Enhanced nearby and vanishing cycles in dimension one and Fourier transform

Enhanced ind-sheaves provide a suitable framework for the irregular Riemann-Hilbert correspondence. In this paper, we give some precisions on nearby and vanishing cycles for enhanced perverse objects in dimension one. As an application, we give a topological proof of the following fact. Let $\mathcal M$ be a holonomic algebraic $\mathcal D$-module on the affine line, and denote by ${}^{\mathsf{L}}\mathcal M$ its Fourier-Laplace transform. For a point $a$ on the affine line, denote by $\ell_a$ the corresponding linear function on the dual affine line. Then, the vanishing cycles of $\mathcal M$ at $a$ are isomorphic to the graded component of degree $\ell_a$ of the Stokes filtration of ${}^{\mathsf{L}}\mathcal M$ at infinity.

1. Introduction 1.1.Let X be a smooth complex curve and F a perverse sheaf on X. Recall that, near a singularity a ∈ X, F admits a quiver description in terms of its nearby and vanishing cycles Ψ a (F ) and Φ a (F ).Let S a X and S * a X be the circles of tangent and cotangent directions at a, respectively.Using the canonical and variation maps Ψ a (F ) c / / Φ a (F ), one may upgrade the vector spaces Ψ a (F ) and Φ a (F ) to local systems on S a X and S * a X, with monodromies 1 − vc and 1 − cv, respectively.Then, one has where ν sph {a} and µ sph {a} denote the traces on S a X and S * a X of Sato's specialization ν {a} and microlocalization µ {a} functors, respectively.1.2.Let X rb a be the real oriented blow-up of X with center a, and consider the natural embeddings (1.2) Recall that one has ν sph {a} (F ) ≃ i −1 Rj * j −1 a F, where i −1 , Rj * and j −1 a denote the external operations for sheaves.
1.3.Let M be a (not necessarily regular) holonomic D X -module, let a ∈ X be one of its singularities, and let F := DR(M) be its de Rham complex, which is a perverse sheaf.If M is regular, the classical Riemann-Hilbert correspondence implies that M can be reconstructed near a from the quiver description of F .If M is irregular, a result of Deligne and Malgrange (see [7]) implies that M can be reconstructed near a by further considering the so-called Stokes filtration 1 Ψ • a (F, M) of Ψ a (F ), indexed by Puiseux germs, defined as follows.Let (a, θ, f ) be a Puiseux germ, that is, a holomorphic function f on a small sector around θ ∈ S a X, which admits a Puiseux series expansion at a.For (a, θ, g) another germ, the order relation g θ f means that Re(g −f ) is bounded from above on a small sector around θ.Then, an element u of the stalk Ψ f a (F, M) θ is a section of the de Rham complex of M in a sectorial neighborhood of θ such that e −f u has tempered growth at a.It turns out that the graded component Ψ f a (F, M) is a locally constant sheaf on S a X.
1.4.In [2] we established an extension of the classical Riemann-Hilbert correspondence to the irregular case, in the framework of enhanced indsheaves, which has the advantage of working in any dimension.More precisely, there is a quasi-commutative diagram Mod hol (D X ) / / E-Perv(I C X ). ( Here, ι embeds regular holonomic D-modules into holonomic D-modules which are not necessarily regular, e ι embeds perverse sheaves into enhanced ones2 (see Definition 3.4), and DR E is an enhancement 3 of the de Rham functor DR.
1.5.In [4, §6.2] we described the Stokes filtration Ψ • a (F, M) in terms of the enhanced de Rham complex DR E (M).Here, using enhanced specialization and microlocalization from [5], and making a more explicit use of the sheafification functor discussed in [6], we propose a description of the Stokes filtration which sheds some light on the geometry underlying these constructions.We also discuss a tempered version of the vanishing cycles Φ a (F ) as follows.
