A Boundedness Theorem for Nearby Slopes of Holonomic D-Modules Jean-Baptiste Teyssier

A Boundedness theorem for nearby slopes of holonomic D-modules Jean-Baptiste Teyssier

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Jean-Baptiste Teyssier. A Boundedness theorem for nearby slopes of holonomic D-modules. 2015. hal-01127706

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HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A BOUNDEDNESS THEOREM FOR NEARBY SLOPES OF HOLONOMIC D-MODULES

Jean-Baptiste Teyssier

Abstract. — Using twisted nearby cycles, we define a new notion of slopes for complex holonomic D-modules. We prove a boundedness result for these slopes, study their functoriality and use them to characterize regularity. For a family of (possibly irregular) algebraic connections Et parametrized by a smooth curve, we deduce under natural conditions an explicit bound for the usual slopes of the differential equation satisfied by the family of irregular periods of the Et. This generalizes the regularity of the Gauss-Manin connection proved by Katz and Deligne. Finally, we address some questions about analogues of the above results for wild ramification in the arithmetic context.

Let V be a smooth algebraic variety over a finite field of characteristic p 0, and let U be an open subset in V such that D : V U is a normal crossing divisor.ą Let ℓ be a prime number different from p. Using“ restrictionz to curves, Deligne defined [Del11] a notion of ℓ-adic local system on U with bounded ramification along D. Such a definition is problematic to treat functoriality questions: the direct image of a local system is not a local system any more, duality does not commute with restriction in general. In this paper, we investigate the characteristic 0 aspect of this problem, that is the Question 1.— Let X be a complex manifold. Can one define a notion of holonomic DX -module with bounded irregularity which has good functoriality properties? In dimension 1, to bound the irregularity number of a D-module with given generic rank amounts to bound its slopes. Let M be a holonomic DX -module and let Z be a hypersurface of X. Mebkhout [Meb90] showed that the irregularity complex IrrZ M p q of M along Z is a perverse sheaf endowed with a R 1 increasing locally finite filtration ą by sub-perverse sheaves IrrZ M r . In dimension one, this construction gives back the usual notion of slope modulop qp theq change of variable r 1 r 1 . The analytic slopes of M along Z are the r 1 for which the supportsÝÑ of{p the´ gradedq pieces of ą IrrZ M r r 1 are non empty. p Thep existenceqp qq ą of a uniform bound in Z is not clear a priori. We thus formulate the following 2 J.-B. TEYSSIER

Conjecture 1.— Locally on X, the set of analytic slopes of a holonomic DX -module is bounded.

This statement means that for a holonomic DX -module M, one can ﬁnd for every point in X a neighbourhood U and a constant C 0 such that the analytic slopes of M along any germ of hypersurface in U are C.ą On the other hand, Laurent deﬁned algebraicď slopes using his theory of micro- characteristic varieties [Lau87]. From Laurent and Mebkhout work [LM99], we know that the set of analytic slopes of a holonomic D-module M along Z is equal to the set of algebraic slopes of M along Z. Since micro-characteristic varieties are invariant by duality, we deduce that analytic slopes are invariant by duality.

The aim of this paper is to define a third notion of slopes and to investigate some of its properties. The main idea lies in the observation that for a germ M of DC-module at 0 C, the slopes of M at 0 are encoded in the vanishing of certain nearby cycles. P We show in 2.3.1 that r Q 0 is a slope for M at 0 if and only if one can find a germ P ě N of meromorphic connection at 0 with slope r such that ψ0 M N 0. We thus introduce the following definition. Let X be a complexp b manifoldq‰ and let b M be an object of the derived category D X of complexes of DX -modules with holp q bounded and holonomic cohomology. Let f OX . We denote by ψf the nearby cycle 1 P functorp q associated to f. We define the nearby slopes of M associated to f to be nb the set Slf M complement in Q 0 of the set of rationals r 0 such that for every germ N of meromorphicp q connectioně at 0 with slope r, we haveě

(0.0.1) ψf M f `N 0 p b q» O Let us observe that the left-hand side of (0.0.1) depends on N only via C,0 OC,0 N, and that nearby slopes are sensitive to the reduced structure of div f, whereasb the p analytic and algebraic slopes only see the support of div f. Twisted nearby cycles appear for the first time in the algebraic context in [Del07]. Deligne proves in loc. it. that for a given function f, the set of OC,0-differential modules N such that ψ M f N 0 is finite. f ` p The main result of thisp paperb is anq‰ affirmative answer to conjecture 1 for nearby slopes, that is the

Theorem 1.— Locally on X, the set of nearby slopes of a holonomic D-module is bounded.

This statement means that for a holonomic DX -module M, one can ﬁnd for every point in X a neighbourhood U and a constant C 0 such that the nearby slopes of ą M associated to any f OU are C. For meromorphic connectionsP ď with good formal structure, we show the following reﬁnement p1qFor general references on the nearby cycle functor, let us mention [Kas83],[Mal83],[MS89] and [MM04]. A BOUNDEDNESS THEOREM FOR NEARBY SLOPES OF HOLONOMIC D-MODULES 3

Theorem 2.— Let M be a meromorphic connection with good formal structure. Let D be the pole locus of M and let D1,...,Dn be the irreducible components of D. We denote by ri M Q 0 the highest generic slope of M along Di. Then, the nearby p q P ě slopes of M are r1 M rn M . ď p q`¨¨¨` p q The main tool used in the proof of theorem 1 is a structure theorem for formal meromorphic connections first conjectured in [CS89], studied by Sabbah [Sab00] and proved by Kedlaya [Ked10][Ked11] in the context of excellent schemes and analytic spaces, and independently by Mochizuki [Moc09][Moc11b] in the algebraic context. Let us give some details on the strategy of the proof of theorem 1. A dévissage carried out in 3.1 allows one to suppose that M is a meromorphic connection. Using Kedlaya-Mochizuki theorem, one reduces further the proof to the case where M has good formal structure. We are thus left to prove theorem 2. We resolve the singularities of Z. The problem that occurs at this step is that a randomly chosen embedded resolution p : X X will increase the generic slopes of M in a way that cannot be controlled. WeÝÑ show in 3.2.2 that a fine version of embedded resolution r [BM89] allows to control the generic slopes of p`M in terms of the sum r1 M p q` rn M and the multiplicities of p˚Z. A crucial tool for this is a theorem [Sab00, ¨¨¨Ì 2.4.3]p provedq by Sabbah in dimension 2 and by Mochizuki [Moc11a, 2.19] in any dimension relating the good formal models appearing at a given point with the generic models on the divisor locus. Using a vanishing criterion 2.4.1, one finally proves (0.0.1) for r r1 M rn M . ąMp Dq`¨¨¨`b p q M M Let hol X and let us denote by D the dual complex of . The nearby slopes satisfyP thep followingq functorialities

