Kashiwara conjugation and the enhanced Riemann-Hilbert correspondence

We study some aspects of conjugation and descent in the context of the irregular Riemann-Hilbert correspondence of D'Agnolo-Kashiwara. First, we give a proof of the fact that Kashiwara's conjugation functor for holonomic D-modules is compatible with the enhanced De Rham functor. Afterwards, we work out some complements on Galois descent for enhanced ind-sheaves, slightly generalizing results obtained in previous joint work with Barco, Hien and Sevenheck. Finally, we show how local decompositions of an enhanced ind-sheaf into exponentials descend to lattices over smaller fields. This shows in particular that a structure of the enhanced solutions of a meromorphic connection over a subfield of the complex numbers has implications on its generalized monodromy data (in particular, the Stokes matrices), generalizing and simplifying an argument given in our previous work.


Introduction
In general, a Riemann-Hilbert correspondence is an equivalence of categories between some category of differential systems (such as integrable connections, meromorphic connections, or D-modules) and some category of topological objects (such as local systems, perverse sheaves, Stokes-filtered local systems, or enhanced indsheaves).In the context of differential equations in one or several complex variables (i.e. in the theory of connections or D-modules on a complex algebraic variety or complex manifold), these topological objects are a priori defined over the field of complex numbers (as their field of coefficients).This being said, three questions immediately come to our mind: (1) How does the Riemann-Hilbert functor behave with respect to the natural complex conjugation on the target?(2) How can we detect if the target object is already defined over a subfield of C? (3) What implications does such a structure over a smaller field have?The first question was studied by M. Kashiwara in [Kas86] in the case of regular holonomic D-modules.At the time, it was known that the de Rham functor induces an equivalence between the derived category of regular holonomic D-modules and the derived category of constructible sheaves (see [Kas84]), and in [Kas86] a conjugation functor on the category of D-modules was defined that corresponds to complex conjugation on constructible sheaves via this equivalence.Around 30 years later, in [DK16], A. D'Agnolo and M. Kashiwara generalized Kashiwara's equivalence from [Kas84] to (not necessarily regular) holonomic D-modules.It is the aim of the first part (Section 3) of this article to show that the conjugation functor of [Kas86] is -not surprisingly -still compatible with this new equivalence of categories (Theorem 3.9).The author's research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), Projektnummer 465657531.The author was also supported by the grant G0B3123N from the Fonds voor Wetenschappelijk Onderzoek -Vlaanderen (FWO, Research Foundation -Flanders).
Indeed, many parts of the proof are due to important previous studies of holonomic D-modules and the Hermitian duality functor, notably by C. Sabbah, K. Kedlaya and T. Mochizuki.In the second part (Section 4) of the paper, we give some complements on a study of the second question that has been initiated in [BHHS23].In loc.cit., together with D. Barco, C. Sevenheck and M. Hien, we studied the question of when topological data associated to hypergeometric differential systems are defined over a subfield of C. For this, we used a technique called Galois descent, whose underlying idea is the following: Given a finite Galois extension L/K and an object over L (e.g. a sheaf of L-vector spaces), then in order to find a structure of this object over K (i.e. an object over K that yields our given object after extending scalars), it should be enough to find suitable isomorphisms with all its Galois conjugates.In our joint work, we developed some statements in this direction for sheaves and enhanced ind-sheaves.In particular, we showed that Galois descent is possible for R-constructible enhanced ind-sheaves concentrated in one degree on compact spaces.We will reformulate and slightly generalize this statement here, dropping the compactness assumption (Theorem 4.9).
Finally, in the last part (Section 5), we give a consequence of the existence of Kstructures (for an arbitrary subfield K ⊂ C) in the irregular setting, thus addressing Question (3): We show that a K-structure on an enhanced ind-sheaf associated to a meromorphic connection on a Riemann surface via the irregular Riemann-Hilbert correspondence implies that its Stokes matrices (and indeed all its generalized monodromy data) can be defined over K.In the case of hypergeometric systems, this was already done in [BHHS23,§5].Our proof here is, however, valid in general and simplifies significantly the quite involved and slightly artificial argument given in our previous work.To this purpose, we study the topological counterpart of meromorphic connections on complex curves (already introduced in [DK18]), which we call enhanced ind-sheaves of meromorphic normal form, and prove that this property descends to a K-lattice (Theorem 5.8).Our argument makes use of the theory of subanalytic sheaves.
Acknowledgements.I would like to thank Andrea D'Agnolo and Claude Sabbah for several helpful discussions and hints, in particular concerning the part on Hermitian duality.I also thank Pierre Schapira for some useful conversations.I am grateful to Philip Boalch for his explanations that brought new insights into the Stokes phenomenon to me.I want to thank Davide Barco, Marco Hien and Christian Sevenheck who, in many discussions related to our common work [BHHS23], inspired me to develop further results in this direction.I am also grateful to Takuro Mochizuki for answering my questions during the preparation of [BHHS23], the answers to which have inspired this work in various places.Finally, I thank the Simons Center for Geometry and Physics, Stony Brook, for hosting me in the course of the program "The Stokes phenomenon and its Applications in Mathematics and Physics" in May and June 2023, during which parts of the research for this work have been performed.

Background and notation
In this work, we want to study objects related to Riemann-Hilbert correspondences for holonomic D-modules, and we will mainly use the approach via (enhanced) indsheaves here, great parts of which have been developed in [KS01] and [DK16].We will therefore need some analytic and topological notions, which we briefly recall in the following, referring to the existing literature for further details.
2.1.Topological spaces and manifolds.Even if not strictly necessary in all places, we will assume all our topological spaces to be good, i.e.Hausdorff, locally compact, second countable and of finite flabby dimension.This is especially important when it comes to the construction of proper direct images and exceptional inverse images, and since we are mainly interested in all these objects in the context of the Riemann-Hilbert correspondence on complex manifolds, goodness is not a restriction.
For a complex manifold X with structure sheaf O X , we can consider its complex conjugate manifold X, having the same underlying topological space, but the structure sheaf O X is the sheaf of antiholomorphic functions (with respect to the original complex structure).Also, X has an underlying real analytic manifold, which is the same as that of X.It is often denoted by X R to emphasize the real structure, but we will mostly just denote it by X if it is clear from the context.Note also that a morphism f : X → Y of complex manifolds induces a morphism of complex manifolds f : X → Y (which is the same as a morphism of the underlying topological spaces).
2.2.Holonomic D-modules.We briefly recall some notions and notation from the theory of D-modules, and we refer to standard references such as [Bjö93], [HTT08] and [Kas03] for details.We mostly use the notation from the last of these three references.
Let X be a complex manifold with structure sheaf O X .We denote by D X the sheaf of (non-commutative) rings of linear partial differential operators with coefficients in O X on X.The category of (left) D X -modules is denoted by Mod(D X ), and its bounded derived category by D b (D X ).We denote by Mod coh (D X ) the full subcategory of coherent D X -modules, by Mod hol (D X ) the full subcategory of holonomic D X -modules, and by Mod rh (D X ) the full subcategory of regular holonomic D X -modules.We moreover denote by D We write D X for the duality functor for D X -modules, and if f : An important basic example of a meromorphic connection which is not regular holonomic is the following: Let X be a complex manifold, D ⊂ X a normal crossing divisor and ϕ ∈ Γ(X; O X ( * D)), then the exponential D X -module E ϕ is defined by E ϕ := (D X /ann(ϕ))( * D), where ann(ϕ) is the (left) ideal of D X of operators annihilating ϕ.
