# Three results on holonomic D-modules

Claude Sabbah

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Claude Sabbah. Three results on holonomic D-modules. 2025. hal-05060791v2

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Distributed under a Creative Commons Attribution 4.0 International License
# THREE RESULTS ON HOLONOMIC $D$-MODULES

## CLAUDE SABBAH

ABSTRACT. In this text, we illustrate the use of local methods in the theory of (irregular) holonomic $D$-modules.

I. (The Euler characteristic of the de Rham complex) We show the invariance of the global or local Euler characteristic of the de Rham complex after localization and dual localization of a holonomic $D$-module along a hypersurface, as well as after tensoring with a rank one meromorphic connection with regular singularities.

II. (Local generic vanishing theorems for holonomic $D$-modules) We prove that the natural morphism from the proper pushforward to the total pushforward of an algebraic holonomic $D$-module by an open inclusion is an isomorphism if we first twist the $D$-module structure by suitable closed algebraic differential forms.

III. (Laplace transform of a Stokes-filtered constructible sheaf of exponential type) Motivated by the construction in [YZ24], we propose a slightly different construction of the Laplace transform of a Stokes-perverse sheaf on the projective line and show directly that it corresponds to the Laplace transform of the corresponding holonomic $D$-module via the Riemann-Hilbert-Birkhoff-Deligne-Malgrange correspondence. This completes the presentation given in [Sab13, Chap. 7], where only the other direction of the Laplace transformation is analyzed. We also compare our approach with the construction made previously in [YZ24].

## CONTENTS

<table><tr><td>Introduction</td><td>2</td></tr><tr><td><i>Part 1. The Euler characteristic of the de Rham complex</i></td><td>2</td></tr><tr><td>1. Introduction to Part 1</td><td>2</td></tr><tr><td>2. Proof of Theorem 1.6</td><td>4</td></tr><tr><td>3. Reduction of the proof of Theorem 1.3 to that of Theorem 1.6</td><td>7</td></tr><tr><td><i>Part 2. Local generic vanishing theorems for holonomic D-modules</i></td><td>8</td></tr><tr><td>4. Local and relative generic vanishing theorems</td><td>8</td></tr><tr><td>5. A criterion for local vanishing</td><td>11</td></tr><tr><td>6. Proof of Theorem 4.5</td><td>13</td></tr><tr><td><i>Part 3. Laplace transform of a Stokes-filtered constructible sheaf of exponential type</i></td><td>16</td></tr><tr><td>7. Introduction to Part 3</td><td>16</td></tr><tr><td>8. The Laplace transformation for Stokes-filtered local systems of exponential type</td><td>20</td></tr><tr><td>9. The Riemann-Hilbert correspondence</td><td>30</td></tr><tr><td><i>References</i></td><td>36</td></tr></table>

2020 Mathematics Subject Classification. 14F10, 14F17, 32C38, 34M40, 35A27.

Key words and phrases. Euler characteristic, de Rham complex, holonomic $D$-module, irregular singularity, vanishing theorems, Laplace transformation, Stokes-filtered local system, Stokes-perverse sheaf, regular singular connection, irregular connection of exponential type.
INTRODUCTION

The theory of holonomic $\mathcal{D}_X$-modules (with possibly irregular singularities) on a complex manifold $X$ has been developed thanks to the fundamental result of K. Ked-laya [Ked11] and its algebraic analogue due to T. Mochizuki [Moc09b, Moc11b]. The latter makes it possible to manipulate Stokes structures in dimension $\ge 2$, as already illustrated in several works, such as, in particular, [Moc14, Moc11a, Sab17, Tey23], and of course by the construction of the Riemann-Hilbert correspondence [DK16].

In this note, we give a few more examples of uses of the methods developed in these articles in relation to questions raised or resolved in Gabriel Ribeiro's thesis [Rib24] and the article [YZ24] by Tony Yue Yu and Shaowu Zhang. Our aim is to illustrate the use (or show how to avoid it) of Stokes structures in dimension $\ge 2$. The main results are Theorems 1.1 and 1.3 in Part 1, Theorem 4.5 in Part 2, and Theorem 7.3 in Part 3.

Part 1. The Euler characteristic of the de Rham complex

1. INTRODUCTION TO PART 1

Let $X$ be a compact complex manifold of dimension $n$ and let $j: U \to X$ be the inclusion of the complement of a reduced divisor $D$ in $X$, with complementary inclusion $i: D \to X$. Let $\mathcal{D}_X$ denote the sheaf of holomorphic differential operators on $X$. For a holonomic (left) $\mathcal{D}_X$-module $\mathcal{M}$ (more generally a bounded complex of $\mathcal{D}_X$-modules with holonomic cohomology, i.e., an object of $\mathcal{D}_{\text{hol}}^b(\mathcal{D}_X)$), we consider its de Rham complex ${}^p\text{DR}(\mathcal{M})$ (with $\Omega_X^n \otimes \mathcal{M}$ in degree zero for a holonomic $\mathcal{D}_X$-module), which is known to be a complex with $\mathbb{C}$-constructible cohomology (and a perverse sheaf if $\mathcal{M}$ is a holonomic $\mathcal{D}_X$-module).

Let $\chi_{\text{dR}}(X, \mathcal{M})$ denote the global Euler-Poincaré characteristic of the de Rham complex ${}^p\text{DR}(\mathcal{M})$:

$$ \chi_{\text{dR}}(X, \mathcal{M}) := \sum_k (-1)^k \dim \mathbf{H}^k(X, {}^p\text{DR}(\mathcal{M})). $$

To $\mathcal{M}$ are associated its localized module (or complex) $\mathcal{M}(\ast D) = \mathcal{D}_X(\ast D) \otimes_{\mathcal{D}_X} \mathcal{M}$ along $D$, which is known to be holonomic (or with holonomic cohomology), and its dual-localized module (or complex) $\mathcal{M}(!D)$ defined by duality from $\mathcal{M}(\ast D)$. Letting $\mathcal{D}$ denote the duality functor for coherent $\mathcal{D}_X$-modules, we know that $\mathcal{D}$ preserves the category of (bounded complexes of) holonomic $\mathcal{D}_X$-modules. We then set

$$ \mathcal{M}(!D) := \mathcal{D}[(\mathcal{D}\mathcal{M})(*D)]. $$

There exists a natural morphism $\mathcal{M}(!D) \to \mathcal{M}(\ast D)$ whose cone has cohomology supported on $D$.

By a meromorphic flat bundle on $(X, D)$ we mean a holonomic $\mathcal{D}_X$-module $\mathcal{M}$ which is a vector bundle with integrable connection on $U = X \setminus D$ and satisfies $\mathcal{M} = \mathcal{M}(\ast D)$. The dual $\mathcal{M}^{\vee}$ of such a meromorphic flat bundle $\mathcal{M}$ is the meromorphic flat bundle $\mathcal{M}^{\vee} := (\mathcal{D}\mathcal{M})(\ast D)$.

**Theorem 1.1.** Let $\mathcal{M}$ be an object of $\mathcal{D}_{\text{hol}}^b(\mathcal{D}_X)$ and let $\mathcal{N}$ be a meromorphic flat bundle of rank one on $X$ with poles along $D$ at most and having regular singularities along $D$. Then the global Euler-Poincaré characteristic of the de Rham complexes of

$$ \mathcal{M}(!D), \mathcal{M}(\ast D), \text{ and } \mathcal{M} \otimes \mathcal{N} $$

coincide.
Remark 1.2.

(1) (Algebraic version) If $U$ is smooth and quasi-projective, a similar result holds in the framework of algebraic $\mathcal{D}_U$-modules, and is obtained from Theorem 1.1 by choosing any smooth projectivization $X$ of $U$ and by making use of GAGA for algebraic $\mathcal{D}_X$-modules. For a coherent algebraic $\mathcal{D}_U$-module $M$ are defined the proper and total pushforwards $\mathcal{D}_{j!}M$ and $\mathcal{D}_{j_*}M$, which are holonomic if $M$ is so.

It states, for an algebraic $\mathcal{D}_U$-module $M$ and an algebraic vector bundle $N$ of rank one on $U$ with a flat connection having a regular singularity at infinity, that

$$ (1.2*) \qquad \chi_{\mathrm{dR},c}(U, M) = \chi_{\mathrm{dR}}(U, M) = \chi_{\mathrm{dR}}(U, M \otimes N), $$

with (due to a GAGA type argument for the second equalities)

$$ \begin{aligned} \chi_{\mathrm{dR},c}(U, M) &:= \sum_k (-1)^k \dim \mathcal{H}^k(X, {}^p\mathrm{DR}(\mathcal{D}_{j!}M)) \\ &= \sum_k (-1)^k \dim \mathcal{H}^k(X^{\mathrm{an}}, {}^p\mathrm{DR}^{\mathrm{an}}(\mathcal{D}_{j!}M)), \end{aligned} $$

$$ \begin{aligned} \chi_{\mathrm{dR}}(U, M) &:= \sum_k (-1)^k \dim \mathcal{H}^k(X, {}^p\mathrm{DR}(\mathcal{D}_{j_*}M)) \\ &= \sum_k (-1)^k \dim \mathcal{H}^k(X^{\mathrm{an}}, {}^p\mathrm{DR}^{\mathrm{an}}(\mathcal{D}_{j_*}M)). \end{aligned} $$

(If $N$ has rank $r$, then $\chi_{\mathrm{dR}}(U, M \otimes N) = r \cdot \chi_{\mathrm{dR}}(U, M)$.)

(2) The first equality of (1.2*) is proved by Gabriel Ribeiro in the appendix of his PhD thesis [Rib24] and the second equality is mentioned as a conjecture, with a nice algebraic proof only in the case where $D = \emptyset$. In this part, we show how local methods allow to prove the theorem in the analytic setting, so that the theorem in the algebraic setting follows by a GAGA type argument. The proof is of a transcendental nature, and relies on the notion of irregularity complex, due to Z. Mebkhout [Meb90, Meb04], and on the theorem of Kedlaya and Mochizuki [Ked10, Ked11, Moc09a, Moc11b] on the normal form of meromorphic flat bundles after blowing ups.

We will thus focus on a local version of the theorem. For a holonomic $\mathcal{D}_X$-module $\mathcal{M}$ and a point $x \in X$, the local Euler characteristic of the de Rham complex at $x$ is the number

$$ \chi_{\mathrm{dR},x}(\mathcal{M}) = \sum_k (-1)^k \dim \mathcal{H}^k({}^p\mathrm{DR}(\mathcal{M})_x) =: \chi_x({}^p\mathrm{DR}(\mathcal{M})). $$

**Theorem 1.3.** Let $\mathcal{M}$ be an object of $\mathcal{D}_{\mathrm{hol}}^{\mathrm{b}}(\mathcal{D}_X)$ and let $\mathcal{N}$ be a meromorphic flat bundle of rank one on $X$ with poles along $D$ at most and having regular singularities along $D$. Then, for each $x \in X$, the local Euler-Poincaré characteristic at $x$ of the de Rham complexes of

$$ \mathcal{M}(!D), \quad \mathcal{M}(\ast D), \quad \text{and} \quad \mathcal{M} \otimes \mathcal{N} $$

coincide.

*Proof that Theorem 1.3 implies Theorem 1.1.* For any complex $\mathcal{F}$ with $\mathbb{C}$-constructible cohomology on $X^{\mathrm{an}}$, whose cohomology is locally constant on the strata of a (Whitney) stratification $(X_\alpha)_\alpha$ of $X$ by locally closed smooth subvarieties, the
Euler characteristic

$$ \chi(\mathcal{F}_x) = \sum_k \dim \mathcal{H}^k(\mathcal{F}_x) $$

does not depend on the choice of $x$ in $X_\alpha$ and is denoted by $\chi_\alpha(\mathcal{F})$. We then have

$$ (1.4) \qquad \chi(X, \mathcal{F}) = \sum_{\alpha} \chi_{\alpha}(\mathcal{F}) \cdot \chi(X_{\alpha}). $$

In particular, for two such complexes $\mathcal{F}_1, \mathcal{F}_2$, we have $\chi(X, \mathcal{F}_1) = \chi(X, \mathcal{F}_2)$ as soon as $\chi(\mathcal{F}_{1,x}) = \chi(\mathcal{F}_{2,x})$ for all $x$ in $X$: indeed, we can find a stratification $(X_\alpha)_\alpha$ suitable for both, and the condition is equivalent to $\chi_\alpha(\mathcal{F}_1) = \chi_\alpha(\mathcal{F}_2)$ for all $\alpha$; we then apply Formula (1.4). $\square$

*Remark 1.5.* For a meromorphic flat bundle on $(X, D)$, Theorem 1.3 amounts to the equality, for any $x \in X$,

$$ \chi_{\mathrm{dR},x}(\mathcal{M}) = \chi_{\mathrm{dR},x}(\mathcal{M}^{\vee}) = \chi_{\mathrm{dR},x}(\mathcal{M} \otimes \mathcal{N}). $$

In other words, we claim that $\chi_{\mathrm{dR},x}(\mathcal{M}(!D)) = \chi_{\mathrm{dR},x}(\mathcal{M}^{\vee})$. This equality holds because

* we have by definition $\mathcal{M}(!D) = \mathcal{D}(\mathcal{M}^{\vee})$;

* let $\mathcal{D}_{\mathrm{PV}}$ be the Poincaré-Verdier duality on the category of bounded complexes with constructible cohomology on $X^{\mathrm{an}}$; then we have

$$ \chi_x(\mathcal{D}_{\mathrm{PV}}(\mathcal{F})) = \chi_x(\mathcal{F}); $$

* for a holonomic bounded complex $\mathcal{N}$, we have $^p\mathrm{DR}(\mathcal{D}\mathcal{N}) \simeq \mathcal{D}_{\mathrm{PV}}(^p\mathrm{DR}(\mathcal{N}))$.

We will show in Section 3 how to reduce the proof of Theorem 1.3 to that of the following theorem. The notion of good formal structure on a meromorphic flat bundle and that of a good meromorphic flat bundle is recalled in Section 2.b.

**Theorem 1.6.** Assume that $D$ is a divisor with simple normal crossings in $X$ and that $\mathcal{M} = \mathcal{M}(*D)$ is a good meromorphic flat bundle on $(X, D)$. Let $\mathcal{N}$ be as in Theorem 1.1. Then, for any $x \in X$, the following local Euler characteristics coincide:

$$ (1.6*) \qquad \chi_{\mathrm{dR},x}(\mathcal{M}) = \chi_{\mathrm{dR},x}(\mathcal{M}^{\vee}) = \chi_{\mathrm{dR},x}(\mathcal{M} \otimes \mathcal{N}). $$

Given a locally constant sheaf $\mathcal{V}$ on $U = X \setminus D$, with $D$ being a normal crossing divisor, the Euler characteristic number $\chi((\mathcal{R}j_*\mathcal{V})_x)$ vanishes at every point $x \in D$. Using the irregularity complex $\mathrm{Irr}_D(\mathcal{M})$ of $\mathcal{M}$ along $D$ as introduced by Mebkhout [Meb90, Meb04], we define the irregularity number of $\mathcal{M}$ at $x \in D$ as

$$ \mathrm{irr}_x(\mathcal{M}, D) := \chi(\mathrm{Irr}_D(\mathcal{M})_x). $$

Then the statement of Theorem 1.6 amounts<fnref n="1" /> to the equalities of irregularity numbers

$$ (1.7) \qquad \mathrm{irr}_x(\mathcal{M}, D) = \mathrm{irr}_x(\mathcal{M}^{\vee}, D) = \mathrm{irr}_x(\mathcal{M} \otimes \mathcal{N}, D) \quad \forall x \in D. $$

## 2. PROOF OF THEOREM 1.6

Under the assumptions of Theorem 1.6, we will prove the equalities (1.7). We first recall some parts of Section 2 of [Sab17].

<footnote n="1">¹In [Meb90, Meb04], the complex $\mathrm{Irr}_D(\mathcal{M})$ is defined in term of the non shifted de Rham complex $\mathrm{DR}(\mathcal{M})$, while here we use the shifted one; nevertheless, the same shift, with respect to (1.6*), occurs in the terms of (1.7).</footnote>
2.a. *Setting.* Let $X$ be a smooth complex variety, let $D$ be a divisor with simple normal crossings in $X$, set $U = X \setminus D$, i.e., we assume that each irreducible component $D_i$ ($i \in J$) of $D$ is smooth. For any subset $I \subset J$, we set $D_I = \bigcap_{i \in I} D_i$ and $D_I^{\circ} = D_I \setminus \bigcup_{j \notin I} D_j$. We denote the codimension of $D_I^{\circ}$ by $\ell$, that we regard as a locally constant function on $D_I^{\circ}$ (which can have many connected components), and by $\iota_I : D_I^{\circ} \hookrightarrow D$ the inclusion.

If $Z$ is any locally closed subvariety of $X$, we denote by $\mathcal{O}_{\hat{Z}}$, the formal completion of $\mathcal{O}_X$ with respect to the ideal sheaf $\mathcal{I}_Z$. We regard $\mathcal{O}_{\hat{Z}}$ as a sheaf on $Z$.

Given $x_o \in D$, there exists a unique subset $I \subset J$ such that $x_o \in D_I^{\circ}$, and we will be mostly interested in the cases where $Z$ is either the point $x_o \in D$ or the locally closed subset $D_I^{\circ}$. We will denote by $\mathcal{O}_{\hat{Z}}(*D)$ the sheaf $\mathcal{O}_{X|Z}(*D) \otimes_{\mathcal{O}_{X|Z}} \mathcal{O}_{\hat{Z}}$, where as usual $\mathcal{O}_{X|Z}$ (resp. $\mathcal{O}_{X|Z}(*D)$) denotes the sheaf-theoretic restriction to $Z$ of the sheaf $\mathcal{O}_X$ (resp. the sheaf $\mathcal{O}_X(*D)$ of rational functions on $X$ with poles at most on $D$).

For every $I \subset J$, we consider the sheaf $\mathcal{O}_{\widehat{D_I}}$ on $D_I^{\circ}$. We also regard it as a sheaf on $X$ by extending it by zero. We then set $\mathcal{D}_{\widehat{D_I}} = \mathcal{O}_{\widehat{D_I}} \otimes_{\mathcal{O}_X} \mathcal{D}_X$, and $\mathcal{M}_{\widehat{D_I}} := \mathcal{D}_{\widehat{D_I}} \otimes_{\mathcal{D}_X} \mathcal{M}$.

