Remarks on rigid irreducible meromorphic connections on the projective line

We illustrate the Arinkin-Deligne-Katz algorithm for rigid irreducible meromorphic bundles with connection on the projective line by giving motivicity consequences similar to those given by Katz for rigid local systems.


Introduction
Let k : U ֒→ P 1 be the inclusion of a proper Zariski open subset of the complex projective line P 1 and let (V, ∇) be an irreducible algebraic bundle (1) of rank r with connection on U .We say that (V, ∇) is rigid if any other (V ′ , ∇) on U having at each puncture x ∈ P 1 U a formal structure isomorphic to that of (V, ∇) satisfies (V ′ , ∇) ≃ (V, ∇).It is proved ( [Kat96,BE04]) that, on P 1 , this property is equivalent to cohomological rigidity, i.e., the rigidity index rig(V, ∇) := 2 − h 1 dR (P 1 , k †+ End(V, ∇)) is equal to 2, where k †+ denotes the minimal extension in the sense of D-modules.
The Arinkin-Deligne-Katz algorithm [Ari10], which relies on the property that the rigidity index is preserved by Fourier transformation ( [BE04]) provides an inductive way of checking whether a given irreducible (V, ∇) is rigid by means of successive specific transformations: (V, ∇) is rigid if and only if the sequence of transformations in the algorithm reaches the trivial rank-one bundle with connection (O U ′ , d) on some open subset U ′ ⊂ P 1 .On the other hand, there is a one-to-one correspondence between irreducible bundles with connection (V, ∇) on some Zariski open subset of P 1 and irreducible holonomic D-modules M on P 1 by the inverse functors "middle extension" and "restriction to a suitable Zariski open set", and the algorithm works with the latter objects.
For N ∈ N * , we say that a bundle with connection (V, ∇) (or its middle extension M to P 1 ) is N -quasi-unipotent if the eigenvalues of the formal monodromy at each x ∈ P 1 U belong to µ N (N -th roots of the unity).
The results of this note concern quasi-unipotent rigid irreducible bundles with connection on some proper Zariski open subset U P 1 .They consist of applications of the Arinkin-Deligne-Katz algorithm.A first application has already been given in [Sab18], where it is shown that any rigid irreducible (V, ∇) on U (without the condition of quasi-unipotency) underlies an irregular mixed Hodge module structure which is pure (of weight zero, say).
The motivation for this question came from various recent talks by Michael Groechenig, Aaron Landesman and Daniel Litt on their respective works [EG18] and [LL22].Of course, the technique used here on P 1 does not extend to higher dimensions, but it opens the way to questions in higher dimensions in the setting of irregular singularities.In another direction, the finiteness result of Haoyu Hu and Jean-Baptiste Teyssier [HT22] looks promising.
Rank one.Any rank-one bundle with connection (V, ∇) on U is isomorphic to (O U , ∇ reg + dϕ), where ∇ reg is a connection having regular singularities on P 1 U and ϕ is a regular function on U .It is clearly irreducible and is cohomologically rigid (because End(V, ∇) = (O U , d)).That it is rigid is seen as follows (a special case of the criterion mentioned above): if (V ′′ , ∇) := (V, ∇) ∨ ⊗ (V ′ , ∇) is a rank-one local system which has regular singularity at each x ∈ P 1 U and has trivial local monodromy there, it extends to a trivial bundle with connection on P 1 .
It is quasi-unipotent if and only if (O U , ∇ reg ) is so, and this amounts to the property that a suitable tensor power

Finiteness Definition 1.1 (Minimally ramified polar part). Let p be an integer
1 and let ϕ ∈ C((t 1/p ))/C[[t 1/p ]] be a nonzero ramified polar part of ramification order p.We say that ϕ is minimally ramified if it is not the pullback of a ramified polar part in If ϕ is a minimally ramified polar part, it yields a p-dimensional C((t))-vector space with connection, that we denote by El(ϕ), obtained as the pushforward by t 1/p → t of (C((t 1/p )), d + dϕ).By the Levelt-Turrittin theorem, any finite-dimensional C((t))-vector space with connection can be written in a unique way as the direct sum of terms El(ϕ) ⊗ R ϕ , where ϕ runs in a finite set of minimally ramified polar parts and R ϕ is a finite-dimensional C((t))-vector space with regular singular connection (see [BE04], see also [Sab08]).The minimally ramified polar parts entering the Levelt-Turrittin decomposition are called the exponential factors (or its irregular values) of the C((t))-vector space with connection.