The classification of holonomic D-modules -in particular in the case of irregular singularities in higher dimensions -has been a difficult problem.In the one-dimensional case, it turned out that the Stokes phenomenon is the key ingredient to achieve a complete description of connections with irregular singular points.Very roughly, the Stokes phenomenon in this context is the observation that a meromorphic connection decomposes as a direct sum of certain "elementary" connections (namely, exponential D-modules) in sufficiently small sectors with vertex at the singular point, but this decomposition might not exist globally in an open neighbourhood of the singularity.It was conjectured that a similar statement holds for meromorphic connections in higher dimensions, i.e. that such local decompositions still exist (as in Definition 2.1 below), at least after suitable blow-ups of the singular locus and de-ramification.A proof was given under certain assumptions in the two-dimensional case in [Sab00], and finally for the general case by K. Kedlaya [Ked10,Ked11] and T. Mochizuki [Moc09,Moc11a] (see also [Maj84] for the asymptotic analysis).
This classification leads to a useful technique for proving statements about holonomic D-modules, and we will briefly recall it here for later use.If X is a complex manifold and D ⊂ X a normal crossing divisor, we denote by X the real oriented blow-up of X along the components of D, and we denote by A mod X the sheaf of functions which are holomorphic on X \ ∂ X having moderate growth along ∂ X (see e.g.[DK16, Notation 7.2.1] for a more precise definition, where this sheaf is denoted by A X , or also [Moc14, §4.1.2]).For a D X -module M, we write We give the following definition of a good normal form, combining elements from [DK16, Definition 7.3.3]and [Sab00, §2.1] (cf.also [Ito20]).
• for any p ∈ D, there exists an open neighbourhood U ⊂ X and a finite good set of functions has an open neighbourhood V ⊂ ̟ −1 (U ) ⊂ X on which there exists an isomorphism Here, the set {ϕ i | i ∈ I} is called good if the divisors associated to the functions ϕ i and ϕ i − ϕ j that are not elements of Γ(U ; O X ), for i, j ∈ I, i = j, are supported on D and non-positive.
Using this notion of good normal form, one can deduce from the above-mentioned classification of holonomic D-modules the following fundamental lemma.(This is a slight variation of [DK16, Lemma 7.3.7]).
Lemma 2.2 (cf.[DK16, Lemma 7.3.7]).Let P X (M) be a statement concerning a complex manifold X and an object M ∈ D b hol (D X ).Then, P X (M) is true for any complex manifold X and any M ∈ D b hol (D X ) if all of the following conditions are satisfied: (a) hol (D X ) and both P X (M ′ ) and P X (M ′′ ) are true, then P X (M) is true.
) has a good normal form along a normal crossing divisor D ⊂ X, then P X (M) is true.
2.3.Sheaves.Let k be a field and let X be a topological space.We denote by Mod(k X ) the category of sheaves of k-vector spaces on X and by D b (k X ) its bounded derived category.For a k-vector space V , denote by V X the constant sheaf on X with stalk V , and an object F ∈ Mod(k X ) is called a local system (of k-vector spaces) if it is locally isomorphic to a constant sheaf.If X is a real analytic (resp.complex) manifold, we have the full subcategory The six Grothendieck operations for sheaves are denoted by RHom, ⊗, Rf * , f −1 , Rf ! and f ! for f : X → Y a morphism.
For details on the theory of sheaves of vector spaces and constructibility, we refer to the standard literature, such as [KS90] or [Dim04].
2.4.Ind-sheaves and subanalytic sheaves.Let X be a good topological space.In [KS01], M. Kashiwara and P. Schapira introduced and studied the category I(k X ) of ind-sheaves on X as a generalization of the category Mod(k X ), and its bounded derived category D b (Ik X ).There is a fully faithful and exact embedding ι : Mod(k X ) ֒→ I(k X ).Since this embedding does not commute with inductive limits in general but the functor ι is sometimes suppressed in the notation, one uses the notation "lim − → " (instead of just lim − → ) for the inductive limit in the category of ind-sheaves.The functor ι has an exact left adjoint α : I(k X ) → Mod(k X ), and this functor admits a fully faithful and exact left adjoint giving another embedding β : Mod(k X ) ֒→ I(k X ).
It was also shown that if X is a real analytic manifold, the category of "subanalytic ind-sheaves" (called "ind-R-constructible ind-sheaves" in [KS01]) is equivalent to the category of sheaves on the (relatively compact) subanalytic site X c sa (see [KS01, Theorem 6.3.5] and cf.[Pre08] for a more detailed study of subanalytic sheaves).
Let us briefly recall this version of the subanalytic site: Open sets of X c sa are relatively compact open subanalytic subsets of X.A covering of an open subset U in X c sa is a covering U = i∈I U i by open sets U i in X c sa admitting a finite subcover.For a more detailed study of subanalytic sets, we refer to [BM88] and the references therein.We denote the subcategory of I(k X ) consisting of subanalytic ind-sheaves by I suban (k X ).
In some regards, subanalytic sheaves behave differently from sheaves on a usual topology.For example, if we are given a filtrant inductive system F j ∈ I suban (k X ), j ∈ J, and U ∈ X c sa , we have The following lemma is also easily proved using the finiteness of the gluing procedure for subanalytic sheaves (see [KS01, Proposition 6.4.1]).
Lemma 2.3.Let F ∈ I suban (k X ) and let A be a k-algebra.Then for U ⊆ X a relatively compact subanalytic open subset, we have Γ(U ; Some important examples of ind-sheaves are the following: On a real analytic manifold X, one has the ind-sheaf C ∞,t X of tempered complex-valued smooth functions (i.e.smooth functions with moderate growth near the boundary of their domain) and the ind-sheaf Db t X of tempered complex-valued distributions (i.e.distributions extending to the whole space).If X is a complex manifold, it is in particular a real analytic manifold, and these two ind-sheaves are modules over βD X and βD X .Their Dolbeault complexes are isomorphic and one defines them to be the complex of tempered holomorphic functions (1) 2.5.Enhanced ind-sheaves.In [DK16] and [DK19], A. D'Agnolo and M. Kashiwara extended the theory further, introducing and studying the category of enhanced ind-sheaves on a so-called bordered space.We recall very few basics here and refer to loc.cit.for more details (see also [KS16] for an exposition).
A bordered space X = (X, X) is a pair of good topological spaces such that X ⊆ X is an open subspace.In particular, every good topological space X can be considered a bordered space (X, X).A morphism f : (X, X) → (Y, Y ) is a continuous map X → Y such that the morphism Γ f → X induced by the projection to the first factor is proper.Here, We say that X is a real analytic bordered space if X is a real analytic manifold and X ⊆ X is a subanalytic open subset.Morphisms of real analytic bordered spaces are additionally required to be subanalytic in the sense that Γ f is a subanalytic subset of X × Y .(Everything we do for real analytic manifolds or bordered spaces could also be done slightly more generally for subanalytic spaces or bordered spaces, see e.g.[KS90, Exercise 9.2] and [DK19, §3.1] for these notions.) For the rest of this subsection, let X = (X, X) be a real analytic bordered space.
