequence of 2.5.3.We now prove that (2.5.5) is an isomorphism.By a standard Galois argument, we can suppose that M is unramified.Injectivity of (2.5.5) is obvious so we are left to prove surjectivity.Since this is a local statement on BP , Mochizuki's asymptotic development theorem [Moc11b, 3.2.10]reduces the question to the case where M is split unramified.We thus treat that case and borrow the notations from 1.3.We put S " pr0, rrˆI 1 q ˆpr0, rrˆI 2 q where I 1 , I 2 are intervals.Sections of St ăD M on BS :" S X BD are automorphisms of M on S X pXzDq of the form Id `f where p a f i b " 0 unless (2.5.6) e a´b P ΓpBS, A ăD q Sections of St ăD M over BS i :" BS X B∆ Di,P " pt0u ˆI1 q ˆps0, rrˆI 2 q are automorphisms of M on S X pXzDq of the form Id `f where p a f i b " 0 unless (2.5.7) e a´b P ΓpBS i , A ăD q We thus have to show that for every distinct irregular values a, b, the conditions (2.5.7) and (2.5.6) are equivalent for a small enough choice of S. A change of variable reduces the problem to the case where a ´b " 1{x α y β where pα, βq P N ˆN˚.Since A ăD is a sheaf, condition (2.5.6) trivially implies condition (2.5.7).Suppose that e 1{x α y β P ΓpBS i , A ăD q.At the cost of shrinking S, this means that for every N P N, every ą 0, there exists a constant C ą 0 such that for px, yq P ps0, rrˆI 1 q ˆps , rrˆI 2 q we have |e 1{x α y β | ď C|x| N Writing x " pr 1 , θ 1 q and y " pr 2 , θ 2 q, this means In particular, α ą 0 and cospαθ 1 `βθ 2 q ă 0 for every pθ 1 , θ 2 q P I 1 ˆI2 .Let c ą 0 such that cospαθ 1 `βθ 2 q ă ´c on I 1 ˆI2 .Then, we have | ď e ´c{|x| α |y| β on S. Since α ą 0 and since β ą 0 from our choice of component D i , we deduce that (2.5.7) holds, which proves the equivalence between conditions (2.5.7) and (2.5.6).
Putting 2.5.3 and 2.5.4 together gives the following Proposition 2.5.8.-The functor H 1 pBD, St ăD M q is an affine scheme of finite type over C.
We have the following Proposition 2.5.9.-For very P P SingpDq and every component D i of D passing through P , the restriction morphism (2.5.1) is a closed immersion.
Proof.-We can suppose that M is unramified in a neighbourhood of P .If not all the two by two differences of M's irregular values at P have poles along D i , then the proof of 2.5.4 shows that (2.5.1) is an isomorphism, so 2.5.9 is true in that case.Let us suppose that the difference of any two distinct irregular values for M at P has poles along D i .Let St M P be the Stokes sheaf of M on BP as defined in [Tey17].Then, St ăD M is distinguished in St M P .We thus have an exact sequence of sheaves of algebraic groups on BP At the cost of restricting ∆ Di,P , any section of St M P extends to B∆ Di,P .We thus have an adjunction morphism (2.5.10)St M P / / ι ´1 P j i,P ˚j´1 i,P St M P " ι ´1 P j i,P ˚j´1 i,P St ăD M Hence, there is a factorization (2.5.11) From an argument similar to that in 2.5.4, the map (2.5.10) is an isomorphism of sheaves on BP .Hence, the vertical arrow in (2.5.11) is an isomorphism of schemes.To prove 2.5.9, it is enough to prove that there is an exact sequence of pointed functors (2.5.12) The complex of sheaves which is surjective at the level of R-points for every R P C-alg.From [BV89, 2.7.3], the morphism (2.5.16) admits a section.Composing this section with gives rise to a commutative triangle of functors The algebraic group ΓpU, Qq acts on Z 1 pU, Qq.Let (2.5.17)G ÝÑ Z 1 pU, Qq be the morphism of schemes obtained by restricting the action of G to the trivial cocycle.Since H 0 pBP, Qq » 0, the morphism (2.5.17) is a monomorphism.The diagram (2.5.15) thus splits into cartesian diagrams of functors We are thus left to show that (2.5.17) is a closed immersion.From the general theory of algebraic group actions, (2.5.17) factors into Let pmp0q, . . ., mpLq, mpL `1qq be an auxiliary sequence for the good set of irregular values of M at P .If i denotes the smallest index for which mpiq does not have poles along every component of D passing through P , then St ăD M|BP " St M P piq .From the description 1.6.1 of the quotients of the level filtration on St M P , we deduce that Q is a product of Stokes sheaves.Since Stokes sheaves are sheaves of unipotent algebraic groups, we conclude that G is a unipotent algebraic group.This concludes the proof of 2.5.9.