1.6.Let k be a field, and consider the natural embedding e ι : D b (k X ) , of sheaves into ind-sheaves into enhanced indsheaves.Recall that e ι has a left quasi-inverse sh called sheafification functor.We say that an enhanced ind-sheaf K is of sheaf type if it lies in the essential image of e ι. 1.7.Let K ∈ E-Perv(I k X ), and let a ∈ X be a singularity of K.The nearby and vanishing cycles of K are defined as follows.Consider the bordered analogue of (1.2) where (X \ {a}) ∞ denotes the bordered space (X \ {a}, X).We set Here, Ei −1 , Ej * and Ej −1 a denote the external operations for enhanced ind-sheaves, and Eν {a} is the natural enhancement of Sato's specialization.Further, for (a, θ, f ) a Puiseux germ, locally at θ set , where K(f ) is the twist of K by an enhanced ind-sheaf which encodes the exponential growth e f (see Definition 4.1).Finally, set , where Eµ {a} is the natural enhancement of Sato's microlocalization.
As it turns out, Eν {a} (K) and Eµ {a} (K) are of sheaf type.
1.8.If k = C, and K = DR E (M) is the enhanced de Rham complex of a holonomic D X -module M then, setting F := DR(M) ≃ sh(K), one has by definition Moreover, one has is the formal D-module at a associated with M, ⊗ D is the inner product for D X -modules, and L f is a regular holonomic D Xmodule.
Set K = DR E (M) and L f = DR(L f ).Then, we have We give an application of the above constructions to the study of the Fourier-Laplace transform in dimension one.
Let V be a one-dimensional complex vector space, and V * the dual vector space.Set V ∞ := (V, P), where P = V ∪ {∞} is the projective compactification, and similarly define V * ∞ and P * .Let K ∈ E-Perv(I k V∞ ), and set L K := L K [1].(Here, the shift ensures compatibility with the Riemann-Hilbert correspondence.)Assume that L K is an enhanced perverse ind-sheaf on V * ∞ .For a ∈ P, let (a, θ, f ) be a Puiseux germ on P such that f is unbounded and not linear (modulo bounded functions).Then its Legendre transform (b, η, g) is a Puiseux germ on P * of the same kind.The stationary phase formula states that there is an isomorphism This is a classical result for holonomic D-modules, and we gave a proof for enhanced ind-sheaves in [4].
1.11.Here we consider the case of linear Puiseux germs, excluded from (1.4), which goes as follows.For a ∈ V, denote by ℓ a the corresponding linear function on V * .Consider the natural identifications S ∞ P * ≃ S * a V and S * b V * ≃ S ∞ P, for b ∈ V * .Then, there are isomorphisms where r is the antipodal map.
Our proof of (1.5) proceeds as follows.The second isomorphism is obtained from the first one by interchanging V and V * , and replacing K by L K.After translation from a to 0, the first isomorphism reads By definition, this is implied by the isomorphism We prove the above isomorphism using the so-called smash functor of [1, §6], in its enhanced version from [5, §6].
1.12.Concerning related literature, the D-module counterpart of (1.5) is proved in [12] when k = C and K = DR E (M) is the enhanced de Rham complex of a holonomic algebraic D V -module M which is regular at finite distance, and has only linear exponential factors at infinity.Note that, in this case, L M satisfies the same conditions.In the framework of enhanced ind-sheaves, a proof of (1.5) is given in [1], in the case where K = eι(F ), for F a perverse sheaf4 on V ∞ .
See [14] for a recent thorough treatment of the Fourier-Laplace transform of holonomic algebraic D-modules on the affine line.
1.13.The contents of this paper are as follows.
After recalling some notations in Section 2, we recall in Section 3 the notion of perverse enhanced ind-sheaf on a complex analytic curve.For such a perverse object, we show that its specialization and microlocalization are perverse sheaves in the classical sense.
In Section 4 we discuss nearby and vanishing cycles along the lines presented in §1.7 above.
In Section 5 we apply our constructions to the Fourier-Laplace transform in dimension one.In particular, we give a proof of (1.5).
Finally, we present in the Appendix an alternative description of vanishing cycles in terms of blow-up transforms.In this setting, both nearby and vanishing cycles are realized on the circle of normal directions S a X.