If $\varphi$ (resp. $\hat{\varphi}$) is a section of $\mathcal{O}_X(*D)$ (resp. of $\mathcal{O}_{\hat{Z}}(*D)$), we denote by $\mathcal{E}^{\varphi}$ (resp. $\mathcal{E}^{\hat{\varphi}}$) the module with connection $(\mathcal{O}_X(*D), \text{d} + \text{d}\varphi)$ (resp. $(\mathcal{O}_{\hat{Z}}(*D), \text{d} + \text{d}\hat{\varphi}$). It only depends on the class, also denoted by $\varphi$ (resp. $\hat{\varphi}$), of $\varphi$ (resp. $\hat{\varphi}$) modulo $\mathcal{O}_X$ (resp. $\mathcal{O}_{\hat{Z}}$).

2.b. *A reminder of the notion of good formal structure.* Let $\mathcal{M}$ be a meromorphic flat bundle on $(X, D)$. We say that $\mathcal{M}$ is good, or has a good formal structure along $D$ if, for any $x_o \in D$, there exists a local ramification $\rho_{d_I}$ of multi-degree $d_I$ around the branches $(D_i)_{i \in I}$ passing through $x_o$ (hence inducing an isomorphism above $D_I^{\circ}$ in the neighborhood of $x_o$) such that the pullback of the formal flat rational bundle $\mathcal{M}_{\widehat{x_o}} := \mathcal{O}_{\widehat{x_o}} \otimes_{\mathcal{O}_{x_o,x_o}} \mathcal{M}_{x_o}$ by this ramification decomposes as the direct sum of formal elementary flat $\mathcal{O}_X(*D)$-modules $\mathcal{E}^{\hat{\varphi}} \otimes \mathcal{R}_{\hat{\varphi}}$, as defined below.

We denote by $\text{nb}(x_o)$ a small analytic open neighborhood of $x_o$ in $X$ above which the ramification is defined, and we denote by $x_o'$ the pre-image of $x_o$, so the ramification is a finite morphism $\rho_{d_I} : \text{nb}(x_o') \to \text{nb}(x_o)$. It induces a one-to-one map above $D_I^{\circ} \cap \text{nb}(x_o)$. We also set $D' = \rho_{d_I}^{-1}(D \cap \text{nb}(x_o))$, so that $D'_I$ maps isomorphically to $D_I \cap \text{nb}(x_o) = D_I^{\circ} \cap \text{nb}(x_o)$.

In the above decomposition, $\hat{\varphi}$ varies in a good finite subset $\hat{\Phi}_{x_o} \subset \mathcal{O}_{\widehat{x_o}'}(*D')/\mathcal{O}_{\widehat{x_o}'}$ and $\hat{\mathcal{R}}_{\hat{\varphi}}$ is a free $\mathcal{O}_{\widehat{D_I}}(*D')$-module with an integrable connection having a regular singularity along $D'$. In other words, we do not distinguish between $\hat{\varphi}$ and $\hat{\psi}$ in $\mathcal{O}_{\widehat{x_o}'}(*D')$ if their difference has no poles along $D'$. Goodness means here that for any pair $\hat{\varphi} \neq \hat{\psi} \in \hat{\Phi}_{x_o} \cup \{0\}$, the difference $\hat{\varphi} - \hat{\psi}$ can be written as $x^{-e}\hat{u}(x)$, with $e \in \mathbb{N}^{\#I}$ and $\hat{u} \in \mathcal{O}_{\widehat{x_o}'}$ satisfying $\hat{\eta}(0) \neq 0$ (cf. [Sab00, §I.2.1]). By [Ked11, Prop. 4.4.1 & Def. 5.1.1] (cf. also [Sab00, §I.2.4] and [Moc11a, Prop. 2.19]), the $\hat{\varphi}$'s are convergent, i.e., the set $\hat{\Phi}_{x_o}$ is the formalization at $x_o$ of a finite subset $\Phi_{x_o} \subset \Gamma(\text{nb}(x_o'), \mathcal{O}_{\text{nb}(x_o')(*D')}/\mathcal{O}_{\text{nb}(x_o')})$, and the decomposition extends in a neighborhood of $x_o'$, that is, it holds for the pullback by $\rho_{d_I}$ of $\mathcal{M}_{\widehat{D_I},x_o}$ and induces the original one after taking formalization at $x_o'$.
It is also important to notice the following openness property. The components $\mathcal{E}\hat{\varphi} \otimes \hat{\mathbb{R}}_{\hat{\varphi}}$ are the formalization at $x'_o$ of local analytic meromorphic flat bundles $\mathcal{E}\varphi \otimes \mathbb{R}_{\varphi}$ defined in some analytic neighborhood of $x'_o$. Let us decompose $D'$ into its irreducible components $D'_i$ with $i \in I$. Then, for any $x' \in D'$ in a sufficiently small analytic neighborhood nb($x'_o$) of $x'_o$, the formalization $(\rho_{\hat{d}_i}^{+} \mathcal{M})_{\hat{x}'}$ decomposes with the components $(\mathcal{E}\varphi \otimes \mathbb{R}_{\varphi})_{\hat{x}'}$ for $\varphi \in \Phi_{x_o}$.

2.c. *Omitting the Stokes filtration.* Let us fix a subset $I \subset \{1, \dots, \ell\}$ and focus on proving (1.7) at $x \in D_I^o$. One can express the complex $\text{Irr}_D(\mathcal{M})$ in the neighborhood of $x$ by means of the Stokes filtration of $\mathcal{M}$ in the neighborhood of $x$, and more precisely by means of the sheaf of rapid decay flat sections of $\mathcal{M}$ on the real blow-up of $D$ along its irreducible components (cf. [Sab93, Cor. 2.11] and [Sab17, Prop. 3.1]). This reduces the computation of the various Euler characteristics to a topological computation. However, the Stokes filtration can make a direct topological computation complicated. Fortunately, [Sab17, Th. 1.2] allows us to do as if the Stokes filtration were trivial. More precisely, in the neighborhood of $D_I^o$, there exists a meromorphic flat bundle with trivial Stokes structure along $D$, that is, the meromorphic flat bundle decomposes locally (after a possible local ramification) in a way similar to that of the formal decomposition of $\mathcal{M}$, and whose irregularity complex along $D_I^o$ is isomorphic to that of $\mathcal{M}$. A fortiori, one can compute the various irregularity numbers of (1.7) at $x \in D_I^o$ by means of such a meromorphic flat bundle.

2.d. *Reduction to the case “good of rank one”*. Let $\rho$ be a local ramification as in Section 2.b. By [Meb04, Prop. 3.6-4] we are reduced to proving (1.7) for $\rho^*\mathcal{M}$. By the previous reduction, we can assume that $\rho^*\mathcal{M}$ has a convergent decomposition mimicking its good formal decomposition. In such a case, $\rho^*\mathcal{M}$ is a successive extension of good meromorphic flat bundles of rank one. Since the irregularity number behaves additively in extensions, we can assume from start that $\mathcal{M}$ is a good meromorphic flat bundle of rank one.

2.e. *Computation of the irregularity of a good meromorphic flat bundle of rank one.*
Let us be given

• a meromorphic function<fnref n="2" />

$$ \varphi(x) = u(x)/x^e, \quad \text{with } u \text{ holomorphic invertible and } e = (e_1, \dots, e_\ell) \in \mathbb{N}^\ell, $$

• a multi-index $\alpha = (\alpha_1, \dots, \alpha_\ell) \in \mathbb{C}^\ell$,

and we will compute the irregularity number at the origin of the germ of trivial bundle of rank one

$$ (\mathcal{E}, \nabla) = \left( \mathbb{C}\{x_1, \dots, x_n\}[(x_1 \cdots x_\ell)^{-1}], d + d\varphi + \sum_{i=1}^\ell \alpha_i \frac{dx_i}{x_i} \right). $$

<footnote n="2">²If $e = 0 \in \mathbb{N}^\ell$, we can choose $u \equiv 1$.</footnote>
We note that, if $(\mathcal{N}, \nabla) = (\mathbb{C}\{x_1, \dots, x_n\}[(x_1 \cdots x_\ell)^{-1}], d + \sum_{i=1}^\ell \beta_i dx_i/x_i)$,

$$ (\mathcal{E}, \nabla)^{\vee} = \left( \mathbb{C}\{x_1, \dots, x_n\}[(x_1 \cdots x_\ell)^{-1}], d - d\varphi - \sum_{i=1}^{\ell} \alpha_i \frac{dx_i}{x_i} \right), $$

$$ (\mathcal{E}, \nabla) \otimes (\mathcal{N}, \nabla) = \left( \mathbb{C}\{x_1, \dots, x_n\}[(x_1 \cdots x_\ell)^{-1}], d + d\varphi + \sum_{i=1}^{\ell} (\alpha_i + \beta_i) \frac{dx_i}{x_i} \right). $$

The equalities (1.7) for $\mathcal{M} = (\mathcal{E}, \nabla)$ are an immediate consequence of the first part of the next lemma.

**Lemma 2.1.** *The irregularity number of $(\mathcal{E}, \nabla)$ at the origin only depends on the family $(e_1, \dots, e_\ell)$ (with the convention that $e = 0$ if $\varphi \equiv 0$). More precisely, we have*

$$ \mathrm{irr}_0(\mathcal{E}, \nabla) = \prod_{i=1}^{\ell} (-e_i). $$

*Proof.* Let $\varpi$ denote the real blowup of the components of $D$, let $\mathcal{E}$ denote the rank-one local system on $\varpi^{-1}(0) \simeq (S^1)^\ell$ consisting of horizontal sections of $(\mathcal{E}, \nabla)$ and let $\mathcal{E}_{\le 0} \subset \mathcal{E}$ be the subsheaf of those which have moderate growth in the neighborhood of $D$. We set $\mathcal{E}^{>0} = \mathcal{E}/\mathcal{E}_{\le 0}$. Then, by [Sab17, Prop. 3.1], we have $\mathrm{Irr}_D(\mathcal{E}, \nabla)[1] = \mathcal{R}\varpi_*\mathcal{E}^{>0}$, and therefore, as $\chi((S^1)^\ell, \mathcal{E}) = 0$, we find $\mathrm{irr}_0(\mathcal{E}, \nabla) = \chi((S^1)^\ell, \mathcal{E}_{\le 0})$. The sheaf $\mathcal{E}_{\le 0}$ has the following description:

(i) if $\varphi \equiv 0$, $\mathcal{E}_{\le 0}$ is the locally constant sheaf of rank 1 and monodromy $\exp(-2\pi i \alpha_i)$ in direction $i$;

(ii) if $\varphi \ne 0$, $\mathcal{E}_{\le 0}$ is the extension by zero of the restriction of the above locally constant sheaf to the open subset $(S^1)_{\varphi,e}^\ell \subset (S^1)^\ell$ defined by the inequations

$$ \mathrm{Re}(\varphi(0) \exp(-e_i \theta_i)) > 0, \quad i = 1, \dots, \ell. $$

The irregularity number is then obtained as follows:

(i) In this case, we have

$$ \mathrm{irr}_0(\mathcal{E}, \nabla) = \chi((S^1)^\ell, \mathcal{E}_{\le 0}) = \chi((S^1)^\ell) = 0, $$

whatever $\alpha_i$ are.

(ii) The open set $(S^1)_{\varphi,e}^\ell$ is the product $\prod_{i=1}^\ell S_{i,\varphi,e_i}^1$, where each $S_{i,\varphi,e_i}^1$ is

• a disjoint union of $e_i$ open intervals of $S^1$ if $m_i \ne 0$,

• the circle $S^1$ if $e_i = 0$.

The Künneth formula yields

$$ \chi((S^1)^\ell, \mathcal{E}_{\le 0}) = \prod \chi_c(S_{i,\varphi,e_i}^1). $$

• If $e_i = 0$ for some $i$, then $\chi_c(S_{i,\varphi,e_i}^1) = \chi(S^1) = 0$, hence $\chi((S^1)^\ell, \mathcal{E}_{\le 0}) = 0$.

• Otherwise, $\chi_c(S_{i,\varphi,e_i}^1) = -e_i$.

□

### 3. REDUCTION OF THE PROOF OF THEOREM 1.3 TO THAT OF THEOREM 1.6

Although this reduction is very standard, we give a detailed proof for the sake of completeness. Given the additivity of $\chi_{dR}$ and $\chi_{dR,c}$ by exact sequences, we can assume that $\mathcal{M}$ is a holonomic $\mathcal{D}_X$-module. Then we argue by induction on the dimension $d$ of the support $Z$ of $\mathcal{M}$. Since $\mathcal{M}(!D) = [\mathcal{M}(*D)](!D)$ and $\mathcal{M} \otimes \mathcal{N} = \mathcal{M}(*D) \otimes \mathcal{N}$, we can assume that $\mathcal{M} = \mathcal{M}(*D)$. The assertion tautological if
dim Supp $\mathcal{M} = 0$, and we assume it to be true if dim Supp $\mathcal{M} < d$. As the assertion is local, we can assume that there exists a hypersurface $H$ in $X$ containing $D$ such that $Z^\circ := Z \setminus (Z \cap H)$ is smooth of pure dimension $d$, that no irreducible component of $Z$ of dimension $d$ is contained in $H$, and that the restriction of $\mathcal{M}$ to $Z^\circ$ in the sense of $\mathcal{D}$-modules (via Kashiwara's equivalence) is a holomorphic vector bundle over $Z^\circ$ with integrable connection. By induction, it is enough to prove the theorem for the localized module $\mathcal{M}(\ast H)$.

Let $\pi : X' \to X$ be a projective morphism such that

* $X'$ is smooth of pure dimension $d$,

* $H' := \pi^{-1}(H)$ is a divisor with normal crossings,

* $\pi : X'^o := X' \setminus H' \to Z^\circ$ is an isomorphism.

Such a morphism exists, by resolving the singularities of $(Z_d, H \cap Z_d)$, where $Z_d$ is the union of the pure $d$-dimensional components of $Z$. The inverse image $\pi_*\mathcal{M}(\ast H)$ of $\mathcal{M}(\ast H)$ is thus a meromorphic flat bundle on $(X', H')$. Furthermore, by the theorem of Kedlaya-Mochizuki [Ked11, Th. 8.1.3], we can moreover assume that $\pi_*\mathcal{M}(\ast H)$ is a good meromorphic flat bundle on $(X', H')$.

Set $D' = \pi^{-1}(D)$. Then $\mathcal{M}(\ast H)$, $\mathcal{M}(\ast H)(!D)$ and $\mathcal{M}(\ast H) \otimes \mathcal{N}$ can be recovered respectively from $\pi_*\mathcal{M}(\ast H)$, $[\pi_*\mathcal{M}(\ast H)](!D')$ and $[\pi_*\mathcal{M}(\ast H)] \otimes \pi_*\mathcal{N}$ as their pushforward $\pi_*\pi_*\mathcal{M}(\ast H)$, etc. As a consequence, $^p\mathrm{DR}(\mathcal{M}(\ast H)) \simeq \mathcal{R}\pi_*^p\mathrm{DR}(\pi_*\mathcal{M}(\ast H))$, etc., and the constructible function $x \mapsto \chi_{dR,x}(^p\mathrm{DR}(\mathcal{M}(\ast H))$, etc., is equal to the pushforward, in the sense of [Mac74], of the constructible function $x' \mapsto \chi_{dR,x'}(^p\mathrm{DR}(\pi_*\mathcal{M}(\ast H)))$, etc.

We conclude that it is enough to prove Theorem 1.3 in the setting of Theorem 1.6, and this has been achieved in Section 2. $\square$

## Part 2. Local generic vanishing theorems for holonomic $\mathcal{D}$-modules

### 4. LOCAL AND RELATIVE GENERIC VANISHING THEOREMS

In his PhD thesis [Rib24], Gabriel Ribeiro suggests the statement of a “relative vanishing theorem”, that we recall in Section 4.b. We give here a complete proof of a “local generic vanishing theorem” that implies the corresponding “relative” statement, and that we regard as a corollary of a more general result stated in Section 4.a: the natural morphism from the proper pushforward to the total pushforward of an algebraic holonomic $\mathcal{D}$-module by an open inclusion is an isomorphism if we first twist the $\mathcal{D}$-module structure by suitable closed algebraic differential forms.

4.a. *A general result.* Let $U$ be a smooth complex algebraic variety and let $M_U$ be a quasi-coherent (left) $\mathcal{D}_U$-module, that we regard as a quasi-coherent $\mathcal{O}_U$-module with integrable connection $\nabla$. Furthermore, let $\eta^o = (\eta_1^o, \dots, \eta_r^o)$ be a finite family of regular closed one-forms on $U$. For any $a \in \mathbb{C}^r$, we consider the twisted $\mathcal{D}_U$-module

$$ \mathcal{M}_{U, \eta^o, a} := (\mathcal{M}_U, \nabla + \sum_i a_i \eta_i^o). $$

This construction can be extended to any object $M_U \in \mathcal{D}_{\mathrm{coh}}(\mathcal{D}_U)$, and the $(\eta^o, a)$ twist commutes with taking cohomology. If $M_U$ belongs to $\mathcal{D}_{\mathrm{hol}}^b(\mathcal{D}_U)$, so does $M_{U, \eta^o, a}$. Let $X$ be any smooth compactification of $U$, that is, a smooth proper complex algebraic variety that contains $U$ as an open dense subvariety and let $j : U \hookrightarrow X$ denote the open inclusion. We say that $X$ is a good compactification
of U if moreover $D := X \smallsetminus U$ is a divisor with simple normal crossings in X (i.e., each irreducible component of D is smooth). Good compactifications exist: starting from any Nagata compactification of U, one blows up above $X \smallsetminus U$ according to Hironaka's theorem on embedded resolution of singularities.

For any smooth compactification $X$ of $U$, we denote by $\mathbb{D}j_*$, resp. $\mathbb{D}j_!$, the direct image, resp. the direct image with proper support (i.e., $\mathbb{D}j_! = \mathcal{D}\mathbb{D}j_*\mathcal{D}$, where $\mathcal{D}$ is the duality functor). For $\mathcal{M} \in \mathcal{D}_{\text{hol}}^{\text{b}}(\mathcal{D}_U)$, there exists a natural morphism (cf. [HTT08, Th.3.2.16])

$$ (4.1) \qquad \mathbb{D}j_*\mathcal{M}_{U,\eta^o,a} \to \mathbb{D}j_*\mathcal{M}_{U,\eta^o,a}, $$

where both terms belong to $\mathcal{D}_{\text{hol}}^{\text{b}}(\mathcal{D}_X)$. We say that relative generic vanishing holds for $(X, \mathcal{M}_U, \eta^o)$ if there exists a Zariski dense open subset $V \subset \mathbb{C}^r$ such that, for each $a \in V$, (4.1) is an isomorphism. Let us start by noticing:

**Lemma 4.2.** Assume that a smooth compactification $X'$ of $U$ dominates $X$, i.e., there exists a proper modification $\pi: X' \to X$ which is the identity on $U$. If relative generic vanishing holds for $(X', \mathcal{M}_U, \eta^o)$, then it holds for $(X, \mathcal{M}_U, \eta^o)$.