Theorem 3.— i For every f OX , we have p q P Slnb DM Slnb M f p q“ f p q ii Let p : X Y be a proper morphism and let f OY such that p X is not p q ÝÑ1 P p q contained in f ´ 0 . Then p q nb nb Slf p M Slfp M p ` qĂ p q Let us observe that ii is a direct application of the compatibility of nearby cycles with proper direct imagep q [MS89]. It is an interesting problem to try to compare nearby slopes and analytic slopes. This question won’t be discussed in this paper, but we characterize regular holonomic D-modules using nearby slopes. M Db Theorem 4.— A complex hol X is regular if and only if for every quasi- finite morphism ρ : Y X withP Y ap complexq manifold, the set of nearby slopes of ÝÑ ρ`M is contained in 0 . t u For an other characterization of regularity (harder to deal with in practice) using RHom and the solution functor, we refer to [Tey14]. Let us give an application of the preceding results. Let U be a smooth complex algebraic variety and let E be an algebraic connection on U. We denote by Hk U, E dRp q the kth de Rham cohomology group of E, and by V the local system of horizontal 4 J.-B. TEYSSIER sections of Ean on U an. If E is regular, Deligne proved [Del70] that the canonical comparison morphism (0.0.2) Hk U, E Hk U an, V dRp q ÝÑ p q is an isomorphism. If E is the trivial connection, this is due to Grothendieck [Gro66]. In the irregular case, (0.0.2) is no longer an isomorphism. It can happen that k E k an V HdR U, is non zero and H U , is zero, which means that there are not enough p q an p q rd E topological cycles in U . The rapid decay homology Hk U, ˚ needed to rem- edy this problem appears in dimension one in [BE04] andp in higherq dimension in [Hie07][Hie09]. It includes cycles drawn on a compactification of U an taking into account the asymptotic at infinity of the solutions of the dual connection E˚. By Hien duality theorem, we have a perfect pairing

k rd (0.0.3) : H U, E H U, E˚ C ż dRp qˆ k p q ÝÑ k rd For ω H U, E and γ H U, E˚ , the complex number ω is a period for E. P dRp q P k p q γ Let f : X S be a proper and generically smooth morphism,ş where X denotes an ÝÑ 1 algebraic variety and S denotes a neighbourhood of 0 in AC. Let U be the complement of a normal crossing divisor D of X such that for every t 0 close enough to 0, Dt ‰ is a normal crossing divisor of Xt. Let E be an algebraic connection on U. Let us 1 denote by D1,...,Dn the irreducible components of D meeting f ´ 0 and let ri E p q p q be the highest generic slope of E along Di. As an application of theorem 2, we prove the following

Theorem 5.— If E has good formal structure along D and if the fibers Xt, t 0 of 2 ‰ f are non characteristic at infinityp q for E, then the periods of the family Et t 0 are solutions of a system of linear polynomial differential equations whose slopesp q at‰0 are r1 E rn E . ď p q`¨¨¨` p q In the case where E is the trivial connection, we recover that the periods of a proper generically smooth family of algebraic varieties are solutions of a regular singular differential equation with polynomial coefficients [Kat70][Del70].

The role played in this paper by nearby cycles has Verdier specialization [Ver83] and moderate nearby cycles [DK73, XIII] as ℓ-adic counterparts. Let V be an algebraic scheme over a perfect field k of characteristic p 0, and let V be a compactification of V . Let j : V V be the canonical inclusion,ą and f O . Let S be the ÝÑ P V strict henselianization of O 1 , denote by η the generic point of S and let us choose Ak,0 a geometric point η over η. Let P be the wild ramification group of π1 η, η . Define p q V S : V A1 S, fS : V S S the base change of f to S, fη : V η η the restriction “ ˆ k ÝÑ ÝÑ of fS over η and ι : V S V the canonical morphism. Let F be a ℓ-adic complex on ÝÑ 3 V with bounded constructible cohomology. We say that r Q 0 is a nearby slopep q P ě p2qthis is for example the case if D is smooth and if the fibers of f are transverse to D. p3qOr a Verdier slope if Verdier specialization is used instead of moderate nearby cycles. A BOUNDEDNESS THEOREM FOR NEARBY SLOPES OF HOLONOMIC D-MODULES 5

for F associated to V,f if one can ﬁnd a constructible ℓ-adic sheaf N on ηét with slope r such that p q P ψf ι˚j!F f ˚N 0 p p b η qq ‰ Hence, conjecture 1 has a ℓ-adic analogue that may be worth investigating. This is the following

Question 2.— Is it true that the set of nearby slopes of F is bounded?

That nearby slopes do not depend on the choice of a compactiﬁcation is not clear to the author. Nor that a smoothness assumption on V is needed. In any case, this leads to a notion of ℓ-adic tame complex in the sense of Verdier specialization or moderate nearby cycles. As an analogue of the regularity of OX in the theory of D-modules, we raise the following

Question 3.— Is it true that the constant sheaf Qℓ on V is tame in the sense of moderate nearby cycles? That is, that for every couple V,f as above and for every p q constructible ℓ-adic sheaf N on ηét with slope 0, we have ą P ψf ι˚j!Q f ˚N 0 ? p p ℓ b η qq » Conjecture 1 ﬁrst appears in [Tey15]. This paper grew out an attempt to prove it. I thank Pierre Deligne, Zoghman Mebkhout and Claude Sabbah for valuable comments on this manuscript and Marco Hien for mentioning [HR08], which inspired me a statement in the spirit of theorem 5 and reignited my interest for a proof of theorem 1. This work has been achieved with the support of Freie Universität/Hebrew University of Jerusalem joint post-doctoral program and ERC 226257 program. I thank Hélène Esnault and Yakov Varshavsky for their support.

