Stokes-Filtered Local Systems Along a Divisor with Normal Crossings

  • Claude Sabbah
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2060)

Abstract

We construct the sheaf \(\mathcal{I}\) to be considered as the index sheaf for Stokes filtrations. This is a sheaf on the real blow-up space of a complex manifold along a family of divisors. We will consider only divisors with normal crossings. The global construction of \(\mathcal{I}\) needs some care, as the trick of considering a ramified covering cannot be used globally. The important new notion is that of goodness. It is needed to prove abelianity and strictness in this setting, generalizing the results of Chap. 3.

9.1 Introduction

After having introduced the real blow-up spaces in the previous chapter, we now extend the notion of a Stokes-filtered local system in higher dimension, generalizing the contents of Chaps. 2 and 3. For this purpose, we first define the sheaf \(\mathcal{I}\) of ordered abelian groups, which will serve as the indexing sheaf for the Stokes filtrations. The general approach of Chap. 1 will now be used, and the sheaf \(\mathcal{I}\) will be constructed in a way similar to that used in Remark 2.23. The present chapter was indeed the main motivation for developing Chap. 1. We will make precise the global construction of the sheaf \(\mathcal{I}\), since the main motivation of this chapter is to be able to work with Stokes-filtered local systems globally on the real blow-up space \(\widetilde{X}({D}_{j\in J})\).

However, the order of a local section φ of \(\mathcal{I}\) is strongly related to the asymptotic behaviour of \(\exp \varphi \) on \(\partial \widetilde{X}({D}_{j\in J})\). This behaviour is in general difficult to analyze, unless φ behaves like a monomial with negative exponents. For instance, the asymptotic behaviour of \(\exp ({x}_{1}/{x}_{2})\) near \({x}_{1} = {x}_{2} = 0\) is not easily analyzed, while that of \(\exp (1/{x}_{1}{x}_{2})\) is easier to understand. This is why we introduce the notion of pure monomiality. Moreover, given a finite subset Φ of local sections of \(\mathcal{I}\), the asymptotic comparison of the functions \(\exp \varphi \) with \(\varphi \in \Phi \) leads to considering the condition “good”, meaning that each nonzero difference \(\varphi - \psi \) of local sections of Φ is purely monomial. The reason for considering differences \(\varphi - \psi \) of elements of Φ is, firstly, that it allows one to totally order the elements of Φ with respect to the order of the pole and, secondly, that these differences are the exponential factors of the endomorphism of a given Stokes-filtered local system having Φ as exponential factors, and the corresponding Stokes-filtered local system is essential in classification questions. Of course, pure monomiality and goodness are automatically satisfied in dimension one.

The analogues of the main results of Chaps. 2 and 3 are therefore proved assuming goodness. We will find this goodness assumption in various points later on, and the notion of Stokes filtration developed in this text always assumes goodness.

9.2 The Sheaf \(\mathcal{I}\) on the Real Blow-Up (Smooth Divisor Case)

Let X be a smooth complex manifold and let D be a smooth divisor in X. Let \(\varpi :\widetilde{ X}(D) \rightarrow X\) be “the” real blow-up space of X along D (associated with the choice of a section \(f : {\mathcal{O}}_{X} \rightarrow {\mathcal{O}}_{X}(D)\) of \(L(D)\) defining D, see Sect. 8.2). We also denote by \(\widetilde{\imath },\widetilde{j}\) the inclusions \(\partial \widetilde{X}(D) `\rightarrow \widetilde{ X}(D)\) and \({X}^{{_\ast}} `\rightarrow \widetilde{ X}(D)\).

In order to construct the sheaf \(\mathcal{I}\), we will adapt to higher dimensions the construction of Remark 2.23. We consider the sheaf \(\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}}\) and its subsheaf \({(\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}})}^{\mathrm{lb}}\) of locally bounded functions on \(\widetilde{X}\). We will construct \(\mathcal{I}\) as a subsheaf of the quotient sheaf \(\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}}/{(\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}})}^{\mathrm{lb}}\) (which is supported on \(\partial \widetilde{X}\)). It is the union, over \(d \in {\mathbb{N}}^{{_\ast}}\), of the subsheaves \({\mathcal{I}}_{d}\) that we define below.

Let us start with\({\mathcal{I}}_{1}\). There is a natural inclusion \({\varpi }^{-1}{\mathcal{O}}_{X}({_\ast}D) `\rightarrow \widetilde{ {j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}}\), and \({\varpi }^{-1}{\mathcal{O}}_{X} = {\varpi }^{-1}{\mathcal{O}}_{X}({_\ast}D) \cap {(\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}})}^{\mathrm{lb}}\), since a meromorphic function which is bounded in some sector centered on an open set of D is holomorphic. We then set \(\widetilde{{\mathcal{I}}}_{1} = {\varpi }^{-1}{\mathcal{O}}_{X}({_\ast}D) \subset \widetilde{ {j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}}\) and \({\mathcal{I}}_{1} = {\varpi }^{-1}({\mathcal{O}}_{X}({_\ast}D)/{\mathcal{O}}_{X}) \subset \widetilde{ {j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}}/{(\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}})}^{\mathrm{lb}}\).

Locally on D, we can define ramified coverings \({\rho }_{d} : {X}_{d} \rightarrow X\) of order d along D, for any d. Let \(\widetilde{{\rho }}_{d} :\widetilde{ {X}}_{d} \rightarrow \widetilde{ X}\) be the corresponding covering. The subsheaf \(\widetilde{{\mathcal{I}}}_{d} \subset \widetilde{ {j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}}\) of d-multivalued meromorphic functions on \(\widetilde{X}\) is defined as the intersection of the subsheaves \(\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}}\) and \(\widetilde{{\rho }}_{d,{_\ast}}\widetilde{{\mathcal{I}}}_{\widetilde{{X}}_{d},1}\) of \(\widetilde{{\rho }}_{d,{_\ast}}\widetilde{{j}}_{d,{_\ast}}{\mathcal{O}}_{{X}_{d}^{{_\ast}}} =\widetilde{ {j}}_{{_\ast}}{\rho }_{d,{_\ast}}{\mathcal{O}}_{{X}_{d}^{{_\ast}}}\). As above, we have \(\widetilde{{\mathcal{I}}}_{d} \cap {(\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}})}^{\mathrm{lb}} =\widetilde{ {j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}}\cap \widetilde{ {\rho }}_{d,{_\ast}}{\varpi }_{d}^{-1}{\mathcal{O}}_{{X}_{d}}\). The sheaf\({\mathcal{I}}_{d}\) is then defined as the quotient sheaf \(\widetilde{{\mathcal{I}}}_{d}/\widetilde{{\mathcal{I}}}_{d} \cap {(\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}})}^{\mathrm{lb}}\). This is a subsheaf of \(\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}}/{(\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}})}^{\mathrm{lb}}\).

The locally defined subsheaves \(\widetilde{{\mathcal{I}}}_{d}\) (and thus \({\mathcal{I}}_{d}\)) glue together as a subsheaf of \(\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}}\), since the local definition does not depend on the chosen local ramified d-covering. Similarly, \({\mathcal{I}}_{d}\) exists as a subsheaf of \(\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}}/{(\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}})}^{\mathrm{lb}}\) all over D.

Definition 9.1 (Case of a smooth divisor). 

The sheaf \(\widetilde{\mathcal{I}}\) (resp. \(\mathcal{I}\)) is the union of the subsheaves \(\widetilde{{\mathcal{I}}}_{d}\) (resp. \({\mathcal{I}}_{d}\)) of \(\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}}\) (resp. \(\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}}/{(\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}})}^{\mathrm{lb}}\)) for \(d \in {\mathbb{N}}^{{_\ast}}\).

Definition 9.2 (\(\mathcal{I}\) as a sheaf of ordered abelian groups). 

The sheaf \(\widetilde{{\imath }}^{-1}\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}}\) is naturally ordered by setting \({(\widetilde{{\imath }}^{-1}\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}})}_{\leq 0} =\log {\mathcal{A}}_{\,\widetilde{X}(D)}^{\mathrm{mod}\,D}\) (see Sect. 8.2). In this way, \(\widetilde{\mathcal{I}}\) inherits an order: \(\widetilde{{\mathcal{I}}}_{\leq 0} =\widetilde{ \mathcal{I}} \cap \log {\mathcal{A}}_{\,\widetilde{X}(D)}^{\mathrm{mod}\,D}\). This order is not altered by adding a local section of \({(\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}})}^{\mathrm{lb}}\), and thus defines an order on \(\mathcal{I}\).

Lemma 9.3.

For any local ramified covering\({\rho }_{d} : ({X}_{d},D) \rightarrow (X,D)\)of order d along D, \(\widetilde{{\rho }}_{d}^{-1}\widetilde{{\mathcal{I}}}_{d}\)is identified with\({\varpi }_{d}^{-1}{\mathcal{O}}_{{X}_{d}}({_\ast}D)\)and\(\widetilde{{\rho }}_{d}^{-1}{\mathcal{I}}_{d}\)with\({\varpi }_{d}^{-1}({\mathcal{O}}_{{X}_{d}}({_\ast}D)/{\mathcal{O}}_{{X}_{d}})\). This identification is compatible with order.

Proof.

The proof is completely similar to that given in Remark 2.23. \(\square \)

9.3 The Sheaf \(\mathcal{I}\) on the Real Blow-Up (Normal Crossing Case)

Let us now consider a family \(({D}_{j\in J})\) of smooth divisors of X whose union D has only normal crossings, and the corresponding real blowing-up map \(\varpi :\widetilde{ X}({D}_{j\in J}) \rightarrow X\). We will consider multi-integers \(d \in {({\mathbb{N}}^{{_\ast}})}^{J}\). The definition of the sheaves \(\widetilde{{\mathcal{I}}}_{d}\) and \({\mathcal{I}}_{d}\) is similar to that in dimension one.

Let us set \(1 = (1,\ldots,1)\) (#J terms) and \(\widetilde{{\mathcal{I}}}_{1} = {\varpi }^{-1}{\mathcal{O}}_{X}({_\ast}D) \subset \widetilde{ {j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}}\). Let us fix \({x}_{o} \in D\), let us denote by \({D}_{1},\ldots,{D}_{\mathcal{l}}\) the components of D going through xo, and set \(\widetilde{{x}}_{o} \in {\varpi }^{-1}({x}_{o}) \simeq {({S}^{1})}^{\mathcal{l}}\). Then a local section of \({\varpi }^{-1}{\mathcal{O}}_{X}({_\ast}D)\) near \(\widetilde{{x}}_{o}\) is locally bounded in the neighbourhood of \(\widetilde{{x}}_{o}\) if and only if it is holomorphic in the neighbourhood of xo. In other words, as in the smooth case, \({\varpi }^{-1}{\mathcal{O}}_{X}({_\ast}D) \cap {(\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}})}^{\mathrm{lb}} = {\varpi }^{-1}({\mathcal{O}}_{X})\).

We locally define \(\widetilde{{\mathcal{I}}}_{d}\) near xo, by using a ramified covering \({\rho }_{d}\) of (X, xo) along (D, xo) of order \(d = ({d}_{1},\ldots,{d}_{\mathcal{l}})\), by the formula \(\widetilde{{\mathcal{I}}}_{d} :=\widetilde{ {\rho }}_{d,{_\ast}}[{\varpi }_{d,{_\ast}}{\mathcal{O}}_{{X}_{d}}({_\ast}D)] \cap \widetilde{ {j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}}\), and \({\mathcal{I}}_{d}\) by \({\mathcal{I}}_{d} :=\widetilde{ {\mathcal{I}}}_{d}/\widetilde{{\mathcal{I}}}_{d} \cap {(\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}})}^{\mathrm{lb}}\).

