jbteyssier.comjbteyssier.com/papers/jbteyssier_ModuliSingularities.pdf · MODULI OF STOKES TORSORS AND SINGULARITIES OF DIFFERENTIAL EQUATIONS by Jean-Baptiste Teyssier Abstract.—

MODULI OF STOKES TORSORS AND SINGULARITIES OFDIFFERENTIAL EQUATIONS

by

Jean-Baptiste Teyssier

Abstract. — LetM be a meromorphic connection with poles along a smooth divisorD in a smooth algebraic variety. Let SolM be the solution complex of M. Weprove that the good formal structure locus of M coincides with the locus where therestrictions to D of SolM and Sol EndM are local systems. By contrast to the verydifferent natures of these loci (the first one is defined via algebra, the second oneis defined via analysis), the proof of their coincidence is geometric. It relies on themoduli of Stokes torsors.

The main problematic of this paper is to understand how the geometry of the Stokesphenomenon in any dimension sheds light on the interplay between the singularitiesof a differential equation and the singularities of its solutions.

Consider an algebraic linear system M of differential equations with n variablesBX

Bxi“ ΩiX i “ 1, . . . , n

where Ωi is a square matrix of size r with coefficients into the ring Crx1, . . . , xnsrx´1n s

of Laurent polynomials with poles along the hyperplane D in Cn given by xn “ 0.At a point away from D, the holomorphic solutions of the system M are fullyunderstood by means of Cauchy’s theorem. At a point of D, the situation is muchmore complicated. It is still the source of challenging unsolved problems. We call Dthe singular locus of M. Two distinguished open subsets of D where the singularitiesof M are mild can be defined.

First, the set GoodpMq of good formal structure points of M is the subset ofD consisting of points P at the formal neighbourhood of which M admits a goodformal structure. For P being the origin, and modulo ramification issues that willbe neglected in this introduction, this means roughly that there exists a base changewith coefficients in CJx1, . . . , xnKrx´1

n s splitting M as a direct sum of well-understoodsystems easier to work with.

Good formal structure 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 structure at every point of D. Thus, the

2 J.-B. TEYSSIER

set GoodpMq is a non trivial invariant of M. As proved by André [And07], the setGoodpMq is the complement in D of a Zariski closed subset F of D either purelyof codimension 1 in D or empty. Traditionally, F is called the Turning locus of M,by reference to the way the Stokes directions of M move along a small circle in Dgoing around a turning point. In a sense, the good formal structure locus ofM is the open subset of D where the singularities of the system M are assimple as possible.

To define the second distinguished subset of D associated to M, let us view Mas a D-module, that is a module over the Weyl algebra of differential operators. Letus denote by SolM the solution complex of the analytification of M. Concretely,H0 SolM encodes the holomorphic solutions of our differential system while thehigher cohomologies of SolM keep track of higher Ext groups in the category ofD-modules. As proved by Kashiwara [Kas75], the complex SolM is perverse.From a theorem of Mebkhout [Meb90], the restriction of SolM to D, that is, theirregularity complex of M along D, denoted by Irr˚DM in this paper, is also perverse.In particular, pSolMq|D is a local system on D away from a closed analytic subset ofD. The smooth locus of pSolMq|D denotes the biggest open in D on which pSolMq|Dis a local system. In a sense, the smooth locus of pSolMq|D is the open subsetof D where the singularities of the (derived) solutions of M are as simpleas possible.

As observed in [Tey13], the open set GoodpMq is included in the smooth lo-cus of pSolMq|D and pSol EndMq|D. The reverse inclusion was conjectured in[Tey13, 15.0.5]. Coincidence of GoodpMq with the smooth locus of pSolMq|Dand pSol EndMq|D seems surprising at first sight, since goodness is an algebraicnotion whereas SolM is transcendental. The main goal of this paper is to prove viageometric means the following

Theorem 1. — Let X be a smooth complex algebraic variety. Let D be a smoothdivisor in X. Let M be a meromorphic connection on X with poles along D. Then, thegood formal structure locus of M is the locus of D where pSolMq|D and pSol EndMq|Dare local systems.

Other criteria detecting good points of meromorphic connections are available inthe literature. Let us mention André’s criterion [And07, 3.4.1] in terms of speciali-sations 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 the ringof formal power series. This criterion is numerical in nature. By contrast, the newcriterion given by Theorem 1 is transcendental. Its sheaf theoretic flavour makes itpossible to track the turning points in the cohomology of the irregularity complex.For an application of this observation, let us refer to Theorem 2 below.

The main tool at stake in the proof of Theorem 1 is geometric, via moduli of Stokestorsors [Tey19]. For a detailed explanation of the line of thoughts that brought theminto the picture, let us refer to 2.1. Before stating an application of Theorem 1 (seeTheorem 2 below), we explain how these moduli are used by giving the main ingre-dients of the proof of Theorem 1 in dimension 2. In that case, we have to show the

MODULI OF STOKES TORSORS AND SINGULARITIES OF DIFFERENTIAL EQUATIONS 3

goodness of a point 0 P D given that pSolMq|D and pSol EndMq|D are local systemsin a neighbourhood of 0. The main problem is to extend the good formal structureof M across 0. This decomposition can be seen as a system of linear differentialequations 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 modulispace X parametrizing very roughly systems defined in a neighbourhood of ∆ andformally 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 aneighbourhood of ∆˚ and formally isomorphic to M|∆˚ along ∆˚. Two distinguishedpoints of Y are M|∆˚ and N . Restriction from ∆ to ∆˚ provides a morphism ofalgebraic varieties res : X ÝÑ Y. The problem of extending N is then the problemof proving that res hits N . The moduli X and Y have the wonderful property thatthe tangent map TM res of res at M is exactly the map

Γp∆,H1 Sol EndMq ÝÑ Γp∆˚,H1 Sol EndMq

associating to s P Γp∆,H1 Sol EndMq the restriction of s to ∆˚. In this geometricpicture, the smoothness of pH1 Sol EndMq|D around 0 thus translates into the factthat TM res is an isomorphism of vector spaces. Since X and Y are smooth, wededuce that res is étale at the point M. Thus, the image of res in Y contains a nonempty open set. We prove furthermore (see Theorem 3 below) that res is a closedimmersion, so its image is closed in Y. Since Y is irreducible, we conclude that res issurjective, which proves the existence of the sought-after extension of N .

Let us now describe an application of Theorem 1. Let X be a smooth variety over afinite field of characteristic p ą 0. Let ` ‰ p be a prime number. As proved by Deligne[EK12], there is only a finite number of semi-simple `-adic local systems on X withprescribed rank, bounded ramification at infinity and up to a twist by a charactercoming from the base field. A natural question is to look for a differential analogue ofthis finiteness result. Let X be a smooth complex proper algebraic variety. Let M bea meromorphic connection on X. In this situation, H. Esnault and A. Langer askedwhether it is possible to control the resolution of turning points of M by means ofX, the rank of M and the irregularity of M. In dimension 2, this question amountsto bound the number of blow-ups needed to eliminate the turning points of M. Tothe author’s knowledge, this question is still widely open. If such a bound exists indimension 2, the number of turning points of M should in particular be bounded bya quantity depending only on the surface X, the rank of M and the irregularity ofM. As an application of Theorem 1, we give such a bound in a relative situation,thus providing the first evidence for a positive answer to H. Esnault and A. Langer’squestion. This is the following

Theorem 2. — Let S be a smooth complex algebraic curve. Let 0 P S. Let p : C ÝÑS be a relative smooth proper curve of genus g. Let M be a meromorphic connectionon C with poles along the fibre C0 of p above 0. Let rDpMq be the highest genericslope of M along C0. Then, the number of turning points of M along C0 is boundedby 8prankMq2pg ` 1qrDpMq.

4 J.-B. TEYSSIER

To prove Theorem 2, the main tools are Theorem 1 and a new boundedness resultfor nearby slopes [Tey16] suggested by the `-adic picture [HT18]. See remark 7.1.4for details.

A crucial step in the proof of Theorem 1 is to understand the geometry of therestriction map for Stokes torsors. This is achieved by Theorem 3 below. To state it,let X be a smooth complex algebraic variety. Let D be a normal crossing divisor inX. Let M be a meromorphic connection on X with poles along D. Suppose that Mhas good formal structure in the sense of Mochizuki 1.5. Let pD : rX ÝÑ X be thefibre product of the real blow-ups of X along the components of D. For every subsetA Ă D, put BA :“ p´1

D pAq. Let StăDM be the Stokes sheaf of M (see section 2.3 fordetails). This is a sheaf of complex unipotent algebraic groups on BD. Then, we havethe following

Theorem 3. — Let U Ă V Ă D be non empty open subsets in D such that V isconnected. Then, the natural morphism

H1pBV,StăDM q // H1pBU,StăDM q

is a closed immersion of affine schemes of finite type over C.

Let us finally give an application of Theorem 3 to degenerations of irregular singu-larities. Let X be a smooth algebraic variety and let D be a germ of smooth divisorat 0 P X. Let M be a germ of meromorphic connection defined in a neighbourhood ofD in X and with poles along D. Motivated by Dubrovin’s conjecture and the studyof Frobenius manifolds, Cotti, Dubrovin and Guzzetti [CDG19] studied how muchinformation on the Stokes data of M can be retrieved from the restriction of M to asmooth curve C transverse to D and passing through 0.

Under the assumption thatMpD splits as a direct sum of regular connections twisted

by meromorphic functions a1, . . . , an P OXp˚Dq with simple poles along D, theyproved that the Stokes data of the restriction M|C determine in a bijective way theStokes data of M in a small neighbourhood of 0 in D. This is striking, since thenumerators of the ai ´ aj may vanish at 0, thus inducing a discontinuity at 0 in theconfiguration of the Stokes directions. Using different methods, this was reproved bySabbah in [Sab, Th 1.4]. In this paper, we give a short conceptual proof of a strongerversion of Cotti, Dubrovin and Guzzetti’s injectivity theorem: we don’t make anyassumption on the shape of M

pD, nor do we suppose that D is smooth, nor do weassume that C is transverse to D. The price to pay for this generality is the use ofresolution of turning points, as proved in the fundamental work of Kedlaya [Ked11]and Mochizuki [Moc11b]. The intuition that the techniques developed in this papercould be applied to the questions considered by Cotti, Dubrovin and Guzzetti is dueto C. Sabbah.

To state our result, let us recall that a M-marked connection is the data of a pairpM, isoq where M is a germ of meromorphic connection with poles along D definedin a neighbourhood of D in X, and where iso : M

pD ÝÑ MpD is an isomorphism of

formal connections.


Theorem 4. — Let X be a germ of smooth algebraic variety around a point 0. Let Dbe a germ of divisor passing through 0. Let M be a germ of meromorphic connectionat 0 with poles along D. Let C be a smooth curve passing through 0 and not containedin any of the irreducible components of D. If pM1, iso1q and pM2, iso2q are M-markedconnections such that

pM1, iso1q|C » pM2, iso2q|C

then pM1, iso1q and pM2, iso2q are isomorphic in a neighbourhood of 0.

Let us give an outline of the paper. In section 1, we introduce some backgroundmaterial on asymptotic analysis and on the Stokes sheaf. In section 2, we introducethe sheaf of relative Stokes torsors and prove its constructibility. In section 3, weprove the representability of the moduli of Stokes torsors. We then prove Theorem3. In section 4, we interpret the tangent spaces and the obstruction theory for thesemoduli in a transcendental way via the solution complexes for connections. We thenprove Theorem 4. In section 5, we show how to reduce the proof of Theorem 1 toextending the good formal model ofM across the point 0 under study. In section 6, weshow that the sought-after extension exists provided that the moduli of Stokes torsorsassociated to a resolution of the turning point 0 for M satisfies suitable geometricconditions. Finally, we show that these geometric conditions are always satisfied whenthe hypothesis of Theorem 1 are satisfied, thus concluding the proof of Theorem 1.Section 7 is devoted to the proof of Theorem 2. We collect in an appendix someelementary facts about torsors and Stokes filtered local systems. Note that our use ofStokes filtered local systems in this paper is a purely technical detour to obtain thetriviality criterion 8.4.1.

Acknowledgement. — I thank Y. André, P. Boalch, M, Brion, H. Esnault, F. Lo-ray, C. Sabbah, T. Saito, C. Simpson and T. Mochizuki for interesting discussionsand constructive remarks on a first draft of this work. I thank C. Sabbah for sharingwith me the intuition that the techniques developed in this paper could be appliedto the questions considered in [CDG19]. I thank H. Hu for stimulating exchangeson nearby slopes. I thank N. Budur and W. Veys for constant support during thepreparation of this paper. I thank an anonymous referee for useful remarks and forpointing out the discrepancy between the notion of good formal structure of Ked-laya and that of Mochizuki. This work has been funded by the long term structuralfunding-Methusalem grant of the Flemish Government. I thank KU Leuven for pro-viding outstanding working conditions. This paper benefited from a one month stayat the Hausdorff Research Institute for Mathematics, Bonn. I thank the HausdorffInstitute for providing outstanding working conditions.