Review on enhanced ind-sheaves
We recall here some notions and results, mainly to fix notations, referring to the literature for details.In particular, we refer to [9] for sheaves, to [15] (see also [8,3]) for enhanced sheaves, to [10] for ind-sheaves, to [2] (see also [11,3,6]) for bordered spaces and enhanced ind-sheaves, and to [5] for enhanced specialization and microlocalization.
In this paper, k denotes a base field.
2.1.Ind-sheaves and bordered spaces.A good space is a topological space which is Hausdorff, locally compact, countable at infinity, and with finite soft dimension.Let M be a good space.
Denote by Mod(k M ) the category of sheaves of k-vector spaces on M, and by D b (k M ) its bounded derived category.For f : M − → N a morphism of good spaces, denote by ⊗, f −1 , Rf ! and RHom , Rf * , f ! the six operations.Denote by D M the Verdier dual.
For S ⊂ M locally closed, we denote by k S the extension by zero to M of the constant sheaf on S with stalk k.
A bordered space is a pair M = (M, C) with M an open subset of a good space C. We set be the full subcategory of sheaves on • M whose support is relatively compact in M. We denote by Mod(I k M ) the category of ind-sheaves on M, that is the category of ind-objects with values in Mod c (k M ).We denote by D b (I k M ) the bounded derived category of ind-sheaves of k-vector spaces on M, and by ⊗, the natural embedding, by α M its left adjoint, and we set RHom : M ), we often write simply F instead of ι M F in order to make notations less heavy.
Recall that ι commutes with R (2.1) 2.2.Enhanced ind-sheaves.Denote by t ∈ R the coordinate on the affine line, consider the two-point compactification R := R ∪ {−∞, +∞}, and set R ∞ := (R, R).For M a bordered space, consider the projection Denote by the bounded derived category of enhanced ind-sheaves of k-vector spaces on M. Denote by Recall that e commutes with Rf !! , f −1 and f !, but it does not commute in general with Rf * .
The functor e M has as a left quasi-inverse the sheafification functor We call sh M (K) the sheaf associated with K. We say that writing for short {t M such that the property holds on the associated bordered space 3. Specialization and microlocalization.Let N be a smooth manifold, V − → N an R-vector bundle, and SV its fiberwise sphere compactification given by SV : The enhanced Fourier-Sato transforms , are the integral transforms with kernel, respectively, whose objects are conic for the natural action of the group object (R × >0 ) ∞ .Recall that L ( * ) and ( * ) ∧ agree on conic objects.
Let M be a smooth manifold, N ⊂ M a submanifold, and denote by the normal and conormal bundles.Consider the normal deformation p nd : M nd N − → M with center N, and the associated commutative diagram of bordered spaces S h h P P P P P P P P P P where (M nd N ) ∞ is the bordered compactification of p nd , and Ω:=s −1 nd (R >0 ).Sato's specialization and microlocalization functors have natural enhancements

Denoting by
• T N M the complement of the zero-section, and setting 6 What we denote here by in the notations of [5] We set We similarly define Eµ sph N .Consider the real oriented blowup p rb : M rb N − → M with center N, and the associated commutative diagram of bordered spaces One has an associated functor (2.4) 2.4.Constructibility.Let M be a subanalytic bordered space.We denote by