1. Notations We collect here a few deﬁnitions used all along this paper. The letter X will denote a complex manifold.

1.1. For a morphism f : Y X with Y a complex manifold, we denote by f ` : b D b D ÝÑ b D b D Dhol X Dhol Y and f : Dhol Y Dhol X the inverse image and p q ÝÑ p q ` p q ÝÑ p q direct image functors for D-modules. We note f : for f ` dim Y dim X . r ´ s M Db O Hk M HkM 1.2. Let hol X and f X . From ψf f `N ψf f `N for every k, weP deducep q P p b q » p b q

nb nb k (1.2.1) Slf M Slf H M p q“ ďk p q Slnb M : Slnb M Slnb M Let us deﬁne f OX f . The elements of are the nearby M p q “ P p q Db p q Db slopes of . For S Q 0Ť, we denote by hol X S the full subcategory of hol X of complexes whose nearbyĂ ě slopes are in S. p q p q 6 J.-B. TEYSSIER

b D b 4 1.3. Let us denote by DR : Dhol X Dc X, C the de Rham functor p q and by b b p q ÝÑ p q Sol : D DX D X, C the solution functor for holonomic DX -modules. holp q ÝÑ cp q

1.4. For every analytic subspace Z in X, we denote by iZ : Z ã X the canonical inclusion. The local cohomology triangle for Z and M Db X Ñreads P holp q / / 1 / (1.4.1) RΓ Z M / M / RM Z ` / r s p˚ q b D M It is a distinguished triangle in Dhol X . The complex RΓ Z is the local algebraic cohomology of M along Z and RMp Zq is the localization rofs M along Z. p˚ q 1.5. Let M be a germ of meromorphic connection at the origin of Cn. Let D be the

M M O n M M pole locus of . For x D, we define x : C ,x OCn,x . We say that has good formal structure if P “ b x p (1) D is a normal crossing divisor. (2) For every x D, one can find coordinates x1,...,xn centred at x with D defined P p q by x1 xi 0, and an integer p 1 such that if ρ is the morphism x1,...,xn p ¨ ¨ ¨ p “ ě p q ÝÑ x1,...,xi , xi 1,...,xn , we have a decomposition p ` q ϕ (1.5.1) ρ`Mx E Rϕ » b ϕ OCnàD OCn x P p˚ q{ ϕ where E OCn,x D , d dϕ and Rϕ is a meromorphic connection with regular singularity“ along p D.p˚ q ` q p (3) For all ϕ OCn D OCn contributing to (1.5.1), we have div ϕ 0. P p˚ q{ ď Let us remark that classically, one asks for condition (3) to be also true for the differences of two ϕ intervening in (1.5.1). We won’t impose this extra condition in this paper.

1.6. Let M be a meromorphic connection on X such that the pole locus D of M has only a ﬁnite number of irreducible components D1,...,Dn. For every i 1,...,n, M M “ we denote by rDi the highest generic slope of along Di. We deﬁne the divisor of highest genericp slopesq of M by

rD1 M D1 rD M Dn Z X Q p q `¨¨¨` n p q P p q

2. Preliminaries on nearby cycles in the case of good formal structure 2.1. Let n be an integer and take i NJ1,nK. The support of i is the set of k J1,nK P P such that ik 0. If E J1,nK, we deﬁne iE by iEk ik for k E and iEk 0 if k E. ‰ Ă “ P “ R p4qIn this paper, we follow Hien’s convention [Hie09] according to which for a holonomic module M, the complex DR M is concentrated in degrees 0,..., dim X. A BOUNDEDNESS THEOREM FOR NEARBY SLOPES OF HOLONOMIC D-MODULES 7

1 2.2. Let R be a regular C t -differential module, and take ϕ C t´ . For every n 1, we define ρ : t tppp qqx and P r s ě ÝÑ “ El ρ, ϕ, R : ρ Eϕ R p q “ `p b q If R is the trivial rank 1 module, we will use the notation El ρ, ϕ . In general, El ρ, ϕ, R has slope ord ϕ p. The C x -modules of type El ρ,p ϕ, Rq for variable ρ,p ϕ, R areq called elementary{ modulespp.qq From [Sab08, 3.3],p we knowq that every Cp x -differentialq module can be written as a direct sum of elementary modules. pp qq 2.3. Dimension 1. — In this paragraph, we work in a neighbourhood of the origin 0 C. Let x be a coordinate on C. Take p 1 and define ρ : x t xp. P ě ÝÑ “ Proposition 2.3.1.— Let M be a germ of holonomic D-module at the origin. Let r 0 be a rational number. The following conditions are equivalent ą (1) The rational r is not a slope for M at 0. (2) For every germ N of meromorphic connection of slope r p, we have { ψρ M ρ`N 0 p b q» Proof. — Since ψ is not sensitive to localization and formalization, one can work formally at 0 and suppose that M and N are differential C x -modules. pp qqp1 1 Let us prove 2 1 by contraposition. Define ρ1 : u u x, ϕ u C u´ p q ùñ p q ÝÑ “ p qP r s with q ord ϕ u and R a C u -regular module such that El ρ1, ϕ u , R is a non zero elementary“ p factorq 2.2 of Mpp qqwith slope r q p. Define p p q q “ { N : ρ El ρ1, ϕ u El ρρ1, ϕ u “ ` p ´ p qq “ p ´ p qq The module N has slope q pp1 r p. A direct factor of ψρ M ρ`N is { “ { p b q ϕ ϕ ψρ ρ1 E R ρ`N ψρ ρ1 E R ρ` El ρρ1, ϕ u p `p b qb qq » p `p b qb p ´ p qqq ϕ ψρ ρ1 E R ρρ1 ` El ρρ1, ϕ u » p `p b b p q p ´ p qqq ϕ ψρρ1 E R ρρ1 ` El ρρ1, ϕ u » p b b p q p ´ p qqq where the last identification comes from the compatibility of ψ with proper direct image. By [Sab08, 2.4], we have

ϕ ζu ρρ1 ` El ρρ1, ϕ u E´ p q p q p ´ p qq » 1 ζppà 1 “

So ψρρ1 R is a direct factor of ψρ M ρ`N of rank np rg R 0, and 2 1 is proved. p b q p qą p q ùñ p q Let us prove 1 2 . Let N be a C t -diﬀerential module of slope r p. Then p q ùñ p q pp qq { ρ`N has slope r. Thus, the slopes of M ρ`N are 0. Hence, it is enough to show the following b ą

Lemma 2.3.2.— Let M be a C x -diﬀerential module whose slopes are 0. pp qq ą Then, we have ψρM 0. » 8 J.-B. TEYSSIER

By Levelt-Turrittin decomposition, we are left to study the case where M is a ϕ 1 direct sum of modules of type E R, where ϕ C x´ and where R is a regular C x -module. The hypothesis onb the slopes of MP impliesr s ϕ 0, and the expected vanishingpp qq is standard. ‰

2.4. A vanishing criterion. — Let M be a germ of meromorphic connection at the origin 0 Cn. We suppose that M has good formal structure at 0. Let D be the P pole locus of M. Let ρp be a ramiﬁcation of degree p along the components of D as in (1.5.1).