1. The Stokes sheaf. Global aspects

1.1. Geometric setup. — In this subsection, we introduce basic notations. In thispaper, a regular pair pX,Dq will be the data of a smooth complex algebraic varietyX and of a strict normal crossing divisor D in X. For a quasi-coherent sheaf F on

6 J.-B. TEYSSIER

X, we denote by F|D the sheaf of germs of sections of F along D. Let D1, . . . , Dm

be the irreducible components of D. For I Ă J1,mK, set

DI :“č

iPI

Di and D˝I :“ DIzď

iRI

Di

1.2. Functions with asymptotic expansion along D. — For i “ 1, . . . ,m, letrXi ÝÑ X be the real blow-up of X along Di. Let pD : rX ÝÑ X be the fibre productof the rXi, i “ 1, . . . ,m above X. For every subset A Ă D, put BA :“ p´1

D pAq. LetιA : BA ÝÑ BD be the canonical inclusion.

Let A be the sheaf of functions on BD admitting an asymptotic expansion alongD [Sab00]. For a closed subset Z in D, let A

pZ be the completion of A along thepull-back by pD of the ideal sheaf of Z. Put AăZ :“ KerpA ÝÑ A

pZq. When Z “ D,the sheaf AăD can be concretely described locally as follows (see [Sab00, II 1.1.11]for a proof). Let px1, . . . , xnq be local coordinates centred at 0 P D such that D isdefined around 0 by x1 ¨ ¨ ¨xl “ 0 for some l P J1,mK. Then, the germ of AăD atQ P B0 is given by those holomorphic functions u defined over the trace on XzD ofa neighbourhood Ω of Q in rX, and such that for every compact K Ă Ω, for everypN1, . . . , Nlq P N

l, there exists a constant CK,N ą 0 satisfying

(1.2.1) |upxq| ď CK,N |x1|N1 ¨ ¨ ¨ |xl|

Nl for every x P K X pXzDq

From the formula (1.2.1), we deduce the following elementary lemma.

Lemma 1.2.2. — Let ρ : Y ÝÑ X be a cyclic Galois cover of X ramified along D.Put E “ ρ´1pDq. Let rρ : rY ÝÑ rX be the map induced by ρ at the level of the realblow-up. Then, the canonical map AăD ÝÑ rρ˚AăE induces an isomorphism betweenAăD and the sheaf of invariants of rρ˚AăE under the Galois group of ρ.

1.3. Good formal structure. — Let pX,Dq be a regular pair. Meromorphicconnections in this paper will be supposed to be flat. Let P be a point in D. Anelementary local model at P denotes a meromorphic connection N of the form

N “à

aPIP

Ea bRa

where IP is a finite set in OX,P p˚Dq, where Ea “ pOX,P p˚Dq, d´daq and where Ra isa regular singular meromorphic connection on X with poles along D. If furthermorethe following conditions are satisfied(1) For an element a in IP , if a does not belong to OX,P , then the divisor of a isanti-effective with support in D,(2) For an element a, b in IP , if a ´ b does not belong to OX,P , then the divisor ofa´ b is anti-effective with support in D,we say that N is a good elementary local model at P . Let M be a meromorphicconnection on X with poles along D. Let Y be the stratum of D containing P .Following [Ked10, 6.2.3], we say that M has an elementary local model at P if at


the cost of shrinking X, there exists an elementary local model N at P and anisomorphism of connections

(1.3.1) OzX|Y

bOX M » OzX|Y

bOX N

If furthermore N is a good elementary local model at P , we say that M has a goodelementary local model at P . We say that M has a good formal structure at P if thereexists a cyclic Galois cover of some neighbourhood of P , ramified along D, on whichthe pull-back of M admits a good elementary local model at some inverse image ofP . If this is true for every point P in D, we say that M has a good formal structure.

1.4. Irregular values. — Let pX,Dq be a regular pair. Let M be a meromorphicconnection on X with poles along D. If M has a good elementary local model atevery point P in D, the images by OXp˚Dq ÝÑ OXp˚DqOX of the finite sets IPappearing in 1.3 organize into a subsheaf of OXp˚DqOX on D. This is the sheaf ofirregular values of M. Let us denote it by I. We say that I is very good if for everypoint P in D, the difference of any two distinct elements of IP has poles along everycomponent of D passing through P . These definitions extend in a straighforward wayto the case where M has good formal structure. See [Sab12, 9.c] for details.

1.5. Mochizuki’s definition of good formal structure and its use in thepaper. — As pointed out in [Ked10, 4.3.3], there is a small discrepancy betweenthe notion of good formal structure in [Ked10] and that from [Moc11b]. Mochizukifurther requires that the sets IP appearing in the decomposition (1.3.1) satisfy theextra assumption that the OX -modules generated by the differences a´b, a, b P IP notlying in OX are totally ordered under containment. If M has good formal structurein the sense of Mochizuki, then M has good formal structure in the sense of Kedlaya.Due to [Ked10, 4.3.1], if M and EndM have good formal structure in the sense ofKedlaya, then M has good formal structure in the sense of Mochizuki.

Note that these notions coincide when D is smooth. Hence, this discrepancy isinvisible in the statement of Theorem 1. However, the notion of goodness used inTheorem 3, Theorem 5 and Theorem 6 is that of Mochizuki. It will be needed toensure that for every point P in D, there exists a component Z of D passing throughP such that for every a, b P IP distinct, a ´ b has poles along Z. If not explicitlymentioned otherwise, good formal structure will be taken in the sense of Kedlaya.

1.6. The Stokes sheaf. — Let pX,Dq be a regular pair. Let M be a meromorphicconnection defined on X with poles along D. Suppose that M has good formalstructure. We set

BM “ Abp´1D OX|D p

´1D M

andBM

pD “ ApD bp´1

D OX|D p´1D M

Let DX be the sheaf of differential operators on X. The sheaf A is endowed with anaction of p´1

D DX|D. Hence, so does BM. We can thus form the De Rham complex of

8 J.-B. TEYSSIER

M with coefficients in A as

BM // p´1D Ω1

X|D bp´1D OX|D BM

// ¨ ¨ ¨ // p´1D ΩnX|D bp´1

D OX|D BM

It is denoted by DR BM. Similarly, we denote by DRăDM the De Rham complex ofM with coefficients in AăD.

Let Z be a closed subset of D. Let StăZM be the subsheaf of H0 DR BEndM ofsections asymptotic to the Identity along Z, that is of the form Id`f where f hascoefficients in AăZ . The sheaf StăZM is a sheaf of complex unipotent algebraic groupson BZ. This is the Stokes sheaf of M along Z. For every C-algebra R, the sheaf ofR-points of StăZM is a sheaf of groups on BZ. It is denoted by StăZM pRq. This is theStokes sheaf of M along Z relative to R.

1.7. The Stokes locus. — Let pX,Dq be a regular pair. Let pD : rX ÝÑ X be thereal blow-up of X along D. Let M be a meromorphic connection defined on X withpoles along D. Suppose that M has a good elementary local model at every point ofD. Let I be the sheaf of irregular values of M. Let P be a point in D. Let a, b P IPdistinct. Put Fa,b :“ Repa ´ bq|x´ ordpa´bq| where px1, . . . , xnq are local coordinatescentred at P such that D is given by x1 ¨ ¨ ¨xm “ 0. By definition, the Stokes locusof pa, bq is defined as Fa,b ˝ pD “ 0. The Stokes locus of M is the union of the lociof the form Fa,b ˝ pD “ 0, where a, b are as above. If M is ramified, then at the costof shrinking X, there exists a cyclic Galois cover ρ : Y ÝÑ X ramified along D suchthat ρ`M is unramified. Let rρ : rY ÝÑ rX be the map induced by ρ at the level ofthe real blow-up. Then the Galois group G of ρ acts on I. Hence, the action of G onrY preserves the Stokes locus of ρ`M. Thus, the Stokes locus of ρ`M descends toa closed subset in BD, called the Stokes locus of M. This locus depends only on Mand not on the choice of ρ.

1.8. Some facts on the Stokes sheaf. — Let pX,Dq be a regular pair. Let Mbe a meromorphic connection on X with poles along D. Suppose that M has goodformal structure. As a consequence of lemma 1.2.2, we have the following

Lemma 1.8.1. — Let ρ : Y ÝÑ X be a cyclic Galois cover of X ramified alongD. Put E “ ρ´1pDq. Let rρ : rY ÝÑ rX be the map induced by ρ at the level of thereal blow-up. Then, rρ´1 StăDM » StăEρ`M and the canonical map StăDM ÝÑ rρ˚ StăEρ`Minduces an isomorphism between StăDM and the sheaf of invariants of rρ˚ StăEρ`M underthe Galois group of ρ.

Lemma 1.8.2. — The Stokes sheaf of M is constructible with respect to the strati-fication of BD induced by the Stokes locus of M.

Proof. — From lemma 1.8.1, we can suppose that M is unramified. The question islocal on BD. From Mochizuki’s asymptotic development theorem [Sab12, 12.5], wecan further suppose that M is a good elementary local model at a point P . In thatcase, let us write

M “à

aPIEa bRa


where I is a good set of irregular values at P , where Ea “ pOXp˚Dq, d ´ daq andwhere Ra is regular with poles along D. Let ia : Ea b Ra ÝÑ N be the canonicalinclusion and let pa : N ÝÑ Ea bRa be the canonical projection. Sections of StăDMon an open set S are automorphisms of M on S X pXzDq of the form Id`f wherepafib “ 0 unless

(1.8.3) ea´b P ΓpS,AăDqLemma 1.8.2 then follows from the observation that the condition (1.8.3) is constanton each stratum of the stratification of BD induced by the Stokes locus of pa, bq.

Lemma 1.8.4. — Let pX,Dq be a germ of regular pair at a point P . Let D1, . . . , Dm

be the components of D. Let M be a meromorphic connection on X with poles alongD. Suppose that M has good formal structure. Let ρ : Y ÝÑ X be a cyclic Galoiscover of X ramified along D such that ρ`M is unramified. Suppose that the differenceof any two distinct irregular values for M at P has poles along ρ´1pDmq. Put I “J1,m´ 1K. Then, the adjunction morphism

(1.8.5) ι´1DI

StăDM// ι´1DIιD˝I˚ι

´1D˝I

StăDM

is an isomorphism.

Proof. — From lemma 1.8.1, we can suppose that M is unramified. Injectivity isobvious, so we are left to prove surjectivity at a point Q in BP . This is a local questionaround Q. From Mochizuki’s asymptotic development theorem [Sab12, 12.5], we canthus suppose that M is a good elementary local model. Let us write

M “à

aPIEa bRa

where I is a good set of irregular values at P , where Ea “ pOXp˚Dq, d´daq and whereRa is regular with poles along D. Let ia : Ea bRa ÝÑ N be the canonical inclusionand let pa : N ÝÑ Ea bRa be the canonical projection. Let S be a neighbourhoodof Q in BX of the form

pr0, rrÎ1q ˆ ¨ ¨ ¨ ˆ pr0, rrÎmq ˆ∆

where r ą 0, where ∆ is a ball in Cn´m centred at 0 and where I1, . . . , Im are closedintervals in S1. To prove the surjectivity of (1.8.5) at Q, it is enough to show that atthe cost of shrinking S, the restriction morphism

ΓpS X BD,StăDM q ÝÑ ΓpS X BD˝I ,StăDM q

is a bijection. Sections of StăDM on S X BD are automorphisms of M on S X pXzDqof the form Id`f where pafib “ 0 unless

(1.8.6) ea´b P ΓpS X BD,AăDqSections of StăDM on

S X BD˝I “ pt0u ˆ I1q ˆ ¨ ¨ ¨ ˆ pt0u ˆ Im´1q ˆ ps0, rrÎmq ˆ∆

are automorphisms of M on S X pXzDq of the form Id`f where pafib “ 0 unless

(1.8.7) ea´b P ΓpS X BD˝I ,AăDq

10 J.-B. TEYSSIER

We thus have to show that for every distinct a, b P I, the conditions (1.8.6) and (1.8.7)are equivalent for a small enough choice of S. A change of variable reduces the problemto the case where a´ b “ 1xα1

1 ¨ ¨ ¨xαmm where pα1, . . . , αmq P Nm´1 ˆN˚. Note that

condition (1.8.6) trivially implies condition (1.8.7). Suppose that e1xα11 ¨¨¨xαmm lies in

ΓpS X BD˝I ,AăDq. At the cost of shrinking S, we can suppose that there exists aconstant C ą 0 such that for every

px1, . . . , xnq P ps0, rrÎ1q ˆ ¨ ¨ ¨ ˆ ps0, rrÎm´1q ˆ prr

2, rrÎmq ˆ∆

we have|e1x

α11 ¨¨¨xαmm | ď C|x1| ¨ ¨ ¨ |xm´1|

Writing xi “ pri, θiq for i “ 1, . . . ,m, this means

ecospα1θ1`¨¨¨`αmθmqrα11 ¨¨¨rαmm ď Cr1 ¨ ¨ ¨ rm´1

In particular, αi ą 0 for i “ 1, . . . ,m ´ 1 and cospα1θ1 ` ¨ ¨ ¨ ` αmθmq ă 0 for everypθ1, . . . , θmq P I1 ˆ ¨ ¨ ¨ ˆ Im. At the cost of shrinking S further, there exists c ą 0such that cospα1θ1 ` ¨ ¨ ¨ ` αmθmq ă ć on I1 ˆ ¨ ¨ ¨ ˆ Im. Then, we have

|e1xα11 ¨¨¨xαmm | ď eć|x1|

α1 ¨¨¨|xm|αm

on S. Since αi ą 0 for i “ 1, . . . ,m, we deduce that (1.8.7) holds. This provesthe equivalence between conditions (1.8.6) and (1.8.7) and thus finishes the proof oflemma 1.8.4.