Since H 1 pBD, St ăD M q is a limit of morphisms of the form (2.5.1) and (2.5.2), lemmas 2.5.3 and 2.5.9 give the following Corollary 2.5.18.-For every P P D, the restriction morphism 2.6.Stokes torsors and marked connections.-Let us recall that a Mmarked connection is the data of a couple pM, isoq where M is a germ of meromorphic connection with poles along D defined in a neighbourhood of D in X, and where iso : M p D ÝÑ M p D is an isomorphism of formal connections.We denote by Isom iso pM, Mq the St ăD M pCq-torsor of isomorphisms between BM and BM which are asymptotic to iso along D.
The proof of the following statement was suggested to me by T. Mochizuki.I thank him for kindly sharing it.When D is smooth, It it was known to Malgrange [Mal83b].See also [Sab02,II 6.3].
Lemma 2.6.1.-The map associating to every isomorphism class of M-marked connection pM, isoq the St ăD M pCq-torsor Isom iso pM, Mq is bijective.
Proof.-Let us construct an inverse.Take T P St ăD M pCq and let g " pg ij q be a cocycle for T associated to a cover pU i q iPI of BD.Let L be the Stokes filtered local system on BD associated to M. Set L i :" L |Ui .Then, g allows to glue the L i into a Stokes filtered local system L T on BD independent of the choice of g.From the irregular Riemann-Hilbert correspondence [Moc11a,4.11],L T is the Stokes filtered local system associated to a unique (up to isomorphism) good meromorphic connection M T defined in a neighbourhood of D and with poles along D. By construction, the isomorphism L T |Ui ÝÑ L |Ui corresponds to an isomorphism BM T |Ui ÝÑ BM |Ui .We thus obtain a formal isomorphism iso i : BM T , p D|Ui, ÝÑ BM p D|Ui .On U ij , the discrepancy between iso i and iso j is measured by the asymptotic of g ij along D. By definition, this asymptotic is Id.Hence, the iso i glue into a globally defined isomorphism BM T , p D ÝÑ BM p D .Applying p ˚thus yields an isomorphism iso : It is then standard to check that the map T ÝÑ pM T , isoq is the sought-after inverse.
2.7.Proof of Theorem 3. -We are now in position to prove Theorem 3, whose notations we use.Let π : Y ÝÑ X be a resolution of the turning point 0 for M. Such a resolution exists by works of Kedlaya [Ked10] and Mochizuki [Moc09].Set E :" π ´1pDq.At the cost of blowing up further, we can suppose that the strict transform C 1 of C is transverse to E at a point P in the smooth locus of E.
2.8.Obstruction theory and tangent space.-Let us compute the obstruction theory of H 1 pBD, St ăD M q at a point T 0 P H 1 pBD, St ăD N pCqq.We fix a morphism of infinitesimal extensions of C-algebras such that I is annihilated by Ker R 1 ÝÑ C. In particular, I 2 " 0 and I is endowed with a structure of C-vector space, which we suppose to be finite dimensional.Let T P H 1 pBD, St ăD M pRqq lifting T 0 .Choose a cover U " pU i q iPK of BD such that T comes from a cocycle g " pg ij q i,jPK .Set L i pRq :" Lie St ăD M pRq |Ui .The identifications allow to glue the L i pRq into a sheaf of R-Lie algebras over BD denoted by Lie St ăD M pRq T and depending only on T and not on g.For t " pt ijk q P Č2 pU, Lie St ăD M pRq T q, we denote by s ijk the unique representative of t ijk in ΓpU ijk , L i pRqq.Then pdtq ijkl " t jkl ´tikl `tijl ´tijk " rg ij s jkl g ´1 ij ´sikl `sijl ´sijk s We have the following Lemma 2.8.1.-There exists obpT q P I b C Ȟ2 pBD, Lie St ăD M pCq T0 q such that obpT q " 0 if and only if T lifts to H 1 pBD, St ăD M pR 1 qq.
