TOPOLOGICAL COMPUTATION OF SOME STOKES PHENOMENA ON THE AFFINE LINE

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ANNALES DE

L’INSTITUT FOURIER

LesAnnales de l’institut Fouriersont membres du Centre Mersenne pour l’édition scientifique ouverte

Andrea D’Agnolo, Marco Hien, Giovanni Morando & Claude Sabbah

Topological computation of some Stokes phenomena on the affine line

Tome 70, n^o2 (2020), p. 739-808.

<http://aif.centre-mersenne.org/item/AIF_2020__70_2_739_0>

Cet article est mis à disposition selon les termes de la licence Creative Commons attribution – pas de modification 3.0 France.

http://creativecommons.org/licenses/by-nd/3.0/fr/

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TOPOLOGICAL COMPUTATION OF SOME STOKES PHENOMENA ON THE AFFINE LINE

by Andrea D’AGNOLO, Marco HIEN, Giovanni MORANDO & Claude SABBAH (*)

Abstract. —LetM be a holonomic algebraic D-module on the affine line, regular everywhere including at infinity. Malgrange gave a complete description of the Fourier–Laplace transform M, including its Stokes multipliers at infinity, inc

terms of the quiver of M. LetF be the perverse sheaf of holomorphic solutions toM. By the irregular Riemann–Hilbert correspondence,^Mcis determined by the enhanced Fourier–Sato transformF^fofF. Our aim here is to recover Malgrange’s result in a purely topological way, by computing F^f using Borel–Moore cycles.

In this paper, we also consider some irregular M’s, like in the case of the Airy equation, where our cycles are related to steepest descent paths.

Résumé. —SoitMunD-module holonome algébrique sur la droite affine, à sin- gularités régulières y compris à l’infini. Malgrange a donné une description complète de son transformé de Fourier–LaplaceM, y compris des multiplicateurs de Stokesc

à l’infini, en termes du carquois deM. SoitF le faisceau pervers des solutions de M. Par la correspondance de Riemann–Hilbert irrégulière,^Mcest déterminé par le transformé de Fourier–Sato enrichiF^fdeF. Notre but est de retrouver le résultat de Malgrange de manière purement topologique, en calculantF^fà l’aide de cycles de Borel–Moore. Nous nous intéressons aussi à d’autresD-modules holonomes ir- réguliersM, tels que celui provenant de l’équation d’Airy, où les cycles que nous considérons sont reliés aux chemins de plus grande pente.

Introduction

Let M be a holonomic algebraic D-module on the affine line V. The Fourier–Laplace transformMcofMis a holonomicD-module on the dual

Keywords:Perverse sheaf, enhanced ind-sheaf, Riemann–Hilbert correspondence, holonomic D-module, regular singularity, irregular singularity, Fourier transform, quiver, Stokes matrix, Stokes phenomenon, Airy equation, Borel–Moore homology.

2020Mathematics Subject Classification:34M40, 44A10, 32C38.

(*) The research of A.D’A. was supported by GNAMPA/INdAM and by grant CPDA159224 of Padova University. The research of M.H. and G.M. was supported by the grant DFG-HI-1475/2-1 of the Deutsche Forschungsgemeinschaft. The research of C.S. was supported by the grant ANR-13-IS01-0001-01 of the Agence nationale de la recherche.

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affine lineV^∗. The Riemann–Hilbert correspondence of [4] (see [8, 15, 19] for expositions) associates toMthe enhanced ind-sheafFof its enhanced holomorphic solutions. By functoriality, the Riemann–Hilbert correspondence interchanges the Fourier–Laplace transform for holonomicD-modules with the Fourier–Sato transform for enhanced ind-sheaves. (This was observed in [18], where the non-holonomic case is also discussed.) As a direct con- sequence, if F is defined over a subfield of C, then so is the enhanced ind-sheafK associated with the Fourier–Laplace transform ofM.

In [21], Malgrange gave a complete description ofMcifMhas only regular singularities, including at infinity. It turns out thatMchas 0 and∞as its only possible singularities, and is regular at 0. The exponential components ofMcat∞are of linear type, with coefficients given by the singularities of M. Moreover, the Stokes multipliers ofMcat ∞are described in terms of the quiver ofM. In such a case, the regular Riemann–Hilbert correspondence of [13, 22] associates toM the perverse sheafF of its holomorphic solutions. AsMccan be reconstructed fromK, our aim here is to recover Malgrange’s result in a purely topological way, by giving a complete description of the enhanced Fourier–Sato transform ofF.

Before going into more detail, let us recall the classical local classification of a meromorphic connection at a singular point. The main idea is to examine the growth-properties of its solutions. There are various ways to implement this idea. One of them leads to the notion of a Stokes structure, a filtered local systems on a circle around the singular point (due to Deligne–Malgrange, see [6, 21, 29]). Another possibility is to compute for- mal power series solutions and then use some machinery (multi-summation methods) to lift these to asymptotic solutions on sectors centered at the point (see [1, 27]). In the present case, since the exponents of the exponential factors are linear, it is known that one can work with two sectors of width slightly bigger thanπ. In the Deligne–Malgrange presentation, this means that the Stokes filtration can be trivialized on these sectors. Hence, the information can be completely encoded by two matrices, governing the transition of the asymptotic fundamental solutions or the glueing of the trivialized filtered local systems. Classically, such a pair of matricesS+, S₋ is called the sequence of Stokes multipliers of the system. It is certainly the most explicit way to encode the desired data. Of course, one has to be careful keeping track of all choices involved.