2. Stokes torsors

2.1. Why moduli of Stokes torsors?— Let us explain in this subsection how themoduli of Stokes torsors were found to be relevant to the proof of Theorem 1. We usethe notations from the introduction and work in dimension 2. We suppose that 0 P Dlies in the smooth locus of pSolMq|D and pSol EndMq|D, and we want to prove that0 is a good formal structure point for M.

From a theorem of Kedlaya [Ked10][Ked11] and Mochizuki [Moc09][Moc11b],our connection M acquires good formal structure at any point after pulling-back by asuitable 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 reducesin this case to the following

Question. — Given two good meromorphic connections M and N with poles alongthe coordinate axis in C2 and formally isomorphic at 0, is it true that

(2.1.1) dimpH1 Sol EndMq0 “ dimpH1 Sol EndN q0 ?

It turns out that each side of (2.1.1) appeared as dimensions of moduli spaces ofStokes torsors constructed by Babbitt-Varadarajan in [BV89]. These moduli wereassociated with germs of meromorphic connections in dimension 1. Babbitt andVaradarajan proved that they are affine spaces. This suggested the existence of amoduli X with two points P,Q P X such that the left-hand side of (2.1.1) would be


dimTPX and the right-hand side of (2.1.1) would be dimTQX . The equality (2.1.1)would then follow from the smoothness and connectedness of the putative moduli.This is what led to [Tey19], but the question of smoothness and connectedness wasleft open. In the meantime, a positive answer to the above question was given bypurely analytic means by C. Sabbah in [Sab17].

2.2. Relation with [Tey19]. — In [Tey19], a moduli for local Stokes torsors wasconstructed in any dimension. This moduli suffers two drawbacks in view of theproof of Theorem 1. First, the Stokes sheaf used in [Tey19] only makes sense at aneigbourhood of a point, whereas our situation will be global as soon as we applyKedlaya-Mochizuki’s resolution of turning points. Second, the relation between Irreg-ularity and the tangent spaces of the moduli from [Tey19] only holds in particularcases. To convert the hypothesis on Irregularity appearing in Theorem 1 into a geo-metric statement pertaining to moduli of torsors, we need to replace the Stokes sheafStM of a connection M by a subsheaf denoted by StăDM . We will abuse terminology bealso calling the torsors under StăDM Stokes torsors. The sheaf StăDM has the advantageof being globally defined when M is globally defined. Along the smooth locus of D,the sheaf StăDM is the usual Stokes sheaf. The only difference between StM and StăDMappears at a singular point of D.

2.3. The functor of relative Stokes torsors. — We use the notations from 1.6.Let R be a C-algebra. Torsors under StăZM pRq are the Stokes torsors along Z relativeto R. For every subset A Ă Z, let H1pBA,StăZM q be the functor

C-alg ÝÑ Set

R ÝÑ H1pBA,StăZM pRqq

From [Tey19, Th. 1], the functor H1pBP,StăPM q is an affine scheme of finite type overC for every point P in D.

Lemma 2.3.1. — Let P be a point in D. Torsors under StăPM on BP have no nontrivial automorphisms.

Proof. — The lemma 2.3.2 was proved in [Tey19, 1.8.1] in the case whereM is a goodelementary local model. An inspection of the proof, relying on Babbitt-Varadarajanrepresentability theorem in dimension 1 as well as Malgrange-Sibuya theorem, showsthat it carries over verbatim to the case of an arbitrary connection with good formalstructure.

Lemma 2.3.2. — Let A be a subset in D. Torsors under StăDM on BA have no nontrivial automorphisms.

Proof. — Let P be a point in A. It is enough to show that torsors under StăDM on BPhave no non trivial automorphisms. Let T be a StăDM -torsor on BP . Let φ : T ÝÑ Tbe an automorphism of StăDM -torsors. Since AăD is a subsheaf of AăP , there is aninjection ι : StăDM ÝÑ StăPM . To show that φ is the identity of T amounts to show thatthe push-forward ι˚φ : ι˚T ÝÑ ι˚T is the identity of the StăPM -torsor ι˚T . This lastassertion is a consequence of lemma 2.3.1. This finishes the proof of lemma 2.3.2.

12 J.-B. TEYSSIER

As a straighforward consequence of lemma 2.3.2, we deduce the following

Corollary 2.3.3. — Let A be a subset in D. We endow A with the topology inducedby D. Then, the presheaf of functors R1pA˚ StăDM defined as

OpenpAq ÝÑ Set

U ÝÑ H1pBU,StăDM q

is a sheaf of functors. That is, for every cover U of A by open subsets, the first arrowin the following diagram of pointed functors

H1pBA,StăDM q //ś

UPU H1pBU,StăDM q //

// ś

U,V PU H1pBU X BV,StăDM q

is an equalizer.

Remark 2.3.4. — Observe that the sheaf condition in corollary 2.3.3 is still satisfiedif one takes instead of a cover by open subsets in A a cover K by compact subsets K P Ksuch that the associated family of open subsets

˝

K form a cover of A.

As a consequence of 1.8.1, 2.3.4 and 8.1.1, we have the following

Corollary 2.3.5. — Let pX,Dq be a regular pair. Let ρ : Y ÝÑ X be a cyclic Galoiscover of X ramified along D. Let G be the Galois group of ρ. Put E “ ρ´1pDq. LetA be a subset in D. Let M be a meromorphic connection on X with poles along D.Suppose that M has good formal structure. Then, the canonical morphism of functors

H1pBA,StăDM q ÝÑ H1pBρ´1pAq,StăEρ`MqG

is an isomorphism.

2.4. Dévissage of the sheaf of relative Stokes torsors. — The goal of thissubsection is to relate Stokes torsors on a stratum with Stokes torsors on a suitablychosen stratum which is less deep. This will be done in proposition 2.4.2.


be the components of D. Let M be a meromorphic connection on X with poles alongD. Suppose that M has good formal structure. Let I be a subset in J1,mK. For everyelement T in H1pBD˝I ,StăDM q, the sheaf ιD˝I˚T is a ιD˝I˚ι

´1D˝I

StăDM -torsor on BD.

Proof. — Let Q be a point in BD. From lemma 8.1.2, we have to show the existenceof a neighbourhood S of Q in rX such that the Stokes torsors on S X BD˝I are trivial.To do this, we can suppose that Q lies in BP . Let ρ : Y ÝÑ X be a cyclic Galoiscover of X ramified along D such that ρ`M is unramified. Let rρ : rY ÝÑ rX be theGalois cover induced by ρ at the level of the real blow-up. Let U be a neighbourhoodof Q that decomposes rρ. By pulling-back the situation to a connected componentof rρ´1pUq, we reduce to the case where M is unramified. Since we are working ina neighbourhood of Q, Mochizuki’s asymptotic development theorem [Sab12, 12.5]reduces the proof of lemma 2.4.1 to the case where M is a good elementary localmodel. Then, lemma 2.4.1 is a consequence of the triviality criterion 8.4.1.


Proposition 2.4.2. — Let pX,Dq be a germ of regular pair at a point P . LetD1, . . . , Dm be the components of D. Let M be a meromorphic connection on Xwith poles along D. Suppose that M has good formal structure. Let ρ : Y ÝÑ X bea cyclic Galois cover of X ramified along D such that ρ`M is unramified. Supposethat the difference of any two distinct irregular values for M at P has poles alongρ´1pDmq. Put I “ J1,m´ 1K. Then, the restriction morphism

(2.4.3) H1pBDI ,StăDM q // H1pBD˝I ,StăDM q

is an isomorphism of functors.

Proof. — From lemma 2.4.1 and lemma 1.8.4, the functor ι´1DIιD˝I˚ induces a well-

defined morphism H1pBD˝I ,StăDM q ÝÑ H1pBDI ,StăDM q, providing an inverse for(2.4.3).

2.5. The stalks of the sheaf of Stokes torsors. —

Lemma 2.5.1. — Let pX,Dq be a regular pair. Let D1, . . . , Dm be the irreduciblecomponents of D. Let M be a meromorphic connection on X with poles along D.Suppose that M has good formal structure. Let I, J be subsets in J1,mK with J Ă I.Let Z be a manifold in D˝I . Then Z admits a fundamental system of neighbourhoodU in DJ such that the restriction morphism

(2.5.2) H1pBU,StăDM q ÝÑ H1pBZ,StăDM q

is an isomorphism.

Proof. — Let I be the sheaf of irregular values of M. It is enough to prove thatStokes torsors on BZ and their morphisms extend uniquely over a neighbourhoodof Z in DJ depending only on I. To do this, we can suppose that Z is a point Pand that pX,Dq is a germ of regular pair at P . Similarly as in [Tey19, 1.9.1], theconstructibility of StăDM allows to construct a ball U in DJ of radius r ą 0 centred atP and a cover V of BU by subsets V depending only on I, of the form

ź

iPJ

pt0u ˆ Iiq ˆź

iRJ

pr0, rrÎiq ˆ∆

where I1, . . . , Im are closed intervals in S1, where ∆ is the ball of radius r centred at0, and such that V trivializes every torsor under StăDM . At the cost of shrinking V,the constructibility of StăDM further allows to suppose that for every V,W P V, themaps

ΓpV,StăDM q ÝÑ ΓpV X BP,StăDM q

andΓpV XW, StăDM q ÝÑ ΓpV XW X BP,StăDM q

are bijective. For the above choice of ball U , the bijectivity of (2.5.2) follows.

14 J.-B. TEYSSIER

Corollary 2.5.3. — Let pX,Dq be a regular pair. Let M be a meromorphic con-nection on X with poles along D. Suppose that M has good formal structure. Let Ibe a subset in J1,mK. Let iZ : Z ÝÑ D˝I be a manifold in D˝I . Then, the canonicalmorphism

i´1Z R1pD˚ StăDM ÝÑ R1pZ˚ StăDM

is an isomorphism.

Proof. — By definition, the sheaf i´1Z R1pD˚ StăDM is the sheaf associated to the

presheaf

FZ : Open(Z) ÝÑ Set

V ÝÑ limÝÑUĄV

ΓpU,R1pD˚ StăDM q

By taking J to be the empty set in lemma 2.5.1, we observe that the above inductivelimit identifies canonically with H1pBV,StăDM q “ ΓpV,R1pZ˚ StăDM q. Hence, we have

FZ » R1pZ˚ StăDM

From lemma 2.3.3, the presheaf R1pZ˚ StăDM is a sheaf. Corollary 2.5.3 thus follows.

Corollary 2.5.4. — Let pX,Dq be a regular pair. Let M be a meromorphic con-nection on X with poles along D. Suppose that M has good formal structure. Let Pbe a point in D. Then, the stalk of R1pD˚ StăDM at P is canonically identified withH1pBP,StăDM q.

2.6. Constructibility of the sheaf of Stokes torsors. —

Theorem 5. — Let pX,Dq be a regular pair. Let D1, . . . , Dm be the irreducible com-ponents of D. Let M be a meromorphic connection on X with poles along D. Sup-pose that M has good formal structure in the sense of Mochizuki. Then, the sheafR1pD˚ StăDM is constructible on D. More precisely, for every subset I Ă J1,mK, therestriction of R1pD˚ StăDM to D˝I is locally constant.

Proof. — The statement is local along D. Hence, we can suppose that pX,Dq is agerm of regular pair at a point P and that I “ J1,mK. From lemma 2.3.5, we cansuppose that M is unramified. We argue recursively on m. The case where m “ 1will be treated last. Suppose that m ě 2. Since R1pD˚ StăDM is a sheaf, to prove thatits restriction to D˝I “ DI is a local system, it is enough to find a connected openneighbourhood UI of P in D˝I “ DI such that any point Q in UI admits a fundamentalsystem of connected open neighbourhoods VI such that

ΓpUI , R1pD˚ StăDM q ÝÑ ΓpVI , R

1pD˚ StăDM q

is an isomorphism. From corollary 2.5.3, we have to find a connected open neighbour-hood UI of P in D˝I “ DI such that any point Q in UI admits a fundamental systemof connected open neighbourhoods VI such that

H1pBUI ,StăDM q ÝÑ H1pBVI ,StăDM q


is an isomorphism. Since M has good formal structure in the sense of Mochizuki, wecan suppose that the difference of any two distinct irregular values of M at P haspoles along Dm. Put J “ J1,m´1K. Let UI be a small ball in DI centred at P . Fromlemma 2.5.1, we can choose a ball U in DJ centred at P and containing UI such that

H1pBU,StăDM q ÝÑ H1pBUI ,StăDM q

is an isomorphism. Let Q be a point in UI . Let VI be a ball in UI centred at Q. Fromlemma 2.5.1 again, we can choose a ball V in DJ centred at Q with VI Ă V Ă UI andsuch that the morphism

H1pBV,StăDM q ÝÑ H1pBVI ,StăDM q

is an isomorphism. Put U˝J “ U XD˝J and V ˝J “ V XD˝J . Then, we are left to provethat the middle vertical arrow in the commutative diagram

(2.6.1) H1pBUI ,StăDM q

H1pBU,StăDM q

„oo // H1pBU˝J ,StăDM q

H1pBVI ,StăDM q H1pBV,StăDM q„oo // H1pBV ˝J ,StăDM q

is an isomorphism. From lemma 2.4.2, the right horizontal arrows in (2.6.1) areisomorphisms. Hence, we are left to prove that the right vertical arrow in (2.6.1) isan isomorphism. By recursion assumption, the restriction of R1pD˚ StăDM to D˝J is alocal system. We observe that the map V ˝J ÝÑ U˝J is a product of the inclusion of twodiscs ∆˚1 Ă ∆˚2 punctured at 0 with the inclusion of two balls B1 Ă B2 in Cn´m. Inparticular, V ˝J ÝÑ U˝J is a homotopy equivalence. Hence the right vertical arrow in(2.6.1) is an isomorphism. This concludes the reduction of the proof of Theorem 5 tothe case where D is smooth. We now treat the case where D is smooth. The questionis again local on D. Hence, we can suppose that pX,Dq is a germ of smooth divisorat a point P . From lemma 2.3.5, we can suppose that M is unramified. This casewas treated in [Sab02, II 6.3]. Alternatively, since D is smooth, the sheaf of irregularvalues for M is very good. Hence, lemma 8.3.2 reduces the proof of Theorem 5 withD smooth to the analogous statement for marked Stokes filtered local systems. Thiscase follows from Mochizuki’s extension Theorem 4.13 in [Moc11a].