The locally defined subsheaves \(\widetilde{{\mathcal{I}}}_{d}\) glue together all over D as a subsheaf \(\widetilde{{\mathcal{I}}}_{d}\) of \(\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}}\). We also set globally\({\mathcal{I}}_{d} =\widetilde{ {\mathcal{I}}}_{d}/\widetilde{{\mathcal{I}}}_{d} \cap {(\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}})}^{\mathrm{lb}}\).

Definition 9.4.

The subsheaf \(\widetilde{\mathcal{I}} \subset \widetilde{ {j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}}\) is the union of the subsheaves \(\widetilde{{\mathcal{I}}}_{d}\) for \(d \in {({\mathbb{N}}^{{_\ast}})}^{J}\). The sheaf\(\mathcal{I}\) is the subsheaf \(\widetilde{\mathcal{I}}/\widetilde{\mathcal{I}} \cap {(\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}})}^{\mathrm{lb}}\) of \(\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}}/{(\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}})}^{\mathrm{lb}}\).

Definition 9.5.

The order on \(\widetilde{\mathcal{I}}\) is given by \(\widetilde{{\mathcal{I}}}_{\leq 0} :=\widetilde{ \mathcal{I}} \cap \log {\mathcal{A}}_{\,\widetilde{X}({D}_{j\in J})}^{\mathrm{mod}\,D}\). It is stable by the addition of an element of \({(\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}})}^{\mathrm{lb}}\) and defines an order on \(\mathcal{I}\).

Lemma 9.6.

For any local ramified covering\({\rho }_{d} : ({X}_{d},D) \rightarrow (X,D)\)of order\(d\)along \(({D}_{j\in J})\), \(\widetilde{{\rho }}_{d}^{-1}\widetilde{{\mathcal{I}}}_{d}\)is identified with\({\varpi }_{d}^{-1}{\mathcal{O}}_{{X}_{d}}({_\ast}D)\)and\(\widetilde{{\rho }}_{d}^{-1}{\mathcal{I}}_{d}\)with\({\varpi }_{d}^{-1}{(}{\mathcal{O}}_{{X}_{d}}({_\ast}D)/{\mathcal{O}}_{{X}_{d}}{)}\). These identifications are compatible with order.

Proof.

Same proof as in Remark 2.23. \(\square \)

Remark 9.7.

For any subset \(I \subset J\), let DI denote the intersection \({cap}_{j\in I}{D}_{j}\) and set \(D_I^{\circ}=D_I\setminus cup_{j\in J\setminus I}D_j\). Set also \({Y }_{I} = {\varpi }^{-1}({D}_{I}^{\circ }) \subset \partial \widetilde{X}({D}_{j\in J})\). The family \(\mathcal{Y} = {({Y }_{I})}_{I\subset J}\) is a stratification of \(\partial \widetilde{X}\) which satisfies the property (1.43). Moreover, the sheaf \(\mathcal{I}\) is Hausdorff with respect to \(\mathcal{Y}\) (this is seen easily locally on D).

9.4 Goodness

The order on sections of \(\mathcal{I}\) is best understood for purely monomial sections of \(\mathcal{I}\). Let us use the following local notation. We consider the case where \(X = {\Delta }^{\mathcal{l}} \times {\Delta }^{n-\mathcal{l}}\) with base point \(0 = ({0}_{\mathcal{l}},{0}_{n-\mathcal{l}})\), and \({D}_{i} =\{ {t}_{i} = 0\}\) (\(i \in L :=\{ 1,\ldots,\mathcal{l}\}\)) and \(D ={ cup}_{i=1}^{\mathcal{l}}{D}_{i}\). The real blowing-up map
$${\varpi }_{L} :\widetilde{ X}({D}_{i\in L}) = {({S}^{1})}^{\mathcal{l}} \times [0,1{)}^{\mathcal{l}} \times {\Delta }^{n-\mathcal{l}}\rightarrow X = {\Delta }^{\mathcal{l}} \times {\Delta }^{n-\mathcal{l}}$$
is defined by sending \(({e}^{i{\theta }_{j}},{\rho }_{j})\) to \({t}_{j} = {\rho }_{j}{e}^{i{\theta }_{j}}\) (\(j = 1,\ldots,\mathcal{l}\)).
  • In the non-ramified case (i.e., we consider sections of \({\mathcal{I}}_{1}\)), a germ η at \(\theta \in {({S}^{1})}^{\mathcal{l}} = {({S}^{1})}^{\mathcal{l}} \times {0}_{\mathcal{l}} \times {0}_{n-\mathcal{l}} \subset {({S}^{1})}^{\mathcal{l}} \times [0,1{)}^{\mathcal{l}} \times {\Delta }^{n-\mathcal{l}}\) of section of \({\mathcal{I}}_{1}\) is nothing but a germ at 0 ∈ X of section of \({\mathcal{O}}_{X}({_\ast}D)/{\mathcal{O}}_{X}\).

Definition 9.8.

We say that η ispurely monomial at 0 if \(\eta = 0\) or η is the class of \({t}^{-m}{u}_{m}\), with \(\mathbf{m}\in\mathbb{N}^{\ell}\smallsetminus\{0\}\), \({u}_{m} \in {\mathcal{O}}_{X,0}\) and \({u}_{m}(0)\neq 0\). We then set \(m = m(\eta )\) with the convention that \(m(0) = 0\), so that \(m(\eta ) = 0\) iff \(\eta = 0\).

  • For \(\eta \in {\mathcal{O}}_{X,0}({_\ast}D)/{\mathcal{O}}_{X,0}\), written as \({\sum}_{k\in {\mathbb{Z}}^{\mathcal{l}}\times {\mathbb{N}}^{n-\mathcal{l}}}{\eta }_{k}{t}^{k}\), the Newton polyhedron\(NP (\eta ) \subset {\mathbb{R}}^{\mathcal{l}} \times {\mathbb{R}}_{+}^{n-\mathcal{l}}\) is the convex hull of \({\mathbb{R}}_{+}^{n}\) and the octants \(k + {\mathbb{R}}_{+}^{n}\) for which \({\eta }_{k}\neq 0\). Then η is purely monomial if and only if \(NP (\eta )\) is an octant \((-m,{0}_{n-\mathcal{l}}) + {\mathbb{R}}_{+}^{n}\) with \(m \in {\mathbb{N}}^{\mathcal{l}}\).

    If η is purely monomial, we have for every \(\theta \in {({S}^{1})}^{\mathcal{l}}\):
    $$\eta {\leq }_{{}_{\theta }}\! 0\Longleftrightarrow\eta = 0\text{ or }\arg {u}_{m}(0) -{\sum}_{j}{m}_{j}{\theta }_{j} \in (\pi /2,3\pi /2)\,\mathrm{mod}\,2\pi.$$
    (9.9)
    If \(\eta \neq 0\) and η is purely monomial at 0, it is purely monomial on some open set \(Y = {({S}^{1})}^{\mathcal{l}} \times V\) (with \(V\) an open neighbourhood of \({0}_{n-\mathcal{l}}\) in \({\Delta }^{n-\mathcal{l}}\), embedded as \({0}_{\mathcal{l}} \times V \subset [0,1{)}^{\mathcal{l}} \times {\Delta }^{n-\mathcal{l}}\)), and \({Y }_{\eta \leq 0}\) is defined by the inequation \(\arg {u}_{m}(v) -{\sum}_{j}{m}_{j}{\theta }_{j} \in (\pi /2,3\pi /2)\,{\rm mod}\,\,2\pi \), where \(v\) varies in the parameter space \(V\). For \(v\) fixed, it is the inverse image by the fibration map \({({S}^{1})}^{n} \rightarrow {S}^{1}\), \(({e}^{i{\theta }_{1}},\ldots,{e}^{i{\theta }_{n}})\mapsto {e}^{i({m}_{1}{\theta }_{1}+\cdots +{m}_{n}{\theta }_{n})}\), of a set of the kind defined at the end of Example 1.4. This set rotates smoothly when \(v\) varies in \(V\). Similarly, the boundary \(\mathrm{St}(\eta,0)\) of \({Y }_{\eta \leq 0}\) in \(Y\) is the pull-back by the previous map of a subset of \({S}^{1} \times V\) which is a finite covering of \(V\). It has codimension one in \(Y\).
  • The order on \({\mathcal{I}}_{d}\) is described similarly after a ramification which is cyclic of order \({d}_{i}\) around the component \({D}_{i}\) of D.

For non-purely monomial elements, we still have the following (e.g. in the non-ramified case).

Proposition 9.10.

For every\(\varphi,\psi \in \Gamma (U,{\mathcal{O}}_{X}({_\ast}D)/{\mathcal{O}}_{X})\), the set\({Y }_{\psi \leq \varphi }\)issubanalytic in\(Y\)and its boundary\(\mathrm{St}(\varphi,\psi )\)has (real) codimension\(\geq 1\)in \(Y\).

Lemma 9.11.

Let\(\eta \in \Gamma (U,{\mathcal{O}}_{X}({_\ast}D)/{\mathcal{O}}_{X})\)and let\(x \in D\). Then there exists a projective modification\(\epsilon : U^\prime \rightarrow U\)of some open neighbourhood\(U\)of \(x\)in X, which is an isomorphism away from D, such that\(D^\prime := \vert {\epsilon }^{-1}(D \cap U)\vert \)is a reduced divisor with normal crossings and smooth components, and\(\eta \circ \epsilon \)is locally purely monomial everywhere on D′.

Proof (Proof of Proposition 9.10). 

Set \(\eta = \psi - \varphi = {t}^{-m}{u}_{m}\) with \(m\in\mathbb{N}^{\ell}\smallsetminus\{0\}\) and \({u}_{m}\) holomorphic. We have \({Y }_{\psi \leq \varphi } = {Y }_{\eta \leq 0}\). The purely monomial case (\({u}_{m}\neq 0\) everywhere on \(U \cap D\)) has been treated above. The statement is local subanalytic on \(U\), and for any point of \(U\) we replace \(U\) by a subanalytic open neighbourhood of this point, that we still denote by X, on which Lemma 9.11 applies. If we set \(D^\prime ={ cup }_{j}{D^\prime}_{j}\), we have natural real analytic proper maps (see Sect. 8.2) \(\widetilde{\epsilon } :\widetilde{ X}^\prime({D^\prime}_{j\in J}) \rightarrow \widetilde{ X}\). Let us set \(Y ^\prime =\widetilde{ {\epsilon }}^{-1}(Y )\). Then \({Y ^\prime}_{\eta \circ \epsilon \leq 0} =\widetilde{ {\epsilon }}^{-1}({Y }_{\eta \leq 0})\) since \({e}^{\eta }\) has moderate growth near \(\widetilde{{x}}_{o} \in Y\) if and only if \({e}^{\eta \circ \epsilon }\) has moderate growth near any \(\widetilde{{x}^\prime}_{o} \in \widetilde{ {\epsilon }}^{-1}(\widetilde{{x}}_{o})\), by the properness of \(\widetilde{\epsilon }\). As a consequence, the set \(Y \ {Y }_{\eta \leq 0}\) is the push-forward by \(\widetilde{\epsilon }\) of the set \(Y ^\prime \ {Y ^\prime}_{\eta \circ \epsilon \leq 0}\). Using the purely monomial case considered previously, one shows that \(Y ^\prime \ {Y ^\prime}_{\eta \circ \epsilon \leq 0}\) is closed and semi-analytic in \(Y ^\prime\). The subanalyticity of \({Y }_{\eta \leq 0}\) follows then from Hironaka’s theorem [31] on the proper images of sub- (or semi-) analytic sets, and stability by complements and closure. The statement for \(\mathrm{St}(\varphi,\psi )\) also follows. \(\square \)

Proof (Sketch of the proof of Lemma 9.11). 