In the context of the irregular Riemann–Hilbert correspondence of D’Agnolo–Kashiwara, [4], the Stokes multipliers can be obtained from the

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associated enhanced solutions in the way described in [4, §9.8]. In particular, Lemma 9.8.1 in loc. cit. clarifies the ambiguities due to various choices. The relation to the Deligne–Malgrange approach mentioned above is discussed in [15, §8].

Let us describe our results (which were announced in [11]) with a few more details. Let Σ ⊂ V be a finite subset. Denote by Perv_Σ(CV) the abelian category of perverse sheaves ofC-vector spaces onVwith possible singularities in Σ. After suitably choosing a total order on Σ, we can encode an objectF ∈Perv_Σ(CV) in its quiver Ψ,Φc, uc, vc

c∈Σ, where Ψ and Φc

are finite dimensional vector spaces, anduc: Ψ−→Φc andvc: Φc −→Ψ are linear maps such that 1−ucvc is invertible for anyc(or, equivalently, such that 1−vcuc is invertible for anyc).

The main result of this paper is Theorem 5.4 which is the topological counterpart of Malgrange’s result. It describes explicitly the Stokes multipliers of the enhanced Fourier–Sato transformKofFin terms of the quiver ofF. We want to stress that the computation reduces to computation with ordinary perverse sheaves, since our startingF is so. In particular, we can reduce the computation to perverse sheaves which are isomorphic to their maximal extension (Beilinson’s extension) at the singularities Σ.

Recall that the classical Fourier–Sato transform sends Perv_Σ(CV) to Perv_{0}(CV^∗) (see [21]). In Section 6 we also give an explicit description of the quiver of the Fourier–Sato transform of F in terms of that of F. For that purpose, we introduce the smash functor, whose properties are explained in Appendix B. Together with the computation of the Stokes multipliers, this completes the description ofK.

As another example of the method, we give in Section 7 a similar treat- ment for the Airy equation and for some elementary irregular meromorphic connections.

Sections 1–3 recall the formalism of enhanced ind-sheaves together with the Riemann–Hilbert correspondence of [4], and the Fourier transforms in this context. Section 4 makes precise the correspondence which associates a quiver to a perverse sheaf on the affine line.

To conclude this introduction, let us mention some other works where the Stokes multipliers of Mc are computed, for M a holonomic algebraic D-module on the affine line. In the book [21], that we already mentioned, Malgrange discusses in fact the more general case whereM is regular at finite distance, but is allowed an irregularity at infinity of linear type. This condition is stable by Fourier–Laplace transform. In [23], Mochizuki treats the case of a general holonomic algebraicD-module M, providing a way

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to compute the Stokes multipliers ofMcin terms of steepest descent paths.

An explicit description of the Stokes multipliers for some special classes of irregular holonomicD-modules is discussed in [30] and [12].

In the context of Dubrovin’s Conjectures in mirror symmetry, a lot of effort has been put into explicit computations of Stokes multipliers of the associatedD-module on the quantum cohomology of varieties in situations where one has some mirror symmetry statement (e.g. [2, 10, 33]). The mirror of the latter usually is given by the Fourier–Laplace transform of the regular singular Gauss–Manin system of some associated Landau–Ginzburg model. This situation therefore fits well into the framework examined in the present work.

1. Enhanced ind-sheaves

Let us briefly recall from [4] the triangulated category of enhanced ind- sheaves (see [5] for its naturalt-structures). This is similar to a construction in [32] (see [9] for an exposition), in the framework of ind-sheaves from [17].

As we are interested in applications to holonomic D-modules, we restrict here to an analytic framework.

LetM be a real analytic manifold and letkbe a field (when considering the enhanced ind-sheaf attached to aD-module, we setk=C).

1.1. Sheaves

(Refer e.g. to [16].) Denote by D^b(kM) the bounded derived category of sheaves ofk-vector spaces onM. ForS ⊂M a locally closed subset, denote bykS the zero extension toM of the constant sheaf onS with stalk k.

Forf:M −→N a morphism of real analytic manifolds, denote by⊗,f⁻¹, Rf! andRHom,Rf_∗,f^! the six Grothendieck operations for sheaves.

1.2. Convolution Consider the maps

µ, q1, q2:M ×R×R−→M×R

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given by q1(x, t1, t2) = (x, t1), q2(x, t1, t2) = (x, t2), and µ(x, t1, t2) = (x, t1+t2). For F1, F2 ∈ D^b(k_M_×R), the functors of convolution in the variablet∈Rare defined by

F₁⊗^∗ F₂=Rµ_!(q₁⁻¹F₁⊗q₂⁻¹F₂), RHom^∗(F1, F2) =Rq1∗RHom(q₂⁻¹F1, µ^!F2).