3. The geometry of the moduli of Stokes torsors

3.1. Representability by a scheme. — The first goal of this section is to provethe following representability Theorem:

Theorem 6. — Let pX,Dq be a regular pair. Let M be a meromorphic connectionon X with poles along D. Suppose that M has good formal structure in the sense ofMochizuki 1.5. Then, the functor H1pBD,StăDM q is representable by an affine schemeof finite type over C.

Proof. — The idea is to analyse separately the contributions coming from each stra-tum of D. Let D1, . . . , Dm be the components of D. We argue by recursion on the

16 J.-B. TEYSSIER

depth of the deepest stratum of D. The case where D is smooth will be treated last.Let Z be the deepest stratum of D. From lemma 2.5.1, there is an open neighbour-hood U of Z in BD such that the restriction morphism

H1pBU,StăDM q // H1pBZ,StăDM q

is an isomorphism. Put V “ DzZ. From corollary 2.3.3, we have

H1pBD,StăDM q “ H1pBU,StăDM q ˆH1pBpUXV q,StăDM q H1pBV,StăDM q

By recursion assumption, the functors H1pBV,StăDM q and H1pBpU X V q,StăDM q areaffine schemes of finite type over C. Hence, we are left to prove that H1pBZ,StăDM q isan affine scheme of finite type over C. To do this, we can suppose that Z is connected.Hence, at the cost of shrinking the situation to a small enough open neighbourhood ofa connected component of Z, we can suppose that Z is D˝I “ DI for I “ J1,mK. Fromlemma 2.5.3, H1pBD˝I ,StăDM q is the space of sections of the sheaf R1pD˚ StăDM on BD˝I .From Theorem 5, the restriction of R1pD˚ StăDM to D˝I is a local system. Hence, if Bis a small ball in D˝I centred at point P , the functor H1pBD˝I ,StăDM q is the functorof invariants for the action of π1pD

˝I , P q on H1pBB, StăDM q. That is, if pγ1, . . . , γN q

denotes a set of generators for π1pD˝I , P q, the following diagram of functors

(3.1.1) H1pBD˝I ,StăDM q //

H1pBB, StăDM q

pId,γ1,...,γN q

H1pBB, StăDM qDiagonal

// H1pBB, StăDM qN`1

is cartesian. To prove Theorem 6, we are thus left to prove that H1pBB, StăDM q is anaffine scheme of finite type over C. In particular, we can suppose that pX,Dq is agerm of regular pair at P . Since M is good in the sense of Mochizuki, the conditionsof proposition 2.4.2 are satisfied. Put J “ J1,m ´ 1K. From lemma 2.5.1, there is asmall ball U in DJ centred at P such that

H1pBU,StăDM q ÝÑ H1pBB, StăDM q

is an isomorphism. Hence we are left to prove that H1pBU,StăDM q is an affine schemeof finite type over C. Hence, if we put U˝J “ U XD˝J , proposition 2.4.2 implies thatthe restriction morphism

H1pBU,StăDM q ÝÑ H1pBU˝J ,StăDM q

is an isomorphism. By recursion assumption, the functor H1pBU˝J ,StăDM q is an affinescheme of finite type over C. This concludes the reduction of Theorem 6 to the casewhere D is smooth. If D is smooth, we reduce using (3.1.1) and corollary 2.5.1 toprove that for a point P in D, the functor H1pBP,StăDM q is a scheme of affine type overC. Let i : C ÝÑ X be a smooth curve in X transverse to D at P . Let ι : rC ÝÑ rX bethe morphism induced by C at the level of the real blow-up. Observe that ι inducesan isomorphism above P . Since ι´1 StăDM » StăPi`M, we deduce that

H1pBP,StăDM q » H1pBP,StăPi`Mq


Hence, we are left to prove Theorem 6 in the one dimensional case. This case wastreated by Babbit-Varadarajan [BV89]. This finishes the proof of Theorem 6.

Remark 3.1.2. — Note that in the case where M has rank two, the moduli of Stokestorsors whose existence is asserted by Theorem 6 is known to be an affine space[Tey20].

The diagram (3.1.1) in the proof of Theorem 6 gives the following

Proposition 3.1.3. — Let pX,Dq be a regular pair. Let M be a meromorphic con-nection on X with poles along D. Suppose that M has good formal structure in thesense of Mochizuki 1.5. Let D1, . . . , Dm be the components of D. Let I be a subsetof J1,mK. Suppose that D˝I is connected. Let P be a point in D˝I . Then, the naturalmorphism

H1pBD˝I ,StăDM q // H1pBP,StăDM q

is a closed immersion.

We store the following immediate corollary of proposition 3.1.3 for later use.

Corollary 3.1.4. — Let pX,Dq be a regular pair. Let M be a meromorphic con-nection on X with poles along D. Suppose that M has good formal structure in thesense of Mochizuki 1.5. Let D1, . . . , Dm be the components of D. Let I be a subset ofJ1,mK. Let U Ă V Ă D˝I be non empty open subsets in D˝I such that V is connected.Then, the natural morphism

H1pBV,StăDM q // H1pBU,StăDM q

is a closed immersion.

Proof. — Choose a point P in U . Then, there is a factorization

H1pBV,StăDM q

((

// H1pBU,StăDM q

H1pBP,StăDM q

From proposition 3.1.3, the diagonal arrow is a closed immersion between affineschemes. Hence, the horizontal arrow is a closed immersion.

3.2. Passing from one stratum to an other stratum is a closed immersion.— The next proposition is the technical core of this paper.

Proposition 3.2.1. — Let pX,Dq be a germ of regular pair at a point P . LetD1, . . . , Dm be the components of D. Put I “ J1,mK. Let i P I. Then, for a smallenough ball ∆ in Di centred at P , the morphism of schemes

(3.2.2) H1pBP,StăDM q // H1pB∆˚,StăDM q

is a closed immersion, where ∆˚ “ ∆zŤ

jPIztiuDj Ă D˝i .

18 J.-B. TEYSSIER

Proof. — Let us first construct the morphism (3.2.2). From lemma 2.5.1, for a smallenough ball ∆ in Di centred at P , the restriction morphism

H1pB∆,StăDM q ÝÑ H1pBP,StăDM q

is an isomorphism. Then, the morphism (3.2.2) is defined as the composition

H1pBP,StăDM q H1pB∆,StăDM q //„oo H1pB∆˚,StăDM q

Note that both functors appearing in (3.2.2) are affine schemes as a consequence ofTheorem 6. Let j : B∆˚ ÝÑ BD be the canonical inclusion. The sheaf of algebraicgroups ι´1

DiStăDM is distinguished in StăDiM . We thus have an exact sequence of sheaves

of algebraic groups on BDi

1 // ι´1Di

StăDMι // StăDiM

// Q // 1

There is an adjunction morphism

(3.2.3) ι´1P StăDiM

// ι´1P j˚j

´1 StăDiM “ ι´1P j˚j

´1 StăDM

Hence, there is a factorization

(3.2.4) H1pBP,StăDM q

((

ι˚// H1pBP,StăDiM q

H1pB∆˚,StăDM q

From a similar argument to that in lemma 1.8.4, the adjunction morphism (3.2.3) is anisomorphism of sheaves on BP . Hence, the vertical arrow in (3.2.4) is an isomorphismof functors. Hence, H1pBP,StăDiM q is an affine scheme of finite type over C and toprove proposition 3.2.1, it is enough to prove that

ι˚ : H1pBP,StăDM q // H1pBP,StăDiM q

is a closed immersion. From [Fre57, I.2], there is an exact sequence of pointed functors

(3.2.5) H0pBP,Qq // H1pBP,StăDM qι˚// H1pBP,StăDiM q // H1pBP,Qq

Let us prove that H0pBP,Qq is trivial. The complex of sheaves

StăDiM// BEndM

pD// BEndM

xDi

induces a sequence of sheaves

(3.2.6) Q

// BEndMpD

// BEndMxDi

By applying pD˚ and then looking at the germs at P , we deduce from [Sab00, p44]the following sequence

(3.2.7) 0 // H0pBP,Qq // EndMpD,P

// EndMxDi,P


By flatness of EndM over OX , the second map in (3.2.7) is injective. Hence,H0pBP,Qq is trivial. Thus, the following diagram of functors


ι˚

// ˚

H1pBP,StăDiM q // H1pBP,Qq

is cartesian, where ˚ denotes the trivial Q-torsor. If we knew that H1pBP,Qq is ascheme, we would directly obtain that ι˚ is a closed immersion. This question doesnot seem to follow from the use of skeletons in [Tey19]. We will circumvent thisproblem with a group theoretic argument.

From lemma 1.8.2, the sheaf StăDiM is constructible with respect to the stratificationof BD induced by the Stokes locus of M. Hence, the same argument as in [Tey19,1.9.1] applies. In particular, there exists a cover U of BP by open subsets such thatthe morphism of affine schemes

(3.2.9) Z1pU ,StăDiM q // H1pBP,StăDiM q

is surjective at the level of R-points for every C-algebra R. From [BV89, 2.7.3], themorphism (3.2.9) admits a section. Composing this section with

Z1pU ,StăDiM q // Z1pU ,Qq

gives rise to a commutative triangle of functors

(3.2.10) H1pBP,StăDiM q //

''

H1pBP,Qq

Z1pU ,Qq

OO

The algebraic groupGU :“

ź

UPUΓpU,Qq

acts on Z1pU ,Qq. Let(3.2.11) GU ÝÑ Z1pU ,Qqbe the morphism of schemes obtained by restricting the action of GU to the trivialcocycle. Since H0pBP,Qq » 0, the morphism (3.2.11) is a monomorphism. There is acommutative diagram


ι˚

GU

// ˚

H1pBP,StăDiM q // Z1pU ,Qq // H1pBP,Qq

We would like to reduce the problem of proving that ι˚ is a closed immersion to theproblem of proving that (3.2.11) is a closed immersion. To do this, we would like to

20 J.-B. TEYSSIER

fill the left diagram in (3.2.12) into a cartesian square. Note that the right squarein (3.2.12) may not be cartesian since there may be cocycles in Z1pU ,Qq that arecohomologous to the trivial cocycle only after passing to a refinement of U . To treatthis problem, we argue by using the universal torsor under StăDM on BP .

Let T univ be the universal torsor under StăDM on BP . Let A be the ring of func-tions of H1pBP,StăDM q. From the commutativity of (3.2.10), the image γ of T univ inZ1pU ,QpAqq induces the trivial QpAq-torsor. Hence, there exists a refinement V of Usuch that γ|V is cohomologous to the trivial cocycle, that is, such that γ|V lies in theimage of GVpAq ÝÑ Z1pV,QpAqq. Hence, there is a commutative square

(3.2.13) H1pBP,StăDM q //

ι˚

GV

H1pBP,StăDiM q // Z1pV,Qq

This square is cartesian. Indeed, let F be the fibre product of H1pBP,StăDiM q withGV over Z1pV,Qq. By definition, there is a commutative diagram of functors

(3.2.14) H1pBP,StăDM q //

ι˚((

F

H1pBP,StăDiM q

Since the right vertical arrow in (3.2.13) is a monomorphism, F is a sub-functor ofH1pBP,StăDiM q. Hence, all maps in (3.2.14) are inclusions of functors. We are thus leftto prove that F is a sub-functor of H1pBP,StăDM q. This is an immediate consequenceof the fact that H1pBP,StăDM q is the functor of torsors T P H1pBP,StăDiM q inducingthe trivial Q-torsor.

Hence, to prove that ι˚ is a closed immersion, we are left to show that (3.2.11) forV is a closed immersion. From the general theory of algebraic group actions, the map(3.2.11) factors as

GVα // O

β// Z1pV,Qq

where α is faithfully flat, where O is the orbit of the trivial cocycle under GV andwhere β is an immersion of schemes. Since smoothness is a local property for the fppftopology [SPD, 05B5], the smoothness of GV 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 Z1pV,Qq. It is enough to show that O isclosed in Z1pV,Qqred. From Kostant-Rosenlicht theorem [Bor91, I 4.10], it is enoughto show that GV is a unipotent algebraic group, which is a consequence of the factthat the Stokes sheaves are sheaves of unipotent algebraic groups. This concludes theproof of proposition 3.2.1.