Proposition 2.4.1.— Let f OCn,0. Let us deﬁne Z : div f and suppose that Z D. We suppose that for everyP irreducible component “E of Z , we have | |Ă | | rE M rvE f p qď p q Then for every germ N of meromorphic connection at 0 with slopes r, we have ą (2.4.2) ψf M f `N 0 p b q» in a neighbourhood of 0.

n Proof. — Let us choose local coordinates x1,...,xn and a N such that f is the function x xa. Take N with slopes p r. One canq alwaysP suppose that N is a ÝÑ ą k C t -diﬀerential module and p qk where ρ1 : t t decomposes N. pp qq “ ÝÑ The morphism ρp is a ﬁnite cover away from D, so the canonical adjunction morphism / (2.4.3) ρp ρp`M M ` is surjective away from D. So the cokernel of (2.4.3) has support in D. From [Meb04, 3.6-4], we know that both sides of (2.4.3) are localized along D. So (2.4.3) is surjective. We thus have to prove

(2.4.4) ψfρ ρ`M fρp `N 0 p p p b p q q» Since Z D, we have fρp ρ1fρq. So the left hand side of (2.4.4) is a direct sum of k copies| |Ă of “

(2.4.5) ψfρ ρ`M fρp `N q p p b p q q We thus have to prove that (2.4.5) is 0 in a neighbourhood of 0. We have

fρp `N fρq `ρ1`N p q » p q with ρ1`N decomposed with slopes rk. The zero locus of fρq is Z , and if E is an ą | | irreducible component of Z , the highest generic slope of ρ`M along E is | | p rE ρ`M p rE M rk q vE f rk vE fρq p p q“ ¨ p qď ¨ ¨ p q“ ¨ p q Hence we can suppose that ρp id and that N is decomposed. Take “ P t tm N E p q{ R “ b A BOUNDEDNESS THEOREM FOR NEARBY SLOPES OF HOLONOMIC D-MODULES 9 with P t C t satisfying P 0 0, with m r and with R regular. Since ψ is insensitivep q P to formalization,r s onep q can ‰ suppose ą ϕ x M E p q R “ b with ϕ x as in 1.5 3 and R regular. By Sabbah-Mochizuki theorem, the multiplicity p q p q of div ϕ x along a component D1 of D is a generic slope of M along D1. Thus, one ´ p q b can write ϕ x g x x where g 0 0 and where the bi are such that if i Supp a, p q“ p q{ p q‰ P we have bi rai mai. We thus have to prove the ď ă Lemma 2.4.6.— Take g,h OCn,0 such that g 0 0 and h 0 0. Let R P p q ‰ p q ‰ be a regular meromorphic connection with poles contained in x1 xn 0. Take J1,nK ¨ ¨ ¨ “ a,b N such that A : Supp a is non empty and bi ai for every i A. Then P “ ă P g x xb h x xa ψxa E p q{ ` p q{ R 0 p b q» in a neighbourhood of 0.

g x xb h x xa 2.5. Proof of 2.4.6. — We deﬁne M : E p q{ ` p q{ R. Since A is not empty, a change of variable allows one to suppose“ h 1. If Supp bb A, a change of variable shows that 2.4.6 is a consequence of 2.6.1.“ Let i Supp bĂbe an integer such that P i A. Using xi, a change of variable allows one to suppose g 1. Let p1,...,pn N˚ R “ P such that ajpj is independent from j for every j A and pj 1 if j A. Let ρp be the morphism x xp. Like in (2.4.3), we see thatP “ R ÝÑ M / M ρp ρp` ` is surjective. We are thus left to prove that 2.4.6 holds for multi-indices a such that aj does not depend on j for every j A. Let us denote by 1A the characteristic function of A. From [Sab05, 3.3.13], itP is enough to prove

1 xb 1 xa ψ 1A E { ` { R 0 x p b q» Using the fact that R is a successive extension of regular modules of rank 1, one can suppose that R xc, where c CJ1,nK. Let “ P n i / n C H C C HH ˆ HH HH x1A HH H$ C be the inclusion given by the graph of x x1A . Let t be a coordinate on the second factor of Cn C. We have to prove ÝÑ ˆ c 1 xb 1 xa ψt i x E { ` { 0 p `p qq » b a 1A c 1 x 1 x Deﬁne δ : δ t x i x E { ` { and let Vk k Z be the Kashiwara-Malgrange “ p ´ qP `p q J1,nK p q P d ﬁltration on DCn C relative to t. For d N such that x 0 is the pole locus of c 1 xb 1 xa ˆ d P c 1 xb 1 xa “ x E { ` { , the family of sections x generates x E { ` { . For such d, the family 10 J.-B. TEYSSIER

d c 1 xb 1 xa s : x δ generates i x E { ` { . We are left to prove s V 1s. One can always suppose“ that 1 A. `p q P ´ P

b1 a1 1A x1 1s d1 c1 s s s x ts B “ p ` q ´ xb ´ xa ´ B J1,nK We deﬁne M N by Mk max ak,bk for every k J1,nK. We thus have P “ p q P M M M b M a M 1A (2.5.1) x x1 1s d1 c1 x s b1x ´ s a1x ´ s x x ts B “ p ` q ´ ´ ´ B J1,nK We have M a bAc 1A a 1A bAc 1A b m with m N . So “ ` “ ` p ´ q` “ ` ` P M b m x ´ s x ts V 1s “ P ´ Moreover, we have M M M m b m b x x1 1s x1 1x s M1x s x1 1x ` ts M1x ` ts V 1s B “ B ´ “ B ´ P ´ and

M 1A m b 2 1A m b 2 m b m b x x ts x ` tx ˆ s x ` tt s 2x ` ts x ` t t t s V 1s B “ B “ B “ ` p B q P ´ So (2.5.1) gives M a (2.5.2) x ´ s V 1s P ´ Let us recall that i is such that i A and i Supp b. In particular M a i bi 0 R P p ´ q “ ‰ and iδ 0. Applying xi i to (2.5.2), we obtain B “ B M a M a x ´ di ci bi x ´ s bi s V 1s p ` ` q ´ xb P ´ M a b so from (2.5.2), we deduce x ´ ´ s V 1s . We have M a b bA, so by M a b bA P ´ ´ ´ “ ´ multiplying x ´ ´ s by x , we get s V 1s. P ´ 2.6. The aim of this paragraph is to prove the following