By using the resolution of singularities in the neighbourhood of \(x \in U\), we can find a projective modification \({\epsilon }_{1} : {U}_{1} \rightarrow U\) such that the union of the divisors of zeros and of the poles of η form a divisor with normal crossings and smooth components in \({U}_{1}\) (so that, locally in \({U}_{1}\) and with suitable coordinates, \(\eta \circ {\epsilon }_{1}\) takes the form of a monomial with exponents in \(\mathbb{Z}\)). The problem is now reduced to the following question: given a divisor with normal crossings and smooth components \({D}_{1}\) in \({U}_{1}\), attach to each smooth component an integer (the order of the zero or minus the order of the pole of \(\eta \circ {\epsilon }_{1}\)), so as to write \({D}_{1} = {D}_{1}^{+} \cup {D}_{1}^{0} \cup {D}_{1}^{-}\) with respect to the sign of the integer; to any projective modification \({\epsilon }_{2} : {U}_{2} \rightarrow {U}_{1}\) such that \({D}_{2} := {\epsilon }_{2}^{-1}({D}_{1})\) remains a divisor with normal crossings and smooth components, one can associate in a natural way a similar decomposition; we then look for the existence of such an \({\epsilon }_{2}\) such that \({D}_{2}^{-}\) and \({D}_{2}^{+}\) do not intersect.

We now denote \({U}_{1}\) by X and \({D}_{1}\) by \(D ={ cup}_{j\in J}{D}_{j}\). The divisor D is naturally stratified, and we consider minimal (that is, closed) strata. To each such stratum is attached a subset \(L\) of \(J\) consisting of indices \(j\) for which \({D}_{j}\) contains the stratum. Because of the normal crossing condition, the cardinal of this subset is equal to the codimension of the stratum \({D}_{L}\). We set \(D(L) ={ cup}_{j\in L}{D}_{j}\). We will construct the modification corresponding to this stratum with a toric argument. Let us set \(\mathcal{l} = \#L\) and, for each \(j \in L\), let us denote by \({\mathcal{I}}_{j}\) the ideal of \({D}_{j}\) in \({\mathcal{O}}_{X}\).

As is usual in toric geometry (see [14, 24, 74] for instance), we consider the space \({\mathbb{R}}^{\mathcal{l}}\) equipped with its natural lattice \(N := {\mathbb{Z}}^{\mathcal{l}}\) and the dual space \({({\mathbb{R}}^{\vee })}^{\mathcal{l}}\) equipped with the dual lattice \(M\). To each rational cone \(\sigma \) in the first octant \({({\mathbb{R}}_{+})}^{\mathcal{l}}\) we consider the dual cone \({\sigma }^{\vee } \in {({\mathbb{R}}^{\vee })}^{\mathcal{l}}\) and its intersection with \(M\). This allows us to define a sheaf of subalgebras \({\sum}_{m\in {\sigma }^{\vee }\cap M}{\mathcal{I}}_{1}^{{m}_{1}}\cdots {\mathcal{I}}_{\mathcal{l}}^{{m}_{\mathcal{l}}}\) of \({\mathcal{O}}_{X}({_\ast}D(L))\). Locally on \({D}_{L}\), if \({D}_{j}\) is defined by \(\{{x}_{j} = 0\}\), this is \({\mathcal{O}}_{X}{ \otimes }_{\mathbb{C}[{x}_{1},\ldots,{x}_{\mathcal{l}}]}\mathbb{C}[{\sigma }^{\vee }\cap M]\), hence this sheaf of subalgebras corresponds to an affine morphism \({X}_{\sigma } \rightarrow X\). Similarly, to any a fan \(\Sigma \) in the first octant \({({\mathbb{R}}_{+})}^{\mathcal{l}}\) one associates a morphism \({X}_{\Sigma } \rightarrow X\), which is a projective modification if the fan completely subdivides the octant. Moreover, if each cone of the fan is strictly simplicial, the space \({X}_{\Sigma }\) is smooth and the pull-back of the divisor \(D(L)\) has normal crossings with smooth components.

We will now choose the fan \(\Sigma \). To each basis vector \({e}_{j}\) of \({\mathbb{R}}^{\mathcal{l}}\) is attached a multiplicity \({\nu }_{j} \in \mathbb{Z}\), namely the order of \(\eta \circ {\epsilon }_{1}\) along \({D}_{j}\). We consider the trace \(H\) on \({({\mathbb{R}}_{+})}^{\mathcal{l}}\) of the hyperplane \(\{({n}_{1},\ldots,{n}_{\mathcal{l}}) \in {\mathbb{R}}^{\mathcal{l}}\mid { \sum}_{j}{\nu }_{j}{n}_{j} = 0\}\) and we choose a strictly simplicial fan in \({({\mathbb{R}}_{+})}^{\mathcal{l}}\) such that \(H\) is a union of cones of this fan. For each basis vector \(({n}_{1},\ldots,{n}_{\mathcal{l}}) \in {\mathbb{N}}^{\mathcal{l}}\) of a ray (dimension-one cone) of this fan, the multiplicity of the pull-back of \(\eta \circ {\epsilon }_{1}\) along the divisor corresponding to this ray is given by \(\sum\limits_{j}{\nu }_{j}{n}_{j}\). Therefore, for each \(\mathcal{l}\)-dimensional cone of the fan, the multiplicities at the rays all have the same sign (or are zero).

The proof of the Lemma now proceeds by decreasing induction on the maximal codimension \(\mathcal{l}\) of closed strata of D which are contained both in \({D}_{+}\) and \({D}_{-}\). In the space \({\mathbb{R}}^{J}\) we consider the various subspaces \({\mathbb{R}}^{L}\) (\(L \subset J\)) corresponding to these closed strata of D. We subdivide each octant \({({\mathbb{R}}_{+})}^{L}\) by a strict simplicial fan as above. We also assume that the fans coincide on the common faces of distinct subspaces \({\mathbb{R}}^{L}\). We denote by \(\Sigma \) the fan we obtain in this way. In order to obtain such a \(\Sigma \), one can construct a strict simplicial fan completely subdividing \({({\mathbb{R}}_{+})}^{J}\) which is compatible with the hyperplane \(\sum\limits_{j\in J}{\nu }_{j}{n}_{j} = 0\) and with the various octants \({({\mathbb{R}}_{+})}^{L}\) corresponding to codimension \(\mathcal{l}\) strata of D contained in \({D}_{+} \cap {D}_{-}\), and then restrict it to the union of these octants \({({\mathbb{R}}_{+})}^{L}\). We also consider the sheaves \(\sum\limits_{m\in {\sigma }^{\vee }\cap {M}_{J}}\prod\limits_{j\in J}{\mathcal{I}}_{j}^{{m}_{j}} \subset {\mathcal{O}}_{X}({_\ast}D)\) for \(\sigma \in \Sigma \), and get a projective modification \({\epsilon }_{\Sigma } : {X}_{\Sigma } \rightarrow X\). By construction, the maximal codimension of closed strata of \({\epsilon }_{\Sigma }^{-1}(D)\) contained in \({\epsilon }_{\Sigma }^{-1}{(D)}_{+} \cap {\epsilon }_{\Sigma }^{-1}{(D)}_{-}\) is \(\leq \mathcal{l} - 1\). \(\square \)

For a finite set \(\Phi \subset {\mathcal{O}}_{X,0}({_\ast}D)/{\mathcal{O}}_{X,0}\), the notion of pure monomiality is replaced by goodness (see also Remark 11.5(4) below).

Definition 9.12 (Goodness). 

We say that a finite subset Φ of \({\mathcal{O}}_{X,0}({_\ast}D)/{\mathcal{O}}_{X,0}\) isgood if \(\#\Phi = 1\) or, for any \(\varphi \neq \psi \) in Φ, \(\varphi - \psi \) is purely monomial, that is, the Newton polyhedron \(NP (\varphi - \psi )\) is an octant with vertex in \(-{\mathbb{N}}^{\mathcal{l}} \times \{ {0}_{n-\mathcal{l}}\}\) (see Definition 9.8).

Remark 9.13.

Let us give some immediate properties of local goodness (see [81, I.2.1.4]).
  1. 1.

    Any subset of a good set is good, and any subset consisting of one element (possibly not purely monomial) is good. If Φ is good, then its pull-back \({\Phi }_{d}\) by the ramification \({X}_{d} \rightarrow X\) is good for any \(d\). Conversely, if \({\Phi }_{d}\) is good for some \(d\), then Φ is good.

     
  2. 2.

    More generally, let \(f : X^\prime \rightarrow X\) be a morphism of complex manifolds and let \(({D}_{j\in J})\) be a family of smooth divisors in X whose union D is a divisor with normal crossings. We set \(D^\prime = {f}^{-1}(D)\), and we assume that \(D^\prime =cup_{j^\prime\in J^\prime}{D^\prime}_{j^\prime}\) is also a divisor with normal crossings and smooth components \({D^\prime}_{j^\prime\in J^\prime}\). We will usually denote by \(f : (X^\prime,D^\prime) \rightarrow (X,D)\) a mapping satisfying such properties. If \(\Phi \subset {\mathcal{O}}_{X,{x}_{o}}({_\ast}D)/{\mathcal{O}}_{X,{x}_{o}}\) is good, then for any \({x^\prime}_{o} \in {f}^{-1}({x}_{o})\), the subset \({({f}^{{_\ast}}\Phi )}_{{x^\prime}_{o}} \subset {\mathcal{O}}_{X^\prime,{x^\prime}_{o}}({_\ast}D^\prime)/{\mathcal{O}}_{X^\prime,{x^\prime}_{o}}\) is good.

     
  3. 3.

    Any germ \({\varphi }_{0}\) of \({\mathcal{O}}_{X,0}({_\ast}D)/{\mathcal{O}}_{X,0}\) defines in a unique way a germ \({\varphi }_{x} \in {\mathcal{O}}_{X,x}({_\ast}D)/{\mathcal{O}}_{X,x}\) for \(x \in D \cap U\), \(U\) some open neighbourhood of 0. Indeed, choose a lifting \({\varphi }_{0}^{{_\ast}}\) in \({\mathcal{O}}_{X,0}({_\ast}D)\). It defines in a unique way a section \({\varphi }^{{_\ast}}\) of \(\Gamma (U,{\mathcal{O}}_{U}({_\ast}D))\) for \(U\) small enough. Its germ at \(x \in U \cap D\) is denoted by \({\varphi }_{x}^{{_\ast}}\). Its image in \({\mathcal{O}}_{X,x}({_\ast}D)/{\mathcal{O}}_{X,x}\) is \({\varphi }_{x}\). Given two liftings \({\varphi }_{0}^{{_\ast}}\) and \({\varphi }_{0}^{\star }\), their difference is in \({\mathcal{O}}_{X,0}\). Choose \(U\) so that \({\varphi }^{{_\ast}}\) and \(({\varphi }^{{_\ast}}- {\varphi }^{\star })\) are respectively sections of \(\Gamma (U,{\mathcal{O}}_{U}({_\ast}D))\) and \(\Gamma (U,{\mathcal{O}}_{U})\). Then these two liftings give the same \({\varphi }_{x}\).

    Similarly, any finite subset Φ of \({\mathcal{O}}_{X,0}({_\ast}D)/{\mathcal{O}}_{X,0}\) defines in a unique way a finite subset, still denoted by Φ, of \({\mathcal{O}}_{X,x}({_\ast}D)/{\mathcal{O}}_{X,x}\) for any \(x\) close enough to 0.

    Then, if Φ is good at 0, it is good at any point of D in some open neighbourhood of 0. Note however that a difference \(\varphi - \psi \) which is non-zero at 0 can be zero along some components of D, a phenomenon which causes \({\mathcal{I}}^{\acute{\text{e}t}}\) to be non-Hausdorff.

     
  4. 4.