The category E b (Ik X ) of enhanced ind-sheaves on X = (X, X) is a quotient category of D b (Ik X×R ), where R = R ⊔ {±∞}.To be more precise, it is constructed in two steps: First, for any bordered space Y = (Y, Y ), one sets This type of category comes equipped with six Grothendieck operations.One then defines The category E b (Ik X ) has six Grothendieck operations denoted by RIhom + , + ⊗, Ef * , Ef −1 , Ef !! , and Ef !(for morphisms f of bordered spaces).One also has a sheaf-valued hom functor and for any F ∈ D b (k X ) a tensor product functor Here, π : X × R → X is the projection (and π −1 F is to be seen as extended by zero to X × R).
The natural t-structure on the derived category D b (Ik X×R ) induces a t-structure on E b (Ik X ), whose heart is denoted by E 0 (Ik X ) (these are therefore the objects represented by complexes of ind-sheaves concentrated in degree 0).Moreover, a notion of R-constructibility is defined for enhanced ind-sheaves, leading to the full subcategory where {t ≥ a} One has an embedding of the category of sheaves on X into the category of enhanced ind-sheaves on X One obtains a natural duality functor for enhanced ind-sheaves, which is given as A fundamental class of objects is that of so-called exponential enhanced ind-sheaves: , or if there is no confusion about the field of coefficients, we often omit the subscript k and just write E f W |X for this object.
Remark 2.4.For simplicity, we assume X = X.The category E b (Ik X ) can be viewed (via the left adjoints of the quotient functors) as a full subcategory of D b (Ik X×R ).In particular, the image of E 0 R-c (Ik X ) lies in I suban (k X×R ).Hence, the object E f W |X,k can be understood as a sheaf on the subanalytic site is described as follows (we do not write the index k here for better readability): where π 0 (U ∩ {t ≥ −f + a}) is the set of connected components of the intersection U ∩ {t ≥ −f + a} (note that the number of connected components is finite since this intersection is subanalytic and relatively compact).
In particular, we have are given in the obvious way, by a combination of identities, zero maps or diagonal maps k → k M (for M a set).
The sheaves E f W |X are therefore very similar to sheaves of the form k Z on the usual topology, i.e. constant sheaves on a closed subset Z ⊆ X, extended by zero outside Z.For E f W |X , the set Z is to be thought of as a half-space lying over the graph of some real-valued function and considered to be shifted towards "infinitely high real values".
A typical example is the case where X is a disc around a point p in the complex plane, W = X \ {p} or W = S an open sector with vertex p, and f = Re ϕ is the real part of a holomorphic function ϕ : W → C with a pole at p.
We recall the following definition (note that the conventions are different between [Moc22] and [DK18], and we use the latter one).
Definition 2.5 (cf.e.g.[DK18, Notation 3.2.1]).Let X be a bordered space.Let W ⊆ X be an open subset and let f, g : W → R be continuous functions.Then we write i.e. f − g bounded above on each V but unbounded below on some V .
Here, V ⊂⊂ X means "V is relatively compact in X".
If X is a complex manifold and ϕ, ψ : W → C are holomorphic, we will write for short ϕ W ψ instead of Re ϕ W Re ψ (and similarly for ∼ W and ≺ W ).
Lemma 2.6.Let X be a bordered space, let W ⊆ X be open and relatively compact in X, and let f 1 , . . ., f n , g 1 , . . ., g m : W → R be continuous subanalytic functions.Then Morphisms between direct sums of exponentials can therefore -after numbering the direct summands -be represented by matrices (and composition corresponds to matrix multiplication).In particular, if W |X are represented by upper-triangular square matrices.
2.6.Riemann-Hilbert correspondences for analytic D-modules.It is a classical idea to ask if the functor associating to a differential system its solution space is an equivalence.(The question about surjectivity of such a functor for Fuchsian differential equations goes back at least to Hilbert's 21st problem.)Let X be a complex manifold of (complex) dimension d X .Classically, in the theory of D-modules, one studies the following objects: Let M ∈ D b hol (D X ).Denote by Ω X the sheaf of top-degree holomorphic differential forms on X.Then one defines the holomorphic de Rham complex of M by In [Kas84], M. Kashiwara proved that the de Rham functor gives an equivalence It was then natural to search for a generalization of this result to holonomic Dmodules (relaxing the regularity assumption).It was clear that the functor DR X is no longer fully faithful on the category D b hol (D X ), and hence the framework (the functor and the target category) would need to be modified.
Finally, A. D'Agnolo and M. Kashiwara (see [DK16]) were able to establish a fully faithful functor, the enhanced de Rham functor Let us briefly give some details on its construction: Starting from the ind-sheaf of tempered holomorphic functions . This is done by first defining enhanced tempered distributions Db E X and setting O E X := Ω X ⊗ D X Db E X , in analogy to (1).(Recall that X denotes the complex conjugate manifold.)They X and define the functors and . These two functors are related by duality (recall the duality functors for D-modules and enhanced ind-sheaves from Sections 2.2 and 2.5, respectively): For exponential D-modules, the enhanced solution and de Rham functors are explicitly described as follows: Consider a normal crossing divisor D ⊂ X and a function ϕ ∈ Γ(X; O X ( * D)), then (see [DK16, Corollary 9.4.12 and Lemma 9.3.1]).
Remark 2.7.The framework and notation are obviously set up to be similar to the regular case, and we refer to [DK16] and [KS16] for more details.Let us, however, remark that the notation is not self-explanatory here, and quite some simplification in notation has happened between [KS01] and works like [DK16] and [KS16].In particular, whenever a sheaf (as opposed to an enhanced ind-sheaf) appears in the above formulae, it should be read as βπ −1 of this sheaf to make sense of these expressions.For example, in the notation of [KS01], the definition of the enhanced solutions functor reads as 2.7.Galois conjugation.Let L/K be a field extension and let g ∈ Aut(L/K), i.e. g is a field automorphism of L with g| K = id K .Then, for an L-vector space V we can define its g-conjugate L-vector space V g , which has the same underlying abelian group, but the action of L is given by ℓ • v := g(ℓ)v for ℓ ∈ L, v ∈ V .This defines a functor from the category of vector spaces to itself (which sends a morphism to the same set-theoretic map).This conjugation naturally induces a g-conjugation on sheaves of L-vector spaces, as well as on ind-sheaves over L and enhanced ind-sheaves over L, i.e. we have functors for a topological space X (resp.a bordered space X ), and these functors also preserve R-and C-constructibility.
In the case L = C, K = R, there is only one nontrivial element γ ∈ Aut(C/R), which is complex conjugation, and we therefore simply write (•) instead of (•) γ .
If X is a complex manifold and X its complex conjugate, their underlying topological spaces are the same and hence the categories D b (k X ) and D b (k X ) (for a field k) are naturally identified, and similarly for E b (Ik X ) and the associated subcategories of R-and C-constructible objects.We can identify O X = O X and D X = D X , and for a D X -module M, the conjugate M is naturally a D X -module.In this situation, it is not difficult to see that

Conjugation of D-modules and the enhanced de Rham functor
In this section, our aim is to prove an analogue of a result of [Kas86] in the context of the enhanced de Rham functor.Let us briefly recall the idea and argument of the main statement in [Kas86].
The Riemann-Hilbert correspondence for regular holonomic D-modules states that the de Rham functor induces an equivalence of categories On the right-hand side, complex conjugation defines an auto-equivalence.It is therefore natural to ask what the corresponding operation on the left-hand side is.In other words, the question is the following: Given an object is enough here).The key to this result is an intermediate step, namely the definition and study of a functor later baptized the Hermitian duality functor, where Db X denotes the sheaf of distributions.