Proof.-For every i, j P K, let h ij P ΓpU ij , St ăD M pR 1 qq be an arbitrary lift of g ij to R 1 .We can always choose the h ij to satisfy h ii " Id and h ij h ji " Id.Since Lie St ăD M pR 1 q is locally free, We will use both descriptions without mention.We set We now see how the second term of the last line above interacts with the second term of the left-hand side of (2.8.2).
where the last equality comes from I 2 " 0. Hence, the rs ijk s define a cocycle of I b C Lie St ăD M pCq T0 .An other choice of lift gives rise to homologous cocycles.We denote by obpT q the class of prs ijk sq ijk in Ȟ2 pBD, I b C Lie St ăD M pCq T0 q.It is standard to check that obpT q has the sought-after property.
Corollary 2.8.3.-Let pM, isoq be a M-marked connection.Then, the space H 2 pD, Irr D End M qq is an obstruction theory for H 1 pBD, St ăD M q at Isom iso pM, Mq.
Reasoning exactly as in [Tey17, 5.2.1], we prove the following Lemma 2.8.4.-For every M-marked connection pM, isoq, the tangent space of H 1 pBD, St ăD M q at pM, isoq identifies canonically to H 1 pD, Irr D End M q.
3. Moduli of Stokes torsors in the one level case 3.1.Roadmap.-The goal of this section is to describe the moduli of Stokes torsors in the case where the irregular values have only one level.To do this, we compare a relative version of the absolute Stokes groups from [MR91][Lod94] with the relative non abelian cohomology of the Stokes sheaf defined in 2.5.3.For the problem raised by this comparison in the multi-level case, we refer to 3.5.Note that over a smooth base (corresponding in this paper to the case where D is smooth), relative Stokes groups appeared in the one level case in [JMU81] and in more generality in [Boa02].In particular, over a smooth base, they were already considered in the multi-level case in [Boa14].
The reader interested only in the proof of Theorem 1 can skip this part, since it will not be used in the sequel.

Relative Stokes groups.
-We keep the setup and notations from 2.3 and 2.4.We recall that M stands for a good meromorphic connection defined in a neighbourhood of a normal crossing divisor D in an algebraic surface X and with poles along D. Let I be the sheaf of irregular values of M. We first suppose that M is unramified.In that case, I is a subsheaf of O X p˚Dq{O X .The set H ă ab is a smooth C 8 -hypersurface in BD.Let H ă be the union of all H ă ab , a, b P I distinct.Let ı : H ă ÝÑ BD be the inclusion.Let Sto M be the subsheaf of ι ˚ι´1 St ăD M whose germ at θ P BD is Sto M,θ " tg P St ăD M,θ such that for every a, b P I distinct, g ab " 0 unless θ P H ă ab u We call p ˚Sto M the relative Stokes group of M. For a possibly ramified connection M, we define the relative Stokes group of M via Galois descent from the unramified case.
Suppose that D is smooth.Then, for every a, b P I, the Stokes lines of pa, bq are parallel to the anti-Stokes lines of pa, bq.Hence, H ă ab does not meet any Stokes line of pa, bq.Thus, for any θ P BD and any g P Sto M,θ , the section g extends uniquely on a small product ∆ ˆI containing θ, where ∆ is a disc in D centred at ppθq and where I is an interval of S 1 .This product only depends on θ and not on g.We deduce that when D is smooth, p ˚Sto M is a local system on D.
where LR P is the isomorphism constructed by Loday-Richaud [Lod94, II 1.9], and where the upper arrow makes (3.3.2) commutative.To prove 3.3.1,we have to show that the identifications LR P glue into an isomorphism of local systems.This amounts to show that the LR P are compatible with the parallel transports of p ˚Sto M and R 1 p ˚St ăD M .That is, for every P, Q P D and every continuous path γ in D joining P to Q, the following diagram commutes where the horizontal arrows are the parallel transports along γ.This compatibility question is a local question on D. Let us thus suppose that P and Q belong to a small disc ∆ in D. By Galois descent, we can suppose that M is unramified.Via a local rectification B∆ » ∆ ˆS1 as in [Sab02, 6.8], the anti-Stokes hyperplanes of M above ∆ can be pictured as follows.Let us order the connected components of B∆ X H ă cyclically α 1 , . . ., α d and denote by α i pxq the point α i X Bx for every x P ∆.For ą 0 small enough and for i P Z{dZ , consider the open sector S i " ∆ˆsα i ´ , α i`1 ` r.Set S :" pS i q iPZ{dZ .Take g " pg i P Sto M,αipP q q iPZ{dZ .Since has been chosen small enough, g i can be seen as a section of St ăD M above sα i pP q´ , α i pP q` r.By definition, LR P pgq is the Stokes torsor on BP associated to the cocycle g P Z 1 pS X BP, St ăD M q.