Property A (Finiteness).Given integers r, N 1 and a finite set Φ of minimally ramified polar parts, there exists only a finite number of quasi-unipotent rigid irreducible bundles with connection (V, ∇) on U of rank r such that • the order of quasi-unipotency is at most N , • At each x ∈ P 1 U , the exponential factors of (the formalization of ) (V, ∇) at x belong to Φ.
Exponential-geometric origin.Let Z be a smooth complex quasi-projective variety.We say that an algebraic vector bundle with an integrable connection (V, ∇) on Z is of exponential-geometric origin if there exist a Zariski dense open subset j : U ֒→ Z, a morphism f : Y → U from a smooth quasi-projective variety and a regular function ϕ on Y such that j * (V, ∇), regarded as an holonomic D U -module, is a submodule of for some r 1 and some k ∈ Z. (This is an adaptation of the definition of "geometric origin" in [EK21], see also [LL22]).
Roughly speaking, horizontal sections (or solutions) of such a (V, ∇) on U an can be given an integral expression, with the integrand being of the form e ϕ • ω for some algebraic differential form ω (see [HR08]).
Integrality.To any bundle with connection (V, ∇) on U is associated a Stokes-filtered local system (L, L • ) on the oriented real blow-up space P 1 of P 1 at the punctures P 1 U (see [Mal91,Sab13]).The local system L and the terms L • of the Stokes filtration are sheaves of C-vector spaces.We say that the C-Stokes-filtered local system is integral if it comes by extension of scalars from -structure in the sense of Definition 4.5.This definition is thus more precise than simply asking for a Z-structure.
Property C (Integrality).The Stokes-filtered local system associated to any quasiunipotent rigid irreducible bundle with connection (V, ∇) on U P 1 is integral.
Remark 1.2.One can define the notions of irreducibility, rigidity and quasi-unipotency for a Stokes-filtered local system.Due to the Riemann-Hilbert correspondence of Deligne an Malgrange (see [Mal91]), they correspond to those of the associated bundle with connection.The previous proposition can be stated as the property that a quasi-unipotent rigid irreducible Stokes-filtered local system is integral.
Example 1.3.A (possibly confluent) non resonant hypergeometric equation is irreducible and rigid, and is quasi-unipotent if its local exponents belong to 1 N Z for some N ∈ N * .In [DM89] and [Hie22], the authors compute the Stokes matrices of confluent hypergeometric equations and integrality is then clear from their formulas.On the other hand, in [BHHS22], as a particular case of their results, the authors make explicit its exponential-geometric origin and show that the associated enhanced ind-sheaf is defined on a cyclotomic extension of Q.
Example 1.4.In [Jak20], the author classifies rigid irreducible bundles with connection (V, ∇) on G m with an irregular singularity at infinity of slope one at infinity and differential Galois group G 2 .The family he obtains depends on various parameters in C * .Quasi-unipotency is equivalent to these parameters belonging to µ N for some N 1, and Property C claims that, in such a case, the corresponding Stokes-filtered local system is integral.See also [Kat96,§8.4]and [DR10] for the geometric origin in the tame case.

Finiteness
In this section we prove Property A. We consider the data (U, N, Φ) and quasiunipotent rigid irreducible bundles with connection of rank r with these data, i.e., defined on U , quasi-unipotent of order dividing N and with exponential factors contained in Φ.
A first approach to Property A is by noticing that from the data (U, N, Φ), one can cook up only a finite number of possible formal structures [El(ϕ) ⊗ R ϕ ] at each x ∈ P 1 U , with ϕ ∈ Φ, R ϕ being regular and N -quasi-unipotent.For each such data of formal structures at every x ∈ P 1 U , there exists a smooth affine moduli space of finite type over C such that the corresponding irreducible rigid (V, ∇) are isolated points of this space (see [BE04, Proof of Th. 4.10]).Finiteness follows.