One of the key observations in the proof of formula (2) is the fact that one has an isomorphism of functors 1 (note that , and this follows directly from the fact that the Dolbeault complex with distribution coefficients is a resolution of the sheaf of holomorphic functions, i.e.DR X (Db In [Kas86], it was in particular shown that C X (and hence c) preserves regular holonomicity and induces an equivalence of categories if we restrict to regular holonomic D-modules.Moreover, it was a conjecture of M. Kashiwara (see [Kas86, Remark 3.5]) that this is also true if we erase the word "regular" and just restrict to holonomic Dmodules.This conjecture has been proved by C. Sabbah  The idea in this section is now to establish a statement similar to (2) for the enhanced de Rham functor.More precisely, the idea is that the same relation holds if we replace DR X by DR E X and consider M ∈ D b hol (D X ).The functor c will remain unchanged.Due to the results in the holonomic context mentioned above, such a compatibility might not be surprising.On the other hand, the argument for a statement like (3) for the enhanced de Rham and solution functors seems not as direct as in the classical case: The expression that appears is DR E X (Db X ), while the definition of enhanced tempered holomorphic functions is rather "DR We therefore apply a different method of proof here, using the classification of general holonomic D-modules due to C. Sabbah, K. Kedlaya and T. Mochizuki (see Section 2.2), to establish the generalization of (3).This is mainly achieved by combining results about Hermitian duality that have already been proved earlier and that we will briefly review in the next subsection.
3.1.The Hermitian dual.Let X be a complex manifold.In [Kas86], M. Kashiwara introduced the functor Here, Db X is the sheaf of Schwartz distributions on the underlying real manifold of X.It is a module over D X and D X .
This functor has been further studied by other authors, in particular by C. Sabbah and T. Mochizuki, and we recall two important properties that we will use below.
Lemma 3.1 (see [Moc15, §12.5.2]).Let f : X → Y be a morphism of complex manifolds and let M be a holonomic D X -module such that f is proper on supp M. Then there is a natural isomorphism For the next lemma, recall the notations M( * D) and M(!D) from Section 2.2.Moreover, note that if X is a complex manifold and D ⊂ X is a hypersurface, this also defines a hypersurface D ⊂ X in the complex conjugate manifold.Lemma 3.2.Let M be a holonomic D X -module and D ⊂ X a hypersurface.Then there is an isomorphism 1 Comparing this directly with what is written in [Kas86], our formula here differs by a shift.This is because we use a different convention for the de Rham functor (which actually seems to be more common nowadays and is consistent with later works like [DK16] and [KS16], for example).Recall the notation on the real oriented blow-up from Section 2.2.If X is a complex manifold and D ⊂ X is a normal crossing divisor, we also have the normal crossing divisor D ⊂ X (if D is locally given by {z 1 • . . .• z k = 0}, then D is locally given by {z 1 • . . .• z k = 0}) and one has the real blow-up space ̟ : X → X. (Its underlying topological space is naturally identified with that of X and A mod X = A mod X .We will often just write ̟ instead of ̟.)If M is a D X -module, we write to emphasize that we are taking the tensor product with A mod X , the sheaf of antiholomorphic functions with moderate growth at ∂ X.
Lemma 3.3.Let X be a complex manifold, D ⊂ X a normal crossing divisor and ϕ ∈ Γ(X; O X ( * D)).Then there is an isomorphism Proof.We use the notation in [Sab00] without explaining all the details, in particular the functor ).There are isomorphisms where the second isomorphism follows from Lemma 3.2 and the last isomorphism follows from [Sab00, Proposition 3.2.2].On the other hand, we know from [Sab00, p. 69] that The following result is shown in [Sab00, §II.3] (see also [Sab13,§12.6]).It states that a decomposition as in the definition of good normal form induces a similar decomposition for the Hermitian dual, and describes how its exponential factors change under Hermitian duality.
3.2.Hermitian duality and the enhanced de Rham complex.As mentioned above, the main step in proving the desired generalization of (2) is the proof of an analogue of the isomorphism (3) in the setting of the enhanced de Rham and solution functors.This will be done in Proposition 3.7 below.Let us first show the existence of a canonical morphism.
Lemma 3.5.Let X be a complex manifold and M ∈ D b coh (D X ).There is a canonical morphism, functorial in M, In the proof of this statement, we will use the following lemma.We write P := P 1 (R) and P := P 1 (C) for the real and the complex projective line, respectively.Lemma 3.6.Let k : X × P → X × P be the inclusion.Then there is an isomorphism Proof.First, we note that for a right D P -module N we have an isomorphism Using this (in the second isomorphism below), one obtains Here, the last isomorphism follows from [Kas95, Lemma 2.3.2].
Proof of Lemma 3.5.We have and Therefore, it suffices to find a canonical morphism and , which means that it suffices to find a morphism Let us consider the natural morphisms of bordered spaces and , so it is enough to construct a morphism (for any complex manifold X) X×P .Now we note that we have a morphism Hence, all we finally need to construct is a morphism Using a slightly more precise notation (explicitly writing the functor β and the pullback via the projection p : X × P → X, cf.Remark 2.7), this morphism follows from the natural morphism Here, the first arrow is induced by the composition of the natural morphisms βp −1 Db X → p −1 Db t X → Db t X×P (the second of these arrows being given by taking the product of a distribution on X with the constant distribution 1 on P, see e.g.[Hör98, Chap.5]).The second arrow in (5) is given by multiplication of tempered smooth functions with tempered distributions (see [Kas84, Lemma 3.3] and also [DK16, Proposition 5.1.3]).More precisely, we use the natural morphism composed with this multiplication.
The classification of holonomic D-modules now enables us to prove that the morphism from Lemma 3.5 is an isomorphism for any holonomic D-module M.
Proposition 3.7.The morphism (4) ) is an isomorphism for any complex manifold X and any M ∈ D b hol (D X ).Proof.Let us consider the following statement for any complex manifold X and any M ∈ D b hol (D X ): P X (M) : The morphism (4) is an isomorphism.
To show that it holds for any X and M, we will employ Lemma 2.2, and we have to check conditions (a)-(f).
Condition (a) is clear by the compatibility of DR E X and Sol E X with inverse images.(Note that the a-priori existence and canonicity of the morphism (4) is essential here.)Condition (b) is also obvious.Condition (c) holds by the axioms of a triangulated category, and condition (d) is true since the morphism (4) is functorial with respect to direct sums.
Condition (e) is proved as follows: Assume that P X (M) holds, and let f : X → Y be a projective morphism.It also induces a morphism f : X → Y (which we will not distinguish in the notation).Then we have , where the last isomorphism follows from Lemma 3.1.
It remains to check condition (f): Let M ∈ Mod hol (D X ) have a good normal form along a normal crossing divisor D ⊂ X (in particular, M is meromorphic at D). Without recalling the details (which are not essential for understanding the idea of proof), we use here some notations and results for enhanced de Rham and solution complexes on the real blow-up space (see [DK16, §9.2]).