Lemma 2.6.1.— Let α, a NJ1,nK such that Supp α is not empty and Supp α P Ă Supp a. Let R be a regular meromorphic connection with poles contained in x1 xn 0. We have ¨ ¨ ¨ “ 1 xa ψxα E { R 0 p b q» Proof. — Let p1,...,pn be integers such that αipi does not depend of i for every i Supp α (we denote by m this integer) and pi 1 if i Supp α. Let ρp be the P p “ M‰ M morphism x x . Like in (2.4.3), the morphism ρp ρp` is surjective. We ÝÑ ` ÝÑ are left to prove 2.6.1 for α such that αi does not depend of i for every i Supp α. P From [Sab05, 3.3.13], one can suppose αi 1 for every i Supp α. So α a. “ P ď One can suppose R xb where b NJ1,nK. Let “ P n i / n C H C C HH ˆ HH HH xα HH H$ C A BOUNDEDNESS THEOREM FOR NEARBY SLOPES OF HOLONOMIC D-MODULES 11 be the inclusion given by the graph of x xα. Let t be a coordinate on the second factor of Cn C. We have to show ÝÑ ˆ b 1 xa ψt i x E { 0 p `p qq » α b 1 xa J1,nK Deﬁne δ : δ t x i x E { . For c N such that Supp c Supp a Supp b, “ p ´ qP `c p q b 1 xPa Ă c Y the family of sections x generates x E { . For such c, the family s : x δ generates b 1 xa “ i x E { . It is thus enough to show s V 1s. Let us choose i Supp α. We have `p q P ´ P ai α xi is ci bi s s x ts B “ p ` q ´ xa ´ B We have α a. Deﬁne a α a1. From ď “ ` α α α x xi is xi ix s x s xi its ts V 1s B “ B ´ “ B ´ P ´ a1 2α 2α 2α 2 we deduce that ais x x ts V 1s. We also have x ts tx s tt s ` B P ´ B “B “B “ 2ts t t t s V 1s. Since ai 0, we deduce s V 1s and 2.6.1 is proved. ` p B q P ´ ‰ P ´

3. Proof of theorem 1 3.1. Dévissage to the case of meromorphic connections. — Suppose that theorem 1 is true for meromorphic connections for every choice of ambient manifold. M Db Let us show that theorem 1 is true for hol X . We argue by induction on dim X. The case where X is a point is trivial.P Letp usq suppose that dim X 0. We define Y : Supp M and we argue by induction on dim Y . ą Let us“ suppose that Y is a strict closed subset of X. We denote by i : Y X ÝÑ the canonical inclusion. Let π : Y Y be a resolution of the singularities of Y [AHV29] and p : iπ. The regular locusÝÑ Reg Y of Y is a dense open subset in Y and r π is an isomorphism“ above Reg Y . By Kashiwara theorem, we deduce that the cone C of the adjunction morphism / p p:M / M ` has support in Sing Y , with Sing Y a strict closed subset in Y . Let x X and let B 1 P be a neighbourhood of x with compact closure B. Then, p´ B is compact. Since b p q dim Y dim X, theorem 1 is true for p:M D Y . Let Ui be a finite family of ă P holp q p q Y p 1 B i Slnb p M openr sets in covering ´ and such that for everyr , the set : Ui is bounded by a rationnal r . Definep q R max r . pp q| q r i i i By induction hypothesis applied to“ C, one can suppose at the cost of taking a nb smaller B containing x that the set Sl C B is bounded by a rational R1. Take p | q f OB . We have a distinguished triangle P / / 1 / (3.1.1) ψf p p:M f `N / ψf M f `N / ψf C f `N ` / p ` b q p b q p b q By projection formula and compatibility of ψ with proper direct image, (3.1.1) is isomorphic to / / 1 / p ψfp p:M pf `N / ψf M f `N / ψf C f `N ` / ` p b p q q p b q p b q 12 J.-B. TEYSSIER

So we have the desired vanishing on B for r max R, R1 . We are left with the case where dim SuppąM pdim Xq. Let Z be a hypersurface containing Sing M. We have a triangle “ / / 1 / RΓ Z M / M / M Z ` / r s p˚ q By applying the induction hypothesis to RΓ Z M, we are left to prove theorem 1 for M Z . The module M Z is a meromorphicr s connection, which concludes the reductionp˚ q step. p˚ q

3.2. The case of meromorphic connections. — At the cost of taking an open cover of X, let us take a resolution of turning points p : X X for M as given by Kedlaya-Mochizuki theorem. Let D be the pole locus of M.ÝÑ Since p is an isomorphism r above X D, the cone of z / (3.2.1) p p`M / M ` has support in the pole locus D of M. From [Meb04, 3.6-4], the left hand side of (3.2.1) is localized along D. So (3.2.1) is an isomorphism. We thus have a canonical isomorphism p ψfp p`M fp `N ψf M f `N ` p b p q q» p b q Since p is proper, we see as in 3.1 that we are left to prove theorem 1 for p`M. We thus suppose that M has a good formal structure. At the cost of taking an open cover, we can suppose that D has only a finite number of irreducible components. Let S be the divisor of highest generic slopes 1.6 of M. Let S1,...,Sm be the irreducible components of S . Let us prove that Slnb M is bounded by deg S. It is is a local | | p q statement. Let f OX and define Z : div f. Let us denote by Z (resp. S ) the support of Z (resp.P S) and let us admit“ for a moment the validity| of| the following| | Proposition 3.2.2.— Locally on X, one can find a proper birationnal morphism π : X X such that ÝÑ (1) πr is an isomorphism above X Z . 1 1 z| | (2) π´ Z π´ S is a normal crossing divisor. p| |q Y p| |q (3) for every valuation vE measuring the vanishing order along an irreducible compo- 1 nent E of π´ Z , we have p| |q vE S deg S vE f p q ď p q p q Let us suppose that 3.2.2 is true. At the cost of taking an open cover, let us take a morphism π : X X as in 3.2.2. Since condition (1) is true, the cone of the canonical comparisonÝÑ morphism r / (3.2.3) π π`M / M ` has support in Z . Since f `N is localized along Z , we deduce that (3.2.3) induces an isomorphism| | | | / π π`M f `N „ / M f `N p ` qb b A BOUNDEDNESS THEOREM FOR NEARBY SLOPES OF HOLONOMIC D-MODULES 13

Applying ψf and using the fact that π is proper, we see that it is enough to prove

(3.2.4) ψfπ π`M fπ `N 0 p b p q q» for every germ N of meromorphic connection at the origin with slope r deg S. Since 1 ą fπ `N is localized along π´ Z , the left-hand side of (3.2.4) is p q p| |q 1 (3.2.5) ψfπ π`M π´ Z fπ `N pp qp˚ p| |qq b p q q The vanishing of (3.2.5) is a local statement on X. Since (2) and (3) are true, 2.4.1 asserts that it is enough to show that for every irreducible component E of π 1 Z , r ´ we have p| |q 1 rE π`M π´ Z deg S vE fπ pp qp˚ p| |qqq ď p q p q Let us notice that vE fπ vE f . Let P be a point in the smooth locus of E. Let ϕ p q“ p q as in (1.5.1) for M at the point Q : π P . For i 1,...,n, let ti 0 be an equation “ p q “ “ r1 rn of Si in a neighbourhood of Q. Modulo a unit in OX,Q, we have ϕ 1 t t “ { 1 ¨ ¨ ¨ n where ri Q 0. If u 0 is a local equation for E in a neighbourhood of P , we have P ě O “ modulo a unit in X,P 1 Ă ϕπ r1vE t1 rnvE tn “ u p q u p q ϕπ 1 ¨ ¨ ¨ So the slope of E π´ Z along E is r1vE t1 rnvE tn . By Sabbah- p˚ p| |qq M p q`¨¨¨` p q M Mochizuki theorem, ri is a slope of generically along Si, so ri rSi . We deduce that ď p q M 1 M rE π` π´ Z rSi vE ti vE S deg S vE f p p˚ p| |qqq ď ÿi p q p q“ p q ď p q p q This concludes the proof of theorem 1 and theorem 2.