*Proof.* Let $j' : U \hookrightarrow X'$ denote the open embedding. As $\pi$ is proper, we have $\mathbb{D}\pi_! = \mathbb{D}\pi_*$, and the natural morphism $\mathbb{D}j'_*(\cdot) \to \mathbb{D}j'_*(\cdot)$ yields the natural morphism

$$ [\mathbb{D}\pi_! \mathbb{D}j'_*(\cdot) \to \mathbb{D}\pi_* \mathbb{D}j_*(\cdot)] \simeq [\mathbb{D}j_*(\cdot) \to \mathbb{D}j_*(\cdot)]. $$

If the first one is an isomorphism when applied to $\mathcal{M}_{U,\eta^o,a}$, so is the second one. $\square$

Let $(X, D)$ be a good compactification of $U$ and let $\eta$ be a closed logarithmic one-form on $X$ with poles along $D = \bigcup_{i \in J} D_i$. Since $\eta$ is closed, the residue $r_i(\eta)$ of $\eta$ along the smooth part $D_i^o$ of $D_i$ is constant. For any subset $I \subset J$ such that $\bigcap_{i \in I} D_i \neq \emptyset$, set $I(\eta) = \{i \in I | r_i(\eta) \neq 0\}$.

**Definition 4.3** (Non-resonance). We say that $\eta$ is *I-non-resonant* if there does not exist any relation $\sum_{i \in I(\eta)} m_i r_i(\eta) = 0$ with $(m_i)_{i \in I(\eta)} \in \mathbb{N}^{\#I(\eta)} \setminus \{0\}$.

For example, if $\eta$ has a pole along only one irreducible component of $D$, it is *I-non-resonant* for any subset $I$.

**Definition 4.4** ([Sab93, §1.3] and [Moc11b, Def. 2.1.2]). Let $X$ be a good compactification of $U$ and let $\eta$ be a closed rational one-form on $X$ with poles along $D$. We say that $\eta$ is *good wild* along $D$ if the following property is satisfied. For any $x_o \in D$ and a local analytic chart $\text{nb}(x_o) = \Delta_x^\ell \times \Delta_y^{n-\ell}$ such that $\text{nb}(x_o) \setminus D = (\Delta_x^*)^\ell \times \Delta_y^{n-\ell}$, there exist

• a multi-index $e \in \mathbb{N}^\ell$ and an invertible holomorphic function $u(x, y) \in \mathcal{O}(\text{nb}(x_o))$,

• a logarithmic one-form $\omega \in \Gamma(\text{nb}(x_o), \Omega_X^1(\log D))$, such that $\eta|_{\text{nb}(x_o)} = \text{d}(u(x, y)x^{-e}) + \omega$, with the notation $x^e = x_1^{e_1} \cdots x_\ell^{e_\ell}$.

**Theorem 4.5.** Let $\eta_1^o, \dots, \eta_r^o$ be a family of closed one-forms on $U$. Assume that there exists a good compactification $(Y, D_Y)$ of $U$ such that their rational extensions $\eta_1, \dots, \eta_r$ as (closed) rational one-forms on $Y$ with pole along $D_Y$ satisfy either

(i) each $\eta_k$ is a logarithmic one-form with poles in $D_Y$ and for each non-empty intersection $\bigcap_{i \in I} D_{Y,i}$, there exists $k \in \{1, \dots, r\}$ such that $\eta_k$ is *I-non-resonant*, or
(ii) each $η_k$ is good wild along $D_Y$, and for each component $D_{Y,i}$ of $D_Y$, there exists $k \in \{1, \dots, r\}$ such that $η_k$ has a pole of order $≥ 2$ along $D_{Y,i}$.

Then, given any holonomic $\mathcal{D}_U$-module $M_U$ (or any object in $\mathcal{D}_{\text{hol}}^b(\mathcal{D}_U)$), for any compactification $X$ of $U$,

(a) in case (i), there exists a finite set of affine forms $L_{\ell,c}(a) = \sum_k \ell_k a_k + c$, with $\ell, c$ belonging to a finite subset of $(\mathbb{C}^r \setminus \{0\}) \times \mathbb{C}$, defining $V := \mathbb{C}^r \setminus \bigcup_{\ell,c} L_{\ell,c}^{-1}(\mathbb{Z})$,

(b) in case (ii), there exists a Zariski dense open subset $V$ of $\mathbb{C}^r$, which is the complement of a finite number of affine hyperplanes,

such that, for any $a \in V$, the natural morphism (4.1) is an isomorphism.

The proof of case (i) will show more precisely that the coefficients $\ell_k$ are some combinations with integer coefficients of the residues of $η_k$ along $D_i$ (cf. Equation (6.3)).

**Remark 4.6** (Analytic version of Theorem 4.5). Theorem 4.5 is stated in an algebraic setting in order to be compatible with the setting in [Rib24]. However, as in Part 1, the proof is mainly done in an analytic setting, where $(X, D)$ and $(Y, D_Y)$ are compact complex manifolds with a divisor, such that $X \setminus D$ and $Y \setminus D_Y$ are isomorphic. We start from any object $\mathcal{M}$ in $\mathcal{D}_{\text{hol}}^b(\mathcal{D}_X)$ that satisfies $\mathcal{M} = \mathcal{M}(\ast D)$ and the morphism (4.1) is written as

$$ \mathcal{M}(!D) \to \mathcal{M}(\ast D). $$

The family $η^o$ is a family of closed meromorphic one-form on $X$ with poles along $D$ that satisfy the assumptions in Theorem 4.5. Then the conclusion of Theorem 4.5 holds (with the same proof) in this analytic setting.

**Remark 4.7** (Extension to twistor $\mathcal{D}$-modules). The same theorem holds for mixed twistor $\mathcal{D}$-modules in the following way (we refer to [Moc15] for this theory and the corresponding notation). Let $\mathcal{T}$ be a mixed twistor $\mathcal{D}$-module on $X$ which satisfies $\mathcal{T} = \mathcal{T}[\ast D]$. On the other hand, there exists an admissible mixed twistor structure $\mathcal{T}_{\eta,a}$ on $(X, D)$ whose associated $\mathcal{D}_X$-module is $(\mathcal{O}_X(\ast D), \text{d} + \sum a_k \eta_k)$. Then there exists a uniquely defined mixed twistor $\mathcal{D}$-module $(\mathcal{T} \otimes \mathcal{T}_{\eta,a})[\ast D]$ (cf. [Moc15, Prop. 11.3.3]). The morphism (4.1) reads, with the notation of loc. cit.,

$$ (4.8) \qquad (\mathcal{T} \otimes \mathcal{T}_{\eta,a})[!D] \to (\mathcal{T} \otimes \mathcal{T}_{\eta,a})[\ast D]. $$

Since the functor which associates to each twistor $\mathcal{D}$-module on $X$ its underlying holonomic $\mathcal{D}_X$-module is conservative, the conclusion of the theorem also applies to (4.8).

4.b. *Applications.* Let $S$ be a smooth complex variety, let $T = \mathbb{G}_m^n$ be the torus of dimension $n$, and let $U = \mathbb{A}^n$ be the affine space of dimension $n$. We fix coordinates $t_1, \dots, t_n$ on $U$, that we induce on $T$.

(i) For $\lambda \in (\mathbb{C}^*)^n$, let $\mathcal{N}_\lambda = (\mathcal{O}_T, \text{d} + (\sum_k a_k \text{d}t_k / t_k))$, with $\exp -2\pi i a_k = \lambda_k$.

(ii) For $a \in \mathbb{C}^n$, let $\mathcal{N}_a = (\mathcal{O}_U, \text{d} + (\sum_k a_k \text{d}t_k))$.

Let $j: T \times S \hookrightarrow \mathbb{P}^n \times S$, resp. $j: U \times S \hookrightarrow \mathbb{P}^n \times S$, denote the open inclusion. We consider the closed one-forms $η_k = \text{d}t_k / t_k$ (case (i)) and $η_k = \text{d}t_k$ (case (ii)). They satisfy the corresponding conditions in Theorem 4.5: for example, $\text{d}t_k / t_k$ has a nonzero residue only along $\{t_k = 0\}$ and $\{t_k = \infty\}$, which do not intersect, so that $\#I(η_k) = 1$.
Furthermore, in case (i), the coefficients $l_k$, which are linear combinations with integral coefficients of the residues of $\eta_k$ along the components of $D = \mathbb{P}^n \setminus T$, are integers. It follows that $V^* := \exp(-2\pi iV)$ is a Zariski dense open subset of $(\mathbb{C}^*)^n$ (in fact, the complement of a finite union of translated codimension-one subtori of $T$).

**Corollary 4.9 (Local vanishing).** Let $\mathcal{M}_T$ be a holonomic $\mathcal{D}_{T \times S}$-module, resp. $\mathcal{M}_U$ a holonomic $\mathcal{D}_{U \times S}$-module. There exists a Zariski dense open set $V^*$ of $(\mathbb{C}^*)^n$, resp. $V$ of $\mathbb{C}^n$, which is the complement of a finite union of translated codimension-one subtori of $T$, resp. the complement of a finite union of affine hyperplanes, such that for any $\lambda \in V^*$, resp. any $a \in V$, the natural morphism (cf. [HTT08, Th.3.2.16])

$$ \begin{array}{rcl} \text{resp.} & \multicolumn{2}{c}{\mathrm{d}j_!(\mathcal{M}_T \otimes \mathcal{N}_\lambda) \to \mathrm{d}j_*(\mathcal{M}_T \otimes \mathcal{N}_\lambda),} \\ & \multicolumn{2}{c}{\mathrm{d}j_!(\mathcal{M}_U \otimes \mathcal{N}_a) \to \mathrm{d}j_*(\mathcal{M}_U \otimes \mathcal{N}_a),} \\ & & \text{is an isomorphism.} \end{array} $$

*Proof.* By taking $X = \mathbb{P}^n \times S$ and $\eta_k = dt_k/t_k$, resp. $dt_k$, Theorem 4.5 yields the statement of the corollary. $\square$

*Remark 4.10 (Other approaches in the multiplicative case).* The case when $T = \mathbb{G}_m$ (i.e., $n=1$) is proved in [BG12] by using the theory of $b$-functions. The case where $T = \mathbb{G}_m^n$ ($n \ge 1$) should also be obtained from the results in [Mai23] and [Wu21]

**Corollary 4.11 (Relative vanishing).** Let $p$ denote the projection $T \times S \to S$ or $U \times S \to S$. For $\mathcal{M}$ as in the theorem, there exists a Zariski dense open subset $V^*$ of $(\mathbb{C}^*)^n$, resp. $V$ of $\mathbb{C}^n$, which is the complement of a finite union of translated codimension-one subtori of $T$, resp. the complement of a finite union of affine hyperplanes, such that for any $\lambda \in V^*$, resp. any $a \in V$, the natural morphism

$$ \begin{array}{rcl} \text{resp.} & \multicolumn{2}{c}{\mathrm{d}p_!(\mathcal{M}_T \otimes \mathcal{N}_\lambda) \to \mathrm{d}p_*(\mathcal{M}_T \otimes \mathcal{N}_\lambda),} \\ & \multicolumn{2}{c}{\mathrm{d}p_!(\mathcal{M}_U \otimes \mathcal{N}_a) \to \mathrm{d}p_*(\mathcal{M}_U \otimes \mathcal{N}_a),} \\ & & \text{is an isomorphism.} \end{array} $$

*Proof.* Immediate by using that the projection $P: \mathbb{P}^n \times S \to S$ is proper, so that $\mathrm{d}P_! \simeq \mathrm{d}P_*$, and the isomorphisms $\mathrm{d}p_! = \mathrm{d}P_! \circ \mathrm{d}j_!$, $\mathrm{d}p_* = \mathrm{d}P_* \circ \mathrm{d}j_*$.

## 5. A CRITERION FOR LOCAL VANISHING

We use the setting of §2.a and the terminology of §2.b. This section is devoted to the proof of the following criterion.

**Proposition 5.1.** Let $\mathcal{M}_U$ be a flat vector bundle on $U$ and let $X$ be a good compactification of $U$ such that $\mathrm{d}j_*\mathcal{M}_U$ has a good formal structure along the normal crossing divisor $D = X \times U$. Let $d$ be the maximal ramification order of $\mathrm{d}j_*\mathcal{M}_U$ along the irreducible components of $D$. Assume that the monodromy of the formal regular part of $\mathcal{M}_U$ along the smooth part of $D$ has no eigenvalue which is a root of unity of order $\le d$ (e.g. if this formal regular part is zero). Then the natural morphism $\mathrm{d}j_!\mathcal{M}_U \to \mathrm{d}j_*\mathcal{M}_U$ is an isomorphism.

For each irreducible component $D_i$ of $D$ ($i \in J$), there exists a (non-ramified) decomposition $\mathcal{M}|_{\hat{D}_i^\circ} \simeq \mathcal{M}|_{\hat{D}_i^\circ}^{\text{irr}} \oplus \mathcal{M}|_{\hat{D}_i^\circ}^{\text{reg}}$, where the second term has regular singularities along $D_i^\circ$: it is the formal regular part of $\mathcal{M}$ along $D_i^\circ$.
The assumption of the proposition is that the ramification order $d_i$ needed to obtain a formal decomposition is bounded by $d$. Note that, by flatness, the characteristic polynomial of the formal monodromy around $D_i^\circ$ is constant along each connected component of $D_i^\circ$.

**Corollary 5.2.** If the formal monodromy of the regular part of $\mathcal{M}$ along the smooth part of $D$ has no eigenvalue which is a root of unity of order $\le d$, then for each $x_o \in D$ and each $\varphi = x^{-e}u(x) \in \Phi_{x_o}$, the monodromy of the germ $\mathcal{R}_{\varphi,x_o'}$ of regular meromorphic flat bundle around any component $D_i'$ such that $e_i = 0$ has no eigenvalue which is equal to one.

*Proof.* Let us fix $I \subset J$, $x_o \in D_I^\circ$ and $x_o' \in D_I'$. By the openness property, for any $i \in I$ and any $x' \in D_I^\circ \cap \text{nb}(x_o')$, the regular part of $(\rho_{d_i}^+ \mathcal{M})_{\hat{x}'}$ is $\bigoplus_{\varphi} (\mathcal{R}_{\varphi})_{\hat{x}'}$, where $\varphi = x^{-e}u(x) \in \Phi_{x_o}$ is such that $e_i = 0$. The assumption implies that the monodromy of each such $\mathcal{R}_{\varphi}$ around $D_i'$ has no eigenvalue which is equal to one (since the ramified monodromy is the $d_i$-th power of the original one). $\square$

5.a. Reduction to the elementary case. In order to check that $\text{d}_{j_i} \mathcal{M}_U \to \text{d}_{j_*} \mathcal{M}_U$ is an isomorphism, it is enough to do it locally analytically. Let us write this morphism as $\mathcal{M}(!D) \to \mathcal{M}(*D)$. Furthermore, as duality commutes with tensoring with $\mathcal{O}_{\hat{x}_o}$, it is enough to check that, for any $x_o \in D$, the morphism $\mathcal{M}(!D)_{\hat{x}_o} \to \mathcal{M}(*D)_{\hat{x}_o}$ is so, and as localization ($*$D) and dual localization ($!D$) commute with tensoring with $\mathcal{O}_{\hat{x}_o}$, we can start from $\mathcal{M}_{\hat{x}_o}$. Lastly, localization and duality both commute with pullback and pushforward by the multi-cyclic ramification $\rho_{d_i}$, so that it is enough to consider the case where $\mathcal{M}_{\hat{x}_o}$ has a good formal decomposition. In such a case, for each exponential factor $\varphi \in \Phi_{x_o}$, $\mathcal{R}_{\varphi}$ is an extension of meromorphic flat bundles of rank one with a connection having regular singularities along the germ $D_{x_o}$, so that it is an extension of rank-one such objects, whose monodromy around $D_i$ is an eigenvalue of that of $\mathcal{R}_{\varphi}$ around $D_i$. We are thus left with considering the case of rank-one elementary meromorphic connections as in §2.e, with $\varphi \in \Phi_{x_o}$ and the assumption of the theorem implies that, if $e_i = 0$, then the monodromy around $D_i$ is not a root of the unity. Note that, at this step, we can come back to the convergent setting, since the elementary formal objects are convergent.

5.b. *Proof in the elementary case.* We work in the local analytic setting where $X = \Delta^\ell \times \Delta^{n-\ell}$, where $\Delta$ is a small disc in $\mathbb{C}^n$, and $D = \prod_{i=1}^\ell x_i = 0$ (where $x_1, \dots, x_\ell$ are the coordinates on $\Delta^\ell$). With the notation of §2.a, we have set $I = \{1, \dots, \ell\}$. We consider the rank-one meromorphic connection

$$ \mathcal{N} = (\mathcal{O}_X(*D), \text{d} + \text{d}\varphi + \sum_i \alpha_i \text{d}x_i/x_i), $$

with

(a) $\varphi = u/x^e$, where $u$ is a regular function on $X$ which does not vanish anywhere on $X$ and $x^e = x_1^{e_1} \cdots x_\ell^{e_\ell}$, with $e_i \in \mathbb{N}$,

(b) $\alpha_i \in \mathbb{C}$ is such that $(e_i = 0 \implies \alpha_i \notin \mathbb{Z})$.

Recall that $D_i^\circ = D_i \setminus \bigcup_{i' \neq i} D_{i'}$. If $e_i = 0$, $\mathcal{N}$ has regular singularity along $D_i^\circ$ and the assumption on $\alpha_i$ means that its monodromy around $D_i^\circ$ is not equal to the identity. On the other hand, if $e_i \neq 0$, then the regular part of $\mathcal{N}$ along $D_i^\circ$ is zero. We will prove that $\mathcal{N}(!D)_{x_o} \to \mathcal{N}_{x_o}$ is an isomorphism. We note that $\mathcal{N}(!D)$ can be obtained as a successive application of the functors $(!D_i)$ for $i = 1, \dots, \ell$. It is then enough to prove that for each $i \in \{1, \dots, \ell\}$, the natural morphism $\mathcal{N}(!D_i) \to \mathcal{N}$
is an isomorphism: indeed, let us check by induction on $k$ that $\mathcal{N}(!(D_{\le k})) \to \mathcal{N}$ is
then an isomorphism; we write this morphism as the composition

$$
\mathcal{N}(!(D_{\le k-1}))(!D_k) \to \mathcal{N}(!D_k) \to \mathcal{N},
$$

and we see by induction that both morphisms are isomorphisms.