Due to Lemma 3.8 below, there is a natural morphism similar to (4) on the real blow-up We want to show that this is an isomorphism.Locally on X = X (they are identified as real analytic manifolds with corners), M A decomposes as a direct sum of exponential D-modules (E ϕ ) A , and C X (M) A decomposes accordingly into exponential D-modules (E −ϕ ) A (cf. Proposition 3.4), so it suffices to prove this result for M = E ϕ with ϕ ∈ Γ(X; O X ( * D)).(Recall that we have (C X (E ϕ )) A ≃ (E −ϕ ) A by Lemma 3.3.) Then we get by [DK16, Corollary 9.2.3 and Lemma 9.3.1].On the other hand, we similarly have Since conjugation does not affect the real part of a holomorphic function, we see that both sides of (6) are isomorphic in the case M = E ϕ .Applying now π −1 C X\D ⊗ E̟ * to (6), we obtain (see [DK16, Corollary 9.2.3] and , and noting that M (having good normal form) is meromorphic at D, while C X (M) ≃ C X (M)(!D) by Lemma 3.2, we obtain the desired isomorphism.
It remains to prove the following lemma, which has been used in the above proof.Lemma 3.8.Let X be a complex manifold and D ⊂ X a normal crossing divisor and X, X the associated real blow-up spaces along D. Let M ∈ D b coh (D X ).Then there is a canonical morphism Proof.Again, we use the notation in [Sab00], in particular the sheaf First, note that there is a canonical morphism and the fact that Db mod D X is a module over A mod X .It therefore suffices to find a morphism Similarly to the proof of Lemma 3.5, it is enough to give a morphism By a reduction similar to that in Lemma 3.5, such a morphism is eventually induced by a morphism X×P , which is given by multiplication of tempered smooth functions with tempered distributions.We leave details to the reader.
With this in hand, we can now proceed completely analogously to [Kas86] to define the conjugation functor c, the functor corresponding to complex conjugation on constructible sheaves via the Riemann-Hilbert correspondence.
We define (recall the definition of M from Section 2.7).
Theorem 3.9.Let X be a complex manifold and , which follows from Proposition 3.7.
The second claim follows by applying the duality functor (and a shift by −d X ) since complex conjugation commutes with duality (cf.[BHHS23, Lemma 2.3]).
To end this section, let us come back to the proof of Kashiwara's conjecture [Kas86, Remark 3.5].The following theorem is the first and main part of this conjecture.
The other parts of the original conjecture of M. Kashiwara follow from this theorem (see [Sab00, §3.1]).They are stated in the (first part of the) following corollary, and with the help of our results from this section, we give a different way of deducing it from Theorem 3.10.
Corollary 3.11.The functor ) is an equivalence of categories and is its own inverse.
hol (D X ) follows easily from Theorem 3.10 by induction on the amplitude of a holonomic complex.Then we have M) by Theorem 3.9, and by the Riemann-Hilbert correspondence of [DK16], we get the isomorphisms C X (C X (M)) ≃ M and c(c(M)) ≃ M.

Galois descent for enhanced ind-sheaves
In this section, we give some complements on the study of Galois descent for enhanced ind-sheaves done in [BHHS23].We mostly reformulate what has been established there, slightly generalizing the descent statement by removing the compactness assumption that was present in [BHHS23, Proposition 2.15].We mainly study Rconstructible enhanced ind-sheaves here.
The starting point is the functor of extension of scalars: Let X be a bordered space and let L/K be a field extension, then we have the functor Its compatibilities with direct and inverse images have been described in [BHHS23].We restate them here, removing some restrictions that are not necessary.
Lemma 4.1.Let X and Y be bordered spaces and let f : If X and Y are real analytic bordered spaces, f is a morphism of real analytic bordered spaces and

If moreover f is semi-proper, we have an isomorphism
Remark 4.2.The condition for the isomorphism for the direct image Ef * is in particular satisfied if X ⊆ X and Y ⊆ Y are relatively compact.This means that the bordered spaces X and Y are of a particular type, namely they are b-analytic manifolds and F and G are b-constructible, notions that have been investigated in [Sch23].
Such functorialities are, of course, not restricted to the tensor product with a field extension, but this is the case we are interested in here.One could also try to perform a study of functorialities for more general sheaves L X , similar to [HS23].
In order to understand homomorphisms between scalar extensions of enhanced ind-sheaves, we prove the following two statements.
Proposition 4.3.Let L/K be a field extension.Let X be a real analytic bordered space and let Proof.There exists a canonical morphism (from right to left), and it suffices to prove that it is an isomorphism locally on an open cover of X by relatively compact subanalytic subsets.In other words, it suffices to prove the isomorphism under the Let us, without loss of generality, choose them such that K {t≥0} + ⊗ F ≃ F (and similarly for G).Then Here, the first isomorphism follows from [DK16, Lemma 4.5.15], and the second isomorphism uses the definition of RHom E .The third isomorphism follows from [BHHS23, Lemma 2.7] and [DK16, Lemmas 3.3.12and 3.3.7](note that the extended map π : X × R → X is proper).
Remark 4.4.In the above proof, we have used [BHHS23, Lemma 2.7], i.e. the compatibility of RIhom with extension of scalars.The proof of this lemma in loc.cit.relied on a technical lemma whose details were not expanded there.Let us note that an alternative proof of this lemma can be performed by using the results of [HS23].
Corollary 4.5.Let L/K be a field extension.Let X be a real analytic bordered space and let Proof.It follows from [DK19, (2.6.3)](cf.also [DK16, Proposition 3.2.9 and Corollary 3.2.10])that where j X : X → X is the natural morphism of bordered spaces, and analogously over L. By Lemma 4.1 and Proposition 4.3, we know that extension of scalars commutes with Ej X !! and RHom E .It therefore remains to show that it commutes with taking global sections.In the case of a finite field extension, taking global sections commutes with extension of scalars (since finite direct sums are finite direct products and direct images are right adjoints).In the case where X ⊂ X is relatively compact, the sheaf we take sections of has compact support, so global sections are a proper direct image and hence commute with extension of scalars.
Lemma 4.6.Let X be a real analytic bordered space and L/K a field extension.
In particular, extension of scalars is a conservative functor on E b R-c (IK X ).Proof.In the situation of Corollary 4.5, if X ⊂ X is relatively compact, then for a morphism f : F → G, we have that 1 ⊗ f is an isomorphism if and only if f is so: Let 1 ⊗ g 0 + n i=1 ℓ i ⊗ g i be an inverse, where 1, ℓ 1 , . . ., ℓ n are elements of L, linearly independent over K, and g i : G → F are morphisms.Then and hence g 0 • f = id F , and analogously f • g 0 = id G .Now let f : F → 0 be the unique morphism.Then we can cover X by subsets U that are relatively compact in X and on each U ∞ = (U, U ) we get that F | U∞ ≃ 0. (Here, the closure is taken in X.)By [DK21, Proposition 3.8], this suffices to conclude F ≃ 0 globally.
Let us also state the following lemma, whose proof is completely analogous to [Hoh23, Proposition 4.9].Lemma 4.7.Let L/K be a finite Galois extension with Galois group G. Let X be a real analytic bordered space and let We can now give a slightly more general version (without the compactness assumption) of [BHHS23, Proposition 2.15].Recall that if L/K is a finite Galois extension with Galois group G and H ∈ E b (IL X ), then for any g ∈ G one has the g-conjugate H g ∈ E b (IL X ).A G-structure on H is a collection (ϕ g ) g∈G of isomorphisms ϕ g : .12] for this notion).Note that for Proposition 4.8.Let L/K be a finite Galois extension with Galois group G. Let X be a real analytic bordered space and let H ∈ E 0 R-c (IK X ) be equipped with a G-structure.Then there exists through which the given G-structure on H coincides with the natural one on Proof.In [BHHS23, Proposition 2.15], we have proved the result in the case that X = (X, X) is such that X is relatively compact in X. (The part of the statement about the G-structures has not explicitly been mentioned in loc.cit.but is clear from the construction.)Now, if X ⊆ X is not relatively compact, we can cover X by open subsets U i , i ∈ I, all of which are relatively compact in X.On each (U i ) ∞ = (U i , U i ), we can perform the above construction (closures are taken in X here) and obtain induced by the identity on H (which is certainly compatible with the given Gstructure).Since the natural G-structures on both sides correspond to the given one on H, we see that this morphism is compatible with the natural G-structures on both sides.Hence, by Lemma 4.7, it descends to an isomorphism With these results in hand, we can now state Galois descent for R-constructible enhanced ind-sheaves concentrated in one degree as an equivalence of categories.