The convolution product⊗^∗ makesD^b(k_M_×R) into a commutative tensor category, withk_M×{0}as unit object. We will often writek_{t=0} instead of k_M_×{0}, and similarly for k_{_t_>0},k_{_t_60}, etc.

1.3. Enhanced sheaves

(Refer to [4, 9, 32].) Consider the projection M×R−−→^π M.

The triangulated categoryE^b(k_M) of enhanced sheaves is the quotient of D^b(kM×R) by the stable subcategory π⁻¹D^b(kM). It splits as E^b(kM) ' E^b₊(k_M)⊕E^b₋(k_M), where E^b_±(k_M) is the quotient of D^b(k_M_×R) by the stable subcategory of objectsF satisfyingk_{∓t>0}⊗^∗ F '0.

The quotient functor

Q:D^b(kM×R)−→E^b(kM)

has a left and a right adjoint which are fully faithful. Let us denote by eE^b₊(k_M)⊂D^b(k_M_×R) the essential image of E^b₊(k_M) by the left adjoint, that is, the full subcategory whose objectsF satisfyk_{t>0}⊗^∗F 'F. Thus, one has an equivalence

Q:eE^b₊(kM)−−→∼ E^b₊(kM).

The functor

(1.1) ε:D^b(k_M)−→eE^b₊(k_M), F⁰ 7→k_{_t_>0}⊗π⁻¹F⁰.

is fully faithful. Forf: M −→N a morphism of real analytic manifolds, it interchanges the operations⊗,f⁻¹ andRf! with⊗^∗, fe⁻¹ andRfe!, respectively. Here, we set

fe=f×id_R:M×R−→N×R.

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1.4. Ind-sheaves

(Refer to [17].) An ind-sheaf is an ind-object in the category of sheaves with compact support. There is a natural embedding of sheaves into ind- sheaves, and it has an exact left adjointα given by α(“lim

−→”Fi) = lim

−→Fi. The functorαhas an exact fully faithful left adjoint, denotedβ.

Denote byD^b(Ik_M) the bounded derived category of ind-sheaves. Denote by⊗,RIhom,f⁻¹,Rf_∗,Rf!!,f^! the six Grothendieck operations for ind- sheaves.

1.5. Enhanced ind-sheaves

(Refer to [4].) Consider the morphisms

(1.2) M −−−→ⁱ^∞ M×P−−→^π M, M×R−−→^j M ×P,

where P= R∪ {∞} is the real projective line,i_∞(x) = (x,∞), π is the projection, andj is the embedding.

The triangulated categoryE^b(IkM) of enhanced ind-sheaves is defined by two successive quotients ofD^b(Ik_M×P): first by the subcategory of objects of the formRi_{∞ ∗}F, and then by the subcategory of objects of the form π⁻¹F. As for enhanced sheaves, the quotient functor has a left and a right adjoint which are fully faithful. It follows that there are two realizations of E^b(IkM) as a full subcategory ofD^b(IkM×P).

Enhanced ind-sheaves are endowed with an analogue of the convolution functors, denoted⊗⁺ andIhom⁺. Forf:M −→N a morphism of real analytic manifold, one also has external operationsEf⁻¹,Ef_∗,Ef_!!,Ef^!. Here, Ef⁻¹is the functor induced byfe⁻¹at the level of ind-sheaves, and similarly for the other operations.

There is a natural embedding E^b(k_M) ⊂ E^b(Ik_M) induced by Rj_! or, equivalently, byRj_∗. Set

k^E_M = “lim

−→”

a→+∞

Qk_{_t_>_a_}∈E^b(Ik_M).

There is a natural fully faithful functor

e:D^b(Ik_M)−→E^b(Ik_M), F 7→k^E_M ⊗π⁻¹F.

We will need the following two lemmas on compatibility between operations for enhanced ind-sheaves and for usual sheaves with an additional variable.

By [4, Remark 3.3.21, Proposition 4.7.14], one has

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Lemma 1.1. — LetF, F1, F2∈D^b(kM×R)andG∈D^b(kN×R). One has (kÊ_M ⊗⁺QF1)⊗⁺(kÊ_M ⊗⁺QF2)'kÊ_M⊗⁺(QF1+

⊗QF2) 'k^E_M⊗⁺Q(F1 ∗

⊗F2), Ef⁻¹Q(k^E_N⊗⁺G)'k^E_M⊗⁺Qfe⁻¹G.

If moreoverf is proper, then

Ef_!!Q(k^E_M ⊗⁺F)'k^E_N ⊗⁺QRfe_!F.

The categoryE^b(Ik_M) has a natural hom functorRHom^Ewith values in D^b(k_M). By [4, Proposition 4.3.10] and [19, Corollary 6.6.6], one has

Lemma 1.2. — ForF ∈D^b(kM×R)one has

RHom^E(k_{_t_>0},k^E_M ⊗⁺QF)'Rπ_∗(k_{_t_>0}⊗^∗ F).