The image of LR P pgq by the parallel transport of R 1 p ˚St ăD M is the restriction to BQ of the unique T P H 1 pB∆, St ăD M q such that T |BP " LR P pgq.But g extends uniquely into r g " pr g i q P Z 1 pS, St ăD M q.Thus, T is the Stokes torsor associated to r g.Hence, travelling down the diagram (3.3.3)produces the torsor over BQ associated to r g |BQ P Z 1 pS X BQ, St ăD M q.On the other hand, we observe that the parallel transport of g as a section of the relative Stokes sheaf is pr g i,αipQq q i .Since LR Q ppr g i,αipQq q i q is the Stokes torsor associated to r g |BQ , the commutativity of (3.3.3) is proved.-Suppose that M admits a unique level at P .Then, there is a canonical isomorphism Proof.-By Galois descent, we can suppose that M is unramified.We denote by m its level.Let us choose local coordinates centred at P and let us denote by C the diagonal.Then, M C admits only one level.Let I be a cover of S 1 C à la Loday-Richaud for M C .By definition, this is a cover by intervals with non empty triple intersection such that any of these interval contains exactly two consecutive anti-Stokes directions.An element of ΓpBP, Sto M q defines a cocycle in Z 1 pBP pm, Iq, St M q from which we deduce a St M -torsor on BP .The same construction holds on S 1 C .Hence, there is a commutative diagram (3.4.2) From [Lod94], the right vertical map of (3.4.2) is an isomorphism.Taking the invariants under the action of π 1 pTpm, 0q, 0q on the right part of (3.4.2) gives a commutative diagram From 1.7.1, the bottom arrow of (3.4.3) is an isomorphism.Hence, the left vertical arrow of (3.4.3) is an isomorphism.This concludes the proof of 3.4.1.
In simple cases, the previous lemmas tell precisely what the moduli of Stokes torsor looks like.
Corollary 3.4.4.-Suppose that M has rank 2.Then, H 1 pBD, St ăD M q is an affine space.
Proof.-It is enough to show that the morphisms (2.5.2) and (2.5.1) are linear inclusions of affine spaces.Relative Stokes groups are sheaves of unipotent algebraic groups.The underlying scheme of a unipotent algebraic group is an affine space.Then, corollary 3.4.4 is an immediate consequence of the description 3.3.1 and 3.4.1 of Stokes torsors in terms of relative Stokes groups in the one level case.

3.5.
A remark on the multi-level case.-In this subsection, we restrict to the case where D is smooth.The question whether R 1 p ˚St ăD M and p ˚Sto M are isomorphic seems to be fruitful, since it would imply that when D is smooth, moduli of Stokes torsors are affine spaces.This last assertion is known in dimension 1 from [BV89].We thus formulate the following Conjecture.-Suppose that D is smooth.Then, the local systems R 1 p ˚St ăD M and p ˚Sto M are isomorphic.
In the several level case, the main difficulty comes from the fact that the parallel transports for R 1 p ˚St ăD M and p ˚Sto M produce different cocycles that are not equal on the nose, but might be cohomologous.The following picture illustrates this phenomenon.The picture on the left features part of our initial element of the Stokes group above P .In this situation, two anti-Stokes lines L 1 and L 2 intersect once along the path joining P to Q. Let us call x the intersection point.Since anti-Stokes lines are parallel to Stokes lines, there is a neighbourhood Ω of x in BD not meeting any Stokes line coming from the differences of irregular values giving rise to L 1 and L 2 .In particular, g 1 and g 2 extend uniquely into sections r g 1 and r g 2 of St ăD M over Ω.When applying the parallel transport for R 1 p ˚St ăD M , we end up with the cocycle in the upper right picture.The bottom right picture represents the effect of the parallel transport for p ˚Sto M .Finally, one passes from one cocycle to the other by permuting r g 1 and r g 2 .Since the Stokes sheaf is not commutative, it is not a priori clear that these cocycles are cohomologous.
4. Reduction of Theorem 1 to extending the formal model 4.1.Reduction to the dimension 2 case.-In this subsection, we reduce the proof of Theorem 1 to the dimension 2 case.The main tool is André's goodness criterion [And07,3.4.3] in terms of Newton polygons.This reduction does not seem superfluous.Of crucial importance for the sequel of the proof (see 4.3.1)will be indeed the fact that for an unramified meromorphic connection M with poles along a divisor D and for a point 0 P D, the formal model of M splits on a small enough punctured disc around 0. This fact is specific to dimension 2, since it pertains to the property that turning points of connections in dimension 2 are isolated.