It is then enough to check that the jumping indices of the V-filtration along each
component $D_i$ are not integers. Set $I_e = \{i \in \{1, \dots, \ell\} \mid e_i \neq 0\}$.

• If $I_e = \emptyset$, then $d\varphi$ is holomorphic and we can omit it. Then the section $1 \in \mathcal{O}_X(*D)$ satisfies $\partial_{x_i}(1) = (\alpha_i/x_i) \cdot 1$ for all $i$, and our assumption that $\alpha_i \notin \mathbb{Z}$ shows on the one hand that $1$ is a $\mathcal{D}_X$-generator of $\mathcal{N}$ and, on the other hand, that its Bernstein polynomial along $D_i$ is $s - \alpha_i$; the $V$-filtration of $\mathcal{N}$ along $D_i$ is then that generated by $1$, hence has jumping indices in $\alpha_i + \mathbb{Z} \neq \mathbb{Z}$.

• If $I_e = \{1, \dots, \ell\}$, then $\partial_{x_i}(1) = (v_i/x^e) \cdot 1$, where $v_i$ is an invertible holomorphic function, and this shows that $1$ is a $\mathcal{D}_X$-generator of $\mathcal{N}$ with constant Bernstein polynomial along $D_i$ for any $i \in \{1, \dots, \ell\}$; therefore, the $V$-filtration of $\mathcal{N}$ along $D_i$ is constant for any $i$, hence has no jumping indices.

• In the other cases, we have $\partial_{x_i}(1) = (v_i/x^e) \cdot 1$ if $i \in I_e$ and, fixing $i_o \in I_e$, we have

$$
\partial_{x_i}(1) = \left[ \frac{\partial u}{\partial x_i} x^{-e} + \frac{\alpha_i}{x_i} \right] \cdot 1 = \left[ w_{i,i_o} \partial_{x_{i_o}}(1) + \frac{\alpha_i}{x_i} \right] \cdot 1
$$

if $i \notin I_e$, for some holomorphic function $w_{i,i_o}$; hence $(\partial_{x_i} - w_{i,i_o} \partial_{x_{i_o}})(1) = (\alpha_i/x_i) \cdot 1$
and we conclude 1 is a $\mathcal{D}_X$-generator of $\mathcal{N}$; similarly, the $V$-filtration along $D_i$ is
constant if $i \in I_e$ and has jumping indices in $\alpha_i + \mathbb{Z} \neq \mathbb{Z}$ if $i \notin I_e$. $\square$

6. PROOF OF THEOREM 4.5

The proof is by induction on the dimension of the support of $M_U$, the case where
this dimension is zero being straightforward.

**Lemma 6.1.** It is enough to prove the theorem when $M_U$ is a flat vector bundle.

*Proof.* We first notice that it is enough to prove the assertion of the theorem for holonomic $\mathcal{D}_U$-modules $M_U$: by induction on the amplitude of the complex, it also holds for any object of $\mathcal{D}_{\text{hol}}^b(\mathcal{D}_U)$. We thus assume that $M_U$ is a holonomic $\mathcal{D}_U$-module.

Let $Z$ denote the closure of the support of $M_U$, which is a subvariety of $X$, and
let $\pi: Z' \to X$ be a resolution of singularities of $Z$. By the adjunction isomorphism
of [HTT08, Cor. 3.2.15], we have a natural morphism

$$
\mathbb{D}\pi!\pi^+(\mathbb{D}j?\mathcal{M}_{U,\eta^o,a})[-\text{codim } Z] \to \mathbb{D}j?\mathcal{M}_{U,\eta^o,a}, \quad ?=!,*.
$$

Denoting by $j': U' := \pi^{-1}(U) \hookrightarrow Z'$ the inclusion, we find that

$$
\pi^+(\mathbb{D}j?\mathcal{M}_{U,\eta^o,a}) \simeq j'_?\pi^+(\mathcal{M}_{U,\eta^o,a}) \quad \text{and} \quad \pi^+(\mathcal{M}_{U,\eta^o,a}) \simeq (\pi^+\mathcal{M}_U)_{\pi^*\eta^o,a}.
$$

On the other hand, since $e$ is proper, we have $\mathbb{D}\pi_* = \mathbb{D}\pi!$. Therefore, if we have
proved that

$$
\mathbb{D}j'_!(\pi^+\mathcal{M}_U)_{\pi^*\eta^o,a} \to \mathbb{D}j'_*(\pi^+\mathcal{M}_U)_{\pi^*\eta^o,a}
$$

is an isomorphism for $a \in V$, then (4.1) is an isomorphism for $a \in V$.

We can thus assume that $M_U$ is a holonomic $\mathcal{D}_U$-module with support equal
to $U$. By the induction assumption, we can moreover assume that, up to changing
notation for $D$, that $M_U$ is a flat vector bundle on $U$. $\square$
**Lemma 6.2.** Let $\eta^o$ be a closed one form on $U$. Assume that its extension $\eta$ with respect to a good compactification $(X, D)$ of $U$ is good wild. Let $\pi : (X', D') \to (X, D)$ be a modification of good compactifications of $U$ which is the identity on $U$. Then $\pi^*\eta$ is good wild with respect to $(X', D')$. Furthermore, each of the following properties is preserved by $\pi^*$:

(i) $\eta$ is logarithmic along $D$ and for each subset $I$ such that $\bigcap_{i \in I} D_i \neq \emptyset$, $\eta$ is $I$-non-resonant,

(ii) $\eta$ has a pole of order $\geq 2$ along each component of $D$.

*Proof.* Let us fix $x_o \in D$ and let $x'_o \subset \pi^{-1}(x_o)$ be a point on the exceptional divisor of $\pi$. There exist local coordinates $(x'_1, \dots, x'_k, y'_{k+1}, \dots, y'_n)$ on a neighborhood of $x'_o$ in $X'$ and local coordinates $(x_1, \dots, x_\ell, y_{\ell+1}, \dots, y_n)$ on a neighborhood of $x_o$ in $X$ such that, on these neighborhoods, $D = \{x_1 \cdots x_\ell = 0\}$, $D' = \{x'_1 \cdots x'_k = 0\} = \pi^{-1}(D)$, and $x_i \circ \pi$ is a monomial in $x'$ times a unit. With this expression, it is straightforward to check that good wildness is preserved by $\pi^*$. Assume that $x'_o$ is general on the exceptional divisor. Then $k=1$. We write $x_i \circ \pi = x_i^{m_i} u_i(x'_1, y')$, with $m_i \geq 1$ and $u_i$ is a local unit. Writing $\omega = \sum_{i=1}^\ell \omega_i(x, y) dx_i/x_i$, the pullback $\pi^*\eta$ reads $d(\pi^* u x_1^{\ell-\sum m_i e_i}) + (\sum_{i=1}^\ell m_i \omega_i \circ \pi) dx_1'/x_1'$. In case (ii), we have $e_i \neq 0$ for $i=1, \dots, \ell$, and so $\sum m_i e_i \neq 0$, hence the assertion for (ii). In case (i), we have $u \equiv 0$ and $\sum_i m_i \cdot (\omega_i \circ \pi)(0, 0) \neq 0$, and the assertion for (i) also follows. $\square$

6.a. *Proof of Theorem 4.5 in case* (i). Let $(Y, D_Y)$ be the good compactification of $U$ satisfying Condition (i) with respect to $\eta_1^o, \dots, \eta_k^o$. If it is not good for $M_U$ in the sense that $\text{D}j_*M_U = \mathcal{M}$ has a good formal structure along $D_Y$, then, by the Kedlaya-Mochizuki theorem ([Ked11, Th. 8.1.3], [Moc11b]), there exists a good compactification $(X, D)$ of $U$ dominating $(Y, D_Y)$ such that $\mathcal{M} = \text{D}j_*M_U$ has a good formal structure along $D$. We can regard the set $J_Y$ parametrizing the irreducible components of $D_Y$ as a subset of $J$ parametrizing those of $D$ (it parametrizes the proper transforms of the components of $D_Y$). The rational extension $\eta_k$ of $\eta_k^o$ ($k=1, \dots, r$) on $X$ has logarithmic poles along $D$ and is closed, hence has a constant residue $r_{k,i}$ along each $D_i^o$. For $i \in J \setminus J_0$, $r_{k,i}$ is a linear combination with coefficients in $\mathbb{N}$ of some $r_{k,j}$ with $j \in J_0$. The non-resonance assumption on $\eta_1^o, \dots, \eta_k^o$ with respect to $(Y, D_Y)$ implies that it holds for $\eta_1, \dots, \eta_k$ with respect to $(X, D)$, and in particular, for each $i \in J$ there exists $k \in \{1, \dots, r\}$ such that $r_{k,i} \neq 0$.

We claim that there exists a finite set of affine forms $L_{\ell,c}(a) = \sum_k \ell_k a_k + c$, with $\ell, c$ belonging to a finite subset of $(\mathbb{C}^r \setminus \{0\}) \times \mathbb{C})$ such that, if $L_{\ell,c}(a) \notin \mathbb{Z}$ for any $(\ell, c)$, the assumption of the proposition in Section 5 is fulfilled by $M_{\eta,a}$.

Indeed, for each $i \in J$, let $A_i \subset \mathbb{C}$ be a finite set such that the eigenvalues of the formal monodromy of $M$ along $D_i^o$ belong to $\exp(-2\pi i A_i)$. Then those of $M_{\eta,a}$ along $D_i^o$ take the form $\exp(-2\pi i(\alpha + \sum_{k=1}^r a_k r_{k,i}))$ for $\alpha \in A_i$. The condition on $a$, so that the assumptions of the proposition of Section 5 is fulfilled for $M_{\eta,a}$, reads

$$ (6.3) \qquad \alpha + \sum_{k=1}^{r} a_k r_{k,i} \notin \frac{1}{d}\mathbb{Z}, \quad \forall i \in J \text{ and } \forall \alpha \in A_i. $$

This defines the desired family of affine forms $L_{\ell,c}(a)$ on $\mathbb{C}^r$ provided that, for any $i \in J$, the linear form $L_i : a \mapsto \sum_{k=1}^r a_k r_{k,i}$ is not identically zero. This property
follows from the non-vanishing property of some $r_{k,i}$ ($k \in \{1, \dots, r\}$), which in turn follows from the non-resonance condition on $\eta_1^o, \dots, \eta_r^o$.

6.b. *Proof of Theorem 4.5 in case (ii).* As above, we start from a good compactification $(Y, D_Y)$ of $U$ such that such that the finite family $(\eta_1^o, \dots, \eta_r^o)$ satisfies the condition (ii) on $(Y, D_Y)$ and consider a good compactification $(X, D)$ of $U$ dominating $(Y, D_Y)$ such that $M = {}_D j_* M_U$ has a good formal structure along $D$. By Lemma 6.2(ii), Condition (ii) is still satisfied for the rational extensions $\eta$ of $\eta^o$ on $X$. We wish to prove that there exists a Zariski dense open subset $V \subset \mathbb{C}^r$ such that, for any $x_o \in D$, the natural morphism

$$ \mathcal{M}(!D)_{\eta, a, \hat{x}_o} \to \mathcal{M}(*D)_{\eta, a, \hat{x}_o} $$

is an isomorphism if $a \in V$

For any $x_o \in D$ and any $a \in \mathbb{C}^r$, the formal connection $\rho_{\hat{d}_I}^+ \mathcal{M}_{\eta, a, \hat{x}_o}$ is a successive extension of rank-one connections with connection forms (using the notation in §5.b, with coordinates $x', y$ centered at $x_o'$ in the ramified space $\rho_{\hat{d}_I}: \Delta^{\prime \ell} \times \Delta^{n-\ell} \to \Delta^{\ell} \times \Delta^{n-\ell}$) given by the formula

$$ (6.4) \qquad d\varphi + \rho_{\hat{d}_I}^* \sum_i \alpha_i \frac{dx_i}{x_i} + \rho_{\hat{d}_I}^* \sum_{k=1}^r a_k \left( d\left( \frac{u_k(x, y)x^{-e(\eta_k)}}{x^e} \right) + \omega_k \right), $$

with $\varphi = u_\varphi(x', y)x'^{-e(\varphi)} \in \Phi_{x_o}$, $u_\varphi$ and $u_k$ are identically zero or local units, and $\omega_k$ are logarithmic. We write $\rho_{\hat{d}_I}^*(u_k(x, y)x^{-e(\eta_k)}) = u_k'(x', y)x'^{-e'(\eta_k)}$. Note that the condition (a) of being elementary as in §5.b, which holds by assumption if $a=0$, may fail for this form for generic $a$, because the meromorphic function $u_\varphi(x', y)x'^{-e(\varphi)} + \sum_k a_k u_k'(x', y)x'^{-e'(\eta_k)}$ is not in general of the form $u(x', y)/x^e$ with $u(x', y)$ invertible. We will overcome this difficulty by a suitable local proper modification of $\Delta^{\prime \ell} \times \Delta^{n-\ell}$.

We define the Newton polyhedron $\mathrm{NP}_{x_o}(\varphi, \eta)$ as the boundary of the convex hull of the union of the octants $-e(\varphi) + \mathbb{N}^\ell$ and $-e'(\eta_k) + \mathbb{N}^\ell$ ($k=1, \dots, r$) for which the corresponding coefficient $u_k'$ is invertible (i.e., not identically zero). On the other hand, for each $a \in \mathbb{C}^r$, one defines in the usual way the Newton polyhedron $\mathrm{NP}(\varphi + \sum_k a_k(u_k'x'^{-e'(\eta_k)})$ as the convex hull of the octants generated by the exponents of occurring with a non-zero coefficient in the expansion of this meromorphic function. By construction we have the inclusion

$$ \mathrm{NP}(\varphi + \sum_k a_k(u_k'x'^{-e'(\eta_k)})) \subset \mathrm{NP}_{x_o}(\varphi, \eta). $$

**Lemma 6.5.** For any vertex $-e$ of $\mathrm{NP}_{x_o}(\varphi, \eta)$, let $A_e$ be the affine hyperplane of $\mathbb{C}^r$ defined by the equation

$$ \delta_{e(\varphi), e} u_{\varphi}(0, 0) + \sum_{k | e'(\eta_k) = e} a_k u_k'(0, 0) = 0, $$

where $\delta$ is the Kronecker index. If

$$ a \in V(x_o) := \mathbb{C}^r \setminus \bigcup_{\substack{e \text{ vertex} \\ \mathrm{NP}_{x_o}(\varphi, \eta)}} A_e, $$

then the following equality holds:

$$ \mathrm{NP}(\varphi + \sum_k a_k(u_k'x'^{-e'(\eta_k)})) = \mathrm{NP}_{x_o}(\varphi, \eta). $$
*Proof.* By definition, the condition $a \in V(x_o)$ means that, at each vertex $-e$ of $NP_{x_o}(\varphi, \eta)$, the coefficient of $x^{-\ell e}$ in $\varphi + \sum_k a_k(u'_k x'^{-e'(\eta_k)})$ does not vanish at $(0, 0)$. It follows that it is invertible in some neighborhood of $(0, 0)$, and the equality of Newton polyhedra follows. $\square$

Furthermore, by openness of the good formal structure (cf. §2.b), this $V(x_o)$ satisfies the same property for any $x$ in a small neighborhood of $x_o$. By compactness of $X$, we obtain a Zariski dense open subset $V \subset \mathbb{C}^r$, which is the complement of a finite number of affine hyperplanes, such that the previous equality of Newton polyhedra holds at each $x_o \in D$.

On the other hand, Condition (ii) in Theorem 4.5 implies that no hyperplane containing a face of the Newton polyhedron $NP_{x_o}(\eta)$, and similarly for $NP_{x_o}(\varphi, \eta)$, and thus $NP(\varphi + \sum_k a_k(u_k \circ \rho_{d_I} x'^{-e'(\eta_k)})$ for each $a \in V$, contains the origin: indeed, for each $i \in \{1, \dots, \ell\}$, there exists $k$ such that $e'_i(\eta_k) \neq 0$.

We define a local toric proper modification $\pi : (Z, D_Z) \to (\text{nb}(x'_o), D'_I)$, where $(Z, D_Z)$ is good and $D_Z = \pi^{-1}(D'_I)$, by refining the dual fan of the Newton polyhedron $NP_{x_o}(\varphi, \eta)$, in order to obtain a smooth fan subdividing the first octant of $\mathbb{R}^\ell$. By construction, in each chart of $Z$ corresponding to a cone of this fan, the pullback by $\pi^*$ of (6.4), for $a \in V$, is good wild, and has a pole of order at least two along each irreducible component of $D_Z$. We can apply the result of §5.b at each point of $D_Z$ and, by Lemma 4.2, we conclude that the isomorphism (4.1) holds for the connection defined by (6.4). It therefore holds for $\rho_{d_I}^* \mathcal{M}_{\eta, a, \hat{x}_o}$, which is a direct sum of such objects. By the same argument as in §5.a, we deduce that it holds for $\mathcal{M}_{\eta, a, \hat{x}_o}$. The property that (4.1) is an isomorphism holds if and only if it holds formally at each $\hat{x}_o$. Therefore, this concludes the proof of Theorem 4.5 in case (ii). $\square$

### Part 3. Laplace transform of a Stokes-filtered constructible sheaf of exponential type

#### 7. INTRODUCTION TO PART 3

7.a. *Statement of the theorem.* The Laplace transformation $F_\pm$ with kernel $E^{\pm t\tau} := (\mathcal{O}_{A_t^1} \times_{A_\tau^1}, d \pm d(\tau t))$ induces an equivalence between the category of algebraic holonomic $\mathcal{D}_{A_t^1}$-modules $N$ on the affine line $A_t^1$ with coordinate $t$, and that of algebraic holonomic $\mathcal{D}_{A_\tau^1}$-modules $M$, with quasi-inverse $F_\mp$. Let us consider the projections

<figure><img src="image_2.png" /></figure>

so that, for $(N, \nabla) \in \text{Mod}(\mathcal{D}_{A_t^1})$ and $(M, \nabla) \in \text{Mod}(\mathcal{D}_{A_\tau^1})$, we have

$$F_{\pm}(N) = q_{+}(p^{+}N, p^{+}\nabla \pm d(\tau t)),$$

$$F_{\mp}(M) = p_{+}(q^{+}M, q^{+}\nabla \mp d(\tau t)).$$

It is known (cf. e.g. [Mal91, KKP08]) that this equivalence induces an equivalence between the full subcategories

* $\text{Mod}_{\text{holreg}}(\mathcal{D}_{A_t^1})$ consisting of regular holonomic $\mathcal{D}_{A_t^1}$-modules $N$, and
• Mod$_{holexp}$(D$_{A_τ^1}$) consisting of algebraic holonomic D$_{A_τ^1}$-modules M which have a regular singularity at τ = 0, an irregular one of exponential type at τ = ∞, and no other singularity at finite distance.