    A subset Φ is good at 0 if and only if for some (or any) \(\eta \in {\mathcal{O}}_{X,0}({_\ast}D)/{\mathcal{O}}_{X,0}\) the translated subset \(\Phi + \eta \) is good.

     
  5. 5.

    For a good set \(\Phi \subset {\mathcal{O}}_{X,0}({_\ast}D)/{\mathcal{O}}_{X,0}\), and for any fixed\({\varphi }_{o} \in \Phi \), the subset \(\{m(\varphi - {\varphi }_{o})\mid \varphi \in \Phi \} \subset {\mathbb{N}}^{\mathcal{l}}\) is totally ordered (i.e., the Newton polyhedra \(NP (\varphi - {\varphi }_{0})\) form a nested family). Its maximum does not depend on the choice of \({\varphi }_{o} \in \Phi \), it is denoted by \(m(\Phi )\) and belongs to \({\mathbb{N}}^{\mathcal{l}}\). We have \(m(\Phi ) = 0\) iff \(\#\Phi = 1\). We have \(m(\Phi + \eta ) = m(\Phi )\) for any \(\eta \in {\mathcal{O}}_{X,0}({_\ast}D)/{\mathcal{O}}_{X,0}\).

     
  6. 6.
    Assume Φ is good. Let us fix \({\varphi }_{o} \in \Phi \) and set \(m := m(\Phi )\). The set \(\{m(\psi - {\varphi }_{o})\mid \psi \in \Phi \}\) is totally ordered, and we denote by \(\mathcal{l} ={ \mathcal{l}}_{{\varphi }_{o}}\) its submaximum. For any \(\varphi \neq {\varphi }_{o}\) in Φ such that \(m(\varphi - {\varphi }_{o}) = m\), we denote by \({[\varphi - {\varphi }_{o}]}_{\mathcal{l}}\) the class of \(\varphi - {\varphi }_{o}\) in \({\mathcal{O}}_{X,0}({_\ast}D)/{\mathcal{O}}_{X,0}(\sum {\mathcal{l}}_{i}{D}_{i})\). For such a φ, the set
    $${\Phi }_{{[\varphi -{\varphi }_{o}]}_{\mathcal{l}}} :=\{ \psi \in \Phi \mid {[\psi - {\varphi }_{o}]}_{\mathcal{l}} = {[\varphi - {\varphi }_{o}]}_{\mathcal{l}}\} \subset \Phi $$
    is good at 0, and \(m({\Phi }_{{[\varphi -{\varphi }_{o}]}_{\mathcal{l}}}) < m = m(\Phi )\). Indeed, for any \(\psi \in {\Phi }_{{[\varphi -{\varphi }_{o}]}_{\mathcal{l}}}\), \(\psi - {\varphi }_{o}\) can be written \(\varphi - {\varphi }_{o} + {t}^{-\mathcal{l}}{u}_{\psi }(t)\) with \({u}_{\psi }(t) \in \mathbb{C}\{t\}\), and the difference of two such elements \(\psi,\eta \in {\Phi }_{{[\varphi -{\varphi }_{o}]}_{\mathcal{l}}}\) is also written as \({t}^{-\mathcal{l}}v(t)\) with \(v(t) \in \mathbb{C}\{t\}\).
     

Let\(\widetilde{{\Sigma }}_{L} \subset {\mathcal{I}}_{\vert {Y }_{L}}^{\acute{\text{e}t}}\) be a finite covering of \({Y }_{L}\) (see Sect. 9.3). There exists \(d\) such that the pull-back \(\widetilde{{\Sigma }}_{L,d}\) of \(\widetilde{{\Sigma }}_{L}\) by the ramification \(\widetilde{{X}}_{d} \rightarrow \widetilde{ X}\) is a trivial covering of \({({S}^{1})}^{\mathcal{l}} \times {D}_{L}\). Hence there exists a finite set \({\Phi }_{d} \subset {\mathcal{O}}_{{X}_{d}}({_\ast}D)/{\mathcal{O}}_{{X}_{d}}\) such that the restriction of \(\widetilde{{\Sigma }}_{L,d}\) over \({({S}^{1})}^{\mathcal{l}} \times \{ 0\}\) is equal to \({\Phi }_{d} \times {({S}^{1})}^{\mathcal{l}} \times \{ 0\}\). We say that \(\widetilde{{\Sigma }}_{L}\) is good at \(0 \in {D}_{L}\) if the corresponding subset \({\Phi }_{d}\) is good for some (or any) \(d\) making the covering trivial. By the previous remark, if \(\widetilde{{\Sigma }}_{L}\) is good at 0, it is good in some neighbourhood of \(0 \in {D}_{L}\). Moreover, if \(f : X^\prime \rightarrow X\) is as in Remark 9.13(2), if \(\widetilde{{\Sigma }}_{L}\) is good at 0 then \(\widetilde{{f}}^{{_\ast}}(\widetilde{{\Sigma }}_{L})\) is good at each point of \({f}^{-1}(0)\).

Let us now consider a stratified \(\mathcal{I}\)-covering\(\widetilde{\Sigma } \subset {\mathcal{I}}^{\acute{\text{e}t}}\) of \(\widetilde{X}\) (see Definition 1.46).

Lemma 9.14.

If\(\widetilde{{\Sigma }}_{L}\)is good at\(0\! \in \! {D}_{L}\), each\(\widetilde{{\Sigma }}_{I}\)is good on some neighbourhood of 0.

Proof.

Assume first that \(\widetilde{{\Sigma }}_{L} \rightarrow {Y }_{L} = {({S}^{1})}^{L} \times {\Delta }^{n-\mathcal{l}}\) is trivial and thus \(\widetilde{{\Sigma }}_{L} = \Phi \times {Y }_{L}\) for some finite good set \(\Phi \subset {\mathcal{O}}_{X,0}({_\ast}D)/{\mathcal{O}}_{X,0}\). Then, if \({\Delta }^{n}\) is small enough, we have \(\widetilde{\Sigma } =cup_{\varphi \in \Phi }\varphi (\widetilde{{\Delta }}^{n})\) and the assertion follows from Remark 9.13(3).

In general, one first performs a suitable ramification around the components of D to reduce to the previous case. \(\square \)

Lastly, let us consider the global setting, where X is a complex manifold and \(({D}_{j\in J})\) is a family of smooth divisors on X which intersect normally, and the sheaf of ordered abelian groups \(\mathcal{I}\) on \(\partial \widetilde{X}({D}_{j\in J})\) is as in Definitions 9.4 and 9.5. For any nonempty subset \(I\) of \(J\), we set \({D}_{I} ={ cap }_{i\in I}{D}_{i}\) and \(Y_I=\varpi^{-1}(D_I^{\circ})\subset\partial\widetilde X(D_{j\in J})\). The family \({({D}_{I}^{\circ })}_{\varnothing \neq I\subset J}\) is a Whitney stratification of \(D =cup_{j\in J}{D}_{j}\).

Definition 9.15 (Global goodness). 

Let us consider a stratified \(\mathcal{I}\)-covering \(\widetilde{\Sigma } \subset {\mathcal{I}}^{\acute{\text{e}t}}\). We say that it isgood if each \(\widetilde{{\Sigma }}_{I}\) is good at each point of \({D}_{I}^{\circ }\).

9.5 Stokes Filtrations on Local Systems

As above, \(({D}_{j\in J})\) is a family of smooth divisors on X which intersect normally, and the sheaf of ordered abelian groups \(\mathcal{I}\) on \(\partial \widetilde{X}({D}_{j\in J})\) is as in Definitions 9.4 and 9.5.

Definition 9.16 (Stokes-filtered local system). 

Let \(\mathcal{L}\) be a local system of \(k\)-vector spaces on \(\partial \widetilde{X}({D}_{j\in J})\). A Stokes filtration of \(\mathcal{L}\) is a \(\mathcal{I}\)-filtration of \(\mathcal{L}\), in the sense of Definition 1.47. We denote by \((\mathcal{L},{\mathcal{L}}_{\bullet })\) aStokes-filtered local system.

Remark 9.17.

We will freely extend in the present setting the notation of Chap. 2 and use some easy properties considered there. In the non-ramified case for instance, the sheaves \({\mathcal{L}}_{\leq \varphi }\) are \(\mathbb{R}\)-constructible, according to Proposition 9.10.

Definition 9.18 (Goodness). 

We say that aStokes-filtered local system \((\mathcal{L},{\mathcal{L}}_{\bullet })\) is good if its associated stratified \(\mathcal{I}\)-covering \(\widetilde{\Sigma }(\mathcal{L}) \subset {\mathcal{I}}^{\acute{\text{e}t}}\), which is the union of the supports of the various \(gr {\mathcal{L}}_{\vert {Y }_{I}}\) (with \({Y }_{I} = {\varpi }^{-1}({D}_{I}^{\circ })\)), is good.

Theorem 9.19.

Let us fix a good stratified\(\mathcal{I}\)-covering\(\widetilde{\Sigma } \subset {\mathcal{I}}^{\acute{\text{e}t}}\)and let\((\mathcal{L},{\mathcal{L}}_{\bullet })\), \((\mathcal{L}^\prime,{\mathcal{L}^\prime}_{\bullet })\)be Stokes-filtered local systems on\(\widetilde{X}({D}_{j\in J})\)whose associated\(\mathcal{I}\)-stratified coverings\(\widetilde{\Sigma }(\mathcal{L}),\widetilde{\Sigma }(\mathcal{L}^\prime)\)are contained in\(\widetilde{\Sigma }\). Let\(\lambda : (\mathcal{L},{\mathcal{L}}_{\bullet }) \rightarrow (\mathcal{L}^\prime,{\mathcal{L}^\prime}_{\bullet })\)be a morphism of local systems which is compatible with the Stokes filtrations. Then \(\lambda \)isstrict.

Corollary 9.20.

If\(\widetilde{\Sigma }\)is good, the category of Stokes-filtered local systems satisfying\(\widetilde{\Sigma }(\mathcal{L}) \subset \widetilde{ \Sigma }\)isabelian.

This will be a consequence of the following generalization of Theorem 3.5.

Proposition 9.21.

Assume that\(\widetilde{\Sigma }(\mathcal{L}),\widetilde{\Sigma }(\mathcal{L}^\prime) \subset \widetilde{ \Sigma }\)for some good stratified\(\mathcal{I}\)-covering\(\widetilde{\Sigma } \subset {\mathcal{I}}^{\acute{\text{e}t}}\). Let \(\lambda \)be a morphism of local systems compatible with the Stokes filtrations. Then, in the neighbourhood of any point of\(\partial \widetilde{X}({D}_{j\in J})\)there exist gradations of the Stokes filtrations such that the morphism is diagonal with respect to them. In particular, it isstrict, and the natural \(\mathcal{I}\)-filtrations on the local systems\(\ker \lambda \), \(Im \lambda \)and\(Coker \,\lambda \)are good Stokes-filtered local systems. Their associated stratified\(\mathcal{I}\)-coverings satisfy
$$\widetilde{\Sigma }(\ker \,\lambda ) \subset \widetilde{ \Sigma }(\mathcal{L}),\quad \widetilde{\Sigma }(Coker \,\lambda ) \subset \widetilde{ \Sigma }(\mathcal{L}^\prime),\quad \widetilde{\Sigma }(Im \lambda ) \subset \widetilde{ \Sigma }(\mathcal{L}) \cap \widetilde{ \Sigma }(\mathcal{L}^\prime).$$

Preliminary reductions. As in the one-dimensional case, one reduces to the non-ramified case by a suitable \(d\)-cyclic covering. Moreover, the strictness property is checked on the germs at any point of \(\partial \widetilde{X}\), so we can work in the local setting of Sect. 9.3 and restrict the filtered local system to the torus \({({S}^{1})}^{\mathcal{l}}\). We will moreover forget about the term \({\Delta }^{n-\mathcal{l}}\) and assume that  = n.