3.3. Proof of 3.2.2. — At the cost of taking an open cover of X, let us take a ﬁnite sequence of blow-up

pn´1 / pn´2 / / p0 / (3.3.1) πn : Xn / Xn 1 / / X1 / X0 X ´ ¨ ¨ ¨ “ given by 3.15 and 3.17 of [BM89] for Z relatively to the normal crossing divisor S . Let Z i be the strict transform of Z in Xi and let Ci be the center of pi. We | | | | | 1| 1 1 define inductively H0 H and Hi 1 pi´ Hi pi´ Ci for i 1,...,n, where pi´ “ ` “ p qY p q “ denotes the set theoretic inverse image. In particular Hi 1 is a closed subset of Xi 1. We will endow it with its canonical reduced structure. Then,` (3.3.1) satisfies ` i Ci is a smooth closed subset of Z i. p q | | ii Ci is nowhere dense in Z i. p q | | iii Ci and Hi have normal crossing for every i. p q iv Z n Hn is a normal crossing divisor. p q | | Y Since Ci and the components of Hi are reduced and smooth, condition iii means p q that locally on Xi, one can find coordinates x1,...,xk such that Hi is given by the p q equation x1 xl 0 and the ideal of Ci is generated by some xj for j 1,...,k. ¨ ¨ ¨ “ 1 1 “ Using condition i , we see by induction that πn´ Z πn´ S Z n Hn. Proposition 3.2.2p isq thus a consequence of p| |q Y p| |q “ | | Y 14 J.-B. TEYSSIER

Proposition 3.3.2.— Let

pn´1 / pn´2 / / p0 / πn : Xn / Xn 1 / / X1 / X0 X ´ ¨ ¨ ¨ “ be a sequence of blow-up satisfying i , ii and iii . For every irreducible component 1 p q p q p q E of π´ Z , we have n p| |q (3.3.3) vE S deg S vE f p q ď p q p q Proof. — Let S1,...,Sm be the irreducible components of S and let Z1,...,Zm1 | | be the irreducible components of Z. Note that some Zi can be in S . We deﬁne | | ai vZ f 0 and let Zji (resp. Sji) be the strict transform of Zj (resp. Sj ) in Xi. “ i p qą We argue by induction on n. If n 0, E is one of the Zi and then (3.3.3) is obvious. We suppose that (3.3.3) is true“ for a composite of n blow-up and we prove that (3.3.3) is true for a composite of n 1 blow-up. ` Let Cn be the set of irreducible components of n 1 ´ 1 pn 1 pi ´ Ci ´ iď0 p ¨ ¨ ¨ q p q “ Each element E Cn will be endowed with its reduced structure. Condition i implies P p q that the irreducible components of πn˚Z are the Zin and the elements of Cn. Condition ii implies that none of the Zin belongs to Cn. Thus, we have p q π˚Z div fπn a1Z1n am1 Zm1n vE f E n “ “ `¨¨¨` ` p q EÿCn P On the other hand, we have

π˚S rS1 M S1n rS M Smn vE S E n “ p q `¨¨¨` m p q ` p q EÿCn P Let us consider the last blow-up pn : Xn 1 Xn. Let us denote by P the excep- ` ÝÑ tionnal divisor of pn and let En 1 be the strict transform of E Cn in Xn 1. We have ` P `

pn˚Zin Zin 1 αiP with αi N “ ` ` P Since m Hn Sjn E “ Y jď0 EďCn “ P we deduce from condition iii and smoothness of Cn that p q pn˚E En 1 ǫEP with ǫE 0, 1 “ ` ` Pt u and

pn˚Sin Sin 1 ǫiP with ǫi 0, 1 “ ` ` Pt u Hence, we have

πn˚Z aiZin 1 vE f En 1 aiαi ǫEvE f P “ ` ` p q ` ` p ` p qq ÿ EÿCn ÿ EÿCn P P A BOUNDEDNESS THEOREM FOR NEARBY SLOPES OF HOLONOMIC D-MODULES 15 and M M πn˚S rSi Sin 1 vE S En 1 rSi ǫi ǫEvE S P “ p q ` ` p q ` ` p p q ` p qq ÿ EÿCn ÿ EÿCn P P Formula (3.3.3) is true for the Zin 1. By induction hypothesis, formula (3.3.3) is true ` for En 1, where E Cn. We are left to prove that (3.3.3) is true for P . Conditions ` P i and ii imply that one of the αi is non zero, so p q p q deg S aiαi ǫEvE f deg S deg S ǫEvE f p q ´ÿ ` ÿ p q ě p q ` p q ÿ p q M rSi ǫi ǫE deg S vE f ě ÿ p q ` ÿ p q p q M rSi ǫi ǫEvE S ě ÿ p q ` ÿ p q

4. Duality We prove theorem 3 i . Let us denote by D the duality functor for D-modules. There is a canonical comparisonp q morphism / (4.0.4) D M f `N / DM f `DN p b q b On a punctured neighbourhood of 0 C, the module N is isomorphic to a ﬁnite sum of copies of the trivial connection. Thus,P there is a neighbourhood U of Z such that the restriction of (4.0.4) to U Z is an isomorphism. Hence, the cone of (4.0.4) has support in Z. We deduce thatz / D M f `N Z „ / DM f ` DN 0 p p b qqp˚ q b pp qp˚ qq We have DN 0 N ˚, where is the duality functor for meromorphic connection. Note thatp isqp˚ a slopeq» preserving˚ involution. Since nearby cycles are insensitive to localization˚ and commute with duality for D-modules, we have

ψf DM f `N ˚ D ψf M f `N p b q» p p b qq and theorem 3 i is proved. p q

5. Regularity and nearby cycles The aim of this section is to prove theorem 4.

5.1. We will use the following Lemma 5.1.1.— Let F be a germ of closed analytic subspace at the origin 0 Cn. n P Let Y1,...,Yk be irreducible closed analytic subspaces of C containing 0 and such that F Yi is a strict closed subset of Yi for every i. Then, there exists a germ of X hypersurface Z at the origin containing F and such that Z Yi has codimension 1 in X Yi for every i. 16 J.-B. TEYSSIER