We will now prove finiteness as a consequence of the Arinkin-Deligne-Katz algorithm.This methods, being more constructive, is more quantitative, although it uses the equivalence between rigidity and cohomological rigidity shown in [BE04,Th. 4.10].The proof is by induction on the rank r of V, and we denote by (A) r the statement that Property A holds for bundles of rank r.
Proof of (A) 1 .We choose an affine coordinate t on P 1 such that ∞ ∈ U .For each x ∈ P 1 U , we regard ϕ x ∈ Φ as a polynomial in 1/(t − x) with no constant term.Any choice of a family (ϕ x ) x∈P 1 U of elements of Φ (there are finitely many such choices) yields a unique regular function ϕ on U (namely, ϕ(t) = x∈P 1 U ϕ x (t − x)).Given a bundle with connection (V, ∇) of rank one on U with data N, Φ, there exists such a family (ϕ x ) x∈P 1 U such that (V, ∇ − dϕ) has regular singularities at each point x ∈ P 1 U .Since the local monodromies at each such x belong to µ N , there is only a finite number of such bundles with connection.
We now assume r 2 and (A) <r , and we will prove (A) r .We are given (U, Φ, N ) and we will prove (A) r for these data, a statement that we denote by (A) r (U, Φ, N ).
• We can (and will) assume that Given (V, ∇) rigid irreducible of rank r on U , there exists at most one x ∈ P 1 U where (V, ∇) x is special with respect to the Katz algorithm (Case II of [Ari10, §4.1]).There exists a finite number of automorphisms of P 1 sending 3 points of P 1 U to 0, 1, ∞.
It is then enough to show the finiteness (a) of the set of quasi-unipotent rigid irreducible (V, ∇) of rank r having data (U, Φ, N ) having no special point, (b) and of the set of quasi-unipotent rigid irreducible (V, ∇) of rank r having data Proof of (a).In this case, there exists a rank-one algebraic bundle with connection (L, ∇) on U , completely determined by (V, ∇), in particular satisfying (A) 1 (U, −Φ, N ), and χ ∈ µ N , such that MC χ (V ⊗ L, ∇) has rank < r, where MC χ denotes the middle convolution functor.Furthermore, according to the formulas given in [DS13, Prop.1.3.11(arXiv version)], its data can be chosen as (U, Φ, N ).We conclude that (V, ∇) belongs to the set obtained from a finite set of bundles with connections on U (satisfying (A) <r )(U, Φ, N ) by applying MC χ −1 , for some χ ∈ µ N and by tensoring by a rank-one bundle with connection belonging to the finite set of those satisfying (A) 1 (U, Φ, N ).This shows finiteness in Case (a).
Proof of (b).For (V, ∇) having a special point at ∞, the A-D-K algorithm starts by exhibiting a summand El(ϕ) ⊗ R ϕ of (V, ∇) ∞ which is ramified of order 2. It could occur that, writing ϕ as a minimal ramified polar part q j 1 a j z j/p , the leading term a q z q/p has exponent q/p (the slope of ϕ) which is an integer.We then set Φ 1 = Φ ∪ {ϕ 1 }, where ϕ 1 is the leading part of ϕ with integral exponents, or is zero if the slope of ϕ is not an integer.Case II in [Ari10] shows that there exists a rank-one algebraic bundle with connection (L, ∇) on U , completely determined by (V, ∇), in particular satisfying (A) 1 (U, −Φ 1 , N ), such that the Fourier transform gives the precise way (U ′ , Φ ′ , N ′ ) is obtained from the data of (V ⊗ L, ∇) (and thus depends only of these), hence from (U, Φ, N ).One notices that U ′ and Φ ′ both depend on (U, Φ) (but not only on U resp.Φ separately).
We end the proof for Case (b) as in Case (a).

Exponential-geometric origin
In this section, we prove Property B. Let M be the the minimal extension of (V, ∇) on P 1 .It is a quasi-unipotent rigid holonomic D P 1 -module.Let us recall the basic result that we will use for the proofs of Theorems B and C. As the proof in [Sab18] is written in a sketchy way, we give a detailed proof in the appendix.