In the following, we will give the proof for a \(\mathcal{I}\)-local system on the torus \({({S}^{1})}^{n}\) in the non-ramified case, that is, \(\mathcal{I}\) is the constant sheaf with fibre \({\mathcal{P}}_{n} = \mathbb{C}\{{t}_{1},\ldots,{t}_{n}\}\)\([{({t}_{1}\cdots {t}_{n})}^{-1}]/\mathbb{C}\{{t}_{1},\ldots,{t}_{n}\}\). As this sheaf satisfies the Hausdorff property, many of the arguments used in the proof of Theorem 3.5 can be extended in a straightforward way for Proposition 9.21. Nevertheless, we will give the proof with details, as the goodness condition is new here.

Level structure of a Stokes filtration. For every \(\mathcal{l} \in {\mathbb{N}}^{n}\), we define the notion ofStokes filtration of level\(\geq \mathcal{l}\) on \(\mathcal{L}\), by replacing the set of indices \({\mathcal{P}}_{n} = \mathbb{C}[t,{t}^{-1}]/\mathbb{C}[t]\) (\(t = {t}_{1},\ldots,{t}_{n}\)) by the set \({\mathcal{P}}_{n}(\mathcal{l}) := \mathbb{C}[t,{t}^{-1}]/{t}^{-\mathcal{l}}\mathbb{C}[t]\) (with \({t}^{-\mathcal{l}} := {t}_{1}^{-{\mathcal{l}}_{1}}\cdots {t}_{n}^{-{\mathcal{l}}_{n}}\)). We denote by \({[\cdot ]}_{\mathcal{l}}\) the map \(\mathbb{C}[t,{t}^{-1}]/\mathbb{C}[t] \rightarrow \mathbb{C}[t,{t}^{-1}]/{t}^{-\mathcal{l}}\mathbb{C}[t]\). The constant sheaf\(\mathcal{I}(\mathcal{l})\) is ordered as follows: for connected open set \(U\) of \({({S}^{1})}^{n}\) and \({[\varphi ]}_{\mathcal{l}},{[\psi ]}_{\mathcal{l}} \in {\mathcal{P}}_{n}(\mathcal{l})\), we have \({[\psi ]}_{\mathcal{l}} {\leq }_{{}_{U}}\! {[\varphi ]}_{\mathcal{l}}\) if, for some (or any) representatives \(\varphi,\psi \) in \(\mathbb{C}[t,{t}^{-1}]\), \({e}^{\vert t{\vert }^{\mathcal{l}}(\psi -\varphi ) }\) has moderate growth along D in a neighbourhood of \(U\) in X intersected with \({X}^{{_\ast}}\). In particular, a Stokes filtration as defined previously has level \(\geq 0\). level \(\geq 0\).

Lemma 9.22.

The natural morphism\(\mathcal{I} \rightarrow \mathcal{I}(\mathcal{l})\)is compatible with the order.

Proof.

Let \(U\) be a connected open set in \({({S}^{1})}^{n}\) and let \(K\) be a compact set in \(U\). Let \(\eta \in {\mathcal{P}}_{n}\) (or a representative of it). We have to show that if \({e}^{\eta }\) has moderate growth along D on \(\mathrm{nb}(K) \ D\), then so does \({e}^{\vert t{\vert }^{\mathcal{l}}\eta }\). Let \(\epsilon : X^\prime \rightarrow X\) be a projective modification as in the proof of Proposition 9.10, let \(\widetilde{\epsilon }\) be the associated morphism of real blow-up spaces, and let \(K^\prime \subset Y ^\prime\) be the inverse image of \(K\) in \(Y ^\prime\). Then \({e}^{\eta }\) has moderate growth along D on \(\mathrm{nb}(K) \ D\) iff \({e}^{\eta \circ \epsilon }\) has moderate growth along D′ on \(\mathrm{nb}(K^\prime) \ D^\prime \simeq \mathrm{ nb}(K) \ D\). It is therefore enough to prove the lemma when η is purely monomial, and the result follows from (9.9). \(\square \)

Given a Stokes filtration \((\mathcal{L},{\mathcal{L}}_{\bullet })\) (of level \(\geq 0\)), we set
$${\mathcal{L}}_{\leq {[\varphi ]}_{\mathcal{l}}} =\sum\limits_{\psi }{\beta }_{{[\psi ]}_{\mathcal{l}}\leq {[\varphi ]}_{\mathcal{l}}}{\mathcal{L}}_{\leq \psi },$$
where the sum is taken in \(\mathcal{L}\). Then
$${\mathcal{L}}_{<{[\varphi ]}_{\mathcal{l}}} :=\sum\limits_{{[\psi ]}_{\mathcal{l}}}{\beta }_{{[\psi ]}_{\mathcal{l}}<{[\varphi ]}_{\mathcal{l}}}{\mathcal{L}}_{\leq {[\psi ]}_{\mathcal{l}}} =\sum\limits_{\psi }{\beta }_{{[\psi ]}_{\mathcal{l}}<{[\varphi ]}_{\mathcal{l}}}{\mathcal{L}}_{\leq \psi }.$$
We can also pre-\(\mathcal{I}\)-filter \({gr }_{{[\varphi ]}_{\mathcal{l}}}\mathcal{L}\) by setting, for \(\psi \in {\mathcal{P}}_{n}\),
$${({gr }_{{[\varphi ]}_{\mathcal{l}}}\mathcal{L})}_{\leq \psi } = ({\mathcal{L}}_{\leq \psi } \cap {\mathcal{L}}_{\leq {[\varphi ]}_{\mathcal{l}}} + {\mathcal{L}}_{<{[\varphi ]}_{\mathcal{l}}})/{\mathcal{L}}_{<{[\varphi ]}_{\mathcal{l}}}.$$

Proposition 9.23.

Assume\((\mathcal{L},{\mathcal{L}}_{\bullet })\)is a Stokes filtration (of level\(\geq 0\)) and let Φ be the finite set of its exponential factors.
  1. 1.

    For each\(\mathcal{l} \in {\mathbb{N}}^{n}\), \({\mathcal{L}}_{\leq {[\bullet ]}_{\mathcal{l}}}\)defines a Stokes filtration\((\mathcal{L},{\mathcal{L}}_{{[\bullet ]}_{\mathcal{l}}})\)of level\(\geq \mathcal{l}\)on \(\mathcal{L}\), \({gr }_{{[\varphi ]}_{\mathcal{l}}}\mathcal{L}\)is locally isomorphic to\({\oplus}_{\psi,\,{[\psi ]}_{\mathcal{l}}={[\varphi ]}_{\mathcal{l}}}{ gr }_{\psi }\mathcal{L}\), and the set of exponential factors of\((\mathcal{L},{\mathcal{L}}_{{[\bullet ]}_{\mathcal{l}}})\)is\(\Phi (\mathcal{l}) := image (\Phi \rightarrow {\mathcal{P}}_{n}(\mathcal{l}))\).

     
  2. 2.

    For every\({[\varphi ]}_{\mathcal{l}} \in \Phi (\mathcal{l})\), \(({gr }_{{[\varphi ]}_{\mathcal{l}}}\mathcal{L},{({gr }_{{[\varphi ]}_{\mathcal{l}}}\mathcal{L})}_{\bullet })\)is a Stokes filtration and its set of exponential factors is the pull-back of\({[\varphi ]}_{\mathcal{l}}\)by\(\Phi \rightarrow \Phi (\mathcal{l})\).

     
  3. 3.
    Let us set
    $${(}{gr }_{\mathcal{l}}\mathcal{L},{({gr }_{\mathcal{l}}\mathcal{L})}_{\bullet }{)} :={ \oplus }_{{[\psi ]}_{\mathcal{l}}\in \Phi (\mathcal{l})}{(}{gr }_{{[\psi ]}_{\mathcal{l}}}\mathcal{L},{({gr }_{{[\psi ]}_{\mathcal{l}}}\mathcal{L})}_{\bullet }{)}.$$
    Then\(({gr }_{\mathcal{l}}\mathcal{L},{({gr }_{\mathcal{l}}\mathcal{L})}_{\bullet })\)is a Stokes-filtered local system (of level\(\geq 0\)) which is locally isomorphic to\((\mathcal{L},{\mathcal{L}}_{\bullet })\).
     

Proof.

Similar to that of Proposition 3.8. \(\square \)

Remark 9.24.

Similarly to Remark 3.9, we note that, as a consequence of the last statement of the proposition, given a fixed Stokes-filtered local system \(({\mathcal{G}}_{\mathcal{l}},{\mathcal{G}}_{\mathcal{l},\bullet })\) graded at the level \(\mathcal{l} \geq 0\), the pointed set of isomorphism classes of Stokes-filtered local systems \((\mathcal{L},{\mathcal{L}}_{\bullet })\) equipped with an isomorphism \({f}_{\mathcal{l}} : ({gr }_{\mathcal{l}}\mathcal{L},{({gr }_{\mathcal{l}}\mathcal{L})}_{\bullet }){ \sim \atop \rightarrow } ({\mathcal{G}}_{\mathcal{l}},{\mathcal{G}}_{\mathcal{l},\bullet })\) is in bijection with the pointed set \({H}^{1}{(}{({S}^{1})}^{\mathcal{l}},{\mathcal{A}\!\mathit{ut} }^{<0}({\mathcal{G}}_{\mathcal{l}},{\mathcal{G}}_{\mathcal{l},\bullet }){)}\).

Proof (Proof of Proposition 9.21). 

Let Φ be a good finite set in \({\mathcal{P}}_{n}\) such that \(\#\Phi \geq 2\). As in Remark 9.13(5), let us set \(m = m(\Phi ) =\max \{ m(\varphi - \psi )\mid \varphi \neq \psi \in \Phi \}\) and let us fix \({\varphi }_{o} \in \Phi \) for which there exists \(\varphi \in \Phi \) such that \(m(\varphi - {\varphi }_{o}) = m\). The subset \(\{m(\varphi - {\varphi }_{o})\mid \varphi \in \Phi \}\) is totally ordered, its maximum is \(m\), and we denote by \(\mathcal{l}\) its submaximum (while \(m\) is independent of \({\varphi }_{o}\), \(\mathcal{l}\) may depend on the choice \({\varphi }_{o}\)).

Let \(\varphi \in \Phi - {\varphi }_{o}\). If \(m(\varphi ) = m\), then the image of φ in \((\Phi - {\varphi }_{o})(\mathcal{l})\) is nonzero, otherwise φ is zero. For every \(\varphi \in \Phi - {\varphi }_{o}\), the subset \({(\Phi - {\varphi }_{o})}_{{[\varphi ]}_{\mathcal{l}}} :=\{ \psi \in \Phi - {\varphi }_{o}\mid {[\psi ]}_{\mathcal{l}} = {[\varphi ]}_{\mathcal{l}}\}\) is good (any of a good set is good) and \(m({(\Phi - {\varphi }_{o})}_{{[\varphi ]}_{\mathcal{l}}}) \leq \mathcal{l} < m\) (see Remark 9.13(6)).

Corollary 9.25 (of Proposition 9.23). 

Let\((\mathcal{L},{\mathcal{L}}_{\bullet })\)be a good Stokes filtration, let \(\Phi ^{\prime\prime}\)be a good finite subset of\({\mathcal{P}}_{n}\)containing\(\Phi (\mathcal{L},{\mathcal{L}}_{\bullet })\), set\(m\,=\,m(\Phi ^{\prime\prime})\), fix \({\varphi }_{o}\)as above and let\(\mathcal{l}\)be the corresponding submaximum element of\(\{m(\varphi \,-\,{\varphi }_{o})\mid \varphi \,\in \,\Phi ^{\prime\prime}\}\).