Furthermore, it specializes as an equivalence between the respective sub-categories

• Mod$^0_{holreg}$(D$_{A_t^1}$), whose objects N have vanishing global de Rham hypercohomology RΓ(A$_t^1$, $^p$DR(N)), and

• Mod$_{holexp}$(D$_{A_τ^1}$, *0) whose objects M satisfy M ≃ M(*0).

As in [KKP08, YZ24], we will focus on the latter equivalence. The analytic restriction functor

$$ \mathrm{Mod}_{\mathrm{holexp}}(\mathcal{D}_{A_\tau^1}, *0) \xrightarrow{\tilde{i}_\infty^*} \mathrm{Mero}(\infty, \exp) $$

to the category Mero(∞, exp) of analytic germs at ∞ ∈ P$_{τ}$ 1 of meromorphic connections with an irregular singularity of exponential type, is an equivalence, with quasi-inverse functor being Deligne's regular meromorphic extension at τ = 0. Composing F$_{±}$ with this restriction functor yields a pair (F$_{±}$∞, F$_{∓}$∞) of quasi-inverse functors.

For an object N of Mod$_{holreg}$(D$_{A_t^1}$), the analytic de Rham complex F = $^p$DR$^{an}$(N) is known to be a C-perverse complex on $\mathbb{C}_t := A_t^{1,an}$. From the standard algebraic/analytic comparison theorem RΓ(A$_t^1$, $^p$DR(N)) ≃ RΓ($\mathbb{C}_t$, $^p$DR$^{an}$(N)), we deduce that $^p$DR(N) has zero global hypercohomology if and only if $^p$DR$^{an}$(N) has zero global hypercohomology, and this is equivalent to the property that $^p$DR$^{an}$(N) is a constructible sheaf F placed in degree −1, having singularities on a finite subset C ⊂ $\mathbb{C}_t$, with vanishing global hypercohomology (cf. [KKP08]). We denote this category as Perv$^0$(C$_t$, C), which can be defined with any field of coefficients k, and then denoted by Perv$^0$(C$_t$, k, C).

On the other hand, the perverse complex $^p$DR$^{an}$(M) on $\mathbb{C}_τ$ can be equipped with a Stokes structure of exponential type at infinity, by means of the function ($^p$DR≤$_{•}$, $^p$DR<$_{•}$) (the moderate and rapid decay de Rham complexes twisted by an exponential factor, cf. [Sab13, Cor. 5.4]), with exponential factors of the form cτ for c belonging to a finite subset C ⊂ $\mathbb{C}$. We denote by (G, G$_{•}$) this Stokes-C-perverse sheaf (we refer to [Sab13] for this notion), which is an object living on the real blow-up space P̃$_{τ}$ 1 of the projective line P$_{τ}$ 1 at infinity, and by PervSt(P̃$_{τ}$ 1 , C) corresponding category. Our assumption that M ≃ M(*0) is equivalent to $^p$DR$^{an}$(M) ≃ Rj$_{0}$$_{*}$j$_{0}^{-1}$ $^p$DR$^{an}$(M), where j$_{0}$ : $\mathbb{C}_τ^* \to \mathbb{C}_τ$ denotes the open inclusion. We denote by PervSt(P̃$_{τ}$ 1 , C, *0) the full subcategory of PervSt(P̃$_{τ}$ 1 , C) consisting of objects (G, G$_{•}$) satisfying G ≃ Rj$_{0}$$_{*}$j$_{0}^{-1}$G. More precisely, let ϖ : P̃$_{τ}$ 1 → P$_{τ}$ 1 denote the real blowing up of the projective line P$_{τ}$ 1 at τ = ∞ and let us consider the complementary inclusions

$$ A_{\tau}^{1} \xrightarrow{\tilde{j}_{\infty}} \tilde{P}_{\tau}^{1} \xleftarrow{\tilde{i}_{\infty}} S_{\tau}^{1} = \varpi^{-1}(\infty) = \tilde{P}_{\tau}^{1} \setminus A_{\tau}^{1}. $$

Then H − $^{1}$G|$_{C}$ ∗ τ is a local system as well as its pushforward by ˜j$_{∞}$, so that the restriction of the latter by ˜i$_{∞}$, that we denote by L, is a C-local system on S$_{τ}$ 1 . The latter is equipped with a Stokes filtration of exponential type, thereby defining a Stokes-filtered C-local system (L, L$_{•}$) on S$_{τ}$ 1 indexed by a finite set C ⊂ $\mathbb{C}$. We denote by Sto(S$_{τ}$ 1 , C) the corresponding category. For any field of coefficients k, the categories PervSt(P̃$_{τ}$ 1 , k, C) and Sto(S$_{τ}$ 1 , k, C) are also defined. One easily obtains the
equivalence (cf. Section 8.f)

$$ (7.1) \qquad \mathrm{PervSt}(\tilde{\mathbb{P}}_{\tau}^{1}, *0) \xrightarrow[\sim]{\tilde{i}_{\infty}^{-1}[-1]} \mathrm{Sto}(S_{\tau}^{1}), $$

which can be decorated with the field of coefficients $k$ and the set $C$ corresponding to exponential factors.

The Riemann-Hilbert correspondence and the RH-Birkhoff-Deligne-Malgrange correspondence

$$ (7.2) \qquad \begin{array}{ccc} \mathrm{Mod}_{\mathrm{holreg}}^{0}(\mathcal{D}_{\mathcal{A}_{\tau}^{1}}) & \xrightarrow{\stackrel{p}{\mathrm{DR}}^{\mathrm{an}}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!} & \mathrm{Perv}^{0}(\mathbb{C}_{t}) \\ \mathrm{Mod}_{\mathrm{holexp}}(\mathcal{D}_{\mathcal{A}_{\tau}^{1}}, *0) & \xrightarrow[\sim]{(p\mathrm{DR}_{\le \bullet})^{p\mathrm{DR}_{<\bullet}}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!} & \mathrm{PervSt}(\tilde{\mathbb{P}}_{\tau}^{1}, *0) \end{array} $$

are well-known (cf. e.g. [Sab13, Chap. 5] for the second line). We can also restrict the above categories by specifying the singularity set $C$ (first line) or the index set $C$ of exponential factors (second line).

**Theorem 7.3.** For any field $k$, there exist pairs of quasi-inverse functors $(F_{\pm}^{\mathrm{top}}, F_{\mp}^{\mathrm{top}})$ and $(F_{\pm}^{\infty, \mathrm{top}}, F_{\mp}^{\infty, \mathrm{top}})$ and a commutative diagram:

$$ (7.3) \qquad \begin{array}{ccc} \mathrm{Perv}^{0}(\mathbb{C}_{t}, \mathbf{k}, C) & \xleftarrow{F_{\pm}^{\mathrm{top}}} & \mathrm{PervSt}(\tilde{\mathbb{P}}_{\tau}^{1}, \mathbf{k}, C, *0) \\ \xrightarrow{F_{\mp}^{\mathrm{top}}} & \xrightarrow{\sim} & \xleftarrow{\left|\right.} \\ F_{\pm}^{\infty, \mathrm{top}} & \xrightarrow{\sim} & F_{\mp}^{\infty, \mathrm{top}} \\ F_{\mp}^{\infty, \mathrm{top}} & \xrightarrow{\sim} & \mathrm{Sto}(S_{\tau}^{1}, \mathbf{k}, C) \end{array} $$

which, when $k = \mathbb{C}$, is compatible, via the Riemann-Hilbert (and RH-Birkhoff-Deligne-Malgrange) correspondence, with the pairs

$$ (7.3) \quad \begin{array}{ccc} \mathrm{Mod}_{\mathrm{holreg}}^{0}(\mathcal{D}_{\mathcal{A}_{\tau}^{1}}, C) & \xleftarrow{F_{\pm}} & \mathrm{Mod}_{\mathrm{holexp}}(\mathcal{D}_{\mathcal{A}_{\tau}^{1}}, *0, C) \\ \xrightarrow{F_{\mp}} & \xleftarrow{\sim} & \xleftarrow{\left|\right.} \\ F_{\pm}^{\infty} & \xrightarrow{\sim} & F_{\mp}^{\infty} \\ F_{\mp}^{\infty} & \xrightarrow{\sim} & \mathrm{Mero}(\infty, C) \end{array} $$

via the equivalences (7.2).

In [Sab13], the correspondence $p\mathrm{DR}^{\mathrm{an}}(N) \mapsto (\mathcal{G}, \mathcal{G}_{\bullet})$, called topological Laplace transformation, has been made explicit for any perverse complex with coefficients in a field $k$, and it has been shown that, when $k = \mathbb{C}$, it is compatible via the Riemann-Hilbert (and RH-Birkhoff-Deligne-Malgrange) correspondence with the Laplace transformation of regular holonomic $\mathcal{D}_{\mathcal{A}_{\tau}^{1}}$-modules. Making explicit the inverse correspondence and its compatibility with (7.2) has been motivated by the work [YZ24]. This text can then be regarded as a complement to [Sab13, Chap. 7]. Let us also mention that the construction of the pair $(F_{\pm}^{\infty, \mathrm{top}}, F_{\mp}^{\infty, \mathrm{top}})$ is a very special case of the more general construction done by T. Mochizuki in [Moc18], and was initiated by B. Malgrange in [Mal91].

7.b. *Comparison with the approach of [YZ24]*. In [YZ24], Tony Yue Yu and Shaowu Zhang prove a formula for the inverse topological Laplace transformation $(\mathcal{G}, \mathcal{G}_{\bullet}) \mapsto \mathcal{F}$ under the global vanishing assumption that we also adopt here. Let us emphasize that the main nice idea of [YZ24] is to use co-Stokes structures, that are easier to handle. We propose here an approach of their formula that we show to be
compatible with the Riemann-Hilbert correspondence (in [YZ24], this compatibility is obtained in an indirect way). We make full use of the notion of Stokes structure in dimension $> 1$ (namely, in dimension two).

In loc. cit., the authors first replace the Stokes-filtered local system $(\mathcal{L}, \mathcal{L}_{\bullet})$ on $S_{\tau}^1$ with a co-Stokes-filtered local system $(\mathcal{L}, \mathcal{L}_{<})$ on $S_{\tau}^1$ (i.e., by replacing the order $\le$ with the strict order $<$)—both notions are known to be equivalent—, and then “thicken” this object to produce, for any given $t \in \mathbb{C}_t^*$, a similar object on $\mathbb{C}_{\tau}^*$. This is motivated by the fact that the topological Laplace transform of a perverse complex on $\mathbb{C}_t$ having no global hypercohomology (equivalently, of a constructible sheaf placed in degree one on $\mathbb{C}_t$ having no global cohomology, cf. [KKP08]) can be defined in order to produce such a “thickened” object. The drawback of this approach is that the Stokes-filtered de Rham complex attached to a free $\mathbb{C}[\tau, \tau^{-1}]$-module with connection having a regular singularity at $\tau = 0$ and an irregular one with exponential type at $\tau = \infty$ produces a “non-thickened” Stokes constructible sheaf. In order to obtain a Riemann-Hilbert correspondence, the authors of [YZ24] make use of the Riemann-Hilbert correspondence for the corresponding regular holonomic module obtained by Laplace transform, and the property that Laplace transformation has an inverse transformation both at the topological level (a property that they prove) and at the algebraic/analytic level (a property much easier to prove).

In the presentation we give in Section 9 of this note, the Riemann-Hilbert correspondence is obtained independently of the $\pm$ involutivity of the topological Laplace transformation. For that purpose, we make full use of the notion of a Stokes-filtered local system along a divisor with normal crossings.

### 7.c. Basic lemmas.
We will make use of the following basic lemmas.

**Lemma 7.4.** Let $I$ be a non-empty semi-closed interval in $\mathbb{R}$ and let $\bar{\Delta}_{<}$ be a non-empty closed disc with a closed interval deleted in its boundary. Then, for any field $k$, we have $H^*(I, k) = 0$ and $H^*(\bar{\Delta}_{<}, k) = 0$. $\square$

**Lemma 7.5.** Let $j: U \to Y$, resp. $i: F \to Y$, be complementary open, resp. closed, inclusion of topological spaces. Given a sheaf $\mathcal{H}_U$ on $U$ and a subsheaf $\mathcal{H}_F \subset i^{-1}j_*\mathcal{H}_U$ on $F$, there exists a unique subsheaf $\mathcal{H}$ on $Y$ such that $j^{-1}\mathcal{H} = \mathcal{H}_U$ and $i^{-1}\mathcal{H} = \mathcal{H}_F$. $\square$

### 7.d. Dramatis personae.

* The holonomic $\mathbb{C}[\tau]\langle\partial_{\tau}\rangle$-module $(M, \nabla)$, its Stokes-filtered local system $(\mathcal{L}, \mathcal{L}_{\bullet})$ on $S_{\tau}^1$ at $\tau = \infty$ indexed by $C\tau$ with $C \subset \mathbb{C}$.

* The exponentially twisted $\mathbb{C}[\tau, t]\langle\partial_{\tau}, \partial_t\rangle$-module $(q^+M, \nabla - d(\tau t))$ and its germ $\mathcal{M}$ along $D_{\infty} = \{\infty\} \times \mathbb{C}_t$.

* The Stokes-filtered local system $(\mathcal{L}, \mathcal{L}_{\bullet})$ on $S_{\tau}^1 \times \mathbb{C}_t$ indexed by $\Phi = \{\varphi_c \mid c \in C\}$ with $\varphi_c(\tau, t) = \tau(c - t)$; it is obtained by an exponential twist from $(\mathcal{L}, \mathcal{L}_{\bullet})$; the main character is played by the subsheaf $\mathcal{L}_{<0} \subset \mathcal{L}$.

* The complex blow-up $e: X \to \mathbb{P}_{\tau}^1 \times \mathbb{C}_t$ at the points $(\infty, c)$, the divisor with normal crossings $D = e^{-1}(D_{\infty})$ and the real blow-up $\tilde{X}$ of $X$ along the irreducible components of $D$, with boundary $\tilde{D}$; the pullback germ of $\mathcal{D}_X$-module $\mathcal{M}_X$ along $D$.

* The Stokes-filtered local system $(\mathcal{L}_X, \mathcal{L}_{X, \bullet})$ on $\tilde{D}$ pulled back from $(\mathcal{L}, \mathcal{L}_{\bullet})$ indexed by $\Phi_X = \{\psi_c \mid c \in C\}$; the main character is $\mathcal{L}_{X, \le 0}$ and the main relation is $\mathcal{L}_{< 0} = \mathcal{R}\tilde{e}_{*}\mathcal{L}_{X, \le 0}$, with $\tilde{e}: \tilde{D} \to S_{\tau}^1 \times \mathbb{C}_t$ induced by $e$.
• On the Fourier side, the regular holonomic $\mathbb{C}[t]\langle\partial_t\rangle$-module $N$ and its associated
perverse complex $\mathcal{F} = {}^p\mathrm{DR}^{\mathrm{an}}(N)$.

• If $N$ is the Fourier transform of $M$, we prove the corresponding topological relation $\mathcal{F} \simeq \mathcal{R}\tilde{p}_*\mathcal{L}_{<0} \simeq \mathcal{R}\tilde{\pi}_*\mathcal{L}_{X, \le 0}$ (cf. Diagrams (8.14) and (9.5)).

8. THE LAPLACE TRANSFORMATION FOR STOKES-FILTERED LOCAL SYSTEMS
OF EXPONENTIAL TYPE

Let $k$ be a field. Let us fix the parametrization

$$
\begin{gather*}
\mathbb{R}/2\pi\mathbb{Z} \to S^1 \subset \mathbb{C} \\
\theta \mapsto e^{i\theta}
\end{gather*}
$$

of $S^1$, and we often write $\theta$ instead of $\theta \mod 2\pi$. We will implicitly identify $\mathbb{R}/2\pi\mathbb{Z}$ with $S^1$ via this exponential map. Recall (cf. [Sab13]) that a $k$-Stokes-filtered local system $(\mathcal{L}, \mathcal{L}_{\bullet})$ of exponential type on $S^1$ indexed by a finite set $C \subset \mathbb{C}$ is a locally constant sheaf $\mathcal{L}$ of $k$-vector spaces on $S^1$ equipped of a family (indexed by $C$) of nested subsheaves $\mathcal{L}_{<c} \subset \mathcal{L}_{\le c}$ of $k$-vector spaces such that

• for each $c \in C$, $\mathrm{gr}_c \mathcal{L} := \mathcal{L}_{\le c} / \mathcal{L}_{<c}$ is a $\mathbb{C}$-local system on $S^1$,

• for $c' \neq c$ and $\theta \in \mathbb{R}/2\pi\mathbb{Z}$, we have $\mathcal{L}_{\le c', \theta} < \mathcal{L}_{<c, \theta}$ if $\mathrm{Re}[(c-c')e^{-i\theta}] < 0$ (a property that we denote by $c <_{\theta} c'$, and we set $c \le_{\theta} c'$ if $c = c'$ or $c <_{\theta} c'$),

• for any $\theta \in \mathbb{R}/2\pi\mathbb{Z}$, we have $\mathcal{L}_{c, \theta} \simeq \bigoplus_{c' \le \theta c} \mathrm{gr}_{c'} \mathcal{L}_{\theta}$.

The filtration indexed by $C$ can be extended to a filtration indexed by $\mathbb{C}$ by
setting, for each $t \in \mathbb{C}$ and each $\theta \in \mathbb{R}/2\pi\mathbb{Z}$,

$$
\mathcal{L}_{\le t, \theta} = \sum_{\substack{c \in C \\ c \le \theta t}} \mathcal{L}_{\le c, \theta} \quad \text{and} \quad \mathcal{L}_{< t, \theta} = \sum_{\substack{c \in C \\ c < \theta t}} \mathcal{L}_{\le c, \theta},
$$

and one checks that these formulas define nested subsheaves $\mathcal{L}_{<t} \subset \mathcal{L}_{\le t}$ of $\mathcal{L}$, which
are equal if $t \notin C$.