1.6. R-constructibility

(Refer to [4].) Denote by D^b_R-c(k_M) the full subcategory of objects with R-constructible cohomologies. Using notations (1.2), denote D^b_R-c(k_M×R∞) the full subcategory ofD^b_R-c(k_M_×R) whose objectsFare such thatRj_!F(or, equivalently,Rj_∗F) isR-constructible inM×P. SinceRj! is fully faithful, we will also considerD^b_R-c(k_M_×R_∞) as a full subcategory ofD^b_R-c(k_M_×P).

The triangulated categoryeE^b_R-c(kM) ofR-constructible enhanced sheaves is the full subcategory of D^b_R-c(k_M×R_∞) whose objects F satisfy F ' k_{_t_>0}⊗^∗ F. It is a full subcategory of Ee^b₊(kM). Denote by E^b_R-c(kM) the full subcategory ofE^b₊(k_M) whose objects are isomorphic toQF, for some F∈Ee^b_R-c(k_M).

The category E^b_R-c(IkM) of R-constructible enhanced ind-sheaves is defined as the full subcategory ofE^b(Ik_M) whose objectsKsatisfy the following property: for any relatively compact open subsetU ⊂M there exists F∈Ee^b_R-c(k_M) such that

π⁻¹k_U ⊗K'k^E_M⊗⁺QF.

The objectk^E_M plays the role of the constant sheaf in E^b_R-c(IkM).

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2. Riemann–Hilbert correspondence

We recall here the Riemann–Hilbert correspondence for (not necessarily regular) holonomicD-modules established in [4] (see [5] for itst-exactness).

This is based on the theory of ind-sheaves from [17], and influenced by the works [32] and [3]. One of the key ingredients of its proof is the description of the structure of flat meromorphic connections by [24] and [20].

LetX be a complex manifold. We set for shortdX = dimX.

2.1. D-modules

(Refer e.g. to [14].) Denote by O^X and D^X the rings of holomorphic functions and of differential operators, respectively.

Denote by D^b(D^X) the bounded derived category of left D^X-modules.

Forf:X −→Y a morphism of complex manifolds, denote by⊗^D,Df^∗,Df_∗ the operations forD-modules.

Consider the solution functor

SolX:D^b(D^X)^op−→D^b(C^X), M 7→RHom_D_X(M,O^X).

Denote byD^b_hol(D^X) andD^b_reg-hol(D^X) the full subcategory ofD^b(D^X) of objects with holonomic and regular holonomic cohomologies, respectively, and byD^b_g-hol(D^X) the full subcategory of objects with good and holonomic cohomologies.

LetD ⊂X be a complex analytic hypersurface and denote byO^X(∗D) the sheaf of meromorphic functions with poles alongD. SetU =XrD.

Forϕ∈ O^X(∗D), set

D^Xe^ϕ=D^X/{P;P e^ϕ= 0 onU}, E_U|X^ϕ =D^Xe^ϕ⊗^DO^X(∗D).

2.2. Tempered solutions

(Refer to [17].) By the functorβ, there is a natural notion ofβD^X-module in the category of ind-sheaves, cp. [17, 5.6]. We denote by D^b(ID^X) the corresponding derived category.

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Denote byOX^t ∈D^b(ID^X) the complex of tempered holomorphic functions. It is related to the functorThom of [13] by the relation

αRIhom(F,OX^t )' Thom(F,O^X) for anyF∈D^b_R-c(C^X).

Consider the tempered solution functor

Sol_X^t :D^b(D^X)^op−→D^b(IC^X), M 7→RHom_D_X(M,OX^t ).

2.3. Enhanced solutions

(Refer to [4].) There is a natural notion of D^X-module in the category of enhanced ind-sheaves, and we denote by E^b(ID^X) the corresponding triangulated category.

LetP=C∪ {∞} be the complex projective line, and let i:X×P−→X×P

be the closed embedding. Denote by τ ∈P the affine coordinate, so that τ∈ OP(∗∞). Consider the exponentialDP-moduleE_C|P^τ .

The enhanced solution functor is given by

Sol_X^E:D^b(D^X)^op−→E^b(IC^X), M 7→Qi^!Sol_X^t_×P(M^DEC|P^τ )[2], where^Ddenotes the exterior tensor product forD-modules.

The functorial properties of Sol^E are summarized in the next theorem.

The statements on direct and inverse images are easy consequences of the corresponding results for tempered solutions obtained in [17]. The statement on the tensor product is specific to enhanced solutions and is due to [4, Corollary 9.4.10].

Theorem 2.1. — Letf:X −→Y be a complex analytic map. LetM ∈ D^b_g-hol(D^X),M¹,M²∈D^b_hol(D^X)andN ∈D^b_hol(D^Y). Assume thatsuppM is proper overY. Then one has

Sol_XÊ(Df^∗N)'Ef⁻¹Sol_YÊ(N), Sol_YÊ(Df_∗M)[dY]'Ef_!!Sol_XÊ(M)[dX], Sol_XÊ(M1)⊗ S⁺ ol_XÊ(M2)' Sol_XÊ(M1⊗^DM2).

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Notation 2.2. — LetD⊂Xbe a closed hypersurface and setU=XrD.