Then, for every\({[\varphi ]}_{\mathcal{l}} \in (\Phi ^{\prime\prime} - {\varphi }_{o})(\mathcal{l})\), \(({gr }_{{[\varphi ]}_{\mathcal{l}}}\mathcal{L}[-{\varphi }_{o}],{({gr }_{{[\varphi ]}_{\mathcal{l}}}\mathcal{L}[-{\varphi }_{o}])}_{\bullet })\)is a good Stokes filtration and\({m}_{\max }({gr }_{{[\varphi ]}_{\mathcal{l}}}\mathcal{L}[-{\varphi }_{o}],{({gr }_{{[\varphi ]}_{\mathcal{l}}}\mathcal{L}[-{\varphi }_{o}])}_{\bullet })\! \leq \! \mathcal{l}\! <\! m\).\(\square \)

Let us fix \({\theta }_{o} \in {({S}^{1})}^{n}\) and \({\alpha }_{1},\ldots,{\alpha }_{n} \in {\mathbb{N}}^{{_\ast}}\) such that \(\gcd ({\alpha }_{1},\ldots,{\alpha }_{n}) = 1\). The map \(\theta \mapsto ({\alpha }_{1}\theta + {\theta }_{o,1},\ldots,{\alpha }_{n}\theta + {\theta }_{o,n})\) embeds \({S}^{1}\) in \({({S}^{1})}^{n}\). In the following, \({S}_{\alpha,{\theta }_{o}}^{1}\) denotes this circle.

Let \(\Phi \subset {\mathcal{P}}_{n}\) be good finite set. Let us describe the Stokes hypersurfaces \(\mathrm{St}(\varphi,\psi )\) with \(\varphi \neq \psi \in \Phi \). Since \(\varphi - \psi \) is purely monomial, it is written \({u}_{m}(t){t}^{-m}\) with \(\mathbf{m}=(m_1,\dots,m_n)\in\mathbb{N}^n\smallsetminus\{0\}\) and \({u}_{m}(0)\neq 0\). Then
$$\mathrm{St}(\varphi,\psi ) = \{({\theta }_{1},\ldots,{\theta }_{n}) \in {({S}^{1})}^{n}\mid { \sum }_{ j}{m}_{j}{\theta }_{j} -\arg {u}_{m}(0) = \pm \pi /2\, {\rm mod}\,\,2\pi \},$$
so in particular it is the union of translated subtori of codimension one. As a consequence, the circle \({S}_{\alpha,{\theta }_{o}}^{1}\) intersects transversally every Stokes hypersurface. We callStokes points with respect to Φ the intersection points when \(\varphi,\psi \) vary in Φ.

Lemma 9.26.

Let \(I\)be any open interval of\({S}_{\alpha,{\theta }_{o}}^{1}\)such that, for any\(\varphi,\psi \in \Phi \), \(card (I \cap \mathrm{ St}(\varphi,\psi )) \leq 1\). Then there exists an open neighbourhood\(\mathrm{nb}(I)\)such that the decompositions (1.38) hold on \(\mathrm{nb}(I)\).

Proof.

A proof similar to that of Lemma 3.12 gives that \({H}^{1}(I,{\mathcal{L}}_{<\psi \vert I}) = 0\) for any \(\psi \). We can then lift for any \(\psi \in \Phi \) a basis of global sections of \({gr }_{\psi }{\mathcal{L}}_{\vert I}\) as a family sections of \({\mathcal{L}}_{\leq \psi \vert I}\), which are defined on some \(\mathrm{nb}(I)\). The images of these sections in \({gr }_{\psi }{\mathcal{L}}_{\vert \mathrm{nb}(I)}\) restrict to the given basis of \({gr }_{\psi }{\mathcal{L}}_{\vert I}\) and thus form a basis of \({gr }_{\psi }{\mathcal{L}}_{\vert \mathrm{nb}(I)}\) if \(\mathrm{nb}(I)\) is simply connected, since \({gr }_{\psi }\mathcal{L}\) is a locally constant sheaf. We therefore get a section \({gr }_{\psi }{\mathcal{L}}_{\vert \mathrm{nb}(I)} \rightarrow {\mathcal{L}}_{\leq \psi \vert \mathrm{nb}(I)}\) of the projection \({\mathcal{L}}_{\leq \psi \vert \mathrm{nb}(I)} \rightarrow { gr }_{\psi }{\mathcal{L}}_{\vert \mathrm{nb}(I)}\).

For every \(\varphi \in \Phi \), we have a natural inclusion \({\beta }_{\psi \leq \varphi }{\mathcal{L}}_{\leq \psi } `\rightarrow {\mathcal{L}}_{\leq \varphi }\), and we deduce a morphism \({\oplus}_{\psi \in \Phi }{\beta }_{\psi \leq \varphi }{ gr }_{\psi }{\mathcal{L}}_{\vert \mathrm{nb}(I)} \rightarrow {\mathcal{L}}_{\leq \varphi \vert \mathrm{nb}(I)}\), which is seen to be an isomorphism on stalks at points of \(I\), hence on a sufficiently small \(\mathrm{nb}(I)\), according to the local decomposition (1.38). The same result holds then for any \(\eta \in {\mathcal{P}}_{n}\) instead of φ, since \({\mathcal{L}}_{\leq \eta } =\sum\limits_{\varphi \in \Phi }{\beta }_{\varphi \leq \eta }{\mathcal{L}}_{\leq \varphi }\) and similarly for the graded pieces. \(\square \)

Corollary 9.27.

In the setting of Corollary 9.25,let us set\(m = ({m}_{1},\ldots,{m}_{n})\)and\(m =\sum\limits_{i}{m}_{i}{\alpha }_{i}\). Let \(I\)be any open interval of \({S}^{1}\)of length\(\pi /m\)with no Stokes points as boundary points. Then, if\(\mathrm{nb}(I)\)is a sufficiently small tubular neighbourhood of \(I\), \({(\mathcal{L},{\mathcal{L}}_{\bullet })}_{\vert \mathrm{nb}(I)} \simeq {({gr }_{\mathcal{l}}\mathcal{L},{({gr }_{\mathcal{l}}\mathcal{L})}_{\bullet })}_{\vert \mathrm{nb}(I)}\).

Proof.

By the choice of \(m\) and the definition of \(m\)\(I\) satisfies the assumption of Lemma 9.26 for both \((\mathcal{L},{\mathcal{L}}_{\bullet })\) and \(({gr }_{\mathcal{l}}\mathcal{L},{({gr }_{\mathcal{l}}\mathcal{L})}_{\bullet })\), hence, when restricted to \(\mathrm{nb}(I)\), both are isomorphic to the trivial Stokes filtration determined by \(gr \mathcal{L}\) restricted to \(\mathrm{nb}(I)\). \(\square \)

End of the proof of Proposition 9.21. Let \(\lambda : (\mathcal{L},{\mathcal{L}}_{\bullet }) \rightarrow (\mathcal{L}^\prime,{\mathcal{L}^\prime}_{\bullet })\) be a morphism of Stokes-filtered local systems on \({({S}^{1})}^{n}\) with set of exponential factors contained in \(\Phi ^{\prime\prime}\). The proof that \(\lambda \) is strict and that \(\ker \,\lambda \), \(Im \lambda \) and \(Coker \,\lambda \) (equipped with the naturally induced pre-\({\mathcal{P}}_{n}\)-filtrations) are Stokes filtrations follows from the local decomposition of the morphism, the proof of which is be done by induction on \(m = m(\Phi ^{\prime\prime})\), with \(\Phi ^{\prime\prime} = \Phi \cup \Phi ^\prime\). The result is clear if \(m = 0\) (so \(\Phi ^{\prime\prime} =\{ 0\}\)), as both Stokes filtrations have only one jump. The remaining part of the inductive step is completely similar to the end of the proof of Theorem 3.5 by working on \(\mathrm{nb}(I)\) instead of \(I\), and we will not repeat it. We obtain that, for any such \(I\), \({\lambda }_{\vert \mathrm{nb}(I)}\) is graded, so this ends the proof of the proposition. \(\square \)

9.6 Behaviour by Pull-Back

Let \(f : (X^\prime,D^\prime) \rightarrow (X,D)\) be a mapping as in Remark 9.13(2). According to Sect. 8.2, there is a natural morphism \(\widetilde{f}\,:\,\widetilde{X}^\prime({D^\prime}_{j^\prime\in J^\prime}) \rightarrow \widetilde{ X}({D}_{j\in J})\) lifting \(f : X^\prime\,\rightarrow \,X\). There are natural inclusions \(\widetilde{j} : X \ D = {X}^{{_\ast}} `\rightarrow \widetilde{ X}({D}_{j\in J})\) and \(\widetilde{j}^\prime : X^\prime \ D^\prime = {X}^{{\prime}{_\ast}} `\rightarrow \widetilde{ X}^\prime({D^\prime}_{j^\prime\in J^\prime})\), and we have \(\widetilde{f} \circ \widetilde{ j}^\prime =\widetilde{ j} \circ f\).

Let us describe such a mapping in a local setting: the space \((X,D)\) is the polydisc \({\Delta }^{n}\) with coordinates \(({x}_{1},\ldots,{x}_{n})\) and \(D =\{ {x}_{1}\cdots {x}_{\mathcal{l}} = 0\}\), and similarly for \((X^\prime,D^\prime)\), and \(f(0) = 0\) in these coordinates. We have coordinates \(({\theta }_{1},\ldots,{\theta }_{\mathcal{l}},{\rho }_{1},\ldots,{\rho }_{\mathcal{l}},{x}_{\mathcal{l}+1},\ldots,{x}_{n})\) on \(\widetilde{X}\), and similarly for \(\widetilde{X}^\prime\). In these local coordinates, we set \(f = ({f}_{1},\ldots,{f}_{n})\), with
$${f}_{1}(x^\prime) = {u^\prime}_{1}(x^\prime){x}^{{\prime}{k}_{1} },\ldots,{f}_{\mathcal{l}}(x^\prime) = {u^\prime}_{\mathcal{l}}(x^\prime){x}^{{\prime}{k}_{\mathcal{l}} },$$
(9.28)
where \({u^\prime}_{j}(x^\prime)\) are local units, and \(\mathbf{k}_j=(k_{j,1},\dots,k_{j,\ell^\prime})\in\mathbb{N}^{\ell^\prime}\smallsetminus\{0\}\). We also have \({f}_{j}(0) = 0\) for \(j \geq \mathcal{l} + 1\). We note that the stratum \({D^\prime}_{L^\prime}\) going through the origin in \(X^\prime\) (defined by \({x^\prime}_{1} = \cdots = {x^\prime}_{\mathcal{l}^\prime} = 0\)) is sent to the stratum \({D}_{L}\) going to the origin in X (defined by \({x}_{1} = \cdots = {x}_{\mathcal{l}} = 0\)), maybe not submersively.
When restricted to \({\varpi }^{{\prime}-1}({D^\prime}_{L^\prime})\) defined by the equations \({\rho ^\prime}_{j^\prime}\! =\! 0\), \(j^\prime\! =\! 1,\ldots,\mathcal{l}^\prime\), the map \(\widetilde{f}\) takes values in \({\varpi }^{-1}({D}_{L})\) and is given by the formula (with \({\rho }_{1} = \cdots = {\rho }_{\mathcal{l}} = 0\)):
$$({\theta ^\prime}_{1},\ldots,{\theta ^\prime}_{\mathcal{l}^\prime},{x^\prime}_{\mathcal{l}^\prime+1},\ldots,{x^\prime}_{n^\prime})\mapsto \left (\begin{array}{c} \sum {k}_{1,i}{\theta ^\prime}_{i} +\arg {u^\prime}_{1}(0,{x^\prime}_{\mathcal{l}^\prime+1},\ldots,{x^\prime}_{n^\prime})\\ \vdots \\ \sum {k}_{\mathcal{l},i}{\theta ^\prime}_{i} +\arg {u^\prime}_{\mathcal{l}}(0,{x^\prime}_{\mathcal{l}^\prime+1},\ldots,{x^\prime}_{n^\prime}) \\ {f}_{\mathcal{l}+1}(0,{x^\prime}_{\mathcal{l}^\prime+1},\ldots,{x^\prime}_{n^\prime})\\ \vdots \\ {f}_{n}(0,{x^\prime}_{\mathcal{l}^\prime+1},\ldots,{x^\prime}_{n^\prime}) \end{array} \right )$$
(9.29)

Going back to the global setting, we have a natural morphism \({f}^{{_\ast}} :\widetilde{ {f}}^{-1}\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}} \rightarrow \widetilde{ {j}^\prime}_{{_\ast}}{\mathcal{O}}_{{X}^{{\prime}{_\ast}}}\), which sends \(\widetilde{{f}}^{-1}{(\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}})}^{\mathrm{lb}}\) to \({(\widetilde{{j}^\prime}_{{_\ast}}{\mathcal{O}}_{{X}^{{\prime}{_\ast}}})}^{\mathrm{lb}}\). This morphism is compatible with the order: it sends \(\widetilde{{f}}^{-1}{\mathcal{A}}_{\,\widetilde{X}({D}_{j\in J})}^{\mathrm{mod}\,D}\) to \({\mathcal{A}}_{\,\widetilde{X}^\prime({D^\prime}_{j^\prime\in J^\prime})}^{\mathrm{mod}\,D^\prime}\).