Enhanced perverse ind-sheaves on a curve
In this section we let X be a smooth complex curve.
3.1.Normal form.Consider the real oriented blow-up X rb a of X with center a ∈ X as in (2.3), and the associated natural morphisms where we write for short i = i rb , j = j rb , and The sheaf P SaX of Puiseux germs on S a X is the subsheaf of i −1 j * j −1 a O X whose stalk at θ ∈ S a M are holomorphic functions on small sectors V • ∋ θ admitting a Puiseux expansion at a. We denote by P SaX the quotient of P SaX modulo bounded functions, and we denote by [f ] ∈ P SaX the equivalence class of f ∈ P SaX .
For f = 0, we set ord a (f ) = −n 0 /m if f has a Puiseux expansion n n 0 c n z n/m a with c n 0 = 0, where n, n 0 ∈ Z, m ∈ Z >0 , and z a is a local coordinate at a with z a (a) = 0. We set ord a (0) = −∞.Note that f is bounded if and only if ord a (f ) 0.
for some 2).)One says that K has normal form at I ⊂ S a X if it has normal form at any θ ∈ I.One says that K has normal form at a if it has normal form at S a X.
If K has normal form at a connected open subset I ⊂ S a X, and f ∈ P SaX (I), the number Proof.The statement is a local problem on I.
Let θ ∈ I. Since K has normal form at θ, there is an open neighborhood I θ ∋ θ such that (3.2) holds with V θ • ⊃ I θ .Thus, we can reduce to the case The statement is clear if ord a (h) 0. If ord a (h) > 0, after a change of variable and a ramification we can assume that h(z) = z −1 a .Hence, one concludes using Lemma 3.3.

and consider the embeddings
) is perverse if and only if there exists a discrete subset Σ ⊂ X such that: (a) the full triangulated subcategory of perverse sheaves.
X \ Σ of finite rank, (b) for any n ∈ Z, H n (K) has normal form at any a ∈ Σ.

Denote by E
, and the functor sh Proof.The first statement is clear from the definitions.The second statement follows using Lemma 3.3.
Note that E-Perv(I k X ) is an abelian subcategory of the quasi-abelian heart 1/2 E 0 R-c (I k X ) for the middle perversity t-structure introduced in [3].Note also that, using [4, Proposition 4.1.2](see also [13,Proposition 3.28]), one has Theorem 3.6.The enhanced de Rham functor induces an equivalence between E-Perv(I C X ) and the category of holonomic D X -modules.Proposition 3.7.Let K ∈ E-Perv(I k X ) and a ∈ X a singularity of K. Then both Eν {a} (K) and Eµ {a} (K) [1] are of sheaf type.Moreover they, as well as their associated sheaves, are perverse with the zero-section as their only singularity.
Proof.Consider the morphisms and consider the distinguished triangle (i) Let us show that Eν {a} K is of sheaf type.By Proposition 3.2, there exists a local system L ∈ Mod(k SaX ) such that Eν sph {a} (K) ≃ eι(L [1]).Since u is semiproper, one has by (2.1) ). Hence Eu !! Eu −1 Eν {a} K is of sheaf type.Since any R-constructible enhanced ind-sheaf on a point is of sheaf type, Eo −1 Eν {a} K is of sheaf type.This implies that Eν {a} K is of sheaf type by the distinguished triangle (3.6).
(ii) Let us show that Eν {a} (K) ∈ E b R-c (I k TaX ) is perverse, with {a} as its only singularity.Since ), with L as in (i).
(iii) It remains to show that Eµ {a} (K) [1] is of sheaf type, and that its associated sheaf is perverse.Setting F := sh Eν {a} (K) , this follows from and the fact that the classical Fourier-Sato transform for sheaves preserves the perversity of R × >0 -conic objects, up to shift [1].

Nearby and vanishing cycles
As we mentioned in the Introduction, nearby cycles for enhanced indsheaves were already discussed in [4, §6.2].However, defining them through enhanced specialization, as we do here, sheds some light on the underlying geometry.Moreover, using enhanced microlocalization, we can here also deal with vanishing cycles.In this section we thus recall and complement some results from loc. cit.4.1.Definitions.Let X be a smooth complex curve, and a ∈ X.Consider the natural morphisms associated with the real blow-up X rb a of X with center a as in (3.1): where ( * ) follows from [6, Lemma 3.9].
(ii) Let I ⊂ S a X be an open subset and f ∈ P SaX (I).For U ⊂ X\{a} an open subset such that U • ⊃ I and f extends on U, set Then there are natural morphisms in Let θ ∈ S a X, f ∈ P SaX,θ , and denote by z a a local coordinate at a with z a (a) = 0.  Proposition 4.4.Let K ∈ E-Perv(I k X ).Then (i) Ψ a (K) is a local system on S a X with Stokes filtration Ψ • a (K), and associated graded components Ψ • a (K) which is a local system on S a X; (ii) Φ 0 a (K) is a local system on S * a X.Proof.(i) is a particular case of Lemma 4.3, and (ii) follows from Proposition 3.7.
Refer to Appendix A for an alternative description of the vanishing cycles Φ 0 a (K) as a local system on S a X, via some blow-up transforms.