3.3. Proof of Theorem 3. — Let U Ă V Ă D be non empty open subsets in Dsuch that V is connected. We want to show that the natural morphism

(3.3.1) H1pBV,StăDM q // H1pBU,StăDM q

is a closed immersion of affine schemes of finite type over C. Let A be the set of opensubsets U 1 in V containing U and such that the natural morphism

H1pBU 1,StăDM q ÝÑ H1pBU,StăDM q

is a closed immersion. We want to show that A contains V . Note that A is not emptysince it contains U . Let A1 be a subset of A which is totally ordered for the inclusion.Let R be the ring of functions of H1pBU,StăDM q. For U 1 P A1, let IU 1 be the idealof functions of H1pBU 1,StăDM q in H1pBU,StăDM q. By assumption on A1, the family ofideals pIU 1qU 1PA1 is totally ordered for the inclusion. Hence, I :“

Ť

U 1PIU1is an ideal

in R. Since R is noetherian, there exists U 10 P A1 such that I “ IU 10 . In particular,IU 1 “ IU 10 for every U 1 P A1 containing U 10. Set V 1 :“

Ť

U 1PA1 U1. From lemma 2.3.3,

we deduce

H1pBV 1,StăDM q » limU 1PA1

H1pBU 1,StăDM q

» limU 1PA1,U 10ĂU

1H1pBU 1,StăDM q

» H1pBU 10,StăDM q

Thus, V 1 P A. From Zorn lemma, we deduce that A admits a maximal element W . IfW is closed in V , then we have W “ V by connectedness of V . Suppose now that Wis not closed in V . Let P PW zW and let B be a small ball in V containing P and suchthat H1pBB, StăDM q ÝÑ H1pBP,StăDM q is an isomorphism. Set W 1 :“ W Y B Ă V .We are going to show that W 1 P A, which contradicts the fact that W is maximal inA. From the factorization

(3.3.2) H1pBW 1,StăDM q // H1pBW, StăDM q // H1pBU,StăDM q

we are left to show that the first arrow in (3.3.2) is a closed immersion. From lemma2.3.3, the following diagram

(3.3.3) H1pBW 1,StăDM q //

H1pBB, StăDM q

H1pBW, StăDM q // H1pBpW XBq,StăDM q

is cartesian. Hence, it is enough to show that the right vertical arrow in (3.3.3) is aclosed immersion. Let pPnqnPN be a sequence of points in W converging to P . SinceW is open, the sequence pPnqnPN can be supposed to lie in some D˝i for i P J1,mK. Let∆ Ă B be a small enough neighbourhood of P in Di. Set ∆˚ “ ∆z

Ť

jPIztiuDj Ă D˝i .From our choice for i, the open set W X ∆˚ is not empty. We have the following

22 J.-B. TEYSSIER

commutative diagram

(3.3.4) H1pBB, StăDM q //

H1pB∆˚,StăDM q

H1pBpW XBq,StăDM q // H1pBpW X∆˚q,StăDM q

From proposition 3.2.1, the top horizontal arrow in (3.3.4) is a closed immersion. Fromcorollary 3.1.4, the right vertical arrow in (3.3.4) is a closed immersion. Hence, theleft vertical arrow in (3.3.4) is a closed immersion. Hence, W 1 P A, which contradictsthe fact that W is maximal in A. Thus, W “ V P A, which finishes the proof ofTheorem 3.

4. Stokes torsors and marked connections

4.1. Notations. — For a morphism of smooth complex varieties π : Y ÝÑ X, wedenote by π` the inverse image functor for D-modules and by π` the direct imagefunctor for D-modules. For precise definitions, let us refer to [HTT00].

In this section, pX,Dq will denote a regular pair. Let M be a connection on Xwith poles along D. Suppose that M has good formal structure.

4.2. Definition of marked connections and relation with Stokes torsors. —Let us recall that a M-marked connection is the data of a pair pM, isoq where M isa germ of meromorphic connection with poles along D defined in a neighbourhoodof D in X, and where iso : M

pD ÝÑ MpD is an isomorphism of formal connections.

We denote by IsomisopM,Mq the StăDM pCq-torsor of isomorphisms between BM andBM which are asymptotic to iso along D. The proof of the following statement wassuggested to me by T. Mochizuki. I thank him for kindly sharing it. When D issmooth, it was known to Malgrange [Mal83a]. See also [Sab02, II 6.3].

Lemma 4.2.1. — The map associating to every isomorphism class of M-markedconnection pM, isoq the StăDM pCq-torsor IsomisopM,Mq is bijective.

Proof. — Let us construct an inverse. Take T P StăDM pCq and let g “ pgijq be acocycle for T associated to a cover pUiqiPI of BD. Let L be the Stokes filtered localsystem on BD associated to M. Set Li :“ L|Ui . Then, g allows to glue the Li intoa Stokes filtered local system LT on BD independent of the choice of g. From theirregular Riemann-Hilbert correspondence [Moc11a, 4.11], LT is the Stokes filteredlocal system associated to a unique (up to isomorphism) good meromorphic connectionMT defined in a neighbourhood of D and with poles along D. By construction, theisomorphism LT |Ui ÝÑ L|Ui corresponds to an isomorphism BMT |Ui ÝÑ BM|Ui .We thus obtain a formal isomorphism isoi : BMT , pD|Ui,

ÝÑ BMpD|Ui

. On Uij , thediscrepancy between isoi and isoj is measured by the asymptotic of gij along D.By definition, this asymptotic is Id. Hence, the isoi glue into a globally definedisomorphism BMT , pD ÝÑ BM

pD. Applying pD˚ thus yields an isomorphism iso :


MT , pD ÝÑ MpD. It is then standard to check that the map T ÝÑ pMT , isoq is the

sought-after inverse.

4.3. Proof of Theorem 4. — We are now in position to prove Theorem 4. Let Xbe a germ of smooth algebraic variety around a point 0. Let D be a germ of divisorpassing through 0. Let M be a germ of meromorphic connection at 0 with polesalong D. Let C be a smooth curve passing through 0 and not contained in any of theirreducible components of D. Let pM1, iso1q and pM2, iso2q be M-marked connectionssuch that

pM1, iso1q|C » pM2, iso2q|C

We want to show that pM1, iso1q and pM2, iso2q are isomorphic in a neighbourhoodof 0. Let π : Y ÝÑ X be a resolution of turning points for M around 0. Sucha resolution exists by works of Kedlaya [Ked11] and Mochizuki [Moc11b]. SetE :“ π´1pDq. At the cost of blowing up further, we can suppose that the stricttransform C 1 of C is transverse to E at a point P in the smooth locus of E. Note thatE is connected. From lemma 4.2.1, the π`M-marked connections pπ`M1, π

` iso1q

and pπ`M2, π` iso2q define two C-points of H1pBE,StăEπ`Mq. For i “ 1, 2, the cone of

the canonical comparison morphism

(4.3.1) π`π`Mi ÝÑMi

is supported on D. Note that the right-hand side of (4.3.1) is localized along D.From [Meb04, 3.6-4], the left-hand side of (4.3.1) is localized along D. Hence, themorphism (4.3.1) is an isomorphism. Thus

pπ`π`Mi, π`π

` isoiq » pMi, isoiq

Hence, it is enough to show pπ`M1, π` iso1q » pπ

`M2, π` iso2q. By assumption,

pπ`M1, π` iso1q|C1 » pM1, iso1q|C

» pM2, iso2q|C

» pπ`M2, π` iso2q|C1

Hence, pπ`M1, π` iso1q|C1 and pπ`M2, π

` iso2q|C1 define the same C-point inH1pBP,StăPpπ`Mq|C1

q. Let ι : ĂC 1 ÝÑ rY be the morphism induced by C 1 ÝÑ Y at thelevel of the real blow-up. Observe that ι induces an isomorphism above P . Sinceι´1 StăEπ`M » StăPpπ`Mq|C1

, we have

H1pBP,StăPpπ`Mq|C1q » H1pBP,StăEπ`Mq

Hence, the image of pπ`M1, π` iso1q and pπ`M2, π

` iso2q by the restriction map

(4.3.2) H1pBE,StăEπ`Mq// H1pBP,StăEπ`Mq

are the same. From Theorem 3, the map (4.3.2) is a closed immersion. Hence,pπ`M1, π

` iso1q » pπ`M2, π

` iso2q, which concludes the proof of Theorem 4.

24 J.-B. TEYSSIER

4.4. Obstruction theory and tangent space. — We use the notationsfrom 4.1. Let us compute the obstruction theory of H1pBD,StăDM q at a pointT0 P H

1pBD,StăDN pCqq. We fix a morphism of infinitesimal extensions of C-algebras

R1 ÝÑ R ÝÑ C, I :“ KerR1 ÝÑ R

such that I is annihilated by KerR1 ÝÑ C. In particular, I2 “ 0 and I is endowedwith a structure of C-vector space, which we suppose to be finite dimensional. LetT P H1pBD,StăDM pRqq lifting T0. Choose a cover U “ pUiqiPK of BD such that T comesfrom a cocycle g “ pgijqi,jPK . Set LipRq :“ Lie StăDM pRq|Ui . The identifications

LipRq|Uij„ÝÑ LjpRq|Uij

M ÝÑ g´1ij Mgij

allow to glue the LipRq into a sheaf of R-Lie algebras over BD denoted byLie StăDM pRqT and depending only on T and not on g. For t “ ptijkq P

C2pU ,Lie StăDM pRqT q, we denote by sijk the unique representative of tijk inΓpUijk, LipRqq. Then

pdtqijkl “ tjkl ´ tikl ` tijl ´ tijk

“ rgijsjklg´1ij ´ sikl ` sijl ´ sijks

We have the following

Lemma 4.4.1. — There exists

obpT q P I bC H2pBD,Lie StăDM pCqT0q

such that obpT q “ 0 if and only if T lifts to H1pBD,StăDM pR1qq.

Proof. — For every i, j P K, let hij P ΓpUij ,StăDM pR1qq be an arbitrary lift of gijto R1. We can always choose the hij to satisfy hii “ Id and hijhji “ Id. SinceLie StăDM pR1q is locally free,

I ¨ Lie StăDM pR1q » I bR1 Lie StăDM pR1q » I bC Lie StăDM pCq

We will use both descriptions without mention. We set

sijk :“ hijhjkhki ´ Id P ΓpUijk, I ¨ Lie StăDM pR1qq

We see sijk as a section of I bC LipCq over Uijk and denote by rsijks its class inIbCLie StăDM pCqT0 . We want to prove that the rsijks define a cocycle. As seen above,this amounts to prove the following equality in ΓpUijk, I bC Lie StăDM pCqq

(4.4.2) gijp0qsjklg´1ij p0q ´ sikl ` sijl ´ sijk “ 0


Where gijp0q is the image of gij by R ÝÑ C. We have

gijp0qsjklg´1ij p0q “ hijhjkhklhljhji ´ Id

“ phijhjk ´ hik ` hikqhklhljhji ´ Id

“ phijhjk ´ hikqgklp0qgljp0qgjip0q ` hikhklhljhji ´ Id

“ phijhjk ´ hikqgkip0q ` hikhklhljhji ´ Id

“ phijhjk ´ hikqhki ` hikhklhljhji ´ Id

“ hijhjkhki ` hikhklhljhji ´ 2 Id

We now see how the second term of the last line above interacts with the second termof the left-hand side of (4.4.2).

hikhklhljhji ´ sikl “ hikhklhljhji ´ hikhklhli ` Id

“ hikhklphljhji ´ hliq ` Id

“ gikp0qgklp0qphljhji ´ hliq ` Id

“ gilp0qphljhji ´ hliq ` Id

“ hilhljhji

Hence,

gijp0qsjklg´1ij p0q ´ sikl ` sijl ´ sijk “ hilhljhji ` hijhjlhli ´ 2 Id

“ phijhjlhliq´1 ` hijhjlhli ´ 2 Id

“ phijhjlhliq´1pphijhjlhliq

2 ´ 2hijhjlhli ` Idq

“ phijhjlhliq´1s2

ijl

“ 0

where the last equality comes from I2 “ 0. Hence, the rsijks define a cocycle ofI bC Lie StăDM pCqT0 . An other choice of lift gives rise to homologous cocycles. Wedenote by obpT q the class of prsijksqijk in H2pBD, IbCLie StăDM pCqT0q. It is standardto check that obpT q has the sought-after property.

Corollary 4.4.3. — Let pM, isoq be a M-marked connection. Then, the spaceH2pD, Irr˚D EndMqq is an obstruction theory for H1pBD,StăDM q at IsomisopM,Mq.

Proof. — Set T :“ IsomisopM,Mq. As observed in [Tey19, 5.2], the canonical iden-tification

H0 DRăD EndM „ // Lie StăDM pCqT

induces

HipBD,Lie StăDM pCqT q » HipBD,Lie StăDM pCqT q

» HipBD,H0 DRăD EndMq

» HipBD,DRăD EndMq

» HipD, Irr˚D EndMq

26 J.-B. TEYSSIER

The second identification comes from the fact [Hie09, Prop. 1] that DRăD EndMis concentrated in degree 0. The third identification comes from [Sab17, 2.2]. Then,corollary 4.4.3 follows from lemma 4.4.1.

Reasoning exactly as in [Tey19, 5.2.1], we prove the following

Lemma 4.4.4. — For every M-marked connection pM, isoq, the tangent space ofH1pBD,StăDM q at pM, isoq identifies canonically with H1pD, Irr˚D EndMq.

5. Reduction of Theorem 1 to extending the formal model

5.1. Reduction to the dimension 2 case. — In this subsection, we reduce theproof of Theorem 1 to the dimension 2 case. The main tool is André’s goodnesscriterion [And07, 3.4.3] in terms of Newton polygons. This reduction does not seemsuperfluous. Of crucial importance for the sequel of the proof will be indeed the factthat when X is an algebraic surface and D a smooth divisor in X, then for every point0 P D and every meromorphic connection M on X with poles along D, the formalmodel of M splits on a small enough punctured disc around 0. This fact is specific todimension 2, since it pertains to the property that turning points in dimension 2 areisolated.