We denote by Theorem 4.9.Let L/K be a finite Galois extension.Let X be a real analytic bordered space.Then extension of scalars induces an equivalence of categories Proof.Full faithfulness follows from Lemma 4.7 and essential surjectivity follows from Proposition 4.8.

K-structures and monodromy data
In this section, we are going to study the impact of a K-structure on the generalized monodromy data (in particular Stokes matrices) associated to an enhanced ind-sheaf or a meromorphic connection on a complex curve.The main descent result is Theorem 5.8, and we draw some explicit consequences in the following subsections.Many other parts, in particular the description of generalized monodromy data and meromorphic connections, are mostly a reproduction of well-known facts in order to clarify our setting and language.
Definition 5.1.Let X be a bordered space.If K ⊂ L is a subfield and If such an H K exists, we say that H has a K-structure.
Example 5.2.It is obvious that objects of the form E f W |X have a K-structure for any subfield K ⊂ C (recall these exponential objects from Section 2.5).Similarly, let us consider the following situation: Let us consider smooth algebraic varieties X and S and morphisms X S A 1 ϕ g and the (algebraic) D X -module E ϕ on X as well as its direct image g + E f on S. We can consider these modules (still denoting them by the same symbols) as analytic D-modules on a smooth completion X of X and S of S, respectively, by taking the analytifications of the meromorphic extensions, and we can also extend g as a map ĝ : X → S between these completions.Then, the enhanced ind-sheaf Sol E S (g X (E f ) has a K-structure over any field K ⊂ C. Using Galois descent, we have also deduced certain K-structures of the enhanced solutions of hypergeometric systems in [BHHS23].
In the rest of this section, X will be a Riemann surface (a connected one-dimensional complex manifold) and D ⊂ X a discrete set of points.We write X * := X \ D. Given charts around every point p ∈ D and ε = (ε p ) p∈D , where 0 ≤ ε p ≪ 1 (we just write 0 ≤ ε ≪ 1, and similarly 0 < ε ≪ 1), we set X * ε := X \ p∈D B εp (p), where B εp (p) is the closed ball around p of radius ε p in the given chart around p. (Here, one should read "ε ≪ 1" as "the ε p are sufficiently small", in the sense that B εp (p) does not have a limit point in the boundary of the chart around p and that B εp (p) ∩ B εq (q) = ∅ for p, q ∈ D with p = q.This ensures that X * ε is an open subset of X and that it is homotopy equivalent to X * .) Let k be a field.Recall the notation for enhanced ind-sheaves from Section 2.5, and in particular the natural embedding e k X from sheaves to ind-sheaves as well as the notation for enhanced exponentials.We will simply write E Re ϕ W,k or even E Re ϕ W (if the field k is clear from the context) instead of E Re ϕ W |X,k to simplify notation.By a sector with vertex at p ∈ X, we will mean a subset of X of the following form: Given a local coordinate z in a neighbourhood U of p with z(p) = 0, an angle θ ∈ R/2Z and r, δ ∈ R >0 , we set 5.1.Descent of meromorphic normal form.Let X be a Riemann surface.We first investigate objects motivated by the topological counterpart of meromorphic connections on X via the Riemann-Hilbert correspondence of [DK16].This type of objects is described in the following definition, which is strongly motivated by [Moc22] and [DK18, §6.2].The goal of this subsection is to show that any lattice inherits this property.
Definition 5.3.Let H ∈ E b (Ik X ).We say that H is of meromorphic normal form with respect to D if the following conditions are satisfied: (a) π −1 k X * ⊗ H ≃ H.(b) There exists a local system L on X * (consider it extended by zero to X, i.e. L ∈ Mod(k X )) such that for any 0 < ε ≪ 1, we have ). (c) For any p ∈ D, any local coordinate z in a neighbourhood U of p and any direction θ ∈ R/2πZ, there exist r, δ ∈ R >0 , n ∈ Z >0 , a determination of z 1/n on the sector S θ := S θ (r, δ), a finite set Φ θ ⊂ z −1/n C[z −1/n ] and an integer r ϕ ∈ Z >0 for any ϕ ∈ Φ θ such that (note that the ϕ ∈ Φ θ define holomorphic functions on S θ ) one has an isomorphism In simple terms, an object of meromorphic normal form is one that is localized at D, looks like a local system away from the singularities and satisfies a "Stokes phenomenon" close to the singularity, i.e. it decomposes on small sectors as a direct sum of exponentials.
It is not difficult to see that if H ∈ E b (Ik X ) satisfies the conditions (a), (b) and (c), then it already follows that it is concentrated in degree 0 and R-constructible (cf.[DK16, Lemma 4.9.9]),i.e.H ∈ E 0 R-c (Ik X ).Remark 5.4.Let us make some remarks that will be useful later.First of all, this definition is really just a global reformulation of [DK18, §6.2].The objects defined above can also be described as those enhanced perverse ind-sheaves (see [DK23, Definition 3.4]) satisfying π −1 k X * ⊗ H ≃ H, and shifted by −1.On the other hand, the notion of meromorphic normal form also corresponds to that of "quasi-normal form" (in dimension 1 and for k = C) used in [Ito20].If k = C, objects of meromorphic normal form are the essential image of meromorphic connections on X with poles at D under the enhanced solution functor Sol E X (cf.Proposition 5.16 below).To make it more similar to those definitions, our condition (b) of Definition 5.3 can be reformulated as follows: (b') There is an isomorphism H| X * ≃ k E X * ⊗ π −1 L for a local system L on X * .This is equivalent to (b) since E 0 (Ik X * ) is a stack and hence the isomorphisms on all the X * ε imply the isomorphism on the whole of X * .(We opted for the formulation (b) in our definition in order to express (b) and (c) in a more similar manner.) Since the circle of directions R/2πZ is compact, part (c) of Definition 5.3 can equivalently be formulated as follows: (c') For any p ∈ D and a local coordinate z in a neighbourhood U of p, there exist a positive integer n ∈ Z >0 and a finite number of sectors S p 1 , . . ., S p kp (for some k p ∈ Z >0 ) covering a punctured neighbourhood of p such that for any j ∈ {1, . . ., k p } one has: a determination of z 1/n on the sector S p j , a finite set Φ p j ⊂ z −1/n C[z −1/n ] and an integer r ϕ ∈ Z >0 for any ϕ ∈ Φ p j such that (note that the ϕ ∈ Φ p j define holomorphic functions on S p j ) one has an isomorphism We will assume that the sectors S p 1 , . . ., S p kp are ordered in a counter-clockwise sense around p (with respect to their central directions).Then, for any j ∈ {1, . . ., k p } we can consider the overlap S p j,j+1 := S p j ∩ S p j+1 (where, of course, we count modulo k p , so that k p + 1 := 1).On this overlap we have the two isomorphisms induced by the ones from ( 7) ) r ψ .