Proposition 3.1 ([Sab18, Prop.2.69]).Let M be a quasi-unipotent rigid holonomic D P 1 -module.There exist (a) a smooth projective complex variety X and a strict normal crossing divisor D ⊂ X, together with a subdivisor D 1 ⊂ D, (b) a projective morphism f : X → P 1 , (c) a rational function g on X with poles contained in D and whose pole and zero divisors do not intersect, (d) a locally free rank-one O X ( * D)-module N = N reg with a regular singular meromorphic connection ∇, such that N is of torsion (i.e., N ⊗N ≃ (O X ( * D), d) for some N 1) and M is the image of the natural morphism In loc.cit., M is assumed to be formally unitary and its is shown that N can be chosen the same kind.Applying the same proof, one finds that if M is quasi-unipotent, then N can be chosen to be of torsion (see the appendix).It is therefore enough to show that the right-hand side of (3.1 * ) is of exponential-geometric origin.Setting Y = X D, we can regard (N, ∇) as a rank-one vector bundle with connection on Y and, in the right-hand side of (3.1 * ), regard f as a morphism f : Y → P 1 .Lemma 3.2.There exists a finite morphism ρ : Proof.Let N ∇ be the rank-one local system of horizontal sections of N an on Y an .Since N ∇ is of torsion, the monodromies of N ∇ around the various irreducible components of D are roots of the unity, and there exists, after [Kaw81, Th. 17], a finite morphism X ′′ → X with X ′′ smooth projective and the pullback D ′′ of D being a normal crossing divisor such that the pullback of N ∇ extends as a rank-one local system on X ′′ , which is thus also of torsion.This local system becomes trivial after pullback by some finite étale covering X ′ of X ′′ , and the composition X ′ → X ′′ → X is the desired ρ.
Restricting to the open subset U of P 1 where these holonomic D P 1 -modules are smooth shows that (V, ∇) is of exponential-geometric origin.

Integrality
In this section we prove Property C. Let L be a local system of k-vector spaces on U an .It extends in a unique way as a local system of k-vector spaces on the real blow-up space P 1 of P 1 at each x ∈ P 1 U .We still denote by L this extended k-local system.For each x ∈ P 1 U , let S 1 x denote the fiber at x of the real blowing-up map ̟ : P 1 → P 1 , and let L(x) be the restriction of L to S 1 x .In order to define the notion of a Stokes filtration on each L(x), we first recall the notion of order between ramified polar parts in a specific direction.
Let x,p and we denote by β ψ ϕ the functor composed of the restriction to {ψ ϕ} and extension by zero to S 1 x,p , also denoted Γ {ψ ϕ} ; and β ψ<ϕ has a similar meaning.
A graded k-Stokes-filtered local system index by Φ is a Φ-graded k-local system L = ϕ∈Φ L ϕ on S 1 x,p equipped with the family of nested subsheaves (2) Clearly, the following properties are satisfied: (2) We implicitly add the element −∞ to Φ, which satisfies −∞ < θ ϕ for any ϕ ∈ Φ and any θ, with In such a way, the set {ψ ∈ Φ | ψ < ϕ} is nonempty for any ϕ, and L<ϕ is possibly zero on some open set.
• L ϕ /L <ϕ = L ϕ , • for each θ ∈ S 1 x,p , the family (L ϕ,θ ) ϕ∈Φ is an exhaustive increasing filtration (3) with respect to the partial order θ , • for each θ, we have (4.1) It is harmful to enlarge Φ by adding some ramified polar part η and set L η = 0.In such a way, we can (and will implicitly) assume that Φ is invariant by the automorphisms induced by z → νz with ν p = 1.Definition 4.2.A k-Stokes filtration L(x) • indexed by Φ of the local system L(x) consists of a family (L(x) ϕ ) ϕ∈Φ of subsheaves of k-vector spaces of ρ −1 L(x) such that (1) locally on S 1 x,p , the pair (L(x), L(x) • ) is isomorphic to that of a graded k-Stokes-filtered local system, (2) for any automorphism σ : S 1 x,p x,p induced by z → νz with ν p = 1, and for any ϕ ∈ Φ, the two subsheaves Remark 4.3.From 4.2(1) and the properties of a graded k-Stokes-filtered local system, we deduce that, for each θ ∈ S 1 x,p , the germs L ϕ,θ (ϕ ∈ Φ) are ordered by inclusion according to the partial order θ of their indices.