The Stokes directions of a finite set $C \subset \mathbb{C}$ are the points on $S^1$ consisting of
$e^{i(\pm\pi/2 - \arg(c_i - c_j))}$ for each pair $c_i \neq c_j$ in $C$. In the following, we often omit the
reference to the field $k$, which is fixed. We consider the following categories:

• the category $\mathrm{Sto}(S^1, k, C)$ of Stokes-filtered $k$-local systems $(\mathcal{L}, \mathcal{L}_{\bullet})$ of exponential type on $S^1$ indexed by $C$;

• the category $\mathrm{Perv}^0(\mathbb{C}_t, k, C)[-1]$ of $\mathbb{C}$-constructible sheaves $\mathcal{F}$ of $k$-vector spaces on $\mathbb{C}_t$ with singularities contained in $C$ and satisfying $H^*(\mathbb{C}_t, \mathcal{F}) = 0$. The notation $\mathrm{Perv}^0(\mathbb{C}_t, k, C)[-1]$ is justified by the property (cf. [KKP08]) that $\mathcal{F}$ is an object of this category if and only if $\mathcal{F}[1]$ is a perverse sheaf on $\mathbb{C}_t$ with vanishing global hypercohomology.

The main result of this section consists of the definition of the functor $F_{-}^{\infty, \text{top}}$ (Definition 8.17), in such a way that, together with the functor $F_{+}^{\infty, \text{top}}$ constructed in [Sab13, Chap. 7], we obtain a diagram

$$
(8.1) \qquad \mathrm{Perv}^0(\mathbb{C}_t, k, C) \xrightarrow[F_-^{\infty, \mathrm{top}}]{\qquad} \mathrm{Sto}(S^1, k, C)
$$

which satisfies the next proposition:

**Proposition 8.2.** The functors $F_+^{\infty, \text{top}}$, $F_-^{\infty, \text{top}}$ are quasi-inverse equivalences of categories.
8.a. A preliminary computation for Stokes-filtered local systems of exponential type.

**Lemma 8.3.** For each $t \in \mathbb{C}$, we have $H^k(S^1, \mathcal{L}_{<t}) = 0$ if $k \neq 1$ and

$$ \dim H^1(S^1, \mathcal{L}_{<t}) = \begin{cases} \mathrm{rk} \mathcal{L} & \text{if } t \notin C, \\ \mathrm{rk} \mathcal{L} - \mathrm{rk} \mathrm{gr}_c \mathcal{L} & \text{if } t = c \in C. \end{cases} $$

*Proof.* Given $t \in \mathbb{C}$, let us decompose $S^1$ as $I_1 \cup I_2$, where each $I_i$ is a closed interval of length $\pi + \varepsilon$ with $\varepsilon > 0$ small enough such that each $I_i$ contains exactly one Stokes direction of $C \cup \{t\}$, and its endpoints are not Stokes directions. We denote the inclusion $I_i \hookrightarrow S^1$ by $\iota_i$, and $\iota_{12} : I_1 \cap I_2 \hookrightarrow S^1$. For any complex of sheaves $\mathcal{F}$ on $S^1$ we have the Mayer-Vietoris triangle

$$ \mathcal{F} \to R\iota_{1*} \iota_1^{-1}\mathcal{F} \oplus R\iota_{2*} \iota_2^{-1}\mathcal{F} \to R\iota_{12*} \iota_{12}^{-1}\mathcal{F} \xrightarrow{+1} \mathcal{F}. $$

By considering the associated long exact sequence attached to $R\Gamma(S^1, \bullet)$ of the above triangle with $\mathcal{F} = \mathcal{L}_{<t}$, we are left with showing

* $H^k(I_i, \mathcal{L}_{<t}) = 0$ for any $k$,

* $H^k(I_{12}, \mathcal{L}_{<t}) = 0$ for $k \geq 1$ and $\dim H^0(I_{12}, \mathcal{L}_{<t}) = \mathrm{rk} \mathcal{L}$ if $t \notin C$, or $\mathrm{rk} \mathcal{L} - \mathrm{rk} \mathrm{gr}_c \mathcal{L}$ if $t = c \in C$.

On each interval $I_i$, $(\mathcal{L}, \mathcal{L}_{\bullet})$ is graded (cf. [Sab13, Lem. 3.12]). By considering each homogeneous component separately, we can thus assume that $(\mathcal{L}, \mathcal{L}_{\bullet})$ has only one jumping index $c_o \in \mathbb{C}$. This means that $\mathcal{L} = \mathrm{gr}_{c_o} \mathcal{L}$ and $\mathcal{L}_{<t} = \mathrm{gr}_{c_o} \mathcal{L}$ on the open sub-interval of $I_i$ where $c_o < t$ and 0 otherwise.

If $t \neq c_o$, this interval is non-empty and the first point follows from the vanishing $H_c^*(\{0, 1\}, \mathbf{k}) = 0$. For the second point, one checks that $\mathcal{L}_{<t}|_{I_{12}}$ is a constant sheaf and the sum of its ranks on both connected components of $I_{12}$ are equal to the rank of $\mathcal{L}$.

If $t = c_o$, we have identically $(\mathrm{gr}_{c_o} \mathcal{L})_{<t} = 0$ and there is nothing to prove. $\square$

**Remark 8.4.** The cohomology $H^k(S^1, \mathcal{L}_{\leq c})$, for $c \in C$, behaves a little differently. Indeed, in the decomposition considered in the last part of the proof of the lemma, the component of $\mathcal{L}_{\bullet}$ corresponding to $c \in C$ is a locally constant sheaf on $S^1$ isomorphic to $\mathrm{gr}_c \mathcal{L}$, and we have a non-canonical decomposition

$$ H^1(S^1, \mathcal{L}_{\leq c}) \simeq H^1(S^1, \mathcal{L}_{<c}) \oplus H^1(S^1, \mathrm{gr}_c \mathcal{L}), \quad H^0(S^1, \mathcal{L}_{\geq c}) \simeq H^0(S^1, \mathrm{gr}_c \mathcal{L}). $$

8.b. *Topology of the family of real blow-ups.* For any $a \in \mathbb{C}^*$, we define the open intervals $S_{0<a}^1, S_{0>a}^1 \subset S^1$ by

$$ S_{0<a}^1 = \{\theta \mid \mathrm{Re}(ae^{-i\theta}) > 0\} = \{\theta \mid \arg a - \theta \in (-\pi/2, \pi/2) \mod 2\pi\}, $$

$$ S_{a<0}^1 = \{\theta \mid \mathrm{Re}(ae^{-i\theta}) < 0\} = \{\theta \mid \arg a - \theta \in (\pi/2, 3\pi/2) \mod 2\pi\}, $$

and the respective complementary closed intervals

$$ S_{a \ge 0}^1 = \{\theta \mid \mathrm{Re}(ae^{-i\theta}) \le 0\} = \{\theta \mid \arg a - \theta \in [\pi/2, 3\pi/2] \mod 2\pi\}, $$

$$ S_{0 \le a}^1 = \{\theta \mid \mathrm{Re}(ae^{-i\theta}) \ge 0\} = \{\theta \mid \arg a - \theta \in [-\pi/2, \pi/2] \mod 2\pi\}. $$

If $a = 0$, then by convention $S_{0<a}^1 = S_{0>a}^1 = \emptyset$ and $S_{0 \le a}^1 = S_{0 \ge a}^1 = S^1$.

In the product $S^1 := S^1 \times \mathbb{C}_t$, we define for each $s \in \mathbb{C}$ the subsets

$$ (8.5) \qquad S_{t<s}^1 = \bigsqcup_{t \in \mathbb{C}_t} S_{0<s-t}^1, \qquad S_{t>s}^1 = \bigsqcup_{t \in \mathbb{C}_t} S_{s-t<0}^1, $$

and $S_{t \le s}^1, S_{t \ge s}^1$ similarly. The next lemma is immediate.
**Lemma 8.6.** The subsets $S_{t<s}^1, S_{t>s}^1$ are open in $S^1 \times \mathbb{C}_t$. $\square$

For $s$ fixed, the fiber $S_{t<s,\theta}^1$ over $\theta \in S^1$ is the open half-plane $\text{Re}[(t-s)e^{-i\theta}] < 0$ (bounded by the real line $\text{Re}[(t-s)e^{-i\theta}] = 0$ going through $s$), while $S_{t\ge s,\theta}^1$ is the complementary closed half-plane.

Let $\tilde{\mathbb{P}}_t^1 = \mathbb{C}_t \sqcup S_t^1$ be the real blow-up of $\mathbb{P}^1$ at $t = \infty$. Let $t' = 1/t$ denote the coordinate at $\infty$ on $\mathbb{P}_t^1$. The coordinate on $S_t^1$ is $-\arg t' = \arg t$ if we choose the orientation induced by that of $\mathbb{C}_t$. For any given $s \in \mathbb{C}$ and $t$ with $|t| \gg 0$, we have $\arg(t-s) = \arg t$ on $S_t^1$ since $1/(t-s) = t' \cdot (1/(1-st'))$. We consider the open subsets of $S^1 \times S_t^1$ defined by

$$ (8.7) \qquad \begin{aligned} \tilde{S}_{\infty,<}^1 &= \{(\theta, \arg t) \in S^1 \times S_t^1 \mid \arg t - \theta \in (\pi/2, 3\pi/2) \bmod 2\pi\}, \\ \tilde{S}_{\infty,>}^1 &= \{(\theta, \arg t) \in S^1 \times S_t^1 \mid \arg t - \theta \in (-\pi/2, \pi/2) \bmod 2\pi\}, \end{aligned} $$

and we extend $S_{t<s}^1, S_{t>s}^1$ as open subsets of $S^1 \times \tilde{\mathbb{P}}_t^1$ by setting

$$ (8.8) \qquad \tilde{S}_{t<s}^1 = S_{t<s}^1 \sqcup \tilde{S}_{\infty,<}^1, \qquad \tilde{S}_{t>s}^1 = S_{t>s}^1 \sqcup \tilde{S}_{\infty,>}^1. $$

We define similarly the respective complementary closed subsets $\tilde{S}_{t\ge s}^1$ and $\tilde{S}_{t\le s}^1$ of $S^1 \times \tilde{\mathbb{P}}_t^1$, which are equal to $S_{t\ge s}^1 \sqcup \tilde{S}_{\infty,>}^1$ and $S_{t\le s}^1 \sqcup \tilde{S}_{\infty,<}^1$ respectively, with an obvious notation. We have an exact sequence<fnref n="3" />

$$ (8.9) \qquad 0 \to \mathbf{k}_{\tilde{S}_{t<s}^1} \to \mathbf{k}_{S_{\tau}^1 \times \tilde{\mathbb{P}}_t^1} \to \mathbf{k}_{\tilde{S}_{t\ge s}^1} \to 0. $$

Let $i_s : S^1 \times \{s\} \hookrightarrow S^1 \times \tilde{\mathbb{P}}_t^1$. Then we have the equalities

$$ (8.10) \qquad i_s^{-1} \mathbf{k}_{\tilde{S}_{t<s}^1} = 0 \quad \text{and} \quad i_s^{-1} \mathbf{k}_{\tilde{S}_{t\ge s}^1} = \mathbf{k}_{S^1 \times \{s\}}. $$

We also set, for example,

$$ \tilde{S}_<^1 := (S^1 \times \mathbb{C}_t) \sqcup \tilde{S}_{\infty,<}^1 \subset S^1 \times \tilde{\mathbb{P}}_t^1 \quad \text{and} \quad \tilde{S}_>^1 := (S^1 \times \mathbb{C}_t) \sqcup \tilde{S}_{\infty,>}^1 \subset S^1 \times \tilde{\mathbb{P}}_t^1. $$

Therefore, $\tilde{S}_{t<s}^1 \subset \tilde{S}_<^1$ for each $s \in \mathbb{C}$.

If we identify $\mathbb{C}_t$ with an open disc centered at $s$, the fiber $S_{t<s,\theta}^1$, which is an open half-plane, is identified with an open half-disc. Then $\tilde{S}_{t<s,\theta}^1$ is obtained by adding to it the open interval of $S_t^1$ consisting of points $\arg t$ satisfying $\cos(\arg t - \theta) < 0$. We have thus obtained the following topological description.

**Lemma 8.11.** Let us fix $s \in \mathbb{C}$. Then, for any $\theta \in S^1$, the fibers $\tilde{S}_{t<s,\theta}^1$ and $\tilde{S}_{t>s,\theta}^1$ are homeomorphic to a closed disc with one non-empty closed interval deleted in its boundary. Furthermore, for any $s \ne s' \in \mathbb{C}$, the fiber $\tilde{S}_{s'<t<s,\theta}^1 := \tilde{S}_{s'<t,\theta}^1 \cap \tilde{S}_{t<s,\theta}^1$ is

* homeomorphic to a closed disc with two disjoint non-empty closed intervals deleted in its boundary, if $s' <_\theta s$, and

* is empty otherwise. $\square$

<footnote n="3">³For any locally closed subset $Z \subset X$ of a topological space $X$ and any sheaf $\mathcal{F}$ on $X$, $\mathcal{F}_Z$ denotes the restriction of $\mathcal{F}$ to $Z$ extended by zero.</footnote>
**Cohomological computations.** As an immediate consequence, we can compute the cohomology with compact support of these sets. We find, according to Lemma 7.4:

$$ (8.12) \qquad \begin{aligned} & \mathcal{R}\Gamma_c(\tilde{S}_{t<s,\theta}^1, \mathbf{k}) = 0, \\ & \mathcal{R}\Gamma_c(\tilde{S}_{s'<t<s,\theta}^1, \mathbf{k}) = \begin{cases} \mathbf{k}[-1] & \text{if } s' <_\theta s, \\ 0 & \text{otherwise.} \end{cases} \end{aligned} $$

Let $\tilde{p} : S^1 \times \tilde{\mathbb{P}}_t^1 \to \tilde{\mathbb{P}}_t^1$ denote the projection.

**Lemma 8.13.** We have

$$ \tilde{p}_* \mathbf{k}_{\tilde{S}_{t \ge s}^1} \simeq \mathbf{k}_{\tilde{\mathbb{P}}_t^1}, \quad R^1 \tilde{p}_* \mathbf{k}_{\tilde{S}_{t \ge s}^1} \simeq \mathbf{k}_s, \quad R^k \tilde{p}_* \mathbf{k}_{\tilde{S}_{t \ge s}^1} = 0, \text{ for } k \ge 2. $$

*Proof.* All fibers of $\tilde{p} : \tilde{S}_{t \ge s}^1 \to \tilde{\mathbb{P}}_t^1$ are connected, hence the first point. If $t \neq s$ the fiber of $p$ is a non-empty closed interval of $S^1$, while the fiber over $t = s$ is equal to $S^1$, yielding the last two points. $\square$

8.c. The Laplace transform of a Stokes-filtered local system of exponential type. In order to motivate and justify the construction, we let $\mathbb{C}_\tau$ be the complex line with coordinate $\tau$, let $\mathbb{P}_\tau^1$ denote its projective completion, and we let $\varpi : \tilde{\mathbb{P}}_\tau^1 \to \mathbb{P}_\tau^1$ denote the real blowing up of $\mathbb{P}_\tau^1$ at $\tau = \infty$, with $\varpi^{-1}(\infty) := S_\tau^1 \simeq S^1$. When needed, we use the coordinate $\tau'$ on the chart centered at $\infty$, so that $\tau' = 1/\tau$ on $\mathbb{C}_\tau^*$. We mainly use the polar coordinates for $\tau'$ on $S^1 \times (0, \infty]$, that we denote $(\rho_{\tau'}, \theta)$, i.e., $\tau' = \rho_{\tau'} e^{i\theta}$.

We consider the product $\mathbb{P}_\tau^1 \times \mathbb{C}_t$ and the commutative diagram (8.14):

$$ (8.14) \qquad \begin{tikzcd}[column sep=2.8em, row sep=2.8em] 
    & S_\tau^1 \arrow[r, "\tilde{q}"] \arrow[d, "S_\tau^1 \times \mathbb{C}_t"] & \\
    \tilde{\mathbb{P}}_\tau^1 \arrow[r, "\tilde{q}"] \arrow[d, "S_\tau^1 \times \mathbb{C}_t"] & \tilde{\mathbb{P}}_\tau^1 \times \mathbb{C}_t \arrow[d, "\tilde{p}"] & \\
    \varpi \arrow[r, "q"] \arrow[d, "\varpi \times \mathrm{Id}"] & \varpi \arrow[r, "p"] \arrow[d, "\varpi \times \mathbb{C}_t"] & \mathbb{C}_t \\
    \mathbb{P}_\tau^1 \arrow[r, "q"] \arrow[d, "\mathbb{P}_\tau^1 \times \mathbb{C}_t"] & \mathbb{P}_\tau^1 \times \mathbb{C}_t \arrow[d, "p"] & \\
    \{\infty\} \arrow[r, "q"] \arrow[d, "\{\infty\} \times \mathbb{C}_t"] & D_\infty \arrow[d, "D_\infty"] & \\
  \end{tikzcd} $$

In this diagram, $q, p$ are the natural projections of $\mathbb{P}_\tau^1 \times \mathbb{C}_t$ to the first and second factor respectively, and $\tilde{q}, \tilde{p}$ are the corresponding projections from $\tilde{\mathbb{P}}_\tau^1 \times \mathbb{C}_t$. Since $S_\tau^1 = \varpi^{-1}(\infty) \subset \tilde{\mathbb{P}}_\tau^1$, we also have $\tilde{D}_\infty := S_\tau^1 \times \mathbb{C}_t = (\varpi \times \mathrm{Id})^{-1}(D_\infty) \subset \tilde{\mathbb{P}}_\tau^1 \times \mathbb{C}_t$.

We denote by $\Phi$ the set of meromorphic functions $\varphi_c(\tau', t) = \tau(c-t) = (c-t)/\tau'$ with $c \in C$. The common pole divisor of these functions is $D_\infty = \{\infty\} \times \mathbb{C}_t$. The set $\Phi$ is a good set of meromorphic functions with poles along $D_\infty$, since for any pair $c \neq c' \in C$, the difference $\varphi_c - \varphi_{c'} = (c-c')/\tau'$ is purely monomial, in the sense of [Sab13, Def. 9.8].

We define a Stokes-filtered local system $(\mathcal{L}, \mathcal{L}_\bullet)$ indexed by $\Phi$ on $S_\tau^1 \times \mathbb{C}_t$ by setting, for $\varphi_c \in \Phi$:

$$ \mathcal{L} = \tilde{q}^{-1}(\mathcal{L}), \quad \mathcal{L}_{\le \varphi_c} = \tilde{q}^{-1}(\mathcal{L}_{\le c}) \quad \text{and} \quad \mathcal{L}_{< \varphi_c} = \tilde{q}^{-1}(\mathcal{L}_{< c}). $$
That $(\mathcal{L}, \mathcal{L}_{\bullet})$ is a Stokes-filtered local system is a consequence of the property that, on the one hand, $\mathrm{gr}_{\varphi_c} \mathcal{L} = \tilde{q}^{-1} \mathrm{gr}_c \mathcal{L}$, which is a locally constant sheaf, and on the other hand, for any $(\theta, t) \in S_{\tau}^1 \times \mathbb{C}_t$, we have $\varphi_c \leqslant_{(\theta,t)} \varphi_{c'}$ if and only if $c \leqslant_{\theta} c'$.