I I Proof. — Denote by F (resp. Yi ) the ideal sheaf of F (resp. Yi). By irreducibility, I O n I I Yi,0 is a prime ideal in C ,0. The hypothesis say F Yi for every i. From [Mat80, 1.B], we deduce Ę I I F Yi Ę ďi I I Any function f F not in i Yi defines a hypersurface as wanted. P Ť 5.2. We say that a holonomic module M is smooth if the support Supp M of M is smooth equidimensional and if the characteristic variety of M is equal to the conor- mal of Supp M in X. We denote by Sing M the complement of the smooth locus of M. It is a strict closed subset of Supp M. Let x X and let us define F as the union of Sing M with the irreducible components ofP Supp M passing through x which are not of maximal dimension. Define Y1,...,Yk to be the irreducible components of Supp M of maximal dimension passing through x. From 5.1.1, one can find a hypersurface Z passing through x such that (1) Z Supp M has codimension 1 in Supp M. (2) TheX cohomology modules of HkM are smooth away from Z. (3) dim Supp RΓ Z M dim Supp M. r s ă 5.3. The direct implication of theorem 4 is a consequence of the preservation of regularity by inverse image and the following Db Db Proposition 5.3.1.— We have hol X reg hol X 0 . p q Ă p qt u M Db Proof. — Take hol X reg. We argue by induction on dim X. The case where X is a point is trivial.P Byp arguingq on dim Supp M as in 3.1, we are left to prove 5.3.1 in the case where M is a regular meromorphic connection. Let D be the pole locus of M. Take f OX and let N with slope 0. To prove P ą ψf M f `N 0 p b q» one can suppose using embedded desingularization that D div f is a normal crossing divisor. We then conclude with 2.4.1. `

5.4. To prove the reverse implication of theorem 4, we argue by induction on dim X 1. The case of curves follows from 2.3.1. We suppose that dim X 2 and we takeě M Db M ě M hol X 0 . We argue by induction on dim Supp . The case where Supp is punctualP isp trivial.qt u Suppose that 0 dim Supp M dim X. Since Supp M is a strict closed subset of ă ă X, one can always locally write X X1 D where Dis the unit disc of C and where “ ˆ 1 the projection X1 D X1 is ﬁnite on Supp M. Let i : X1 D X1 P be the canonical immersion.ˆ ÝÑ There is a commutative diagram ˆ ÝÑ ˆ

/ 1 (5.4.1) Supp M / X1 P L ˆ LLL LLL p LL LL& X1 A BOUNDEDNESS THEOREM FOR NEARBY SLOPES OF HOLONOMIC D-MODULES 17

The oblique arrow of (5.4.1) is finite, and p is proper. So the horizontal arrow is 1 proper. Thus, Supp M is a closed subset in X1 P . Hence, M can be extended by 0 1 M ˆ Db 1 to X1 P . We still denote by this extension. It is an object of hol X1 P 0 and weˆ have to show that it is regular. p ˆ qt u Let Z be a divisor in X1 given by the equation f 0 and let ρ : Y X1 be “ ÝÑ a finite morphism. Since p is smooth, the analytic space Y 1 making the following diagram 1 ρ / 1 Y 1 / X1 P ˆ p1 p / Y ρ X1 cartesian is smooth. Moreover ρ1 is finite. By base change [HTT00, 1.7.3], projection formula and compatibility of ψ with proper direct image, we have for every germ N of meromorphic connection with slope 0 ą ψf ρ`p M f `N ψf p1 ρ1`M f `N p ` b q» p ` b q ψf p1 ρ1`M fp1 `N » p `p b p q qq p1 ψfp1 ρ1`M fp1 `N » ` p b p q q 0 » By induction hypothesis p M is regular. Let Y1,...,Yn be the irreducible components ` of Supp M with maximal dimension. Since Sing M Yi is a strict closed subset of Yi X and since a finite morphism preserves dimension, p Sing M p Yi is a strict closed p qX p q subset of the irreducible closed set p Yi . In a neighbourhood of a given point of p Sing M , one can find from 5.2 a hypersurfacep q Z containing p Sing M such that p q 1 p q Z p Yi has codimension 1 in p Yi for every i. So p´ Z contains Sing M and X p q p q p q 1 dim p´ Z Yi dim Z p Yi dim p Yi 1 dim Yi 1 p qX “ X p q“ p q´ “ ´ Since IrrZ˚ is compatible with proper direct image [Meb04, 3.6-6], we have M M IrrZ˚ p Rp Irrp˚´1 Z 0 ` » ˚ p q » Since p is finite over Supp M, we have M M Rp Irrp˚´1 Z p Irrp˚´1 Z ˚ p q » ˚ p q 1 M So for every x p´ Z , the germ of Irrp˚´1 Z at x is a direct factor of the complex M P p q Mp q p IrrZ˚ p p x 0. Thus Irrp˚´1 Z 0. From [Meb04, 4.3-17], We deduce p ˚ ` 1 q p q » p q » that M p´ Z is regular. p˚ p qq To show that M is regular, we are left to prove that RΓ p´1 Z M is regular. From r p qs1 5.3, the nearby slopes of all quasi-finite inverse images of M p´ Z are contained p˚ p qq in 0 . Thus, this is also the case for RΓ p´1 Z M. By construction of Z, t u r p qs dim Supp RΓ p´1 Z M dim Supp M r p qs ă We conclude by applying the induction hypothesis to RΓ p´1 Z M. Let us suppose that Supp M has dimension dim X, and letr Zp beqs a hypersurface as in 18 J.-B. TEYSSIER

5.2. Then M Z is a meromorphic connection with poles along Z. Let us show that M Z is regular.p˚ q By [Meb04, 4.3-17], it is enough to prove regularity generically alongp˚ Zq . Hence, one can suppose that Z is smooth. By Malgrange theorem [Mal96], one can suppose that Z is smooth and that M Z has good formal structure along p˚ q Z. Let x1,...,xn,t be coordinates centred at 0 Z such that Z is given by t 0 p q p P g x,u uk “ and let ρ : x, u x, u be as in 1.5 for M Z . Let E p q{ R be a factor p q ÝÑ p q p˚ q b of ρ` M0 Z where g 0, 0 0 and where R is a regular meromorphic connection with polesp p˚ alongqq Z. Forp a choiceq‰ of k-th root in a neighbourhood of g 0, 0 , we have x p q k M k E 1 u ψu ?k g ρ` u ?g ` ´ { 0 { p b p { q q» Since nearby cycles commute with formalization, we deduce g uk g uk ψu ρ` M0 Z E´ { ψu ρ`M0 E´ { 0 p p p˚ qq b q» p b q» Thus ψuR 0, so R x0. Hence, the only possiblyx non zero factor of ρ` M0 Z is the regular» factor. So» M Z is regular. We obtain that M is regular byp applyingp˚ qq p˚ q x the induction hypothesis to RΓ Z M. r s