Furthermore, for each ϕ ∈ Φ, there exists a subsheaf L <ϕ well-defined by a formula analogous to (4.1) and gr ϕ L := L ϕ /L <ϕ is a locally constant sheaf on S 1 x,p .As a consequence, (L(x), L(x) • ) is locally isomorphic to the graded k-Stokes-filtered local system (gr L(x), gr L(x) • ).
Given a ramification ρ ′ : z ′ → z = z ′q , the pullback of a k-Stokes-filtered local system (L(x), L(x) • ) indexed by Φ is a k-Stokes-filtered local system indexed by ρ ′ * Φ and, conversely, any k-Stokes-filtered local system indexed by ρ ′ * Φ which is invariant by the automorphisms induced by z ′ → ν ′ z ′ with ν ′q = 1 comes by pullback of a k-Stokes-filtered local system indexed by Φ.
Given two k-Stokes-filtered local systems, we can assume that they are indexed by the same Φ.A morphism of k-Stokes-filtered local systems (L(x), L(x) • ) → (L ′ (x), L ′ (x) • ) is then a morphism between the corresponding k-local systems whose pullback by ρ is compatible with the Stokes filtration, and in particular induces a morphism of the corresponding graded k-local systems.
These notions can be globalized to U : a k-Stokes-filtered local system on U indexed by Φ consists of a k-local system on U an together with a Stokes filtration indexed by Φ on each L(x) for x ∈ P 1 U .A morphism is defined correspondingly.
Theorem 4.4.The category of k-Stokes-filtered local system on U is abelian.
Proof.Since the category of k-local system on U an is abelian, it is enough to consider the category of k-Stokes-filtered local system on S 1 x (x ∈ P 1 U ).This is e.g.[Sab13, Th. 3.1].

Definition 4.5
An x,p , the germs L o (x) ϕ,θ of o-submodules of L o (x) θ are ordered by inclusion according to the partial order θ , (c) for each ϕ ∈ Φ and each σ as in Definition 4.2(2), we have A morphism between k-Stokes-filtered local systems with an o-structure is a morphism between the corresponding o-local systems which preserves the o-filtrations at each x ∈ P 1 U .
One can give an equivalent definition.Remark 4.7.In Definitions 4.5 and 4.6, we do not impose that the sheaves gr ϕ L o (x) are local systems of o-modules on S 1 x,p .This is why we do not use the terminology "o-Stokes-filtered local system".Proof.The category consisting of objects (L o , (L o (x) • ) x∈P 1 U ) satisfying 4.6(i) and (ii), and with morphisms as described above, is an abelian category.The condition that it yields a k-Stokes-filtered local system by tensoring the objects with k does not break abelianity, according to Theorem 4.4 and o-flatness of k.Remark 4.9 (k-Stokes structures).One can define similarly the notion of a C-Stokesfiltered local system with an o = Z[ζ]-structure and obtain the corresponding abelian category.We notice that the latter category is equivalent, by the extension of scalars from k to C, to the abelian category of k-Stokes-filtered local systems with an o-structure.Indeed, it is a matter of proving that, given a family (L k , (L k (x) • ) x∈P 1 U ) satisfying 4.6(i) and (ii) with o replaced with k, and which satisfies the local triviality property 4.2(1) after tensoring it with C, already satisfies it before tensorization.This amounts to proving that, for a sheaf F k of finite-dimensional k-vector spaces on a locally path connected topological space Z, F k is locally constant if and only if However, in the case of confluent hypergeometric systems considered in Example 1.3, the computation of the Stokes matrices after a suitable ramification done in [Hie22] provides Stokes matrices with entries in o if the local formal monodromies belong to o.