Lemma 5.1.1. — The converse inclusion in Theorem 1 is true in any dimension ifit is true in dimension 2.

Proof. — Take n ą 2. We argue recursively by supposing that Theorem 1 holds indimension strictly less than n and we prove that Theorem 1 holds in dimension n.Let X be a smooth complex algebraic variety of dimension n. Let D be a smoothdivisor in X. Let M be an algebraic meromorphic connection on X with poles alongD. Let 0 P D and suppose that Irr˚DM and Irr˚D EndM are local systems in aneighbourhood of 0. If j : XzD ÝÑ X and i : D ÝÑ X are the canonical inclusions,we have a distinguished triangle

j!L // SolM // i˚ Irr˚DM

where L is a local system on the complement of D. Hence, the characteristic cycle ofSolM is supported on the union of T˚XX with T˚DX. From a theorem of Kashiwaraand Schapira [KS90, 11.3.3], so does the characteristic cycle of M. Hence, anysmooth hypersurface transverse to D and passing through 0 is non characteristic withrespect to M in a neighbourhood of 0. Let us choose such a hypersurface Z and letiZ : Z ÝÑ X be the canonical inclusion. From [And07, 3.4.3], the turning locusof M is a closed subset of D which is either empty or purely of codimension 1 inD. Since n ą 2, the hypersurface Z can consequently be chosen such that M andEndM have good formal structure generically along Z X D. The connection i`ZMis a meromorphic connection with poles along Z X D. It satisfies the hypothesis ofTheorem 1 at the point 0. Indeed by Kashiwara’s restriction theorem [Kas95],

Irr˚ZXD i`ZM “ pSol i`ZMq|ZXD » pSolMq|ZXD


and similarly for EndM. Hence, Irr˚ZXD i`ZM and Irr˚ZXD End i`ZM are local systems

in a neighbourhood of 0 in Z X D. By recursion hypothesis, i`ZM is good at 0. Inparticular, the Newton polygon of i`ZM at 0 (which is also the Newton polygon ofM at 0) is the generic Newton polygon of i`ZM along Z XD. From our choice for Z,the generic Newton polygon of i`ZM along Z XD is the generic Newton polygon ofM along D. Hence, the Newton polygon of M at 0 is the generic Newton polygon ofM along D, and similarly with EndM. By a theorem of André [And07, 3.4.1], wededuce that M has good formal structure at 0, which proves lemma 5.1.1.

5.2. Setup and recollections. — From now on, we restrict the situation to di-mension 2. We use coordinates px, yq on A2

C and set Dx :“ ty “ 0u, Dy :“ tx “ 0u.Let D be a neighbourhood of 0 in Dx and let CrDs be the coordinate ring of D. SetD˚ :“ Dzt0u.

Let M be an algebraic meromorphic flat bundle on a neighbourhood of D in A2C

with poles along D. In algebraic terms, MpD defines a CrDsppyqq-differential module.

At the cost of shrinking D if necessary, we can suppose that the restriction M˚ of Mto a neighbourhood of D˚ has good formal structure at every point of D˚.

There is a ramification v “ y1d, d ě 1 and a finite Galois extension LCpxq suchthat the set I of generic irregular values for M lies in FracLpvq. If p : DL ÝÑ D isthe normalization of D in L, the generic irregular values of M are thus meromorphicfunctions on DL Â

1v. We have

(5.2.1) Lppvqq bM »à

aPIEa bRa

where the Ra are regular. Following [And07, 3.2.4], we recall the following

Definition 5.2.2. — We say that M is semi-stable at P P D if(1) We have I Ă CrDLsP ppvqq.(2) The decomposition (5.2.1) descends to CrDLsP ppvqq bM.

In this definition, CrDLsP denotes the localization of CrDLs above P . This is asemi-local ring. Let πa P Lppvqq bEndM be the projector on the factor EabRa. Asexplained in [And07, 3.2.2], the point P is stable if and only if the generic irregularvalues of M and the coefficients of the πa in a basis of EndM belong to CrDLsP ppvqq.Since M has good formal structure at any point of D˚, the generic irregular valuesof M and the coefficients of the πa in a basis of EndM belong to CrDLsP ppvqq forevery P P D˚. Hence, they belong CrD˚Lsppvqq where D

˚L :“ Dzp´1p0q. Thus

(5.2.3) CrD˚Lsppvqq bM » CrD˚Lsppvqq bN ˚L

whereN ˚L “

à

aPIEa bRa

is a germ of meromorphic connection defined on a neighbourhood of D˚L in DL Â1v

and with poles along D˚L. The action of

GalpLCpxqq ˆZdZ

28 J.-B. TEYSSIER

on the left-hand side of (5.2.3) induces an action on N ˚L . Taking the invariants yields

a meromorphic flat bundle N ˚ defined on a neighbourhood Ω of D˚ in A2C. By Galois

descent, (5.2.3) descents to an isomorphism iso˚ between the formalizations of M˚

and N ˚ along D˚.

5.3. Reduction to the problem of extending the formal model. — Thegoal of this subsection is to show that Theorem 1 reduces to prove that the M˚-marked connection pN ˚, iso˚q defined in 5.2 extends into a M-marked connection ina neighbourhood of 0. To do this, we need three preliminary lemmas. The notationsand constructions from 5.2 are in use.

Lemma 5.3.1. — Suppose that N ˚ extends into a meromorphic flat bundle N de-fined in a neighbourhood of D in A2

C and with poles along D. Then, N is semi-stableat 0.

Proof. — It is enough to treat the case where K “ Cpxq and d “ 1. In that case,discussion 5.2 shows that on a neighbourhood Ω of D˚ in A2

C, we have

N ˚ “à

aPIN ˚a

where N ˚a is a meromorphic connection on Ω with poles along D˚ and with single

irregular value a. The open DÂ1C retracts on the small neighbourhood on which N

is defined. Since N is smooth away from D, we deduce that N extends canonicallyinto a meromorphic connection on D Â1

C with poles along D.Let a P I. The restriction of the projector πa to the complement of D˚ in Ω is

a flat section of EndN . Since D˚ ˆ A1C retracts on Ω, parallel transport allows to

extend πa canonically with D˚ ˆ A1C. We still denote by πa this extension. Hence,

N ˚a extends into a meromorphic connection on D˚ ˆ A1

C with poles along D˚. Letγ be a small loop in Ω going around the axis Dy. By assumption, the monodromyof N along γ is trivial. Thus, πa is invariant under the monodromy of EndN alongγ. Hence, πa extends canonically to pD Â1

Cqzt0u. By Hartog’s property, it extendsfurther into a section $a of EndN on D Â1

C.Set Na :“ $apN q Ă N for every a P I. We have $2

a “ $a andř

aPI $a “ IdNbecause these equalities hold on a non empty open set. Hence, N “ ‘aPINa. Since$a is flat, the connection on N preserves each Na. Let us prove that the Na arelocally free as ODÂ1

Cp˚Dq-modules.

Let E be a Deligne-Malgrange lattice [Mal96] for N . Since we work in dimension2, 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, $apEq is a direct factor of E. So $apEq is a vector bundle. Thus,

Na “ $apN q “ $apEp˚Dqq “ p$apEqqp˚Dq

is a locally free ODÂ1Cp˚Dq-module of finite rank with connection extending N ˚

a . Toprove lemma 5.3.1, we are thus left to consider the case where I “ tau.

If I “ tau, then [And07, 3.3.1] implies a P CrDsppyqq. Hence, R :“ EábNpD is a

formal meromorphic connection with poles along D. By assumption, R is generically


regular alongD. From [Del70, 4.1], we deduce thatR is regular. Hence, NpD “ EabR

with R regular, which concludes the proof of lemma 5.3.1.

Lemma 5.3.2. — Let N be a meromorphic flat connection with poles along D. Sup-pose that N is semi-stable at 0 and that Irr˚DN and Irr˚D EndN are local systems ina neighbourhood of 0. Then, N has good formal structure at 0.

Proof. — Let I be the set of irregular values of N at 0. There is a ramificationv “ y1d, d ě 1 and a finite Galois extension LCpxq such that I Ă Lppvqq. LetDL ÝÑ D be the normalization of D in L. At the cost of shrinking D, we cansuppose that every point of D is semi-stable for N . Hence, I Ă CrDLsppvqq and

CrDLsppvqq bN “à

aPIEa bRa

where the connections Ra are regular. As seen in the proof of lemma 5.1.1, the as-sumption on Irr˚D implies that any smooth curve transverse to D is non characteristicfor N . Taking the axis Dy yields

dimH1 Irr˚0 N|Dy “ dimpH1 Irr˚DN q0 “ÿ

aPIpordy aq rkRa

On the other hand, choose a point P P DL above 0. Then, the irregular values ofN|Dy are the apP q, a P I. Thus,

H1 Irr˚0 N|Dy “ÿ

aPIordy apP q rkRa

Hence, ordy apP q “ ordy a for every a P I. In particular, the coefficient function ofthe highest power of 1v contributing to a P I does not vanish at P . Arguing similarlyfor EndN , we obtain that N has good formal structure at 0.

Lemma 5.3.3. — Suppose that Irr˚DM is a local system. For every M-marked con-nection pN , isoq, the complex Irr˚DN is a local system.

Proof. — From [Meb90], the complex Irr˚DN is perverse. To prove that it is alocal system, it is thus enough to prove that the local Euler Poincaré characteris-tic χpD, Irr˚DN q : D ÝÑ Z of Irr˚DN is constant. From the local index theorem[Kas73][Mal81], the local Euler Poincaré characteristic of Irr˚DN depends only onthe characteristic cycle of N . Since the characteristic cycle of N depends only on Nvia N

pD, we haveχpD, Irr˚DN q “ χpD, Irr˚DMq

By assumption, χpD, Irr˚DMq is constant. Hence, χpD, Irr˚DN q is constant, whichfinishes the proof of lemma 5.3.3.

Proposition 5.3.4. — Let D be an open neighbourhood of 0 in an hyperplane of A2C.

Let M be an algebraic meromorphic flat bundle on a neighbourhood of D with polesalong D. Set D˚ “ Dzt0u and let M˚ be the restriction of M to a neighbourhoodof D˚. Let pN ˚, iso˚q be the M˚-marked connection constructed in 5.2. Supposethat Irr˚DM and Irr˚D EndM are local systems in a neighbourhood of 0. Then, ifpN ˚, iso˚q extends into a M-marked connection, M has good formal structure at 0.

30 J.-B. TEYSSIER

Proof. — Let pN , isoq be a M-marked connection extending pN ˚, iso˚q. From lemma5.3.1, the extension N is semi-stable at 0. From lemma 5.3.3, we know that Irr˚DNand Irr˚D EndN are local systems in a neighbourhood of 0. From lemma 5.3.2, wededuce that N has good formal structure at 0. Hence, so does M.

6. Extension via moduli of Stokes torsors

6.1. A geometric extension criterion. — In this subsection, we relate the mod-uli of Stokes torsors to the problem of extending marked connections. Let D be anopen subset of a hyperplane in A2

C. Pick P P D. Set D˚ :“ DztP u. Let M bean algebraic meromorphic flat bundle in a neighbourhood U of D in A2

C and withpoles along D. Let M˚ be the restriction of M to UztP u. Let π : Y ÝÑ A2

C be aresolution of the turning point P for M. Such a resolution exists by works of Kedlaya[Ked10] and Mochizuki [Moc09]. Let ∆ be an open disc of D containing P . Set∆˚ “ ∆ztP u. Set E :“ π´1p∆q and pick Q P ∆˚. Let

Φ : H1pBE,StăEπ`Mq// H1pBQ,Stă∆

M q

be the restriction morphism of Stokes torsors.

Lemma 6.1.1. — Let pN ˚, iso˚q be a M˚-marked connection such that pN ˚Q, iso

˚Qq

lies in the image of Φ. Then, pN ˚, iso˚q extends into an M-marked connection.

Proof. — From lemma 4.2.1, any C-point of H1pBE,StăEπ`Mq comes from a uniqueπ`M-marked connection. Hence, there exists pN 1, iso1q P H1pBE,StăEπ`Mq such thatΦpN 1, iso1q “ pN ˚

P , iso˚P q. From [Meb04, 3.6-4], the D-module N :“ π`N 1 is a

meromorphic connection defined in a neighbourhood of ∆ and and with poles along∆. By flat base change

Np∆ » O

A2C|∆bRπ˚pDXÑA2

CbN 1q

» Rπ˚pOzX|E

bDXÑA2CbN 1q

» Rπ˚pDXÑA2CbN 1

pEq

» π`N 1pE

and similarly Mp∆ » π`pπ

`MqpE . Hence, iso :“ π` iso1 defines an isomorphism

between Np∆ and M

p∆. So pN , isoq is a M-marked connection in a neighbourhood of∆. By definition, the germ of pN , isoq at Q is pN ˚

Q, iso˚Qq. Since R1p∆˚ Stă∆˚

M is alocal system on ∆˚, we deduce

pN , isoq|∆˚ “ pN ˚, iso˚q|∆˚

Hence, the gluing of pN , isoq with pN ˚, iso˚q provides the sought-after extension ofpN ˚, iso˚q into an M-marked connection. So lemma 6.1.1 is proved.

Let us now give a sufficient condition for the surjectivity of Φ in terms of theirregularity complex.