Lemma 4.1.1.-Theorem 1 is true in any dimension if it is true in dimension 2.
Proof.-Take n ą 2. We argue recursively by supposing that Theorem 1 holds in dimension strictly less than n and we prove that Theorem 1 holds in dimension n.Let 0 P D and suppose that Irr D M and Irr D End M are local systems in a neighbourhood of 0. If j : XzD ÝÑ X and i : D ÝÑ X are the canonical inclusions, we have distinguished triangle where L is a local system on the complement of D. Hence, the characteristic cycle of Sol M is supported on the union of T X X with T D X. From a theorem of Kashiwara and Schapira [KS90, 11.3.3],so does the characteristic cycle of M. Hence, any smooth hypersurface transverse to D and passing through 0 is non characteristic with respect to M in a neighbourhood of 0. Let us choose such a hypersurface Z and let i Z : Z ÝÑ X be the canonical inclusion.From [And07, 3.4.3], the turning point locus of M is a closed subset of D which is either empty or purely of codimension 1 in D. Since n ą 2, the hypersurface Z can consequently be chosen such that M and End M have good formal decomposition generically along Z X D. The connection i Z M is a meromorphic connection with poles along Z X D. It satisfies the hypothesis of Theorem 1 at the point 0. Indeed by Kashiwara's restriction theorem [Kas95], marked connection pN ˚, iso ˚q defined in 4.2 extends into a M-marked connection in a neighbourhood of 0. To do this, we need three preliminary lemmas.
Lemma 4.3.1.-Suppose that N ˚extends into a meromorphic flat bundle N defined in a neighbourhood of D in A 2 and with poles along D.Then, N is semi-stable at 0.
Proof.-It is enough to treat the case where K " Cpxq and d " 1.In that case, discussion 4.2 shows that on a neighbourhood Ω of D ˚in A 2 , we have where N å is a meromorphic connection on Ω with poles along D ˚and with single irregular value a.The open D ˆA1 retracts on the small neighbourhood on which N is defined.Since N is smooth away from D, we deduce that N extends canonically into a meromorphic connection on D ˆA1 with poles along D.
Let a P I.The restriction of the projector π a to the complement of D ˚in Ω is a flat section of End N .Since D ˚ˆA 1 retracts on Ω, parallel transport allows to extend π a canonically to D ˚ˆA 1 .We still denote by π a this extension.Hence, N å extends into a meromorphic connection on D ˚ˆA 1 with poles along D ˚.Let γ be a small loop in Ω going around the axis D y .By assumption, the monodromy of N along γ is trivial.Thus, π a is invariant under the monodromy of End N along γ.Hence, π a extends canonically to pD ˆA1 qzt0u.By Hartog's property, it extends further into a section a of End N on D ˆA1 .
Set N a :" a pN q Ă N for every a P I. We have 2 a " a and ř aPI a " Id N because these equalities hold on a non empty open set.Hence, N " ' aPI N a .Since a is flat, the connection on N preserves each N a .Let us prove that the N a are locally free as O DˆA 1 p˚Dq-modules.
Let E be a Deligne-Malgrange lattice [Mal96] for N .Since we work in dimension 2, we know from [Mal96, 3.3.2]that E is a vector bundle.We observe that a stabilizes E away from 0. By Hartog's property, we deduce that a stabilizes E. Hence, a pEq is a direct factor of E. So a pEq is a vector bundle.Thus, N a " a pN q " a pEp˚Dqq " p a pEqqp˚Dq is a locally free O DˆA 1 p˚Dq-module of finite rank with connection extending N å .To prove 4.3.1,we are thus left to consider the case where I " tau.
If I " tau, then [And07, 3. Hence, ord y apP q " ord y a for every a P I.In particular, the coefficient function of the highest power of 1{v contributing to a P I does not vanish at P .Arguing similarly for End N , we obtain that N has good formal decomposition at 0. Proof.-From 4.3.1, the extension N is semi-stable at 0. From 4.3.3,we know that Irr D N and Irr D End N are local systems in a neighbourhood of 0. From 4.3.2,we deduce that N has good formal decomposition at 0. Hence, so does M.