On the other hand, given a locally constant sheaf of free o-modules on a punctured P 1 (without any assumption of irreducibility or rigidity, but one can add them), the computation of the Stokes matrices of the Fourier transform of its associated perverse sheaf on A 1 done in [DHMS20] also provides Stokes matrices with entries in o.Such an example, with o = Z, can be obtained as follows.Let f : Y → A 1 be a regular function on a smooth affine complex variety Y of dimension n.Assume that f is cohomologically tame (in the sense of [Sab06]), so that in particular f has only isolated critical points in Y .Then the Stokes matrices at t = ∞ of the free C[t, t −1 ]-module with connection , d t + f dt can be defined over Z.This result goes back to [Pha85].Note that (V, ∇), which is of exponential-geometric origin by definition (with ϕ = tf on Y × G m ), is known to be semi-simple, but is possibly not rigid.
such that (i) the strict normal crossing divisor D decomposes as H ∪P with P = f −1 (P 1 U ), (ii) the pole divisor P g of g decomposes correspondingly as P g = H ′ ∪ P ′ (and the zero divisor of g does not cut P g ), (iii) the pair (X U , H U ) is smooth over U , i.e., each stratum of its natural stratification is smooth over U .Lemma 4.11.Assume that the data (X, f, D, g) of Proposition 3.1 satisfy the properties (i)-(iii) above, and let N be a torsion locally free O X ( * D)-module with a regular singular meromorphic connection ∇.Let D 1 be a sub-divisor of D. Then each term of (3.1 * ) is a vector bundle with connection on U whose associated C-Stokesfiltered local system admits a Z[ζ]-structure for which the morphism associated to that of The idea of the proof is that the morphism between the C-Stokes-filtered local systems associated to (3.1 * ) can be computed in a purely topological way, by considering suitable real oriented blow-up spaces, from the local system N ∇ .The latter being defined over o = Z[ζ], it follows that the corresponding Stokes-filtered local systems and the morphism between them have an o-structure.
Proof, part one: the local systems on U an .We start with the local systems on U an .Let us decompose and, setting E g = (O X ( * P g ), d + dg), it also reads (4.12) We will compute these perverse sheaves on U an and the morphism between them by means of the real blowing up denote the dense open subset of X U consisting of points in the neighborhood of which the function e −g has moderate growth.One also checks (because of (ii)) that Let α : Y an ֒→ X mod U and β : X mod U ֒→ X U denote the open inclusions and let us set n = dim X.One can define the moderate de Rham complex ).We compute similarly the left-hand side of (4.12).We denote by We then have the corresponding open inclusions α 1 : Y an ֒→ X mod U,1 and β 1 : X mod U,1 ֒→ X U and a standard computation shows that the sheaf be the open inclusion, we have a natural morphism showing that the morphism is also defined over o.
Proof, part 2: the local Stokes structures.Since both terms of (3.1 * ) are holonomic D P 1 -modules, their exponential factors at the points of P 1 U are contained in a finite set of ramified polar parts Φ, of some ramification order p.We will not need to compute explicitly this set and we will only use its existence.
We fix x ∈ P 1 U and restrict the setting over a small disc ∆ centered at x.We restrict all the data of the lemma as analytic data over ∆.In particular, U is replaced with the punctured disc ∆ * .Otherwise, we keep the same notation with this new analytic meaning.
Both terms of (3.1 * ) are holonomic D ∆ -modules and, since we are only interested in computing Stokes filtrations, we consider the associated localized modules that we regard as meromorphic flat bundles (V 1 , ∇) and (V, ∇), i.e., free O ∆ ( * 0)-modules of finite rank with a connection, and the natural morphism (V 1 , ∇) → (V, ∇) between them.They have associated C-Stokes-filtered local systems (L 1 , L 1,• ) and (L, L • ) with exponential factors contained in Φ.The local systems are the restriction to ∆ * of those computed in Part one of the proof.In particular, we already know that they have an o-structure, that we aim at expanding to the whole Stokes structure.For that purpose, we will give a geometric construction of the corresponding Stokes filtrations by means of the maps analogous to α, β, α 1 , β 1 .
We consider the oriented real blow-ups X(P ) and X = X(D) of X along the components of P and D respectively, so that we have a composition and we extend the map f as a continuous map where ∆ is the oriented real blow-up of ∆ at the origin.We denote by X 0 the open subset of X consisting of points in the neighborhood of which e −g has moderate growth, and we keep the similar notation for the maps α, β (see Part one of the proof).