Fourier transform on the affine line
Let K be an enhanced perverse ind-sheaf on the affine line, and assume that so is its shifted enhanced Fourier-Sato transform L K := L K [1].The stationary phase formula provides a relation (see (1.4)) between the graded components of the Stokes filtrations of K and L K, for degrees which are not linear (modulo bounded function).We discuss here the case of linear degrees.5.1.Linear exponential factors.Let z be a coordinate on the complex line V, and w the dual coordinate on V * , so that the pairing V × V * − → C is given by (z, w) → zw.The underlying real vector spaces are in duality by the pairing z, w = Re(zw).Denoting by P = V ∪ {∞} the complex projective line with affine chart V, one has whose objects are of the form Ej −1 K for some K ∈ E-Perv(I k P ).Here, j : V ∞ ֒→ P is the natural morphism.
Consider the enhanced Fourier-Sato transform Here, the shift ensures compatibility with the Riemann-Hilbert correspondence.

Then:
(i) for any a ∈ V, under the canonical identification S ∞ P * ≃ S * a V, there is an isomorphism of local systems , where r denotes the antipodal map.
(i) The translation τ a : V − → V, z → z + a, induces an identification S a V ≃ S 0 V.Moreover, one has Hence, we may assume a = 0.It is then enough to check that there is an isomorphism and Eµ sph {0} (K) are of sheaf type, it is equivalent to prove that there is an isomorphism . This follows from Lemma 5.3 and (5.2) below.5.2.Smash functor.We consider here the smash functor of [1], in its enhanced version from [5], and establish a small additional result needed to complete the proof of Theorem 5.1.
The sphere compactification SV := (R u × V) \ {(0, 0)} /R × >0 of V decomposes as SV = V + ⊔ H ⊔ V − , corresponding to u > 0, u = 0 or u < 0. Let us identify V = V + .Note that H is a real hypersurface of SV.One has a natural identification Recall also that, by [5,Proposition 6.6], for (5.2) Then, with the natural identification where we write for short i = i rb , j = j rb and p = p rb .
When L = Ep −1 K, the above distinguished triangle reads By applying Ei !we get (i).
(ii) Consider the distinguished triangle When L = Ep !K, the above distinguished triangle reads One concludes by applying Ei −1 .
The following result is clear from the definitions and [5,Lemma 4.7].
Proof.Since the proofs are similar, we will only discuss (i).(a) We will construct in part (b) below a natural morphism Note that r • ℓ gives the identification S N (T N M) ≃ S N M. Hence, by definition, (A.1) is written as On one hand, there is a chain of morphisms where we set Here, (1) follows by adjunction from the isomorphism E  −1 E p −1 E * ≃ E p −1 , and (2) uses the fact that Eu −1 ≃ Eu ! .Hence, there is a morphism On the other hand, there is a chain of isomorphisms Here, (3) easily follows using the identification (M ×R) rb N ×R ≃ (M rb N ) ×R, and (4) uses the fact that r and r are proper.
A.2.The case of vector bundles.Let τ : V − → N be a vector bundle, and o : N − → V its zero section.Let , and consider the quotient γ : Proof.Since the proofs are similar, let us only discuss the first isomorphism.
Consider the morphisms where i 0 (x) = (x, 0).One has where ( * ) is due to the fact that γ is an (R × >0 ) ∞ -bundle, and ) ∞ -conic with respect to the action on the second factor of . Since (p, q 1 ) decomposes into (p, q 1 ) : A.3.Blow-up and vanishing cycles.Let X be a smooth complex curve, and a ∈ X.Let z be a coordinate on the complex vector line T a X, and w the dual coordinate on T * a X, so that the pairing T a X × T * a X − → C is given by (z, w) → zw.Then, the isomorphism c : ( a X) ∞ , z → −z −1 does not depend on the choice of the coordinate, and induces a homeomorphism c : S a X ∼ −→ S * a X.Lemma A.7.For K ∈ E b + (I k X ), there is a natural morphism Ec −1 Eµ sph {0} (K) [1] − → Eλ rb {0} (K).Proof.Since c is an isomorphism and γ −1 is fully faithful, it is enough to show that there is a natural morphism Then we obtain (A.4) by applying Eu −1 to (A.5).In fact, on one hand, recalling the notations on the enhanced Fourier-Sato transforms from §2.3, one has [1].
On the other hand, one has G∩(V× In particular, Eλ rb {a} (K) ≃ E λ rb {a} (K) is of sheaf type, and its associated sheaf is a local system.
Proof.(i) Let us show that the first isomorphism follows by duality from the second one.
The underlying real vector spaces to V and V * are in duality by the pairing v, w = Re(zw).For Γ ⊂ V * , the set Γ • = {v ∈ V ; v, w 0 for any w ∈ Γ} is called the polar cone of Γ.
bordered spaces, denote by + ⊗, Ef −1 , Ef !! and RIhom + , Ef * , Ef ! the six operations.Recall that the external operations are induced via Q by the corresponding operations for indsheaves with respect to the morphism f R := f × id R∞ .Denote by D E M the Verdier dual.Denote by RHom E the hom functor taking values in D is finite, does not depend on the choice of θ ∈ I, and only depends on the class [f ] of f .If N ([f ]) > 0, one says that [f ] is an exponential factor of K, and N ([f ]) is called its multiplicity.Proposition 3.2.Let K ∈ E b R-c (I k X ) have normal form at I ⊂ S a X.Then Eν rb {a} (K)| I is of sheaf type.More precisely, Eν rb {a} (K)| I ≃ eι(L) for L ∈ Mod(k I ) a local system of rank N ([0]).