6. Slopes and irregular periods 6.1. Let X be a smooth complex manifold and let D be a normal crossing divisor in X. Define U : X D and let j : U X be the canonical inclusion. Let M be a meromorphic connexoin“ z on X with polesÝÑ along D. D We denote by p : X X the real blow-up of X along D. Let Aă be the sheaf ÝÑ X [Sab00, II] of differentiabler functions on X whose restriction to X areĂ holomorphic and whose asymptotic development along p 1 D are zero. We define the de Rham r ´ complex with rapid decay by p q D D 1 DRă M : Aă ´1O p´ DRX M X “ X bp X 6.2. With the notations inĂ 6.1, if MĂhas good formal structure along D, we define [Hie09, Prop 2] rd 2d k D H X, M : H ´ X, DRă M k p q “ p X q M The left-hand side is the space of cycles with rapidr decayĂ for . For a topological description justifying the terminology, we refer to [Hie09, 5.1].

6.3. Proof of theorem 5. — We denote by j : U X the canonical immersion, k k ÝÑ d : dim X and Sl0 H f E the slopes of H f E at 0. We will also use the letter f for“ the restriction ofpf to`U.q From [HTT00, 4.7.2],` we have a canonical identiﬁcation / (6.3.1) f E an f j E an „ / f an j E an p ` q » p `p ` qq ` p ` q We deduce k k an an Sl0 H f E Sl0 H f j E p ` q“ p ` p ` q q Let x be a local coordinate on S centred at the origin. From 2.3.1, we have k an an nb k an an Sl0 H f j E Slx H f j E p ` p ` q q“ p ` p ` q q A BOUNDEDNESS THEOREM FOR NEARBY SLOPES OF HOLONOMIC D-MODULES 19

nb Hk an E an nb an E an Since Slx f j Slx f j , we deduce from theorem 2 and theorem 3 p ` p ` q q Ă p ` p ` q q Hk E nb E an Sl0 f Slf x j 0, r1 rn p ` qĂ p qpp ` q qĂr `¨¨¨` s k an an We are thus left to relate Sol H f j E to the periods of Et, for t 0 close enough to 0. Such a relationp appears` p ` forq aq special type of rank 1 connecti‰ ons in [HR08]. We prove more generally the following Proposition 6.3.2.— For every k, we have a canonical isomorphism

k an an / k an D an (6.3.3) R f Sol j E „ / R f p DRă j E˚ ˚ p ` q p q˚ X p ` q For t 0 close enough to 0, the fiber of the right-hand sideĂ of (6.3.3) at t is canonically ‰ rd E isomorphic to H2d 2 k Ut, t˚ . ´ ´ p q Proof. — Set M : j E an. Since M has good formal structure, we know from [Moc14, 5.3.1] that“ p ` q D Rp DRă M DR M !D ˚ X » p p qq where M !D : D DM D . WeĂ have DM D M˚ where M˚ is the dual connectionp ofqM“. Thuspp weqp˚ haveqq p qp˚ q“ D Sol M DR DM Rp DRă M˚ » » ˚ X By applying Rf an, we obtain for every k and every tĂ 0 close enough to 0 ˚ ‰ k an / k an D R f Sol M t „ / R f p DRă M˚ t p ˚ q p p q˚ X q Ă 1 6 p q p q k an k an D H X , Sol M t H X , DRă M˚ t p t p q q p t p X q q 2 Ă p q k an H X , Sol Mt 7 p t q p q 3 p q k an M k an Dt M H´ Xt , D Sol t ˚ H Xt , DRă t˚ p q p Xt q Ă 4 ≀ p q 2d 2 k an M rd an M H ´ ´ Xt , DR t ˚ H2d 2 k Xt , t˚ p q ´ ´ p q 5 p q | 2d 2 k / rd H ´ ´ Ut, DR Et ˚ / H Ut, E˚ 8 2d 2 k t p q p q ´ ´ p q By proper base change theorem, the morphisms 1 and 6 are isomorphisms. The morphism 2 is an isomorphism by non charactericityp q hypothesis.p q The morphism 3 is an isomorphismp q by Poincaré-Verdier duality. The morphism 4 is an isomorphismp q p q 20 J.-B. TEYSSIER by duality theorem for D-modules [Meb79][KK81]. The morphism 5 is an isomor- p q phism by GAGA and exactness of jt where jt : Ut Xt is the inclusion morphism. The morphism 8 is an isomorphism˚ by Hien dualityÝÑ theorem. We deduce that 7 is an isomorphism.p q p q k Let e : e1,...,en be a local trivialization of H f E 0 in a neighbourhood “ p q p ` qp˚ q 1 of 0. One can suppose that f is smooth above S˚ : S 0 . Set U ˚ : U f ´ 0 . “ zt u “ zt p qu From [DMSS00, 1.4], we have an isomorphism of left DS-modules k k d 1 H f E S˚ R ` ´ f DRU ˚ S˚ E p ` q| » ˚ { where the right hand side is endowed with the Gauss-Manin connection as defined in [KO68]. We deduce that et t S˚ is an algebraic family of bases for the k d 1 E p q P HdR` ´ Xt, t . At thep costq of shrinking S, Kashiwara perversity theorem [Kas75] shows that the only possibly non zero terms of the hypercohomology spectral sequence pq Hp H q E an Hp q E an E2 Sol ´ f S˚ ` Sol f S˚ “ p ` q| ùñ p ` q| sit on the line p 0. Hence, at the cost of shrinking S again, we have “ Hk E an H0 Hk E an H k E an (6.3.4) Sol f S˚ Sol f S˚ ´ Sol f S˚ p ` q| » p ` q| » p ` q| Since Sol is compatible with proper direct image, we deduce from (6.3.1) and (6.3.4) Hk E an k d 1 E an (6.3.5) Sol f S˚ R´ ` ´ f Sol j p ` q| » ˚ p ` q k an Let s : s1,...,sn be a local trivialization of Sol H f E over an open subset in an “ p q p ` q S˚ . From (6.3.5) and 6.3.2, we deduce that st t S˚ is a continuous family of basis rd E pE q P s for the spaces Hk d 1 Ut, t˚ . The periods of are by definition the coefficients of in e, and theorem` 5´ isp proved.q

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J.-B. Teyssier, Freie Universität Berlin, Mathematisches Institut, Arnimallee 3, 14195 Berlin, Germany ‚ E-mail : [email protected]