As in part one, we decompose D 1 as D ′ 1 ∪ D ′′ 1 , so that the polar components of g contained D 1 are those of D ′ 1 .We consider the open subset and we keep the similar notation for the maps α 1 , β 1 .Similarly, f 1 resp.f denote the restriction of f to X 1, 0 resp.X 0 .From [Moc14, Cor.4.7.5 & Lem.5.1.6]we obtain: Lemma 4.13.There exists a commutative diagram where the vertical morphisms are the natural ones.
Since N ∇ is equipped with the o-structure N ∇ o , all terms and morphisms in the lemma acquire a natural o-structure compatible with that already obtained for L 1 , L.
In order to obtain the o-structure on each L 1, ϕ and L ϕ for any ϕ ∈ Φ ⊂ C((t 1/p ))/C[[t 1/p ]], we consider the diagram where ρ p is the cyclic ramification of order p and X p is a resolution of singularities of the pair (X ′ p , P ).We replace the rational function g on X with g + f * p ϕ on X p , the divisor D with its pullback by X p → X, and N with its pullback N p on X p .Then L 1, ϕ , L ϕ are obtained by the same procedure as that of Lemma 4.13 with these new data, so that the o-structure is obtained in the same way.This concludes the proof of Property C.

4.c. Final remarks on integrality.
There are other possible approaches to o-structures.They all mainly rely on the general Riemann-Hilbert correspondence as developed by D'Agnolo and Kashiwara [DK16] and their subsequent work.As indicated in [DK16, §2], the theory of loc.cit.can be applied to objects defined on o.The point is then to check that the R-constructible enhanced ind-sheaves associated to N ⊗ E ϕ and N(!D 1 ) ⊗ E ϕ and the morphism between them are defined over o.The pushforward of these objects by the projective morphism f provides an o-structure on the enhanced ind-sheaves associated to both terms of (3.1 * ) and the morphism between them.This can be made a little more precise by considering the categories of C-constructible enhanced ind-sheaves of [Ito20], or the characterization of R-constructible enhanced ind-sheaves which come from holonomic D-modules given in [Moc22].
One could also work within the setting of irregular constructible complexes of [Kuw21].Such objects can be defined over the ring o.
In all these theories, the main point is the compatibility of the irregular Riemann-Hilbert correspondence with projective pushforward.This is probably the most delicate point in [DK16], that replaces [Moc14,Cor.Both functors vanish when applied to holonomic D X -modules supported on H (this is clear for the localization functor, and the property for the dual one follows from the fact that duality preserves the support).
Let g be a meromorphic (or rational) function on X.If M is considered as an O Xmodule with an integrable connection ∇, it will be convenient to denote the same O Xmodule with the twisted connection ∇ + dg as E g ⊗ M .Note that this twist contains the localization functor along the pole divisor P of g, so that E g ⊗M = Γ [ * P ] (E g ⊗M ).
We will make use of the relations between various functors described in [Sab18, §1.77].
Let M be rigid irreducible on P 1 and let us assume that there exist X, D, D 1 , f : X → P 1 , g and (N, ∇) as in 3.1(a)-(d) so that M is the image of (3.1 * ).Denoting by P the pole divisor of g, we can assume that D 1 has no component contained in P : indeed, denoting by D 0 the union of those components not contained in P , we have Moreover, if N ∇ is unitary (resp. of torsion) and L is locally formally unitary (resp.formally quasi-unipotent), then so is M ′ and (N ′ , ∇) can be chosen unitary (resp. of torsion).
(ii) Let A 1 t be the chart with coordinate t corresponding to the choice 0, 1, ∞ ∈ P 1 .By (i), we can assume that M = Γ [ * ∞] M. Let M ′ be the Laplace transform of M with respect to this choice.Then Data 3.1(a)-(d) such that M ′ is the image of (3.1 * ) with these data.Moreover, if N ∇ is unitary (resp. of torsion) and L is locally formally unitary (resp.formally quasi-unipotent), then so is M ′ and (N ′ , ∇) can be chosen unitary (resp. of torsion).
These two properties allow us to conclude the proof of the proposition, since any rigid holonomic D P 1 -module can be obtained by applying a sequence of (i) and (ii) to (O P 1 , d), according to the Arinkin-Deligne algorithm, and moreover, if M is locally formally unitary (resp.quasi-unipotent), then the rank-one connections L reg chosen at each step are locally unitary (resp.quasi-unipotent).