Extension via moduli of Stokes torsors
5.1.A geometric extension criterion.-We keep notations from 4.2.We first relate moduli of Stokes torsors to the problem of extending marked connections.Let π : X ÝÑ C 2 be a resolution of the turning point 0 for M. Such a resolution exists by works of Kedlaya [Ked10] and Mochizuki [Moc09].Set E :" π ´1pDq and pick P P D ˚.Let Φ : H 1 pBE, St ăE π `M q / / H 1 pBP, St ăD M q be the restriction morphism of Stokes torsors.
Lemma 5.1.1.-Let pN ˚, iso ˚q be a M ˚-marked connection such that pN P , iso P q lies in the image of Φ.Then, pN ˚, iso ˚q extends into an M-marked connection in a neighbourhood of 0.
Proof.-From 2.6.1, any C-point of H 1 pBE, St ăE π `M q comes from a unique π `Mmarked connection.Hence, there exists pN 1 , iso 1 q P H 1 pBE, St ăE π `M q such that ΦpN 1 , iso 1 q " pN P , iso P q.From [Meb04, 3.6-4], the D-module N :" π `N 1 is a meromorphic connection with poles along D. By flat base change and similarly M p D » π `pπ `Mq p E .Hence, iso :" π `iso 1 defines an isomorphism between N p D and M p D .So pN , isoq is a M-marked connection.By definition, the germ of pN , isoq at P is pN P , iso P q.Since Rp ˚St ăD M is a local system on D ˚, we deduce pN |D ˚, iso |D ˚q " pN ˚, iso ˚q Hence, pN , isoq extends pN ˚, iso ˚q in a neighbourhood of 0. So 5.1.1 is proved.
Combining 4.3.4 with 5.1.1 and the following proposition will finish the proof of Theorem 1.
Proposition 5.1.2.-If the perverse complex Irr D End M is a local system on D, then Φ induces an isomorphism between each irreducible component of H 1 pBE, St ăE π `M q and H 1 pBP, St ăD M q. Proof.
-From [BV89], we know that H 1 pBP, St ăD M q is an affine space.Since affine spaces in characteristic 0 have no non trivial finite étale covers, it is enough to prove that Φ is finite étale.From 2.5.18, the morphism Φ is a closed immersion.We are thus left to show that Φ is étale.
Etale morphisms between smooth schemes of finite type over C are those morphisms inducing isomorphisms on the tangent spaces.Hence, we are left to prove that H 1 pBE, St ăE π `M q is smooth and that Φ induces isomorphisms on the tangent spaces.Let pM, isoq be a π `M-marked connection.From 2.8.3, an obstruction theory to lifting infinitesimally the Stokes torsor of pM, isoq is given by H 2 pE, Irr E End M q » H 2 pD, Irr D π `End M q » 0 The first identification expresses the compatibility of irregularity with proper pushforward.From 4.3.3applied to the End M-marked connection pπ `End M, π `isoq, the perverse complex Irr D π `End M is a local system in a neighbourhood of 0 and concentrated in degree 1.This implies the vanishing.Hence, H 1 pBE, St ăE π `M q is smooth at pM, isoq.From 2.6.1, any C-point of H 1 pBE, St ăE π `M q is of the form pM, isoq.
ăD M Stokes torsors.The sheaf St ăD M has the advantage of being globally defined when M is globally defined.Along the smooth locus of D, the sheaf St ăD M is the usual Stokes sheaf.The only difference between St M and St ăD M appears at a singular point of D.
For a, b P I, the function G a,b :" pa ´bq{|a ´b| induces a C 8 -function BG a,b on BD.The anti-Stokes lines of pa, bq are the connected components of H ă ab :" tθ P BD such that BG a,b pθq P R ´u

3. 3 .
One level along the smooth locus of D. -Lemma 3.3.1.-We suppose that D is smooth and that M admits a unique level.Then, the sheaf p ˚Sto M is canonically isomorphic to R 1 p ˚St ăD M .Proof.-Since D is smooth, the sheaves p ˚Sto M and R 1 p ˚St ăD M are local systems on BD.For every P P D, we have a diagram with canonical vertical arrows (3.3.2)

3. 4 .
One level at a singular point of D. -In this paragraph, we restrict our attention to what happens at a point P P SingpDq.Proposition 3.4.1.