Forϕ∈ O^X(∗D), we set

E^ϕ:=k_{_t+Re_ϕ(x)>0}∈Ee^b_R-c(k_M), E^ϕ:=k^E_M ⊗⁺QE^ϕ∈E^b_R-c(IkM), where we set for short

{t+ Reϕ(x)>0}={(x, t)∈X×R; x∈U, t+ Reϕ(x)>0}. We will also need the following computation from [4, Corollary 9.4.12].

Theorem 2.3. — With the above notations, fork=C, one has Sol_X^E(EU^ϕ|X)'E^ϕ.

2.4. Riemann–Hilbert correspondence

Let us state the Riemann–Hilbert correspondence for holonomic D- modules established in [4, Theorem 9.5.3].

Theorem 2.4. — The enhanced solution functor induces a fully faithful functor

Sol_X^E:D^b_hol(D^X)^op−→E^b_R_-c(IC^X).

Moreover, there is a functorial way of reconstructingM ∈D^b_hol(D^X)from Sol_X^E(M).

This implies in particular that the Stokes structure of a flat meromorphic connectionMis encoded topologically inSol^E(M). For this, we refer to [4,

§9.8] (see also [15, §8]).

2.5. A lemma

We will later use the following remark. LetD⊂X be a closed hypersurface, setU =XrD, and denote byj: U −→X the embedding.

Lemma 2.5. — With the above notations, let M ∈ D^b_hol(D^X) be such that M ' M(∗D). Assume that X is compact. Then there exists F ∈ eE^b_R_-_c(C^U)such thatRe_!F ∈eE^b_R_-_c(C^X)and

Sol_X^E(M)'C^EX

⊗+QRe!F.

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Proof. — SetK =Sol_X^E(M). Since Mis holonomic, K isR-constructible. SinceX is compact, there existsF⁰∈eE^b_R-c(C^X) withK'C^EX

⊗+QF⁰. Since M ' M(∗D), one hasK 'π⁻¹C^U ⊗K ' C^EX

⊗+Q(π⁻¹C^U ⊗F⁰).

HenceF =e⁻¹F⁰ satisfies the assumptions in the statement.

3. Fourier transform

By functoriality, the enhanced solution functor interchanges integral transforms at the level of holonomic D-modules with integral transforms at the level of enhanced ind-sheaves. (This was observed in [18], where the non-holonomic case is also discussed.) We recall here some consequences of this fact, dealing in particular with the Fourier transform.

3.1. Integral transforms Consider a diagram of complex manifolds

(3.1)

S

p

q

X Y.

At the level ofD-modules, the integral transform with kernelL ∈D^b(D^S) is the functor

∗^D◦ L:D^b(D^X)−→D^b(D^Y), M^D◦ L=Dq_∗(L ⊗^DDp^∗M).

At the level of enhanced ind-sheaves, the integral transform with kernel H∈E^b(Ik_S) is the functor

∗⁺◦H:E^b(IkX)−→E^b(IkY), K⁺◦H =Eq_!!(H⊗⁺Ep⁻¹K).

Combining the isomorphisms in Theorem 2.1, one has

Corollary 3.1. — LetM ∈D^b_g-hol(D^X)and L ∈D^b_g-hol(D^S). Assume that p⁻¹supp(M)∩supp(L) is proper over Y. Set K = Sol_X^E(M) and H=Sol_S^E(L). Then there is a natural isomorphism inE^b_R_-c(IkY)

Sol_Y^E(M^D◦ L)'K⁺◦H[dS−dY].

Remark 3.2. — There is a similar statement with the solution functor replaced by the de Rham functor. This has been extended in [18] to the case whereMis good, but not necessarily holonomic.

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3.2. Globally R-constructible enhanced ind-sheaves

Consider the diagram of real analytic manifolds induced by (3.1) S×R

ep

eq

X×R Y ×R,

wherepe=p×idR andeq=q×idR.

The natural integral transform forR-constructible enhanced sheaves with kernelL∈eE^b_R-c(kS) is the functor

∗◦^∗L:eE^b_R-c(k_X)−→eE^b_R-c(k_Y), F◦^∗L=Req!(L⊗^∗ ep⁻¹F).

Combining Lemma 1.1 and Corollary 3.1 we get

Proposition 3.3. — LetM ∈D^b_g-hol(D^X),L ∈D^b_g-hol(D^S), and assume that p⁻¹supp(M)∩supp(L) is proper over Y. Let F ∈ eE^b

R-c(C^X), L ∈ eE^b_R_-c(C^S), and assume that there are isomorphisms

(3.2) Sol_X^E(M)'C^EX

⊗+QF, Sol_S^E(L)'C^ES

⊗+QL.

Then there is a natural isomorphism inE^b

R-c(Ik_Y) Sol_Y^E(M^D◦ L)'C^EY

⊗+Q(F^∗◦L)[dS−dY].

Note that if X and S are compact, then for any M ∈ D^b_hol(D^X) and L ∈D^b_hol(D^S) there existF ∈eE^b_R-c(C^X) andL∈eE^b_R-c(C^S) satisfying (3.2).