Let us show (i).There exists a meromorphic function ψ on P 1 and a rank-one meromorphic connection L reg with regular singularities, such that L = E ψ ⊗ L reg .We can write L = (O P 1 ( * Σ), d + dψ + ω), where Σ is the pole divisor of L and ω is a one-form with at most simple poles at Σ.Moreover, L is locally formally unitary (resp.quasi-unipotent) if and only if L reg is unitary (resp.quasi-unipotent), i.e., the residues of ω at Σ are real (resp.rational).
Since L is O P 1 -flat, M ⊗ L is the image of ] N) ⊗ L, and, since the functors Γ [⋆Σ] (⋆ =!, * ) are exact on the category of holonomic D P 1 -modules, M ′ is the image of We set H = f −1 (Σ) and we decompose D as D 2 ∪ D 3 ∪ D 4 , where D 3 are those components of D which are components of H, D 2 are the components of D 1 which are not components of H, and D 4 are the remaining components.We set g 1 = g + f • ψ.
Firstly, one checks that, for ⋆ = * , !, for ⋆ = * , !, and since f + L reg = Γ [ * H] f + L reg (so that we can replace D 1 with D 2 in the right-hand side), we deduce that M ′ is the image of ( * ) Let e : X ′ → X be a projective modification such that e −1 (D ∪ H) is a divisor with normal crossings and such that the pole and zero divisors of g ′ 1 := e * g 1 do not intersect.For the first condition the blowing ups can be chosen to take place above the union of D ∩ H and of the singular set of H, while for the second condition, since the pole and zero divisors of g 1 intersect at most in D ∩ H, the blowing ups can be chosen to take place above D ∩ H.As a consequence, we can assume that, setting H ′ = e −1 (H), the morphism e : X ′ H ′ → X H is an isomorphism. Since we have, after [Sab18,1.77(vi)], 4.a.Stokes-filtered local systems defined over Z[µ N ].Let us set k = Q[ζ] for some complex N -th root ζ of 1 and some N 1, and let o = Z[ζ] denote its ring of integers.

Definition 4. 6 .
A k-Stokes-filtered local system with an o = Z[ζ]-structure consists of the data (L o , (L o (x) • ) x∈P 1 U ), where (i) L o is a local system of o-modules of finite type on U an , (ii) the families (L o (x) • ) x∈P 1 U ) satisfy 4.5(2b) and (2c), such that, after tensoring with k, Property 4.2(1) is fulfilled.In this presentation, morphisms between k-Stokes-filtered local systems with an o = Z[ζ]-structure are any morphisms between data (L o , (L o (x) • ) x∈P 1 U ).

Lemma 4. 8 .
The category of k-Stokes-filtered local systems with an o-structure is abelian.
Remark 4.10 (o-Stokes structures and Stokes matrices).The Stokes matrices (or Stokes multipliers) of a C-Stokes-filtered local system with an k-structure, equivalently a k-Stokes-filtered local system, are conjugate to matrices having entries in k.On the other hand, in presence of an o-structure, we cannot assert in general the existence of Stokes matrices with entries in o.

4
.b. Proof of Property C. Let us consider the data as in Proposition 3.1.Up to blowing up X, we can achieve the following properties.There exist a Zariski dense open subset U of P 1 and a diagram of H 1 along which g has a pole.Denoting by p DR the analytic de Rham functor (shifted by the ambient dimension), it suffices to show that p DR (3.1 * ) is a morphism of (shifted) local systems on U an defined over o.Due to the commutation of p DR and direct images, this morphism reads over U : 4.7.5 & Lem.5.1.6]used in Lemma 4.13.Appendix: detailed proof of Proposition 3.1 Notation.Let H be a hypersurface in a smooth variety X.We denote by Γ [ * H] the localization functor acting on the category of holonomic D X -modules: for such a D Xmodule M , we have Γ [ * H] M = O X ( * H) ⊗ OX M as an O X -module.Denoting by D the duality functor on holonomic D X -modules, we define the dual localization functor Γ [!H] as DΓ [ * H] D.