Proposition 6.1.2. — With the notations from 6.1, suppose furthermore that theperverse complex Irr˚D EndM is a local system on ∆. Then Φ induces an isomorphismbetween each irreducible component of H1pBE,StăEπ`Mq and H

1pBQ,Stă∆M q.

Proof. — From [BV89], we know that H1pBQ,Stă∆M q is an affine space. Since affine

spaces in characteristic 0 have no non trivial finite étale covers, it is enough to provethat Φ is finite étale. From Theorem 3, the morphism Φ is a closed immersion. Weare thus left to show that Φ is étale.

Etale morphisms between smooth schemes of finite type over C are those mor-phisms inducing isomorphisms on the tangent spaces. Hence, we are left to prove thatH1pBE,StăEπ`Mq is smooth and that Φ induces isomorphisms on the tangent spaces.Let pM, isoq be a π`M-marked connection. From corollary 4.4.3, an obstructiontheory to lifting infinitesimally the Stokes torsor of pM, isoq is given by

(6.1.3) H2pE, Irr˚E EndMq » H2p∆, Irr˚D π` EndMq » 0

The first identification expresses the compatibility of irregularity with proper push-forward. Furthermore, from lemma 5.3.3 applied to the EndM-marked connectionpπ` EndM,π` isoq, the perverse complex Irr˚D π` EndM is a local system concen-trated in degree 1. This implies the vanishing (6.1.3). Hence, H1pBE,StăEπ`Mq issmooth at pM, isoq. From lemma 4.2.1, any C-point of H1pBE,StăEπ`Mq is of the formpM, isoq. Thus, H1pBE,StăEπ`Mq is smooth. Furthermore, we have a commutativediagram

TpM,isoqH1pBE,StăEπ`Mq

//

o

TpMQ,isoQqH1pBQ,Stă∆

M q

o

H1pE, Irr˚E EndMq //

o

pH1 Irr˚D EndMqQ

|

H1p∆, Irr˚D π` EndMq //

o

pH1 Irr˚D EndMqQ

|

H0p∆,H1 Irr˚D π` EndMq // pH1 Irr˚D EndMqQ

The first vertical maps are isomorphisms by lemma 4.4.4. As already seen,Irr˚D π` EndM is a local system concentrated in degree 1. Hence, the last ver-tical and the bottom arrows are isomorphisms. Thus, the tangent map of Φ atpM, isoq is an isomorphism. This finishes the proof of proposition 6.1.2.

6.2. Proof of Theorem 1. — Let X be a smooth complex algebraic variety. LetD be a smooth divisor in X. Let M be an algebraic meromorphic connection withpoles along D.

We first prove the direct inclusion in Theorem 1. Suppose that M has good formalstructure at a closed point P P D. Since the good formal structure locus of M is

32 J.-B. TEYSSIER

open in D [And07], we can suppose at the cost of restricting the situation that Mhas good formal structure along D. By Mebkhout’s theorem [Meb90], the complexesIrr˚DM and Irr˚D EndM are perverse. To prove that they are local systems on D, it isthus enough to prove that their local Euler Poincaré characteristic is constant. Fromthe local index theorem [Kas73][Mal81], the local Euler Poincaré characteristic ofIrr˚DM depends only on the characteristic cycle of M. Since the characteristic cycleofM depends only onM viaM

pD, we are reduced to treat the case whereM “ EabRwhere a P OXp˚Dq is good and where R is a regular singular meromorphic connectionwith poles along D. Since Irr˚D is exact, we can suppose further that the rank of R isone. In that case, a standard computation shows that the characteristic cycle of Mis supported on the union of T˚XX with T˚DX. Hence, any smooth transverse curveto D is non-characteristic for M. Let P P D and let C be a smooth transverse curveto D passing through P . From [Kas95], we have

pIrr˚DMqP » Irr˚P M|C » CordD ar´1s

Hence, the local Euler-Poincaré characteristic of Irr˚DM is constant and similarly forIrr˚D EndM. This finishes the proof of the direct inclusion in Theorem 1.

We now prove the converse inclusion in Theorem 1. From lemma 5.1.1, we cansuppose that X is a surface. Let P P D such that Irr˚DM and Irr˚D EndM are localsystems in a neighbourhood of P in D. At the cost of taking local coordinates aroundP , we can suppose thatD is an open subset of a hyperplane inA2

C. PutD˚ :“ DztP u.

Let M˚ be the restriction of M to a small neighbourhood of D˚ in X. Let pN ˚, iso˚qbe the M˚-marked connection defined in 5.2. Such a connection exists at the costof replacing X by a small enough neighbourhood of P in X. From proposition 5.3.4,we are left to show that pN ˚, iso˚q extends into a M-marked connection. Let ∆ bea small enough disc in D containing P such that Irr˚DM and Irr˚D EndM are localsystems on ∆. Put ∆˚ :“ ∆ztP u. Let π : Y ÝÑ X be a resolution of turning pointsfor M at P . Set E :“ π´1p∆q and pick Q P ∆˚. Let

Φ : H1pBE,StăEπ`Mq// H1pBQ,Stă∆

M q

be the restriction morphism of Stokes torsors. From lemma 6.1.1, to prove thatpN ˚, iso˚q extends into a M-marked connection, it is enough to prove that pN ˚

Q, iso˚Qq

lies in the image of Φ. This is indeed the case by lemma 6.1.2, which finishes the proofof Theorem 1.

7. A boundedness theorem for turning points

7.1. Nearby slopes. — Let X be a smooth complex algebraic variety and letM be an holonomic DX -module. Let f P OX be a non constant function. Letψf be the nearby cycle functor associated to f [Kas83][Mal83b][Meb89][MM04].Following [Tey16], we recall that the nearby slopes of M associated to f are therational numbers r P Qě0 such that there exists a germ N of meromorphic connectionat 0 P A1

C with slope r such that

(7.1.1) ψf pMb f`Nq ‰ 0


We denote by Slnbf pMq the set of nearby slopes of M associated to f . In dimension 1,

the nearby slopes of M associated to a local coordinate centred at a point 0 are theusual slopes of M at 0. See [Tey16, 3.3.1] for a proof. In general, the set Slnb

f pMq isfinite [Del07]. If M is a meromorphic connection, an explicit bound for Slnb

f pMq isgiven in [Tey16] in terms of a resolution of turning points of M. This bound behavespoorly with respect to restriction. We will need a sharper bound in the case where fis a smooth morphism. It will be provided by the following more general proposition.

Proposition 7.1.2. — Let M be a germ of meromorphic connection at 0 P An withpoles along the divisor D given by f :“ x1 ¨ ¨ ¨xd “ 0. Let ri be the highest genericslope of M along xi “ 0. Put rDpMq “ Maxtr1, . . . , rdu. Then,

Slnbf pMq Ă r0, rDpMqs

Proof. — To prove proposition 7.1.2, take r ą rDpMq and let N be a germ of mero-morphic connection at 0 P A1

C with slope r. We want to show the vanishing (7.1.1) ina neigbourhood of 0. By a standard Galois argument, one reduces to the case where rand the ri, i “ 1, . . . , d are integers. Since ψf is a formal invariant, we can further sup-pose that N “ tαE1tr where α P C. Let us accept for a moment that M is generatedas a DX -module by a coherent OX -submodule F stable by frDpMqxiBxi , i “ 1, . . . , dand such that M “ F p˚Dq. Let pe1, . . . , eN q be a generating family for F in a neigh-bourhood of 0. Then, the fαe1frei, i “ 1, . . . , N generateMbf`N as a DX -module.Let ι : Cn ÝÑ CnˆCt be the graph of f . Set δ :“ δpt´fq. Then, the si “ fαe1freiδ,i “ 1, . . . , N generate ι`pM b f`Nq. To show that the germ of ψf pM b f`Nq at0 vanishes, we are thus left to prove that si belongs to V´1pDCnˆCtqι`F for everyi “ 1, . . . , N , where V‚pDCnˆCtq is the Kashiwara-Malgrange filtration on DCnˆCt .For i “ 1, . . . , N , we have

frDpMqx1Bx1si “ frDpMqpα´

r

frqsi `

dÿ

j“1

gjsj ´ frDpMq`1Btsi, gj P OX

Hence,

rsi “ αtrsi ` tr´rDpMq

dÿ

j“1

gjsj ´ frx1Bx1si ´ f

r`1Btsi

Since r ą rDpMq, we have

tr´rDpMq

dÿ

j“1

gjsj P V´1pDCnˆCtqι`F

Note furthermore that

frx1Bx1si “ x1Bx1frsi ´ rf

rsi “ trpx1Bx1 ´ rqsi P V´1pDCnˆCtqsiand that

fr`1Btsi “ Bttr`1si “ pr ` 1qtrsi ` t

rtBtsi P V´1pDCnˆCtqsiHence, si P V´1pDCnˆCtqι`F , which proves the sought-after vanishing. We are thusleft to prove the lemma 7.1.3 below.

34 J.-B. TEYSSIER

Lemma 7.1.3. — Let M be a germ of meromorphic connection at 0 P AnC with polesalong the divisor D given by f :“ x1, ¨ ¨ ¨xd “ 0. Let ri be the highest generic slope ofM along xi “ 0. Suppose that the ri are integers and put rDpMq “ Maxtr1, . . . , rdu.Then, M is generated as a DX-module by a coherent OX-submodule F stable byfrDpMqxiBxi for every i “ 0, . . . , d and such that M “ F p˚Dq.

Proof. — Let E be a lattice in M as constructed by Malgrange in [Mal96]. By con-struction, M “ Ep˚Dq. Since holonomic DX -modules are noetherian, M “ DXf´kEfor k big enough. Let us show that F “ f´kE fits our purpose. For m P E, we have

frDpMqxiBxipf´kmq “ ´kfrDpMqpf´kmq ` f´kpfrDpMqxiBximq

Hence, it is enough to show that E is stable by frDpMqxiBxi , i “ 0, . . . , d. SinceOXan,x is faithfully flat over OX,x for every x P D, we have E “ M X Ean in Man.Hence, it is enough to show that Ean is stable by frDpMqxiBxi , i “ 0, . . . , d. Letj : U ÝÑ Xan be the complement in X of the union of the singular locus of D withthe turning locus of M. By construction of the Deligne-Malgrange lattices, a sectionof M belongs to Ean if and only if its restriction to U belongs to Ean

|U . Hence, we cansuppose that D is smooth and that M has good formal structure along D. We canfurther suppose that M is unramified along D. Since OXan, pD is faithfully flat overOXan,D, we can suppose that M is a good elementary local model, that is

M “à

aPOXan prDpMqDq

Ea bRa

where the Ra are regular meromorphic connections with poles along D. In that case,E is by definition a direct sum of the form

À

Ea where Ea is a Deligne lattice [Del70]in Ra. In that case, the sought-after stability is obvious. This finishes the proof oflemma 7.1.3.

Remark 7.1.4. — The bound for nearby slopes proved in proposition 7.1.2 was sug-gested by the `-adic picture [HT18]. In loc. it. indeed, a similar bound was obtainedfor `-adic nearby slopes of smooth morphisms [Tey15]. In that setting, the main toolsare Beilinson’s and Saito’s work on the singular support [Bei16] and the characteristiccycle [Sai17] for `-adic sheaves, as well as semi-continuity properties [HY17][Hu17]for various ramification invariants produced by Abbes and Saito’s ramification theory[AS02]. From this perspective, proposition 7.1.2 is a positive answer to a local variantfor differential equations of a conjecture in [Lea16] on the ramification of the étalecohomology groups for local systems on the generic fiber of a strictly semi-stable pair.See Conjecture 5.8 from [HT18] for a precise statement.

7.2. Boundedness of the turning locus in the case of smooth proper relativecurves. — This subsection is devoted to the proof of Theorem 2. Let S be a smoothcomplex algebraic curve. Let p : C ÝÑ S be a relative smooth proper curve of genusg. Let M be a meromorphic connection of rank r on C with poles along the fibreC0. Let ZpMq be the subset of points in C0 at which M does not have good formalstructure (that is, the turning locus of M). Let irrC0

M be the generic irregularity of


M along C0. Let rDpMq be the highest generic slope of M along C0. We put

K :“ pSolMq|C0r1s ‘ pSol EndMq|C0

r1s

Then, K is a complex of C-vector spaces on C0 with constructible cohomology. It isconcentrated in degree 0 and 1. The generic rank of K is

rK “ irrC0M` irrC0

EndMď rrDpMq ` r2rDpEndMq

ď 2r2rDpMq

where the last inequality comes from rDpEndMq ď rDpMq. The Euler-Poincarécharacteristic formula [Lau87, Th. 2.2.1.2] applied to K gives

χpC0,Kq “ p2´ 2gqrK ´ÿ

xPSingK

prK ´ dimH0Kxq ` dimH1Kx

where SingK denotes the singular locus of K, that is the subset of points in C0 inthe neighbourhood of which K is not a local system concentrated in degree 0. FromMebkhout perversity theorem [Meb90], the complex K is perverse. In particular,H0K does not have sections with punctual support. Thus,

rK ´ dimH0Kx ě 0

for every x P SingK. From perversity again [Tey13, 13.1.6], the local Euler-Poincarécharacteristic of K at x P SingK differs from its generic value rK . Hence, for x PSingK, the quantity

prK ´ dimH0Kxq ` dimH1Kx

is positive and non zero. It is thus strictly positive. Hence, we have a bound

| SingK| ď p2´ 2gqrK ´ χpC0,Kq

From Theorem 1, the singular points of K are exactly the points in C0 at which Mdoes not have good formal structure. Hence

|ZpMq| ď 2rK ` |χpC0,Kq|

We are now left to bound χpC0,Kq. Since the irregularity complex is compatible withproper push-forward [Meb04, 3.6-6], we have

|χpC0, pSolMq|C0q| “ |χp0, Rp˚pSolMq|C0

q|

“ |χp0, pSol p`Mq|0q|

“ |ÿ

i

p´1qi irr0 Hip`M|

ďÿ

i

irr0 Hip`M

ďÿ

i

rkHip`MˆMax Slnbt pHip`Mq

ďÿ

i

rkHip`MˆMax Slnbt pp`Mq

36 J.-B. TEYSSIER

Since nearby slopes are compatible with proper push-forward [Tey16, Th. 3 piiq], wehave Slnb

t pp`Mq Ă Slnbp pMq. Since p is smooth, proposition 7.1.2 yields

|χpC0, pSolMq|C0q| ď rDpMq

ÿ

i

rkHip`M

For a generic point s P S, we have furthermoreÿ

i

rkHip`M “ÿ

i

dimpSolHip`Mqs

“ÿ

i

dimpHi Sol p`Mqs

“ÿ

i

dimpRip˚ SolMqs

“ÿ

i

dimHipCs,Lq

where L denotes the local system of solutions of M|Cs . Then HipCs,Lq “ 0 for every

i ‰ 0, 1, 2 and we have

dimH0pCs,Lq ď rkL “ rkM|Cs “ r

From Poincaré-Verdier duality, we have

dimH2pCs,Lq “ dimH0pCs,L˚q ď rkL˚ “ rkM|Cs “ r

Finally,

dimH1pCs,Lq “ ´χpCs,Lq ` dimH0pCs,Lq ` dimH2pCs,Lq“ ´χpCs,Cq rkL` dimH0pCs,Lq ` dimH2pCs,Lqď 2rpg ` 1q

Putting everything together yields

|ZpMq| ď 8r2pg ` 1qrDpMq

This finishes the proof of Theorem 2.