In other words, for a fixed $t \in \mathbb{C}$, we have twisted the exponential factors of $(\mathcal{L}, \mathcal{L}_{\bullet})$ by adding $-t/\tau'$ to all of them without changing the corresponding sheaves $\mathcal{L}_{\bullet}$.

The sheaf $\mathcal{L}_{<0}$. Although the family $\Phi \cup \{0\}$ is not good at the points $(\infty, c)$ for $c \in C$ (indeed, for a given $c \in C$, $\varphi_c$ is not purely monomial at $t = c$, in the sense of [Sab13, Def. 9.8]), the subset $\{(\theta, t) | \varphi_c <_{(\theta,t)} 0\}$ is open in $S_{\tau}^1 \times \mathbb{C}_t$, as it contains no point of $S_{\tau}^1 \times \{c\}$. The latter property yields that, if we set for each $(\theta, t) \in S_{\tau}^1 \times \mathbb{C}_t$,

$$ \mathcal{L}_{<0,(\theta,t)} := \sum_{c | \varphi_c <_{(\theta,t)} 0} \mathcal{L}_{<\varphi_c,(\theta,t)} = \bigoplus_{\varphi_c <_{(\theta,t)} 0} \mathrm{gr}_c \mathcal{L}_{\theta}, $$

then $\mathcal{L}_{<0,(\theta,t)}$ is a germ at $(\theta, t)$ of a well-defined sheaf $\mathcal{L}_{<0}$ on $S_{\tau}^1 \times \mathbb{C}_t$.

**Lemma 8.15.** The sheaf $\mathcal{L}_{<0}$ is the unique subsheaf of $\mathcal{L} = \tilde{q}^{-1}\mathcal{L}$ whose restriction to $S_{\tau}^1 \times \{t\}$ is equal to $\mathcal{L}_{<t}$.

*Proof.* For $(\theta, t)$ fixed, we have $\{c | \varphi_c <_{(\theta,t)} 0\} = \{c | c <_{\theta} t\}$, and therefore $\mathcal{L}_{<0,(\theta,t)} = \mathcal{L}_{<t,\theta}$, as desired. On the other hand, two subsheaves $\mathcal{L}', \mathcal{L}''$ of $\mathcal{L} = \tilde{q}^{-1}\mathcal{L}$ coincide if (and only if) their restrictions to $S_{\tau}^1 \times \{t\}$ coincide for each $t$: denoting by $i_t$ the inclusion of the $t$-fiber, and noting that the functor $i_t^{-1}$ is exact, the assumption is that $i_t^{-1}[(\mathcal{L}'+\mathcal{L}'')/\mathcal{L}'] = 0$ for each $t$, and thus $(\mathcal{L}'+\mathcal{L}'')/\mathcal{L}' = 0$. $\square$

**Proposition 8.16** (Pushforward of $\mathcal{L}_{<0}$). The complex $R\tilde{p}_{*}\mathcal{L}_{<0}$ has cohomology in degree one only, and $R^1\tilde{p}_{*}\mathcal{L}_{<0}$ is a constructible sheaf on $\mathbb{C}_t$ with singularities contained in $C$. Furthermore, its global cohomology $H^*(\mathbb{C}_t, R^1\tilde{p}_{*}\mathcal{L}_{<0})$ is zero.

As a consequence (cf. [KKP08]), $R\tilde{p}_{*}\mathcal{L}_{<0}[2]$ is a perverse complex.

*Definition 8.17* (cf. [YZ24]). The topological Laplace transform $F_{-}^{\infty,\mathrm{top}}(\mathcal{L}, \mathcal{L}_{\bullet})$ of the Stokes-filtered local system $(\mathcal{L}, \mathcal{L}_{\bullet})$ is the object $R\tilde{p}_{*}\mathcal{L}_{<0}[2]$ of $\mathrm{Perv}^0(\mathbb{C}_t, k, C)$.

*Proof of the first part of Proposition 8.16.* As $\tilde{p}$ is proper, we can argue fiber by fiber. It is easy to check that $R^k\tilde{p}_{*}\mathcal{L}_{<0}$ is a locally constant sheaf on $\mathbb{C}_t \setminus C$ for any $k$. The first part of the proposition is a then consequence of Lemma 8.15 together with Lemma 8.3. $\square$

*Proof of the second part of Proposition 8.16.* According to the first part, we have to prove the vanishing of the hypercohomology of $R\tilde{q}_{*}R\tilde{p}_{*}\mathcal{L}_{<0}$ on $S_{\tau}^1 \times \mathbb{C}_t$ and, by first switching the pushforward functors, it is enough to prove the vanishing of $R\tilde{q}_{*}\mathcal{L}_{<0}$. In order to argue fiberwise, we need to replace $\tilde{q} : S_{\tau}^1 \times \mathbb{C}_t \to S_{\tau}^1$ with a proper morphism. We thus compactify $\mathbb{C}_t$ by the real blow-up $\tilde{\mathbb{P}}_t^1$ of $\mathbb{P}_t^1$ at $t = \infty$ and we denote by $S_t^1$ the complement of $\mathbb{C}_t$ in $\tilde{\mathbb{P}}_t^1$. Let $j_t : S_{\tau}^1 \times \mathbb{C}_t \hookrightarrow S_{\tau}^1 \times \tilde{\mathbb{P}}_t^1$ denote the open inclusion.

Let $t' = 1/t$ be the coordinate of $\mathbb{P}_t^1$ at infinity. Near $(\infty, \infty)$, each $\varphi_c$ reads $c/\tau' - 1/t'\tau'$ and $\Phi \cup \{0\}$ is good at this point. We note that the set

$$ \{(\theta, \theta_t) \in S_{\tau}^{1} \times S_{t}^{1} | \varphi_c <_{(\theta, \theta_t)} 0\} = \{(\theta, \theta_t) | \theta_t - \theta \in (-\pi/2, \pi/2) \mod 2\pi\} = \tilde{S}_{\infty,>}^{1} $$
**Lemma 8.18.** We have $Rj_{t,*}\mathcal{L}_{<0} = j_{t,*}\mathcal{L}_{<0} := \tilde{\mathcal{L}}_{<0}$ and the latter sheaf satisfies
$$ (8.18*) \qquad \tilde{\mathcal{L}}_{<0}|_{S_{\tau}^{1} \times S_{t}^{1}} = (\tilde{q}_{\infty}^{-1}\mathcal{L})_{\tilde{S}_{\infty,>}^{1}}, $$

i.e., the extension by zero of the restriction of $\tilde{q}_{\infty}^{-1}\mathcal{L}$ to the open subset $\tilde{S}_{\infty,>}^{1}$, where $\tilde{q}_{\infty}$ denotes the projection $S_{\tau}^{1} \times S_{t}^{1} \to S_{\tau}^{1}$.

*Proof.* The statement is local, so we can fix a small product neighborhood $U \times \bar{V}$ of $(\theta, \theta_t)$ in $S_{\tau}^{1} \times \tilde{\mathbb{P}}_{t}^{1}$ and choose a decomposition $\mathcal{L} = \bigoplus_{c \in C} \mathrm{gr}_c \mathcal{L}$ on $U$. Let us set $V = \bar{V} \cap \mathbb{C}_t$. Then, on $U \times V$, we have $\mathcal{L}_{<0}|_{U \times V} \simeq \bigoplus_{c \in C} \mathrm{gr}_c \mathcal{L}|_{U \times \tilde{\mathbb{P}}_{t}^{1}}$, and we can argue with each summand independently. We are thus reduced to showing

$$ (8.19) \qquad \mathbf{R}j_{t,*} \mathbf{k}_{S_{c<t}^{1}}|_{U \times V} \simeq \mathbf{k}_{\tilde{S}_{>^{1}}}|_{U \times \bar{V}}. $$

Choosing $\theta$ in the open interval $U$, we have $\mathbf{k}_{S_{c<t}^{1}}|_{U \times V} \simeq \mathbf{k}_{U} \boxtimes \mathbf{k}_{S_{c<t,\theta}^{1}}|_{V}$. The set $\{t | c <_{\theta} t\}$ is an open half-plane. The closure of the boundary line intersects $S_{t}^{1}$ in two points $\theta_{t,1}$, $\theta_{t,2}$. Away from these boundary points, the isomorphism (8.19) holds on $\bar{V} \cap S_{t}^{1}$, and it remains to be shown that the germ of $Rj_{t,*}\mathbf{k}_{S_{c<t,\theta}^{1}}$ at $\theta_{t,1}$, $\theta_{t,2}$ is zero. We can choose open neighborhoods $\bar{V}_i$ of $\theta_{t,i}$ ($i = 1,2$) in $S_{\tau}^{1} \times \tilde{\mathbb{P}}_{t}^{1}$ to be homeomorphic to a closed disc with $\theta_{t,i}$ on its boundary, and $\mathbf{k}_{S_{c<t,\theta}^{1}}$ is the constant sheaf on an open half-disc of $\bar{V}_i$, extended by zero on the boundary line (a diameter of $\bar{V}_i$). The desired assertion amounts to $H^*(\bar{V}_i, \mathbf{k}_{S_{c<t,\theta}^{1}}) = 0$, which is the content of Lemma 7.4. $\square$

**Lemma 8.20.** We have $R\tilde{q}_{*}\mathcal{L}_{<0} = 0$.

*Proof.* Abusing notation, we also denote by $\tilde{q}$ the projection $S_{\tau}^{1} \times \tilde{\mathbb{P}}_{t}^{1} \to S_{\tau}^{1}$. According to Lemma 8.18, the assertion amounts to $R\tilde{q}_{*}\tilde{\mathcal{L}}_{<0} = 0$. Now that $\tilde{q}$ is made proper, we can argue fiberwise, and we fix $\theta \in S_{\tau}^{1}$. We can thus assume that $(\mathcal{L}, \mathcal{L}_{\bullet})$ is graded, so that $\tilde{\mathcal{L}}_{<0,\theta}$ can be decomposed with respect to the chosen grading, and we can argue with each $c \in C$ separately.

Let us fix $c_o \in C$, and assume that $\mathcal{L}_{\theta} = \mathrm{gr}_{c_o} \mathcal{L}_{\theta}$. Then, using the notation of (8.8), $\mathcal{L}_{<0,\theta} \simeq \mathrm{gr}_{c_o} \mathcal{L}_{\theta} \otimes \mathbf{k}_{\tilde{S}_{c_o<t,\theta}^{1}}$ (i.e., the constant sheaf with fiber $\mathrm{gr}_{c_o} \mathcal{L}_{\theta}$ on $\tilde{S}_{c_o<t,\theta}^{1}$, extended by zero). The conclusion follows from (8.12). $\square$

$$ \text{The lemma yields } \mathbf{R}\Gamma(\mathbb{C}_t, \mathbf{R}\tilde{p}_{*}\mathcal{L}_{<0}) = \mathbf{R}\tilde{q}_{*}\mathbf{R}\tilde{p}_{*}\mathcal{L}_{<0} \simeq \mathbf{R}\tilde{p}_{*}\mathbf{R}\tilde{q}_{*}\mathcal{L}_{<0} = 0. \quad \square $$

We end this section by showing how to recover the Stokes-filtered local system $(\mathcal{L}, \mathcal{L}_{\bullet})$ on $S_{\tau}^{1}$ from the pair $(\tilde{\mathcal{L}} = \tilde{q}^{-1}\mathcal{L}, \tilde{\mathcal{L}}_{<0})$ defined on $S_{\tau}^{1} \times \tilde{\mathbb{P}}_{t}^{1}$.

**Proposition 8.21.** For each $s \in \mathbb{C}_t$, we have

$$ R^k \tilde{q}_* (\tilde{\mathcal{L}}_{<0} \otimes \mathbf{k}_{\tilde{S}_{t \ge s}^1}) = 0 = R^k \tilde{q}_* (\tilde{q}^{-1} \mathcal{L} \otimes \mathbf{k}_{\tilde{S}_{t \ge s}^1}) \quad \text{for } k \neq 0, $$

and the natural morphism

$$ \tilde{q}_* (\tilde{\mathcal{L}}_{<0} \otimes \mathbf{k}_{\tilde{S}_{t \ge s}^1}) \to \tilde{q}_* (\tilde{q}^{-1} \mathcal{L} \otimes \mathbf{k}_{\tilde{S}_{t \ge s}^1}) \simeq \mathcal{L} $$

is an isomorphism onto $\mathcal{L}_{<s}$.

*Proof.* On the one hand, we have a natural isomorphism

$$ \mathbf{R}\tilde{q}_* (\tilde{q}^{-1} \mathcal{L} \otimes \mathbf{k}_{\tilde{S}_{t \ge s}^1}) \simeq \mathcal{L} \otimes \mathbf{R}\tilde{q}_* \mathbf{k}_{\tilde{S}_{t \ge s}^1} $$
and we claim that the natural morphisms $\mathbf{R}\tilde{q}_*\mathbf{k}_{\tilde{S}_{t \ge s}^1} \to \mathbf{R}\tilde{q}_*\tilde{q}^{-1}\mathbf{k}_{S_t^1} \leftarrow \mathbf{k}_{S_t^1}$ are isomorphisms. Indeed, it is enough to check this fiber by fiber. The assertion follows from the property that the germ $\tilde{S}_{t \ge s, \theta}^1$, as well as $\tilde{q}^{-1}(\theta)$, has the topology of a closed disc.

On the other hand, we have $\mathbf{R}\tilde{q}_*(\mathcal{L}_{<0} \otimes \mathbf{k}_{\tilde{S}_{t \ge s}^1}) \simeq \mathbf{R}\tilde{q}_*(\mathcal{L}_{<0} \otimes \mathbf{k}_{\tilde{S}_{t < s}^1}[1])$, because of Lemma 8.20. For $\theta \in S_t^1$, we have $\mathcal{L}_{<0, \theta} \simeq \bigoplus_{c \in C}(\mathrm{gr}_c \mathcal{L}_\theta \otimes \mathbf{k}_{\tilde{S}_{c<t,\theta}^1})$, so that

$$ \mathcal{L}_{<0, \theta} \otimes \mathbf{k}_{\tilde{S}_{t < s, \theta}^1} \simeq \bigoplus_{c \in C} (\mathrm{gr}_c \mathcal{L}_\theta \otimes \mathbf{k}_{\tilde{S}_{c < t < s, \theta}^1}). $$

The term of index $c$ in the sum vanishes iff $c \not\le \theta s$. On the other hand, if $c < \theta s$, we have $\mathbf{R}\Gamma(\tilde{\mathbb{P}}_t^1, \mathbf{k}_{\tilde{S}_{c<t,\theta}^1}) = 0$ by Lemma 7.4 and an exact sequence

$$ 0 \to \mathbf{k}_{\tilde{S}_{c < t < s, \theta}^1} \to \mathbf{k}_{\tilde{S}_{c < t, \theta}^1} \to \mathbf{k}_{\tilde{S}_{t \ge s}^1} \to 0. $$

From the above isomorphisms we obtain

$$
\begin{align*}
\mathbf{R}\Gamma(\tilde{\mathbb{P}}_t^1, \mathcal{L}_{<0,\theta} \otimes \mathbf{k}_{\tilde{S}_{t \ge s}^1}) &\simeq \mathbf{R}\tilde{q}_*(\mathcal{L}_{<0} \otimes \mathbf{k}_{\tilde{S}_{t < s}^1}[1]) \\
&\simeq \bigoplus_{c \in C} (\mathrm{gr}_c \mathcal{L}_\theta \otimes \mathbf{R}\Gamma(\tilde{\mathbb{P}}_t^1, \mathbf{k}_{\tilde{S}_{c < t < s, \theta}^1}[1])) \\
&\simeq \bigoplus_{c < \theta s} (\mathrm{gr}_c \mathcal{L}_\theta \otimes \mathbf{R}\Gamma(\tilde{\mathbb{P}}_t^1, \mathbf{k}_{\tilde{S}_{t \ge s, \theta}^1})) \\
&\simeq \bigoplus_{c < \theta s} \mathrm{gr}_c \mathcal{L}_\theta = \mathcal{L}_{<s,\theta}. \quad \Box
\end{align*}
$$

8.d. The local Laplace transformation on $\mathrm{Perv}^0(\mathbb{C}_t, \mathbf{k}, C)$. Let us recall the construction of the functor $F_+^{\infty, \mathrm{top}}$ done in [Sab13, Chap. 7]. We will work with constructible sheaves instead of perverse sheaves, for the sake of simplicity.

Let $\mathcal{F}$ be a constructible sheaf in $\mathrm{Perv}^0(\mathbb{C}_t, \mathbf{k}, C)[-1]$. Since it is locally constant near $t = \infty$, it extends as a sheaf $\tilde{\mathcal{F}} = j_{t,*}\mathcal{F} \simeq \mathbf{R}j_{t,*}\mathcal{F}$ on $\tilde{\mathbb{P}}_t^1$ which is locally constant near $S_t^1$.

**Proposition 8.22** ([Sab13, Prop. 7.11]). The complex $\mathbf{R}\tilde{q}_*(\tilde{p}^{-1}\tilde{\mathcal{F}} \otimes \mathbf{k}_{\tilde{S}_{t \le s}^1})$ on $S_t^1$ has cohomology in degree one at most for each $s \in \mathbb{C}$, as well as the complex $\mathbf{R}\tilde{q}_*(\tilde{p}^{-1}\tilde{\mathcal{F}} \otimes \mathbf{k}_{\tilde{S}_<^1})$. Moreover, $\mathcal{L}' := \mathbf{R}^1\tilde{q}_*(\tilde{p}^{-1}\tilde{\mathcal{F}} \otimes \mathbf{k}_{\tilde{S}_<^1})$ is a locally constant sheaf on $S_t^1$ and $\mathcal{L}'_s := \mathbf{R}^1\tilde{q}_*(\tilde{p}^{-1}\tilde{\mathcal{F}} \otimes \mathbf{k}_{\tilde{S}_{t \ge s}^1})$ defines a co-Stokes filtration indexed by $C$. $\square$

From the exact sequence (8.9) and since $\tilde{\mathcal{F}}$ has vanishing global cohomology, we deduce

$$ (8.23) \qquad \mathcal{L}'_s \simeq \tilde{q}_*(\tilde{p}^{-1}\tilde{\mathcal{F}} \otimes \mathbf{k}_{\tilde{S}_{t \ge s}^1}) \quad \text{and} \quad \mathbf{R}^k \tilde{q}_*(\tilde{p}^{-1}\tilde{\mathcal{F}} \otimes \mathbf{k}_{\tilde{S}_{t \ge s}^1}) = 0 \text{ for } k \ge 1. $$

**Corollary 8.24.** For $s \in \mathbb{C}_t$ fixed, we have $\mathbf{H}^k(S_t^1, \mathcal{L}'_s) = 0$ for $k \ne 1$ and a canonical identification $\mathbf{H}^1(S_t^1, \mathcal{L}'_s) \simeq \mathcal{F}_s$. In particular, $\mathrm{dim} \, \mathcal{F}_s = \mathrm{rk} \, \mathcal{L}'$ if $s \notin C$.