3.3. Fourier–Laplace transform

Let V be a finite dimensional complex vector space, denote by P its projective compactification, and setH=P r V.

Definition 3.4. — LetD^b_hol(DV∞)be the full triangulated subcategory ofD^b_hol(DP)whose objectsMsatisfyM ' M(∗H).

Let V^∗ be the dual vector space ofV, denote by P^∗ its projective com- pactifications, and setH^∗=P^∗r V^∗. The pairing

V×V^∗−→C, (z, w)7→ hz, wi defines a meromorphic function onP×P^∗ with poles along

(P×H^∗)∪(H×P^∗) = (P×P^∗)r(V×V^∗).

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Consider the projections

(3.3)

P×P^∗

P P^∗. Definition 3.5. — Set

L=E_V×V^h^z,w^∗ⁱ_|P×P^∗, L^a=E_V^−h^∗_×V|P^w,zⁱ^∗_×P.

(Beware that the signs in±hw, ziare sometimes reversed in the literature.) The Fourier–Laplace transform ofM ∈D^b_hol(DV∞)is given by

M^∧=M^D◦ L ∈D^b_hol(DV^∗∞).

The inverse Fourier–Laplace transform ofN ∈D^b_hol(DV^∗∞)is given by N^∨=N ^D◦ L^a ∈D^b_hol(DV∞).

Theorem 3.6. — The Fourier–Laplace transform gives an equivalence of categories

∧:D^b_hol(DV∞)−−→∼ D^b_hol(DV^∗∞).

A quasi-inverse is given byN 7→ N^∨.

This result is classical, see [21, App. 2]. The idea of the proof is as follows. Denoting byD_Vthe Weyl algebra, there is an equivalence of categories D^b_hol(DV)'D^b_hol(DV∞). Under this equivalence, the Fourier–Laplace transform is induced by the algebra isomorphismD_V'D_V^∗given byzi7→ −∂wi,

∂zi 7→wi.

Using the Riemann–Hilbert correspondence and a result of [32], we give an alternative topological proof of Theorem 3.6 in Remark 3.12 below.

3.4. Enhanced Fourier–Sato transform

Recall that j:V−→Pdenotes the embedding

Definition 3.7. — LetEe^b_R_-_c(k_V_∞)be the full triangulated subcategory ofeE^b_R_-c(k_V)whose objectsF satisfyRe!F ∈Ee^b_R_-c(k_P).

Consider the projections

V×V^∗×R eq

$$

p

ep

{{V×R V V^∗×R.

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Definition 3.8. — Using Notation 2.2, set L=E^h^z,wⁱ, L^a =E^−h^w,zⁱ. The enhanced Fourier–Sato transform ofF ∈eE^b

R-c(k_V∞)is given by F^f=F◦^∗L∈Ee^b_R_-_c(k_V^∗_∞).

The enhanced inverse Fourier–Sato transform ofG∈eE^b_R_-_c(k_V^∗_∞)is given by G^g=G^∗◦L^a∈eE^b_R_-c(k_V_∞).

The transformF 7→F^f has been investigated by Tamarkin [32] (in the more general case of vector spaces over R). He proved in particular the following result.

Proposition 3.9 ([32, Theorem 3.5], see also [18, §5.1]). — The enhanced Fourier–Sato transform gives an equivalence of categories

f: eE^b_R_-_c(k_V∞)−−→∼ eE^b_R_-_c(k_V^∗_∞).

A quasi-inverse is given byG7→G^g[2 dimV].

The following result will be used when dealing with enhanced solutions of regular holonomicD-modules. It was noticed in [3, Proposition A.3]. We letεas in (1.1).

Lemma 3.10. — ForF⁰∈D^b

R-c(k_V∞)in the notation analogous to Def- inition 3.7, one has

ε(F⁰)^f'Req!(L⊗p⁻¹F⁰).

As explained in [4, §3] there is a natural notion of “bordered space”, of whichV∞ = (V,P) is an example. The notion of enhanced ind-sheaves naturally extends to this setting (see [18] or [5]), and there is a natural equivalence of E^b(Ik_V_∞) with the full subcategory of E^b(Ik_P) whose ob- jectsK satisfyK'π⁻¹k_V⊗K. There are a natural functor

Q:eE^b₊(k_V)−→E^b(Ik_V_∞), and a natural objectk^E_V_∞ ∈E^b(Ik_V_∞).

Similarly to Definition 3.8, there is an enhanced Fourier–Sato transform for ind-sheaves

f:E^b(Ik_V_∞)−−→∼ E^b(Ik_V^∗_∞).

By [4, Corollary 9.4.12], one has Sol_V^E_∞_×V∗

∞(L)'E^h^z,wⁱ:=C^EV∞×V^∗∞

⊗+QL.

The next result is then easily checked.

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Lemma 3.11.

(i) LetM ∈D^b_hol(DV∞)andF ∈Ee^b

R-c(CV∞)satisfy (3.4) Sol_V^E_∞(M)'C^E_V∞

⊗+QF.

Then, there is an isomorphism

Sol_V^E^∗_∞(M^∧)'C^E_V^∗_∞⊗⁺QF^f[dimV].