We consider the sheaves \(\widetilde{\mathcal{I}}\) and \(\mathcal{I}\) on \(\widetilde{X}\) and \(\widetilde{\mathcal{I}}^\prime\) and \(\mathcal{I}^\prime\) on \(\widetilde{X}^\prime\), relative to the divisors D and D′.

Proposition 9.30.

The morphism fsends\(\widetilde{{f}}^{-1}\widetilde{\mathcal{I}}\)to\(\widetilde{\mathcal{I}}^\prime\)and induces a morphism\({f}^{{_\ast}} :\widetilde{ {f}}^{-1}\mathcal{I} \rightarrow \mathcal{I}^\prime\), which is compatible with the order. Moreover, if\(\widetilde{f} :\widetilde{ X}^\prime \rightarrow \widetilde{ X}\)is open, the morphism fis injective and strictly compatible with the order.

Remark 9.31.

If \(\dim X = 1\), then \(\widetilde{f}\) is open. Indeed, there are local coordinates in \(X^\prime\) where f is expressed as a monomial. The assertion is easy to see in this case.

Proof (Proof of Proposition  9.30). 

Let us prove the first statement. It is clear that f ∗  sends \(\widetilde{{f}}^{-1}\widetilde{{\mathcal{I}}}_{1}\) to \(\widetilde{{\mathcal{I}}^\prime}_{1}\). In general, we note that the assertion is local on X and \(X^\prime\) and, given a local ramified covering \({\rho }_{d} : {X}_{d} \rightarrow X\), there is a commutative diagram for a suitable \(d^\prime\) (this is easily seen in local coordinates in X and \(X^\prime\) adapted to D and D′). The morphism \({\rho }_{d^\prime}^{{\prime}-1}{f}^{{_\ast}} : {\rho }_{d^\prime}^{{\prime}-1}{f}^{-1}{\mathcal{O}}_{{X}^{{_\ast}}} \rightarrow {\rho }_{d^\prime}^{{\prime}-1}{\mathcal{O}}_{{X}^{{\prime}{_\ast}}}\) is identified with \({g}^{{_\ast}} : {g}^{-1}{\mathcal{O}}_{{X}_{d}^{{_\ast}}} \rightarrow {\mathcal{O}}_{{X}_{d^\prime}^{{\prime}{_\ast}}}\) since \({\rho }_{d}\) and \({\rho ^\prime}_{d^\prime}\) are coverings, and f ∗  is recovered from \({g}^{{_\ast}}\) as the restriction of \({\rho ^\prime}_{d^\prime,{_\ast}}({g}^{{_\ast}})\) to \({\mathcal{O}}_{{X}^{{\prime}{_\ast}}} \subset {\rho ^\prime}_{d^\prime,{_\ast}}{\mathcal{O}}_{{X}_{d^\prime}^{{\prime}{_\ast}}}\).

As we know that \({g}^{{_\ast}}\) sends \(\widetilde{{g}}^{-1}{\varpi }_{d}^{-1}{\mathcal{O}}_{{X}_{d}}({_\ast}D)\) to \({\varpi }_{d^\prime}^{{\prime}-1}{\mathcal{O}}_{{X^\prime}_{d^\prime}}({_\ast}D^\prime)\), we conclude by applying \({\rho ^\prime}_{d^\prime,{_\ast}}\) and intersecting with \(\widetilde{{j}^\prime}_{{_\ast}}{\mathcal{O}}_{{X}^{{\prime}{_\ast}}}\) that f ∗  sends \(\widetilde{{f}}^{-1}\widetilde{{\mathcal{I}}}_{d}\) to \(\widetilde{{\mathcal{I}}^\prime}_{d^\prime}\), hence in \(\widetilde{\mathcal{I}}^\prime\).

For the injectivity statement, it is enough to prove that, if \(\widetilde{f} :\widetilde{ X}^\prime \rightarrow \widetilde{ X}\) is open, then \({f}^{{_\ast}} :\widetilde{ {f}}^{-1}\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}}/\widetilde{{f}}^{-1}{(\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}})}^{\mathrm{lb}} \rightarrow \widetilde{ {j}^\prime}_{{_\ast}}{\mathcal{O}}_{{X}^{{\prime}{_\ast}}}/{(\widetilde{{j}^\prime}_{{_\ast}}{\mathcal{O}}_{{X}^{{\prime}{_\ast}}})}^{\mathrm{lb}}\) is injective. Let \(y^\prime \in \partial \widetilde{X}^\prime\) and set \(y =\widetilde{ f}(y^\prime) \in \partial \widetilde{X}\). Note first that \(\partial \widetilde{X}^\prime =\widetilde{ {f}}^{-1}(\partial \widetilde{X})\). If \(V (y^\prime)\) is an open neighbourhood of \(y^\prime\) in \(\widetilde{X}^\prime\), then \(\widetilde{f}(V (y^\prime))\) is an open neighbourhood of \(y\) in \(\widetilde{X}\) by the openness assumption, and we have
$$\widetilde{f}(V (y^\prime)) \partial \widetilde{X} =\widetilde{ f}{(}V (y^\prime) \widetilde{ {f}}^{-1}(\partial \widetilde{X}){)} =\widetilde{ f}(V (y^\prime) \ \partial \widetilde{X}^\prime).$$
If \(\lambda \) is a local section of \(\widetilde{{f}}^{-1}\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}}\) at \(y^\prime\), it is defined on such a \(V (y^\prime)\). If \({f}^{{_\ast}}\lambda \) is bounded on \(V (y^\prime) \ \partial \widetilde{X}^\prime\), then \(\lambda \) is a bounded section of \({\mathcal{O}}_{{X}^{{_\ast}}}\) on the open set \(\widetilde{f}(V (y^\prime) \ \partial \widetilde{X}^\prime) =\widetilde{ f}(V (y^\prime)) \ \partial \widetilde{X}\), hence is a local section of \(\widetilde{{f}}^{-1}{(\widetilde{{j}}_{{_\ast}}{\mathcal{O}}_{{X}^{{_\ast}}})}^{\mathrm{lb}}\).

Let us show the strictness property. It means that, for any \(y^\prime \in \widetilde{ {f}}^{-1}(y)\), \(\varphi \circ f{\leq }_{{}_{y^\prime}}\! 0\) implies \(\varphi {\leq }_{{}_{y}}\! 0\) if \(\varphi \in {\mathcal{I}}_{\widetilde{X},y}\). Let \(h\) be a local equation of D and set \(h^\prime = h \circ f\). The relation \(\varphi \circ f {\leq }_{{}_{y^\prime}}\! 0\) means \(\vert {e}^{\varphi } \circ f\vert \leq \vert h^\prime{\vert }^{-N} = \vert h \circ f{\vert }^{-N}\) for some \(N \geq 0\) on \(V (y^\prime) \ \partial \widetilde{X}^\prime\). We then have \(\vert {e}^{\varphi }\vert \leq \vert h{\vert }^{-N}\) on \(\widetilde{f}(V (y^\prime)) \ \partial \widetilde{X}\), hence \(\varphi {\leq }_{{}_{y}}\! 0\) according to the openness of \(\widetilde{f}\). \(\square \)

In general, the map f ∗  may not be injective. Indeed, (in the local setting) given \(\varphi \in {\mathcal{O}}_{X,0}({_\ast}D)/{\mathcal{O}}_{X,0}\), \({f}^{{_\ast}}\varphi \) may have no poles along D′ near \(0 \in X^\prime\). More precisely, let φ be a local section of \(\mathcal{I}\) at 0 ∈ X and let \(\widetilde{{\Sigma }}_{\varphi } \subset {\mathcal{I}}^{\acute{\text{e}t}}\) be the image by φ of a small open neighbourhood of 0. Then f ∗  induces a map from \(\widetilde{{f}}^{-1}\widetilde{{\Sigma }}_{\varphi }\) to \({\mathcal{I}}^{{\prime}\acute{\text{e}t}}\) whose image is equal to \(\widetilde{{\Sigma }}_{{f}^{{_\ast}}\varphi }\).

With a goodness assumption (which is automatically satisfied in dimension one) we recover the injectivity.

Lemma 9.32.

Assume that\(\varphi \in {\mathcal{O}}_{X,0}({_\ast}D)/{\mathcal{O}}_{X,0}\)is purely monomial. Then,
$${f}^{{_\ast}}\varphi = 0\Longrightarrow\varphi = 0.$$

Proof.

If \(\varphi \neq 0\) let us set \(\varphi = U(x)/{x}^{m}\), where \(U(x)\) is a local unit and \(\mathbf{m}=(m_1,\dots,m_{\ell})\in\mathbb{N}^{\ell}\smallsetminus\{0\}\). Using the notation above for f, we have \({f}^{{_\ast}}\varphi = U^\prime(x^\prime)/{x}^{{\prime}m^\prime}\), where \(U^\prime = {f}^{{_\ast}}U/{u}_{1}^{{\prime}{m}_{1}}\cdots {u}_{\mathcal{l}}^{{\prime}{m}_{\mathcal{l}}}\) is a local unit and \(m^\prime = {m}_{1}{k}_{1} + \cdots + {m}_{\mathcal{l}}{k}_{\mathcal{l}}\). Then \(\mathbf{m}^\prime\in\mathbb{N}^{\ell^\prime}\smallsetminus\{0\}\), so \({f}^{{_\ast}}\varphi \neq 0\). \(\square \)

Corollary 9.33.

Let φ be a local section of\(\mathcal{I}\)at 0 ∈ X which is purely monomial. Then fin injective on\(\widetilde{{f}}^{-1}{\Sigma }_{\varphi }\).\(\square \)

We also recover a property similar to strictness.

Lemma 9.34.

With the same assumption as in Lemma 9.32, if\({f}^{{_\ast}}\varphi \leq 0\)(resp. \({f}^{{_\ast}}\varphi < 0\)) at\(({\theta ^\prime}_{o},0) \in \partial \widetilde{{X}^\prime}_{\vert {D}_{L^\prime}}\), then\(\varphi \leq 0\)(resp. \(\varphi < 0\)) at\(({\theta }_{o},0) =\widetilde{ f}({\theta ^\prime}_{o},0)\).

Proof.