We can therefore identify the index sets Φ p j and Φ p j+1 (see [Moc22,Lemma 3.25], cf. also [DK18, Corollary 5.2.3]), and also r ϕ = r ψ via this identification.More precisely, this identification is given by holomorphic continuation of functions in counterclockwise direction from S p j to S p j+1 .We will hence consider all the direct sums in the isomorphisms (7) to be indexed by the same set In particular, this also means that Φ p is closed under analytic continuation around the circle, i.e. if ϕ(z −1/n ) ∈ Φ p , then ϕ(e Denoting by r the rank of the local system L from (b), it is also clear that r = ϕ∈Φ p r ϕ for any p.Our goal is to show that the property of being of meromorphic normal form descends to a lattice.To do this, let us prepare some lemmas.Essentially, the two lemmas below are concerned with the descent of direct sums of exponentials (Lemma 5.6) and descent of enhanced local systems (Lemma 5.7) separately, and we will combine them to obtain our desired result (Theorem 5.8).
Remark 5.5.To prepare for the argument that follows, let us make the following easy observation: Let X be an open disc around a point p ∈ C, let S ⊂ X be an open sector with vertex p, and let us consider holomorphic functions ϕ, ψ : S → C given by elements ϕ, ψ ∈ z −1/n C[z −1/n ] for some choice z −1/n of an n-th root of a local coordinate z at p. We assume that ψ ≺ S ϕ, i.e.Re(ϕ − ψ) is bounded from below near p, but unbounded from above near p.Then: • Certainly, U does not intersect {t ≥ − Re ψ + a} for any a > 0, so E Re ψ S (U ) = 0. On the other hand, for any a ∈ R there exists x ∈ V such that Re ϕ(x) − Re ψ(x) > a, and hence U intersects any {t ≥ − Re ϕ + a}.Note that V can be chosen such that this intersection has only one "relevant" connected component (i.e. one connected component with p in its boundary, which "survives" for a → ∞), and so E Re ϕ S (U ) = k.
Lemma 5.6.Let L/K be a field extension and X a Riemann surface.Let S ⊂ X be an open sector with vertex at some p ∈ X and let Φ ⊂ z −1/n C[z −1/n ] be a finite set of holomorphic functions on S for some n-th root z −1/n of a local coordinate z at p.Then, up to an automorphism of ϕ∈Φ (E Re ϕ S,L ) rϕ , any K-lattice Proof.It is clear from Lemma 4.6 that F K is concentrated in degree 0. Let us assume for simplicity that r ϕ = 1 for any ϕ ∈ Φ, and let us think of E Re ϕ S,L and F K as subanalytic sheaves on X × R (cf.Remark 2.4).Then for any open U ⊆ X × R, the sections of F K on U must be a K-lattice of the sections of ϕ∈Φ E Re ϕ S,L on U (see Lemma 2.3).
Fix a numbering on the elements of Φ = {ϕ 1 , . . ., ϕ n }.For any relatively compact subanalytic open V ⊆ S, the space of sections of ϕ∈Φ E Re ϕ S,L on V × R is L n .A K-lattice for this space of sections is given by a vector space n j=1 K • v j for some v j ∈ L n (linearly independent over L).Since S is connected and restriction maps of ϕ∈Φ E Re ϕ S,L (recall that they mainly consist of projections and inclusions) need to be compatible with those of F K , it is not difficult to see that one can choose the same vectors v j for each such V , and this lattice will also determine the sections of F K on all other open subsets of X × R.
We will show that a suitable choice of the vectors v 1 , . . ., v n yields a matrix (v 1 , . . ., v n ) that defines an automorphism of ϕ∈Φ E Re ϕ S,L (cf.Lemma 2.6).Under (the inverse of) this automorphism, the lattice F K (with basis given by v 1 , . . ., v n ) will then correspond to the lattice ϕ∈Φ E Re ϕ S,L (whose basis is given by the standard basis vectors e 1 , . . ., e n ).
Assume that there exists an open subset S ′ ⊆ S such that ϕ ℓ ≺ S ′ ϕ k for some k, ℓ ∈ {1, . . ., n}.By possibly shrinking S ′ , we can assume that we have a total ordering ϕ j1 ≺ S ′ . . .≺ S ′ ϕ jn , where {j 1 , . . ., j n } = {1, . . ., n}.Now we see (as in Remark 5.5) that there are relatively compact subanalytic open sets U 1 , . . ., U n ⊂ X × R with U j ⊂ U j ′ whenever ϕ j ′ ≺ S ′ ϕ j , and such that E Re ϕ j ′ S,L (U j ) = 0 if ϕ j ′ ≺ S ′ ϕ j and E Re ϕ j ′ S,L (U j ) = L if ϕ j S ′ ϕ j ′ , and restriction maps between these U j are given by projections.This means that for any i ∈ {1, . . ., n}, the vectors pr ≥ji (v 1 ), . . ., pr ≥ji (v n ) span a K-vector space of dimension i, since they form a K-lattice of sections over U i .(Here, pr ≥ji is the projection to the components j i , . . ., j n ∈ {1, . . ., n}.) We can therefore assume without loss of generality that, for any i ∈ {2, . . ., n}, the j i -th component of v 1 , . . ., v i−1 vanishes, as shown by the following argument: First of all, the j n -th entries of v 1 , . . ., v n span a K-vector space of dimension 1, meaning that at least one of them is nonzero and all the nonzero ones are K-multiples of each other.We can hence renumber the v j such that the j n -th entry of v n is nonzero, and we can subtract suitable K-multiples of v n from v 1 , . . ., v n−1 in order to obtain vectors v ′ 1 , . . ., v ′ n which define the same K-lattice as v 1 , . . ., v n but satisfy the following properties: The j n -th entry of v ′ n is nonzero and the j n -th entries of v ′ 1 , . . ., v ′ n−1 vanish.Next, consider the two-dimensional vectors consisting of the j n−1th and j n -th entries of v ′ 1 , . . ., v ′ n .Since they form a K-vector space of dimension 2 and the j n -th entries of v ′ 1 , . . ., v ′ n−1 vanish by construction, the j n−1 -th entry of at least one of the vectors v ′ 1 , . . ., v ′ n−1 must be nonzero, and all the others are K-multiples of this nonzero entry.Proceeding in a similar way as above, we obtain vectors v ′′ 1 , . . ., v ′′ n that generate the same K-lattice as v ′ 1 , . . ., v ′ n and satisfies the following properties: The j n -th entry of v ′′ n is nonzero (in fact, v ′′ n = v ′ n ) and the j n−1 -th entry of v ′′ n−1 is nonzero, while the j n -th entries of v ′′ 1 , . . ., v ′′ n−1 and the j n−1 -th entries of v ′′ 1 , . . ., v ′′ n−2 vanish.Continuing this process shows that we can indeed assume the j i -th entry of v 1 , . . ., v i−1 to be zero, for every i ∈ {2, . . ., n}.In other words, the matrix (v 1 , . . ., v n ) is upper-triangular with invertible diagonal entries when we consider its rows ordered as in Definition 5.3(c) for H, we see that, due to the L-linearity of ξ, the object ξ(π −1 K S θ ⊗ H K ) ⊂ ϕ∈Φ θ (E Re ϕ S θ ,L ) rϕ is a K-lattice.We therefore see that, by Lemma 5.6, the isomorphism above comes, after composing it with an automorphism of the right-hand side, from an isomorphism This completes the proof.For any p ∈ D irr , we choose a local coordinate at p. We then choose a collection of real numbers ε = (ε p ) p∈D with 0 ≤ ε ≪ 1 such that the following holds: For any p ∈ D irr , we have ε p > 0 and B εp (p) \ {p} ⊂ kp j=1 S p j , and for p ∈ D reg , we have ε p = 0 (so that B εp (p) = {p}, independently of the choice of a chart around p).This determines X * ε ⊂ X.By Definition 5.3 (b) and (c), it is clear that there is a local system L on X * such that we can choose an isomorphism (10) γ : π −1 k X * ε ⊗ H ∼ −→ e k X (L) (Note that this does not depend on the concrete choice of ε.Note also that, here again, we tacitly consider the local system L on X * extended by zero as a constructible sheaf on X.) In view of wanting to use Lemma 2.6, we also fix the following data: (i) For any p, a numbering (total order) on Φ p and, for any ϕ ∈ Φ p , a numbering on the factors of the power (E Re ϕ S p j ) rϕ , so that there is a total order on the summands of monodromy data less ambiguous and more natural, but we will not insist on these choices here and only use the existence of these sectors and isomorphisms (see e.g.[Boa01,Boa21] and references therein for a thorough study of intrinsic generalized monodromy data for meromorphic connections).Let us just remark that with appropriate choices of sectors and isomorphisms, for example in the case of unramified exponents of one level, one gets Stokes matrices with a certain block-triangular structure and the diagonal blocks of all but one Stokes matrix around each point p ∈ D are identities.