(ii) LetM ∈D^b_reg-hol(DV∞)and setF⁰=Sol_P(M)|V. Then, there is an isomorphism

Sol_V^E^∗_∞(M^∧)'C^E_V^∗_∞⊗⁺QReq!(L⊗p⁻¹F⁰)[dimV].

Analogous results hold for∧andfreplaced by∨andg, respectively.

Note that, in view of Lemma 2.5, for any M ∈ D^b_hol(DV∞) there is an F∈Ee^b_R-c(CV∞) satisfying (3.4).

Remark 3.12. — Lemma 3.11(i) and Proposition 3.9 imply Theorem 3.6, due to the Riemann–Hilbert correspondence. In fact, with notations as in (3.4), one has

Sol_V^E_∞(M^∧∨)'C^E_V∞

⊗+QF^fg[2 dimV] 'C^EV∞

⊗+QF ' Sol_V^E_∞(M).

HenceM^∧∨' M. One similarly getsN^∨∧' N forN ∈D^b_hol(DV^∗∞).

4. Perverse sheaves on the affine line

The results in this section concern perverse sheaves on the affine line and their quivers. These are classical and well-known results (see e.g. [7]).

However, we present them here from a point of view slightly different than usual, better suited to our needs. In particular, we give a purely topological description of Beilinson’s maximal extension.

From now on, letVdenote a complex affine line.

4.1. Constructible complexes and perverse sheaves

Let Σ ⊂ V be a discrete subset and let us denote by D^b_C-c(k_V,Σ) the full triangulated subcategory of D^b(k_V) whose objects F have cohomology sheaves which are C-constructible with respect to the stratification

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(VrΣ,Σ). In particular,H^j(F)_|VrΣis a locally constant sheaf. The abelian category Perv_Σ(k_V) of Σ-perverse sheaves is the heart of the middle per- versity t-structure on D^b_C-c(k_V,Σ). Note that k_V[1] and k_Σ are objects of Perv_Σ(k_V) and, for an object F of Perv_Σ(k_V), the shifted restriction F_|VrΣ[−1] is a locally constant sheaf that we usually denote byL.

Definition 4.1. — LetΣ, X⊂V. We say thatX isΣ-negligible if it is disjoint fromΣ, locally closed, simply connected, and ifRΓ_c(V;k_X)'0.

Lemma 4.2. — Let Σ ⊂ V be a discrete subset, and let X ⊂ V be Σ-negligible. IfF ∈D^b_C_-c(k_V,Σ), then

RΓ_c(V;k_X⊗F)'0.

Lemma 4.3. — Let Σ ⊂ V be a discrete subset, and let Bc ⊂ V be a convex open neighborhood of c ∈ Σ such that Bc ∩Σ = {c}. If F ∈ D^b_C_-c(k_V,Σ), then

RΓ_c(V;k_{_c_}⊗F)'i⁻¹_c F 'RΓ(Bc;F), RΓ_c(V;k_B_c⊗F)'i_c^!F 'RΓ_{c}(V;F), whereic:{c} −→Vdenotes the embedding.

We will also use the following general lemma.

Lemma 4.4. — If`is a semi-open interval, X is a manifold and Lis a locally constant sheaf onX×`, thenRΓc(X×`;L) = 0.

4.2. From perverse sheaves to quivers

Assume that Σ ⊂V is finite. We will denote byQuiv_Σ(k) the category whose objects are the tuples

(Ψ,Φc, uc, vc)_c∈Σ,

where Ψ and Φc are finite dimensional k-vector spaces, and uc: Ψ−→Φc

andvc: Φc−→Ψ arek-linear maps such that 1−ucvc is invertible for anyc (or, equivalently, such that 1−vcuc is invertible for any c). Morphisms in Quiv_Σ(k) are the natural ones. Our aim in this section is to present a functor

Q^(α,β)_Σ :Perv_Σ(k_V)−→Quiv_Σ(k)

which, as we will recall in the next section, provides an equivalence of categories. This functor depends on choices (α, β) that we fix now, for the remaining part of this section.

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ci−1 ci ci+1

Bci B^>ci

ci−1 cici+1

B_c⁶_i ci−1 cici+1

`^×_c_i ci−1 ci ci+1

Figure 4.1. The setsBci,B_c^>_i,B_c⁶_i, and`^×_c_i.

Convention 4.5. — We fixα∈V r{0}andβ ∈V^∗r{0}such that Rehα, βi= 0,

(4.1)

Rehc−c⁰, βi 6= 0, ∀c, c⁰∈Σ, c6=c⁰. (4.2)

Notation 4.6.

(i) TheR-linear projection

pβ:VR−→R, z7→Rehz, βi

enables one, according to (4.2), to define a total order on Σ by the formula

c <βc⁰ if pβ(c)< pβ(c⁰).