Lemma 3 . 5 .
the full triangulated subcategory of C-constructible enhanced ind-sheaves.(ii) We say that K ∈ E b R-c (I k X ) is an enhanced perverse ind-sheaf if there exists a discrete subset Σ ⊂ X such that: (a) H n Ei −1 a K = 0 for any n > 0 and a ∈ Σ; (b) H n Ei ! a K = 0 for any n < 0 and a ∈ Σ; (c) K| X\Σ ≃ eι(L[1]), for L ∈ Mod(k X\Σ ) a local system of finite rank, (d) H −1 (K) has normal form at any a ∈ Σ. Denote by E-Perv(I k X ) ⊂ E b C-c (I k X ) the full subcategory of enhanced perverse ind-sheaves.The functor e ι : D b and is a local system of rank N ([f ]) on I. Recall the notion of a Stokes filtration, e.g. from [4, §6.1].

Remark 5 . 2 .
With notations as in §1.4,for k Appendix A. Vanishing cycles by blow-up transform A.1.Blow-up transforms.Let M be a real analytic manifold, and N ⊂ M a smooth submanifold.As in (2.3) consider the real oriented blowup M rb N of M with center N, and the associated commutative diagram of bordered spaces We denote by λ rb N and λ rb N the analogous functors for sheaves.Note that one has e • λ rb N ≃ Eλ rb N • e, e • λ rb N ≃ E λ rb N • e, and similarly for e replaced by ǫ, ǫ + or ǫ − .Remark A.2.Note that Eλ rb N ≃ E λ rb N in general, as shown by the following example.(See however Proposition A.8.) For M = R x and N = {0}, one has M rb N ≃ {x 0} ⊔ {x 0}.Restricted to the left component, the maps i and p are the embeddings {0} i − → {x 0} p − → R.Then, for F = k {x>0} , one has