Definition 5.13.Let L be a field and let K ⊂ L be a subfield.Let H ∈ E b R-c (IL X ) be of meromorphic normal form.We say that generalized monodromy data associated to H (as in Definition 5.10) have entries in K if all the matrices Σ p j ∈ L r×r and C p ∈ L r×r have entries in K and the local system L ∈ Mod(L X * ) is defined over K, i.e. there exists a local system L K ∈ Mod(K X * ) and an isomorphism L ≃ L X ⊗ KX L K .
We say that H admits generalized monodromy data with entries in K (or there exist generalized monodromy data for H with entries in K) if there exist suitable choices with respect to which the generalized monodromy data associated to H have entries in K.
From what we proved above, we can now deduce the following statement.
Corollary 5.14.Let L/K be a field extension, and let H ∈ E b R-c (IL X ) be of meromorphic normal form.Assume that H has a K-structure H K ∈ E b R-c (IK X ).Then there exist generalized monodromy data for H with entries in K.
Proof.By Theorem 5.8, if H = π −1 L X ⊗ π −1 KX H K , then H K is also of meromorphic normal form.Therefore, H admits generalized monodromy data that are induced by generalized monodromy data of H K , and hence the matrices Σ p j , C p have entries in K and also the local system is defined over K.We briefly recall why Sol E X (M) is of meromorphic normal form: First of all, M( * D) ≃ M implies that π −1 C X * ⊗ Sol E X (M) ≃ Sol E X (M).Since a meromorphic connection is generically an integrable connection, we also have, for any 0 < ε ≪ 1, an isomorphism π −1 X * ε ⊗ Sol E X (M) ≃ e C X (L), where L is the local system of solutions L := Sol X * ε (M| X * ε ), extended by 0 to X. Let p ∈ D, then the Levelt-Turrittin theorem gives us a decomposition (up to ramification) of the formalization of the stalk of M at p where ρ(t) = t n is a ramification map in a small neighbourhood of p, the ϕ i (z −1 ) ∈ z −1 C[z −1 ] are (pairwise distinct) polar parts of Laurent series in a local coordinate z at p and the R i are regular holonomic D X -modules.By the Hukuhara-Turrittin theorem, locally on sufficiently small sectors around p, this decomposition lifts to an analytic decomposition of M, which is usually formulated as a statement on the real blow-up space of X at the points of D. We will not go into too much detail here, and rather refer to the existing literature on asymptotic expansions in this context (see e.g.[Was65], [Mal91]).
we denote by Df * (resp.Df * ) the direct image for D X -modules (resp.the inverse image for D Y -modules) along f .If D ⊂ X is a hypersurface, we denote by O X ( * D) the sheaf of meromorphic functions with poles on D at most.It is a left D X -module and for M ∈ D b hol (D X ) we denote its localization at D by M( * D) := M ⊗ OX O X ( * D).We also introduce the notation M(!D) := D X ((D X M)( * D)).We call M ∈ Mod hol (D X ) a meromorphic connection along D if sing supp(M) = D and M( * D) ≃ M, where sing supp(M) denotes the singular support of M.
Proposition 3.4 ([Sab00]).Let D ⊂ X be a normal crossing divisor and M a holonomic D X -module with a good normal form along D. If p ∈ D and {ϕ i | i ∈ I} is a good set of functions in a neighbourhood of p as in Definition 2.1 such that every point of ̟ −1 (p) has an open neighbourhood V ⊂ X with an isomorphism For any relatively compact subanalytic open subset U ⊆ X × R such that E Re ψ S (U ) = 0, we have E Re ϕ S (U ) = 0. To see this, note that E Re ψ S (U ) = 0 means in particular that U intersects any {t ≥ − Re ψ + a} nontrivially.Now let us consider an arbitrary b ∈ R, then there exists a ∈ R such that Re(ϕ − ψ) > b − a, so {t ≥ − Re ϕ + b} ⊇ {t ≥ − Re ψ + a}.This means that also E Re ϕ S (U ) = 0. • There exists a relatively compact subanalytic open subset U

5. 2 .)
Generalized monodromy data as gluing data for enhanced ind-sheaves.Given an enhanced ind-sheaf H ∈ E b R-c (Ik X ) ofmeromorphic normal form with respect to D ⊂ X, we can describe it purely in terms of linear algebra data.Let us recall the description of these so-called generalized monodromy data here (in the case k = C, they are what is often called Stokes data).Since H is of meromorphic normal form, any p ∈ D admits data as in Remark 5.4, in particular we can fix a finite collection of open sectors S p 1 , . . ., S p kp and isomorphisms (9)ξ p j : π −1 k S p j ⊗ H rϕ for j ∈ {1, . . ., k p }.Definition 5.9.Let H be of meromorphic normal form.We denote by D reg := {p ∈ D | Φ p = {0}} ⊆ D the set of regular singularities of H.We also set D irr := D \ D reg and call its elements irregular singularities of H.
j. (ii) A point x ∈ X * ε , a basis of the stalk L x , and for each p ∈ D irr a point y p ∈ S p 1 ∩ X * ε and a path in X * ε from x to y p .(Indeed, instead of choosing y p and such a path, it suffices to fix the homotopy class of this path in X * among all paths starting in x and ending in an arbitrary point of the contractible set S p 1 .)Together with the above γ, this gives in particular an isomorphism π−1 k S p 1 ∩X * ε ⊗ H ≃ r m=1 E 0 S p 1 ∩X * εfor any p.

5. 3 .
Meromorphic connections and Stokes data.Let now M be a meromorphic connection on X with poles at D, i.e. a holonomic D X -module such that M( * D) ≃ M and sing supp M = D.
RHom D P (O P , Db t X×P )[1]by Lemma 3.6 (observing that we can clearly replace P by P there).