We will enumerate the elements of Σ as c1<βc2<β. . . <βcn, and we will set

ri=pβ(ci), r0=−∞, rn+1= +∞ (i= 1, . . . , n), so that−∞=r₀< r₁<· · ·< rn< r_n+1= +∞. (ii) Fori= 1, . . . , n, the open bands

Bci(β) :=p⁻¹_β ( (ri−1, ri+1) ) coverVand satisfyBci(β)∩Σ ={ci}. We set

B^>_c_i(β) :=p⁻¹_β ( (ri, ri+1) ), B_c⁶_i(β) :=p⁻¹_β ( (ri−1, ri] ),

`ci(α) :=ci+R>0α, `^×_c_i(α) :=ci+R>0α.

Note that the half lines`ci(α) are disjoint, and set

`_Σ(α) :=`c1(α)∪`c2(α)∪ · · · ∪`cn(α).

We will write for shortBci,B_c⁶_i,`^×_c_i, etc., instead ofBci(β),B_c⁶_i(β),

`^×_c_i(α), etc. These sets are pictured in Figure 4.1.

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Definition 4.7. — Let c ∈ Σ and F ∈ D^b

C-c(k_V,Σ). The complexes of nearby cycles at c, vanishing cycles at c, and global nearby cycles are respectively defined by the formulas

Ψ^(α,β)c (F) = Ψc(F) :=RΓ_c(V;k_`× c ⊗F), Φ^(α,β)c (F) = Φc(F) :=RΓ_c(V;k_`_c⊗F),

Ψ^(α,β)(F) = Ψ(F) :=RΓ_c(V;k_Vr_`_Σ⊗F)[1].

We construct below the morphisms ucc : Ψc(F) −→ Φc(F) and vcc : Φc(F)−→Ψc(F) and we define the monodromy as

Tcc:= 1−vccucc: Ψc(F)−→Ψc(F).

On the one hand, the exact sequence 0−→k_`^×

c −→k`c −→k_{c}−→0 enables us to obtain the distinguished triangle definingucc:

Ψc(F)−−−→^u^cc Φc(F)−→i⁻¹_c F −−−→⁺¹ . In order to define the morphism

vcc: Φc(F)−→Ψc(F), we consider the exact sequences

0−→k_B^>

c −→k_B>

c∪`^×_c −→k_`^×

c −→0, 0−→k_B>

c −→k_B_c_r_`_c −→k_B6

cr`c−→0.

(4.3)

SinceB_c⁶r`c andB^>_c ∪`^×_c are Σ-negligible, they give rise to the isomorphisms

(4.4) Ψc(F)−−→∼ RΓc(V;k_B>

c ⊗F)[1]−−→∼ RΓc(V;k_B_c_r_`_c⊗F)[1].

On the other hand, the exact sequence

0−→kB_cr`_c−→kB_c −→k`_c −→0 gives rise to the morphism

vcc: Φc(F)−→RΓ_c(V;k_B_c_r_`_c⊗F)[1]_(4.4)' Ψc(F).

For each c∈Σ, we have a short exact sequence

(4.5) 0−→kB_cr`_c−→k_Vr`_Σ −→k_(Vr_`_Σ_)r_B_c−→0,

and we note that (V r`Σ)rBcis Σ-negligible. We deduce an isomorphism

(4.6)c Ψc(F)−−−→_(4.4)^∼ RΓc(V;kBcr`c⊗F)[1]

−−→∼ RΓ_c(V;k_Vr_`_Σ⊗F)[1] = Ψ(F)

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c

`c

c

Bcr`c V r`Σ

Figure 4.2. The sets`c,Bcr`c andV r`Σappearing in(4.6)c. for eachc∈Σ. The subsets appearing in this construction are sketched in Figure 4.2.

Let us now apply this to perverse sheaves.

Lemma 4.8. — LetF be an object of Perv_Σ(k_V). Recalling that L de- notes the local systemF|VrΣ[−1], we have

Ψc(F)'H⁰Ψc(F) =H⁰RΓ_c(`^×c;F) =H_c¹(`^×c;L), Φc(F)'H⁰Φc(F) =H⁰RΓ_c(`c;F),

Ψ(F)'H⁰Ψ(F) =H_c²(V r`Σ;L),

and these are exact functors fromPerv_Σ(k_V)to finite dimensionalk-vector spaces.

Lemma 4.9. — Let F, G ∈ Perv_Σ(k_V) and let ϕ, ψ : F −→ G be two morphisms. Assume thatϕandψcoincide onVrΣand thatΦc(ϕ) = Φc(ψ) for everyc∈Σ. Then ϕ=ψ.

Proof. — The image F⁰ of ϕ−ψ is a sub-perverse sheaf of G. By the first assumption, it is supported on Σ. By exactness of Φc for perverse sheaves, the second assumption gives ΦcF⁰ = 0, hence F⁰ = 0. Therefore,

ϕ−ψ= 0.

Definition 4.10. — LetF∈Perv_Σ(k_V). ThequiverofF is the datum Q^(α,β)_Σ (F) = (Ψ(F),Φc(F), uc, vc)_c_∈Σ,

with

uc:=ucc◦(4.6)c−1, vc:= (4.6)c◦vcc.

Note that the choice ofαandβ induces an orientation ofV'R² by the ordered basis given byαand a vector tangent at time zero to the rotation of`^×_c around c in the direction ofB_c^<. (The orientation class depends on the sign of Imhα, βi.)