8. Appendix

8.1. Torsors. — In this subsection, we collect elementary facts and definitions ontorsors under a sheaf of groups.

Let X be a topological space. Let G be a sheaf of groups on X. We recall thata torsor under G is a sheaf F on X endowed with a left action of G such that thereexists a cover U by open subsets of X such that for every U P U , there exists anisomorphism of sheaves F|U » G|U commuting with the action of G, where G acts onitself by multiplication on the left. We denote by TorspX,Gq the category of G-torsorson X. It is a standard fact that isomorphism classes of G-torsors are in bijection withH1pX,Gq, the set of non abelian cohomology classes of G.

The following lemma is due to Babbit-Varadarajan [BV89, 1.3.3].


Lemma 8.1.1. — Let p : Y ÝÑ X be a Galois finite covering of compact metricspaces with Galois group G. Let G be a sheaf of groups on X. Suppose that thep˚G-torsors have no non trivial automorphisms. Then, the canonical morphism

H1pX,Gq ÝÑ H1pY, p˚GqG

is bijective.

In [BV89, 1.3.3], the condition on p˚G is expressed in terms of the triviality ofthe set of global sections of the twist [BV89, 1.3] of p˚G by a p˚G-torsor T . Observethat this set is also the set of automorphisms of T .

Lemma 8.1.2. — Let p : Y ÝÑ X be a morphism of topological spaces. Let G be asheaf of groups on Y . Let T be a G-torsor on Y . Suppose that X admits a cover Uby open subsets such that for every U P U , the torsor T is trivial on p´1pUq. Thenp˚T is a p˚G-torsor on X.

8.2. Recollections on Stokes filtered local systems. — The goal of this sub-section is to recall the notion of Stokes filtered local systems. Note that our use ofStokes filtered local systems in this paper is a purely technical detour to obtain thetriviality criterion 8.4.1. Hence, we don’t claim for completeness in these recollectionsand they can be omitted in a first reading. For more background material, let us referto [Moc11a] and [Sab12].

We fix a germ of regular pair pX,Dq at a point P and denote by pD : rX ÝÑ X thereal blow-up of X along D. Let I be a good sheaf of irregular values in OXp˚DqOX .Let us recall the following fact [Moc11a, 3.5].

Fact 8.2.1. — For every point Q in BD, there is an open neighbourhood UQ of Q inBD such that for every point Q1 in UQ, the induced map

pIpDpQq,ďQq ÝÑ pIpDpQ1q,ďQ1q

is well-defined and order preserving.

The following definition appears in the absolute case in [Moc11a, 3.6].

Definition 8.2.2. — Let R be a ring. Let U be a locally connected subset in BD.An I-Stokes filtered local system on U is the data of a local system L of projective R-modules of finite type on U , and for every point Q in U , a split pIpDpQq,ďQq-filtrationLď,Q on LQ by projective submodules. We further require the following compatibilityconditions when Q varies. For every point Q in U , for every neighbourhood UQ of Qas in 8.2.1 such that UQ X U is connected, the filtration Lď,Q1 on LQ1 is induced bythat on LQ via pIpDpQq,ďQq ÝÑ pIpDpQ1q,ďQ1q.

Remark 8.2.3. — Let U be a locally connected subset in BD. Put Z “ pDpUqand suppose that I|Z is constant. Then, the notion of I|Z-graded local system on Umakes sense. Observe that any I|Z-graded local system on U gives rise to an I|Z-Stokes filtered local system on U . On the other hand, the grading operation Gr from[Moc11a, 3.6] is a well defined functor converting I|Z-Stokes filtered local systems on

38 J.-B. TEYSSIER

U into I|Z-Stokes graded local systems on U . The case of interest to us will be thecase where Z lies in a stratum of BD or when I is very good.

The following is a variant of a proposition due to Mochizuki [Moc11a, 3.16].

Proposition 8.2.4. — Let pX,Dq be a germ of regular pair at a point P . LetD1, . . . , Dm be the components of D. Let I be a subset in J1,mK. Let I Ă OXp˚DqOX

be a good sheaf of irregular values. Let Q be a point in BP . Let S be a neighbourhoodof Q in rX of the form

mź

i“1

pr0, rrÎiq ˆ∆

where r ą 0, where I1, . . . , Im are closed intervals in S1, and where ∆ is a ball inCn´m centred at 0. We have

S X BD˝I “ź

iPI

pt0u ˆ Iiq ˆź

iRI

ps0, rrÎiq ˆ∆

Let R be a ring. Then, at the cost of shrinking S, every I-Stokes filtered local systemon S X BD˝I relative to R comes from an I-Stokes graded local system on S X BD˝I .

Proof. — By shrinking S enough, we can achieve that the Stokes locus of I in SXBD˝Iis homotopic to the union of a finite number of hyperplane sections of a cube in R2n.By shrinking S further, we can achieve that for every a, b P I distinct, the Stokeslocus of pa, bq does not meet S X BD˝I or divides S X BD˝I into two regions. Then,proposition 8.2.4 is a consequence of [Moc11a, 3.16].

Remark 8.2.5. — Note that proposition 3.16 from [Moc11a] is proved in the casewhere the base ring is C. The proof is purely algebraic and carries over to the relativecase. Note also that Mochizuki requires the subset S to live above a point in D. How-ever, an inspection of Mochizuki’s proof shows that it only pertains to the combinatoricof the jumps of the orderings in I. Hence, it carries over to any situation homotopicto that of a sheaf of finite ordered sets J on a cube C in R2n whose underlying sheafof sets J set is constant and such that for any distinct global sections a, b of J set,either the order between a and b is constant on C or there exists a hyperplane H inR2n dividing C into two regions C1 and C2 such that a is smaller than b on C1, a isbigger than b on C2, and a and b cannot be compared at points in H. This is the casein 8.2.4 when S is small enough.

The relationship between Stokes filtered local systems and Stokes torsors will bemade via the notion of marked Stokes filtered local systems, that we now introduce.

Definition 8.2.6. — Let U be a locally connected subset in BD. Put Z “ pDpUqand suppose that I|Z is constant. Let R be a ring. Let pL,Lďq be an I-Stokes filteredlocal system on U relative to R. A pL,Lďq-marked Stokes filtered local system on Uis the data of a pair ppL,Lďq, isoq where pL,Lďq is an I-Stokes filtered local systemon U and where iso is an isomorphism between GrL and GrL.


8.3. Relation with Stokes torsors. — Let pX,Dq be a germ of regular pair at apoint P . Let D1, . . . , Dm be the components of D. Let I be a subset in J1,mK. LetM be a good elementary local model on X with poles along D, that is

M “à

aPIEa bRa

where I is a good set of irregular values at P , where Ea “ pOXp˚Dq, d ´ daq andwhere Ra is a regular singular meromorphic connection on X with poles along D.For a P I, we denote by ξIpaq the truncation of a which consists in keeping only themonomials in a having poles along every component Di, i P I. Put

MpIq “à

aPIEξIpaq bRa

With the help of proposition 8.2.4, the lemma 8.3.1 that follows allows to transfersplitting statements for Stokes filtered local systems to triviality statements for Stokestorsors.

Lemma 8.3.1. — Let pL,Lďq be the Stokes filtered local system on BD associatedto MpIq. Let U be a locally connected subset in BD˝I . Let R be a C-algebra. LetpLpRq, LpRqďq be the Stokes filtered local system relative to R induced by pL,Lďq.Then, there is a canonical bijection between H1pU,StăDM pRqq and the set of isomor-phisms classes of pLpRq, LpRqďq-marked Stokes filtered local systems on U . Via thisidentification, the trivial torsor under StăDM pRq corresponds to ppLpRq, LpRqďq, idq.

Proof. — Observe that the restriction of StăDM to BD˝I is the sheaf StăDMpIq. Hence, weare left to construct a bijection betweenH1pU,StăDMpIqpRqq and the set of isomorphismsclasses of pLpRq, LpRqďq-marked Stokes filtered local systems on U . Note that thesheaf ξIpIq of irregular values of MpIq is very good on D˝I . Then, lemma 8.3.1 isconsequence of the next lemma 8.3.2.

Lemma 8.3.2. — Let pX,Dq be a germ of regular pair at a point P . Let M bea meromorphic connection on X with poles along D. Suppose that M has a goodelementary local model at every point. Suppose that the sheaf of irregular values ofM is very good. Let pL,Lďq be the Stokes filtered local system on BD associated toM. Let U be a locally connected subset in BD. Let R be a C-algebra. Then, there isa canonical bijection between H1pU,StăDM pRqq and the set of isomorphism classes ofpLpRq, LpRqďq-marked Stokes filtered local systems on U . Via this identification, thetrivial torsor under StăDM pRq corresponds to ppLpRq, LpRqďq, idq.

Proof. — Let ppL,Lďq, isoq be a pLpRq, LpRqďq-marked Stokes filtered local sys-tem on U . Consider the sheaf IsomisopL, LpRqq on U whose sections on anopen subset V in U is the set of isomorphisms of Stokes filtered local systemsf : pL,Lďq ÝÑ pLpRq, LpRqďq on V such that Gr f “ iso. Since L is locally iso-morphic to the Stokes filtered local system associated to GrL, and similarly with L,the sheaf IsomisopL, LpRqq is a torsor for the action of StăDM pRq on IsomisopL, LpRqqby post-composition. On the other hand, let T be an element in H1pU,StăDM pRqq.Choose a cover V “ pViqiPK of U such that T comes from a cocycle g “ pgijqi,jPK . Set

40 J.-B. TEYSSIER

Li :“ LpRq|Ui . The identifications gij : Li|Uij ÝÑ Lj|Uij allow to glue the Li into aStokes filtered local system pL,Lďq on U . Since the gij lie in StăDM pRq, the graded ofthe identity morphisms Li ÝÑ LpRq|Ui glue into an isomorphism iso : GrL ÝÑ GrL.Hence, ppL,Lďq, isoq defines an pLpRq, LpRqďq-marked Stokes filtered local systemson U whose isomorphism class does not depend on any choice. It is then a standardcheck to verify that the two constructions above are mutually inverse bijections.

8.4. Trivialization of Stokes torsors. — As a consequence of the relationship8.3.1 between Stokes filtered local systems and Stokes torsors, the variant 8.2.4 ofMochizuki’s splitting criterion gives the following triviality criterion for torsors underthe Stokes sheaf of a good elementary local model.


be the components of D. Let I be a subset in J1,mK. Let M be a good elementary localmodel on X with poles along D. Let Q be a point in BP . Let S be a neighbourhood ofQ in rX of the form

mź

i“1

pr0, rrÎiq ˆ∆

where r ą 0, where I1, . . . , Im are closed intervals in S1, and where ∆ is a ball inCn´m centred at 0. We have

S X BD˝I “ź

iPI

pt0u ˆ Iiq ˆź

iRI

ps0, rrÎiq ˆ∆

Then, at the cost of shrinking S, Stokes torsors on S X BD˝I are trivial.

Proof. — We use the notations from 8.3. Let pL,Lďq be the ξIpIq-Stokes filteredlocal system on BD associated with MpIq. From lemma 8.3.1, we are left to showthat at the cost of shrinking S, the pL,Lďq-marked Stokes filtered local systems onS X BD˝I are trivial. To do this, it is enough to show that at the cost of shrinkingS, the ξIpIq-Stokes filtered local systems on S X BD˝I come from ξIpIq-graded localsystems on S X BD˝I . Then, lemma 8.4.1 is a consequence of proposition 8.2.4.

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J.-B. Teyssier, Institut de Mathématiques de Jussieu. 4 place Jussieu. Paris.E-mail : [email protected]