Mq |ZXD and similarly for End M. Hence, Irr ZXD M and Irr ZXD End M are local systems in a neighbourhood of 0 in Z X D. By recursion hypothesis, i Z M is good at 0. In particular, the Newton polygon of i Z M at 0 (which is also the Newton polygon of M at 0) is the generic Newton polygon of i Z M along Z X D. From our choice for Z, the generic Newton polygon of i Z M along Z X D is the generic Newton polygon of M along D. Hence, the Newton polygon of M at 0 is the generic Newton polygon of M along D, and similarly with End M. By a theorem of André[And07, 3.4.1],we deduce that M has good formal decomposition at 0, which proves the reduction 4.1.1.4.2.Setup and recollections.-Fromnow on, we restrict to dimension 2. We use coordinates px, yq on A 2 and set D x :" ty " 0u, D y :" tx " 0u.Let D be a neighbourhood of 0 in D x and let CrDs be the coordinate ring of D. Set D ˚:" Dzt0u.Let M be an algebraic meromorphic flat bundle on a neighbourhood of D in A 2 with poles along D. In algebraic terms, M p D defines a CrDsppyqq-differential module.At the cost of shrinking D if necessary, we can suppose that the restriction M ˚of M to a neighbourhood of D ˚has good formal decomposition at every point of D ˚.There is a ramification v " y 1{d , d ě 1 and a finite Galois extension L{Cpxq such that the set I of generic irregular values for M lies in Frac Lpvq.If p : D L ÝÑ D is the normalization of D in L, the generic irregular values of M are thus meromorphic functions on D L ˆA1 v .We have (4.2.1) Lppvqq b M » à aPI E a b R a where the R a are regular.Following [And07, 3.2.4],we recall the following Definition 4.2.2.-We say that M is semi-stable at P P D if (1) We have I Ă CrD L s P ppvqq.(2) The decomposition (4.2.1) descends to CrD L s P ppvqq b M. In this definition, CrD L s P denotes the localization of CrD L s above P .This is a semi-local ring.Let π a P Lppvqq b End M be the projector on the factor E a b R a .As explained in [And07, 3.2.2], the point P is stable if and only if the generic irregular values of M and the coefficients of the π a in a basis of End M belong to CrD L s P ppvqq.Since M has good formal decomposition at any point of D ˚, the generic irregular values of M and the coefficients of the π a in a basis of End M belong to CrD L s P ppvqq for every P P D ˚.Hence, they belong CrD Lsppvqq where D L :" Dzp ´1p0q.Thus (4.2.3) CrD Lsppvqq b M » CrD Lsppvqq b N L where N L " à aPI E a b R a is a germ of meromorphic connection defined on a neighbourhood of D L in D L ˆA1 v and with poles along D L. The action of GalpL{Cpxqq ˆZ{dZ on the left-hand side of (4.2.3) induces an action on N L .Taking the invariants yields a meromorphic flat bundle N ˚defined on a neighbourhood Ω of D ˚in A 2 .By Galois descent, (4.2.3) descents to an isomorphism iso ˚between the formalizations of M ånd N ˚along D ˚. 4.3.Reduction to the problem of extending the formal model.-The goal of this subsection is to show that Theorem 1 reduces to prove that the M ˚-

H 1
3.1] implies a P CrDsppyqq.Hence, R :" E ´a b N p D is a formal meromorphic connection with poles along D. By assumption, R is generically regular along D. From [Del70, 4.1], we deduce that R is regular.Hence, N p D " E a bR with R regular, which concludes the proof of 4.3.1.Lemma 4.3.2.-Let N be a meromorphic flat connection with poles along D. We suppose that N is semi-stable at 0 and that Irr D N and Irr D End N are local systems in a neighbourhood of 0. Then, N has good formal decomposition at 0. Proof.-Let I be the set of irregular values of N at 0. There is a ramification v " y 1{d , d ě 1 and a finite Galois extension L{Cpxq such that I Ă Lppvqq.Let D L ÝÑ D be the normalization of D in L. At the cost of shrinking D, we can suppose that every point of D is semi-stable for N .Hence, I Ă CrD L sppvqq and CrD L sppvqq b N " à aPI E a b R a where the connections R a are regular.As seen in the proof of 4.1.1,the assumption on Irr D implies that any smooth curve transverse to D is non characteristic for N .Taking the axis D y yields dim H 1 Irr 0 N |Dy " dimpH 1 Irr D N q 0 " ÿ aPI pord y aq rk R aOn the other hand, choose a point P P D L above 0.Then, the irregular values of N |Dy are the apP q, a P I. Thus, Irr 0 N |Dy " ÿ aPI ord y apP q rk R a Lemma 4.3.3.-Suppose that Irr D M is a local system.For every M-marked connection pN , isoq, the complex Irr D N is a local system.Proof.-Let χpD, Irr D Mq : D ÝÑ Z be the local Euler-Poincaré characteristic of Irr D M. By local index theorem[Kas73][Mal81], the value of χpD, Irr D Mq at P P D only depends on the multiplicities of the components of the characteristic cycle of M passing through P .These multiplicities can be computed at the level of the formal neighbourhood of P in C 2 .Since M and N are formally isomorphic at P , we have χpD, Irr D Mq " χpD, Irr D N q Hence, χpD, Irr D N q is constant.On the other hand, we know from[Meb90] that Irr D N is perverse.We conclude with the fact that a perverse sheaf with constant local Euler-Poincaré characteristic is a local system [Tey13, 13.1.6].Using notations from 4.2, we are now in position to prove the sought-after Proposition 4.3.4.-Suppose that Irr D M and Irr D End M are local systems in a neighbourhood of 0. If pN ˚, iso ˚q extends into a M-marked connection pN , isoq, then M has good formal decomposition at 0.