We keep the same notation as above, and it is enough to consider the case \({f}^{{_\ast}}\varphi < 0\), according to Lemma 9.32. The assumption can then be written as
$$\arg U^\prime(0) -\langle {m}_{1}{k}_{1} + \cdots + {m}_{\mathcal{l}}{k}_{\mathcal{l}},{\theta ^\prime}_{o}\rangle \in (\pi /2,3\pi /2)\,\mathrm{mod}\,2\pi,$$
where \(\langle \,,\rangle\) is the standard scalar product on \({\mathbb{R}}^{\mathcal{l}^\prime}\). Notice now that \({\theta }_{o,j} =\langle { k}_{j},{\theta ^\prime}_{o}\rangle +\arg {u^\prime}_{j}(0)\) for \(j = 1,\ldots,\mathcal{l}\), so that the previous relation is written as
$$\arg U(0) -\sum {m}_{j}{\theta }_{o,j} \in (\pi /2,3\pi /2)\,\mathrm{mod}\,2\pi,$$
which precisely means that \(\varphi {< }_{{}_{{\theta }_{o}}}\! 0\). \(\square \)

Let now \((\mathcal{L},{\mathcal{L}}_{\bullet })\) be a Stokes-filtered local system on \(\partial \widetilde{X}\). Itspull-back \({f}^{+}(\mathcal{L},{\mathcal{L}}_{\bullet })\) (see Definition 1.33), which is a priori a pre-\(\mathcal{I}\)-filtered local system, is also a Stokes-filtered local system (see Lemma 1.40), and its associated stratified \(\mathcal{I}\)-covering is \({f}^{{_\ast}}(\widetilde{{f}}^{-1}\widetilde{\Sigma }(\mathcal{L}))\) (see Lemma 1.48).

Proposition 9.35.

With the previous assumptions on f, let us assume that\((\mathcal{L},{\mathcal{L}}_{\bullet })\)is good. Then\({f}^{+}(\mathcal{L},{\mathcal{L}}_{\bullet })\)is also good.

Proof.

According to the previous considerations, it remains to check that, if \(\widetilde{\Sigma }\) is a good stratified \(\mathcal{I}\)-covering, then \({f}^{{_\ast}}(\widetilde{{f}}^{-1}\widetilde{\Sigma })\) is also good, and this reduces to showing that, if \(\varphi \in {\mathcal{O}}_{X,0}({_\ast}D)/{\mathcal{O}}_{X,0}\) is purely monomial, then so is \({f}^{{_\ast}}\varphi \), a property that we already saw in the proof of Lemma 9.32. \(\square \)

9.7 Partially Regular Stokes-Filtered Local Systems

In the setting of Sect. 9.2, let \((\mathcal{L},{\mathcal{L}}_{\bullet })\) be a Stokes-filtered local system on \(\partial \widetilde{X}({D}_{j\in J})\) with associated stratified \(\mathcal{I}\)-covering \(\widetilde{\Sigma }\) equal to the zero section of \({\mathcal{I}}_{\vert \partial \widetilde{X}}^{\acute{\text{e}t}}\). In such a case, we will say that \((\mathcal{L},{\mathcal{L}}_{\bullet })\) isregular. Then \((\mathcal{L},{\mathcal{L}}_{\bullet })\) is the graded Stokes-filtered local system with jump at \(\varphi = 0\) only. The category of regular Stokes-filtered local systems is then equivalent to the category of local systems on \(\partial \widetilde{X}({D}_{j\in J})\).

We now consider the case where \((\mathcal{L},{\mathcal{L}}_{\bullet })\) ispartially regular, that is, there exists a decomposition \(J = J^\prime \cup J^{\prime\prime}\) such that its associated stratified covering \(\widetilde{\Sigma }\) reduces to the zero section when restricted over \(D(J^{\prime\prime}) \ D(J^\prime)\) (recall that \(D(I) =cup_{j\in I}{D}_{j}\)). We will set \(D^\prime = D(J^\prime)\) and \(D^{\prime\prime} = D(J^{\prime\prime})\) for simplicity. Near each point of \(D^{\prime\prime} \ D^\prime\) the Stokes-filtered local system is regular. We will now analyze its local behaviour near \(D^{\prime\prime} \cap D^\prime\). We will restrict to a local analysis in the non-ramified case.

According to Sect. 8.2, the identity map X → X lifts as a map \(\pi :\widetilde{ X} :=\widetilde{ X}({D}_{j\in J}) \rightarrow \widetilde{ X}^\prime :=\widetilde{ X}({D}_{j\in J^\prime})\) and we have a commutative diagram The boundary \(\partial \widetilde{X}\) of \(\widetilde{X}\) is \({\varpi }^{-1}(D)\) and \(\partial \widetilde{X}^\prime = {\varpi }^{{\prime}-1}(D^\prime)\).

We now consider the local setting of Sect. 9.4 and we set \(\mathcal{l} = \mathcal{l}^\prime + \mathcal{l}^{\prime\prime}\), \(L^\prime =\{ 1,\ldots,\mathcal{l}^\prime\}\) and \(L^{\prime\prime} =\{ \mathcal{l}^\prime + 1,\ldots,\mathcal{l}\}\). If \((\mathcal{L},{\mathcal{L}}_{\bullet })\) is a non-ramified Stokes-filtered local system with associated \(\mathcal{I}\)-covering equal to \(\widetilde{\Sigma }\), we assume that \(\widetilde{{\Sigma }}_{\vert {D}_{L}}\) is a trivial covering of \({Y }_{L} = {\varpi }^{-1}({D}_{L})\). Then \(\widetilde{\Sigma }\) is determined by a finite set \(\Phi \subset {\mathcal{O}}_{X,0}({_\ast}D)/{\mathcal{O}}_{X,0}\). The partial regularity property means that the representatives φ of the elements of Φ are holomorphic away from D′, that is, have no poles along D′, that is also, \(\Phi \subset {\mathcal{O}}_{X,0}({_\ast}D^\prime)/{\mathcal{O}}_{X,0}\), and Φ defines a trivial stratified \({\mathcal{I}}_{\widetilde{X}^\prime}\)-covering \(\widetilde{\Sigma }^\prime\). The map \(q\) induces an homeomorphism of \({\pi }^{-1}\widetilde{\Sigma }^\prime\) onto \(\widetilde{\Sigma }\).

Proposition 9.37.

In this local setting, the category of non-ramified Stokes-filtered local systems on\(\partial \widetilde{X}{({D}_{j\in L})}_{\vert {D}_{L}}\)with associated stratified\(\mathcal{I}\)-covering contained in \(\widetilde{\Sigma }\)is equivalent to the category of non-ramified Stokes-filtered local systems on\(\partial \widetilde{X}{({D}_{j\in L^\prime})}_{\vert {D}_{L}}\)with associated stratified\(\mathcal{I}\)-covering contained in\(\widetilde{\Sigma }^\prime\), equipped with commuting automorphisms Tk(\(k \in L^{\prime\prime}\)).

Proof.

Let us first consider the following general setting: \(\mathcal{F}\) is a \(\mathbb{R}\)-constructible sheaf on \(Z \times {({S}^{1})}^{k}\), where Z is a nice space (e.g. a subanalytic subset of \({\mathbb{R}}^{n}\)). We denote by \(\pi : Z \times {({S}^{1})}^{k} \rightarrow Z\) the projection and by \(\rho : Z \times {\mathbb{R}}^{k} \rightarrow Z \times {({S}^{1})}^{k}\) the map \((z,{\theta }_{1},\ldots,{\theta }_{k})\mapsto (z,{e}^{i{\theta }_{1}},\ldots,{e}^{i{\theta }_{k}})\). We will also set \(\widetilde{\pi } = \rho \circ \pi \).

Lemma 9.38.

The category of\(\mathbb{R}\)-constructible sheaves\(\mathcal{F}\)on\(Z \times {({S}^{1})}^{k}\)whose restriction to each fibre of π is locally constant is naturally equivalent to the category of\(\mathbb{R}\)-constructible sheaves on Z equipped with commuting automorphisms\({T}_{1},\ldots,{T}_{k}\).

Proof.

Let \({\sigma }_{i}\) (\(i = 1,\ldots,k\)) denote the translation by one in the direction of the \(i\)-th coordinate in \({\mathbb{R}}^{k}\). Then the functor \({\rho }^{-1}\) induces an equivalence between the category of sheaves \(\mathcal{F}\) on \(Z \times {({S}^{1})}^{k}\) and that of sheaves \(\mathcal{G}\) on \(Z \times {\mathbb{R}}^{k}\) equipped with isomorphisms \({\sigma }_{i}^{-1}\mathcal{G}{ \sim \atop \rightarrow } \mathcal{G}\) which commute in a natural way. It induces an equivalence between the corresponding full subcategories of \(\mathbb{R}\)-constructible sheaves which are locally constant in the fibres of π and \(\widetilde{\pi }\).

Let \(\mathcal{G}\) be a \(\mathbb{R}\)-constructible sheaf on \(Z \times {\mathbb{R}}^{k}\). We have a natural (dual) adjunction morphism \(\mathcal{G}\rightarrow \widetilde{ {\pi }}^{!}R\widetilde{{\pi }}_{!}\mathcal{G} =\widetilde{ {\pi }}^{-1}R\widetilde{{\pi }}_{!}\mathcal{G}[k]\) (see [39, Proposition 3.3.2] for the second equality), which is an isomorphism if \(\mathcal{G}\) is locally constant (hence constant) in the fibres of \(\widetilde{\pi }\) (see [39, Proposition 2.6.7]). This shows that (via \({R}^{k}\widetilde{{\pi }}_{!}\) and \(\widetilde{{\pi }}^{-1}\)) the category of \(\mathbb{R}\)-constructible sheaves on \(Z \times {\mathbb{R}}^{k}\) which are constant in the fibres of \(\widetilde{\pi }\) is equivalent to the category of \(\mathbb{R}\)-constructible sheaves on Z. If now \(\mathcal{H}\) is a \(\mathbb{R}\)-constructible sheaf on Z with commuting automorphisms Ti (\(i = 1,\ldots,k\)), it produces a sheaf \(\mathcal{G} =\widetilde{ {\pi }}^{-1}\mathcal{H}\) with commuting isomorphisms \({\sigma }_{i}^{-1}\mathcal{G}\simeq \mathcal{G}\) by composing the natural morphism \({\sigma }_{i}^{-1}\widetilde{{\pi }}^{-1}\mathcal{H}\rightarrow \widetilde{ {\pi }}^{-1}\mathcal{H}\) with Ti. \(\square \)

Let us end the proof of the proposition. We know that the first category considered in the proposition is equivalent to that of Stokes-filtered local systems indexed by Φ. Each \({\mathcal{L}}_{\leq \varphi }\) is locally constant in the fibres of π, due to the local grading property of \((\mathcal{L},{\mathcal{L}}_{\bullet })\). We can therefore apply Lemma 9.38, since each \({\mathcal{L}}_{\leq \varphi }\) is \(\mathbb{R}\)-constructible, to get the essential surjectivity. The full faithfulness is obtained in the same way. \(\square \)

Remark 9.39.

The statement of Proposition 9.37 does not extend as it is to the ramified case. Indeed, even if \(\widetilde{\Sigma }\) is regular along D′, a ramification may be necessary along D′ to trivialize \(\widetilde{{\Sigma }}_{\vert {Y }_{L}}\) (e.g. a local section of \(\widetilde{{\Sigma }}_{\vert {Y }_{L}}\) is written \(a(y^\prime,y^{\prime\prime})/{y}^{{\prime}k}\) for ramified coordinates y′, y′, and a is possibly not of the form \(a(y^\prime,x^{\prime\prime})\)).

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Authors and Affiliations

  • Claude Sabbah
    • 1
  1. 1.CNRS École polytechniqueCentre de mathématiques Laurent SchwartzPalaiseauFrance

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