-The map ϕ : St N piq ÝÑ St N s ÝÑ e Mě se ´Mě
St N |e a pg ´idq has rapid decay for every a with ord a ą mpi ´1qu The sheaf St i N is a sheaf of normal algebraic subgroups of St N .Let us define two diagonal matrices M :" Diagpe a , i P Iq and M ě :" Diagpe a ěmpiq , i P Iq.The sheaf ÝÑ e ´M F ´1sF e M .By definition, ιpΓpS, St N qq is the subgroup of elements g P GL r such that for every a, b P I, mpiq paq ‰ ξ mpiq pbq and ξ mpiq paq ć S ξ mpiq pbq g ab " 0 if a ‰ b and ξ mpiq paq " ξ mpiq pbq Note that if a, b P I with ξ mpiq paq ‰ ξ mpiq pbq, then a ć S b if and only if ξ mpiq paq ć S ξ mpiq pbq Hence, ΓpS, St N piq q identifies with the subgroup of elements g P ιpΓpS, St N qq such that for every a, b P I, g ab " 0 if ξ mpiq paq " ξ mpiq pbq and a ‰ b Let s P ΓpS, St i N q, and let a, b P I with a ‰ b.If ξ mpiq paq " ξ mpiq pbq, then ιpsq ab " e b´a F ´1 a s ab F b " F ´1 a pe b ěmpiq ´aěmpiq s ab qF b By definition, e b ěmpiq ´aěmpiq At the cost of refining U, we can suppose that there exists an isomorphism θ U : BM |U ÝÑ BN |U asymptotic to iso for every U P U.Then, conjugation by θ U provides an isomorphism St M P |U ÝÑ St N P |U carrying St ăD M|U to St ăD N |U .To prove that ΓpU, Qq is unipotent, we can thus suppose that M is good split unramified.
, Qq where α is faithfully flat, where O denotes the orbit of the trivial cocycle under G and where β is an immersion of schemes.Since smoothness is a local property for the fppf topology, smoothness of G implies that O is smooth.By definition, α is an isomorphism at the level of C-points.Hence, α is an isomorphism of varieties.We are thus left to show that O is closed in Z 1 pU, Qq.It is enough to show that O is closed in Z 1 pU, Qq red .From Kostant-Rosenlicht theorem [Bor91, I 4.10], it is enough to show that G is a unipotent algebraic group.Let N be the good split unramified bundle formally isomorphic to M at 0. Let us choose a formal isomorphism iso : M p 0 ÝÑ N p 0 .
jk h ki ´Id P ΓpU ijk , I ¨Lie St ăD M pR 1 qq We see s ijk as a section of I b C L i pCq over U ijk and denote by rs ijk s its class in I b C Lie St ăD M pCq T0 .We want to prove that the rs ijk s define a cocycle.As seen above, this amounts to prove the following equality in ΓpU ijk , I b C Lie St ăDWhere g ij p0q is the image of g ij by R ÝÑ C. We haveg ij p0qs jkl g ´1 ij p0q " h ij hjk h kl h lj h ji ´Id " ph ij h jk ´hik `hik qh kl h lj h ji ´Id " ph ij h jk ´hik qg kl p0qg lj p0qg ji p0q `hik h kl h lj h ji ´Id " ph ij h jk ´hik qg ki p0q `hik h kl h lj h ji ´Id " ph ij h jk ´hik qh ki `hik h kl h lj h ji ´Id " h ij h jk h ki `hik h kl h lj h ji ´2 Id