*Proof.* The last part follows from Lemma 8.3. For the first part, it suffices to show, according to (8.23), that $\mathbf{R}\Gamma(S_t^1, \mathcal{L}'_s) \simeq \mathcal{F}_s[-1]$. We write

$$
\begin{align*}
\mathbf{R}\Gamma(S_{\tau}^{1}, \mathcal{L}'_{<s}) &\simeq \mathbf{R}\Gamma(S_{\tau}^{1}, \mathbf{R}\tilde{q}_{*}(\tilde{p}^{-1}\tilde{\mathcal{F}} \otimes \mathbf{k}_{\tilde{S}_{t \ge s}^{1}})) \\
&\simeq \mathbf{R}\Gamma(S_{\tau}^{1} \times \tilde{\mathbb{P}}_{t}^{1}, (\tilde{p}^{-1}\tilde{\mathcal{F}} \otimes \mathbf{k}_{\tilde{S}_{t \ge s}^{1}})) \\
&\simeq \mathbf{R}\Gamma(\tilde{\mathbb{P}}_{t}^{1}, \mathbf{R}\tilde{p}_{*}(\tilde{p}^{-1}\tilde{\mathcal{F}} \otimes \mathbf{k}_{\tilde{S}_{t \ge s}^{1}})) \\
&\simeq \mathbf{R}\Gamma(\tilde{\mathbb{P}}_{t}^{1}, \tilde{\mathcal{F}} \otimes \mathbf{R}\tilde{p}_{*}\mathbf{k}_{\tilde{S}_{t \ge s}^{1}}).
\end{align*}
$$

As $\mathbf{R}\Gamma(\tilde{\mathbb{P}}_t^1, \tilde{\mathcal{F}}) = 0$, the conclusion follows from Lemma 8.13. $\square$
8.e. *Proof of Proposition 8.2.* Now that we have at our disposal both functors of (8.1), we will prove that they are quasi-inverse from each other.

Given $\mathcal{F}$ in $\mathrm{Perv}^0(\mathbb{C}_t, \mathbf{k}, C)[-1]$ and extended as $\tilde{\mathcal{F}}$ to $\mathbb{P}_t^1$, we reconstruct the sheaf $\mathcal{L}_{<0}$ by varying the parameter $s$ in Proposition 8.22. For that purpose, we consider the product $S_\tau^1 \times \tilde{\mathbb{P}}_t \times \mathbb{C}_s$ and the projections $\tilde{p}_t, \tilde{p}_s, \tilde{q}_s$ to $\tilde{\mathbb{P}}_t, \tilde{\mathbb{P}}_s$ and $S_\tau^1 \times \mathbb{C}_s$ respectively. We consider the subset $\tilde{\mathcal{F}}_{t<s}^1$ of $S_\tau^1 \times \tilde{\mathbb{P}}_t \times \mathbb{C}_s$ whose fiber at $s \in \mathbb{C}_s$ is the set $\tilde{\mathbb{S}}_{t<s}^1$. We also set $\tilde{\mathcal{F}}_<^1 = \tilde{\mathbb{S}}_<^1 \times \mathbb{C}_s$. Therefore, $\tilde{\mathcal{F}}_{t<s}^1 \subset \tilde{\mathcal{F}}_<^1$. Furthermore, the intersection $\tilde{\mathcal{F}}_{t<s}^1 \cap (S_\tau^1 \times S_t^1 \times \mathbb{C}_s)$ is equal to the product $\tilde{\mathbb{S}}_{\infty,<}^1 \times \mathbb{C}_s$.

**Lemma 8.25.** The subset $\tilde{\mathcal{F}}_{t<s}^1$ is open in $S_\tau^1 \times \tilde{\mathbb{P}}_t \times \mathbb{C}_s$. $\square$

Proposition 8.22 in family reads:

**Proposition 8.26.** The complex $R\tilde{q}_{s,*}(\tilde{p}_t^{-1}\tilde{\mathcal{F}}\otimes\mathbf{k}_{\tilde{\mathcal{F}}_{t<s}^1})$ on $S_\tau^1\times\mathbb{C}_s$ has cohomology in degree one at most, as well as the complex $R\tilde{q}_{s,*}(\tilde{p}_t^{-1}\tilde{\mathcal{F}}\otimes\mathbf{k}_{\tilde{\mathcal{F}}_<^1})$. Moreover, the sheaf $\mathcal{L}' = R^1\tilde{q}_{s,*}(\tilde{p}_t^{-1}\tilde{\mathcal{F}}\otimes\mathbf{k}_{\tilde{\mathcal{F}}_<^1})$ is isomorphic to $\tilde{q}_s^{-1}\mathcal{L}'$ (defined by Proposition 8.22) and the sheaf $\mathcal{L}_{x,<0} = R^1\tilde{q}_{s,*}(\tilde{p}_t^{-1}\tilde{\mathcal{F}}\otimes\mathbf{k}_{\tilde{\mathcal{F}}_{t<s}^1})$ satisfies Lemma 8.15. Furthermore, we have a canonical identification

$$R\tilde{p}_{s,*}\mathcal{L}_{x,<0}[1] \simeq \mathcal{F},$$

where $\mathcal{F}$ is now regarded on $\mathbb{C}_s$.

This proposition gives a isomorphism of functors $F_{-}^{\infty, \mathrm{top}} \circ F_{+}^{\infty, \mathrm{top}} \xrightarrow{\sim} \mathrm{Id}$.

*Proof.* The morphisms are defined in a way similar to what is done in Section 8.d. We then argue fiber by fiber with respect to the proper projection $S_\tau^1 \times \tilde{\mathbb{P}}_t^1 \times \mathbb{C}_s \to \mathbb{C}_s$ and apply Proposition 8.22 and Corollary 8.24. $\square$

We now construct an isomorphism of functors $F_{+}^{\infty, \mathrm{top}} \circ F_{-}^{\infty, \mathrm{top}} \xrightarrow{\sim} \mathrm{Id}$. For that purpose, we start from $(\mathcal{L}, \mathcal{L}_{\bullet})$ and define $\mathcal{F} := R\tilde{p}_{*}\mathcal{L}_{<0}[1]$, with $\mathcal{L}_{<0}$ as in Section 8.c. Then $\mathcal{F}$ is an object of $\mathrm{Perv}^0(\mathbb{C}_t, \mathbf{k}, C)[-1]$ (Proposition 8.16). With the previous notation, we then have $\tilde{\mathcal{F}} = R\tilde{p}_{*}\tilde{\mathcal{L}}_{<0}[1]$. We denote by $(\mathcal{L}', \mathcal{L}'_{\bullet})$ the Stokes-filtered local system $F_{+}^{\infty, \mathrm{top}}(\mathcal{F})$.

**Proposition 8.27.** There exists a canonical morphism of Stokes-filtered local systems

$$(\mathcal{L}, \mathcal{L}_{\bullet}) \to (\mathcal{L}', \mathcal{L}'_{\bullet})$$

such that the induced morphism $\mathcal{L}' \to \mathcal{L}$ is an isomorphism of local systems.

Since the category of Stokes-filtered local systems is abelian (cf. e.g. [Sab13, Chap. 3]), that $\mathcal{L} \to \mathcal{L}'$ is an isomorphism implies that the corresponding morphism $(\mathcal{L}, \mathcal{L}_{\bullet}) \to (\mathcal{L}', \mathcal{L}'_{\bullet})$ of Stokes-filtered local systems is an isomorphism. We conclude that there exists an isomorphism of functors $\mathrm{Id} \xrightarrow{\sim} F_{+}^{\infty, \mathrm{top}} \circ F_{-}^{\infty, \mathrm{top}}$.

*Proof of Proposition 8.27.* We first define, for each $s \in \mathbb{C}_t$, a canonical morphism

$$(8.28) \qquad \mathcal{L}_{<s} \to \mathcal{L}'_{<s} := R\tilde{q}_{*}(\tilde{p}^{-1}\tilde{\mathcal{F}} \otimes \mathbf{k}_{\tilde{\mathbb{S}}_{t<s}^{1}})[1].$$

As in (8.23), we write the right-hand side as

$$\mathcal{L}'_{<s} = \tilde{q}_{*}(\tilde{p}^{-1}\tilde{\mathcal{F}} \otimes \mathbf{k}_{\tilde{\mathbb{S}}_{t \ge s}^{1}}) \simeq R\tilde{q}_{*}(\tilde{p}^{-1}\tilde{\mathcal{F}} \otimes \mathbf{k}_{\tilde{\mathbb{S}}_{t \ge s}^{1}}),$$
since we know by (the proof of) Lemma 8.20 that $\tilde{\mathcal{F}} = \mathcal{R}\tilde{p}_*\tilde{\mathcal{L}}_{<0}[1]$ has vanishing global cohomology. On the other hand, since $\tilde{\mathcal{L}}_{<0}$ is an $\mathbb{R}$-constructible sheaf on $S_\tau^1 \times \tilde{\mathbb{P}}_t^1$ and since $\tilde{p}$ is smooth of relative dimension one, we can apply Poincaré bi-duality to the adjunction morphism $\tilde{p}^{-1}\mathcal{R}\tilde{p}_* \to \mathrm{Id}$ to obtain a canonical morphism

$$ (8.29) \qquad \tilde{\mathcal{L}}_{<0} \to \tilde{p}^{-1} \mathcal{R} \tilde{p}_* \tilde{\mathcal{L}}_{<0}[1] = \tilde{p}^{-1} \tilde{\mathcal{F}}. $$

From Proposition 8.21 we obtain the desired morphism (8.28):

$$ (8.30) \qquad \tilde{q}_*(\tilde{\mathcal{L}}_{<0} \otimes \mathbf{k}_{\tilde{S}_{t \ge s}^1}) \to \tilde{q}_*(\tilde{p}^{-1} \tilde{\mathcal{F}} \otimes \mathbf{k}_{\tilde{S}_{t \ge s}^1}). $$

This is a morphism of Stokes-filtered local systems, that is, for any $\theta \in S_\tau^1$, if $s <_\theta s'$, the following diagram commutes:

<figure><img src="image_3.png" /></figure>

We wish to prove that the morphism $\mathcal{L} \to \mathcal{L}'$ that we deduce from these morphisms is an isomorphism. We know from Lemma 8.3 and Corollary 8.24 that $\mathcal{L}$ and $\mathcal{L}'$ have the same rank, so it is enough to prove that it is onto. We can also fix $\theta \in S_\tau^1$, and we note that, for $s \in \mathbb{C}_t$ such that $\tilde{S}_{t \ge s, \theta}^1$ is a closed half-plane not intersecting $C$, we have $\mathcal{L}_{<s, \theta} = \mathcal{L}_\theta$, and $\mathcal{L}'_{<s, \theta} = \mathcal{L}'_\theta$. We then fix such an $s$. The fiber at $\theta$ of the morphism (8.30) reads

$$ \Gamma(\tilde{\mathbb{P}}_t^1, \mathcal{L}_{<0,\theta} \otimes \mathbf{k}_{\tilde{S}_{t \ge s, \theta}^1}) \to \Gamma(\tilde{\mathbb{P}}_t^1, \tilde{\mathcal{F}} \otimes \mathbf{k}_{\tilde{S}_{t \ge s, \theta}^1}) $$

and since the sheaves $\mathcal{L}_{<0,\theta}$ and $\tilde{\mathcal{F}}$ are locally constant on the closed disc $\tilde{S}_{t \ge s, \theta}^1$, the above morphism is nothing but the morphism between their fibers at $s$, which reads

$$ (8.31) \qquad \mathcal{L}_{<s,\theta} \to H^1(S_\tau^1, \mathcal{L}_{<s}) $$

and is the fiber at $(\theta, s)$ of (8.29). We are thus left with proving that the later is onto. By construction, this morphism is the fiber at $\theta$ of the morphism

$$ \mathcal{L}_{<s} \to H^1(S_\tau^1, \mathcal{L}_{<s}) \boxtimes \mathbb{C}_{S_\tau^1} $$

which is dual to the adjunction morphism

$$ \mathbf{H}^0(S_\tau^1, (\mathcal{L}_{<s})^\vee) \boxtimes \mathbb{C}_{S_\tau^1} \to (\mathcal{L}_{<s})^\vee, $$

where $\mathcal{I}^\vee$ denotes the Poincaré dual complex $\mathcal{R}\mathcal{H}\mathit{om}(\mathcal{I}, \mathbb{C}_{S_\tau^1})$. On the other hand, we know (cf. [Sab13, Lem. 2.16]) that the Poincaré dual complex $\mathcal{R}\mathcal{H}\mathit{om}(\mathcal{L}_{<s}, \mathbb{C}_{S_\tau^1})$ is a sheaf (i.e., concentrated in degree zero), so that the adjunction morphism

$$ \mathcal{H}^0(S_\tau^1, (\mathcal{L}_{<s})^\vee) \boxtimes \mathbb{C}_{S_\tau^1} \to (\mathcal{L}_{<s})^\vee $$

is injective. By duality, we obtain that (8.31) is onto, as desired. $\square$
8.f. *Proof of the equivalence* (7.1). In a first step, we work with the real blow-up $\tilde{\mathbb{P}}_\tau^1$ of $\mathbb{P}_\tau^1$ at $\infty$ and 0, so that $\tilde{\mathbb{P}}_\tau^1$ is homeomorphic to a cylinder $S_\tau^1 \times [0, \infty]$. A Stokes constructible sheaf on $\tilde{\mathbb{P}}_\tau^1$ consists of a local system $\tilde{\mathcal{L}}$ on $\tilde{\mathbb{P}}_\tau^1$ together with a Stokes filtration on the local systems $\mathcal{L} = \tilde{\mathbb{I}}_\infty^1 \tilde{\mathcal{L}}$ and $\tilde{\mathbb{I}}_0^1 \tilde{\mathcal{L}}$. In our setting, we consider a Stokes filtration $\mathcal{L}_\bullet$ of exponential type $C$ on $\mathcal{L}$ and a trivial Stokes filtration on $\tilde{\mathbb{I}}_0^1 \tilde{\mathcal{L}}$. Furthermore, due to Lemma 7.5, there exists a unique filtration $\tilde{\mathcal{L}}_\bullet$ of $\tilde{\mathcal{L}}$ such that the restriction of $\tilde{\mathcal{L}}_\bullet$ away from $S_\tau^1 \times \{\infty\}$ is equal to the restriction of $\tilde{\mathcal{L}}_\bullet$, while its restriction to $S_\tau^1 \times \{\infty\}$ is equal to $\mathcal{L}_\bullet$.

Since $\tilde{\mathcal{L}}$ is the pullback of $\mathcal{L}$ by the projection $\tilde{\mathbb{P}}_\tau^1 \to S_\tau^1$, Lemma 7.5 yields an equivalence

$$ (8.32) \qquad (\tilde{\mathcal{L}}, \tilde{\mathcal{L}}_\bullet) \xrightarrow{\tilde{\mathbb{I}}_\infty} (\mathcal{L}, \mathcal{L}_\bullet). $$

**Lemma 8.33.** *The shifted Stokes-constructible sheaf $(\tilde{\mathcal{L}}, \tilde{\mathcal{L}}_\bullet)$[1] is a Stokes-perverse sheaf on $\tilde{\mathbb{P}}_\tau^1$, in the sense of [Sab13, Def. 4.12].*

*Proof.* It is a matter of proving that, for any $s \in \mathbb{C}$, the Poincaré-Verdier dual complexes $\mathcal{D}(\tilde{\mathcal{L}}_{<s}[1])$ and $\mathcal{D}(\tilde{\mathcal{L}}_{\le s}[1])$ have nonzero cohomology in degree one only. This being clear away from $S_\tau^1 \times \{\infty\}$, where both are equal to the shifted local system $\tilde{\mathcal{L}}[1]$, the question is local at a point $(\theta, \infty)$ of $S_\tau^1 \times \{\infty\}$. We can assume that $(\mathcal{L}, \mathcal{L}_\bullet)$ is graded, as well as $(\tilde{\mathcal{L}}, \tilde{\mathcal{L}}_\bullet)$. We consider an open neighborhood of the form $I \times [0, \varepsilon)$ for some small interval $I$ of $S_\tau^1$ containing $\theta$. We set $V = I \times (0, \varepsilon)$, $\bar{V} = I \times [0, \varepsilon)$ and, decomposing $I$ as $I_1 \sqcup I_2 \sqcup \{\theta\}$, where $\theta$ is the common boundary point of the open intervals $I_1, I_2$, we set $\bar{V}_i = V \sqcup (I_i \times \{0\})$, so that $V \subset \bar{V}_i \subset \bar{V}$ ($i = 1, 2$). We are led to consider three cases for either $\tilde{\mathcal{L}}_{<s}[1]$ or $\tilde{\mathcal{L}}_{\le s}[1]$, depending on whether $\theta$ is a Stokes direction for $s$ or not:

(1) the shifted sheaf $k_V[1]$ (i.e., its extension by zero to $I \times \{0\}$),

(2) the shifted sheaf $k_{\bar{V}}[1]$,

(3) the shifted sheaf $k_{\bar{V}_i}[1]$ ($i = 1, 2$).

Then one checks that the Poincaré-Verdier dual of $k_V[1]$ is $k_{\bar{V}}[1]$, and that of $k_{\bar{V}_1}[1]$ is $k_{\bar{V}_2}[1]$ (and conversely). This yields the assertion, which proves the lemma. $\square$

Let us now consider the real blow-up map $\varpi_0 : \tilde{\mathbb{P}}_\tau^1 \to \tilde{\mathbb{P}}_\tau^1$ of $\tilde{\mathbb{P}}_\tau^1$ at $\tau = 0$. Then arguing as in [Sab13, Prop. 4.22], one checks that $(\tilde{\mathcal{G}}, \tilde{\mathcal{G}}_\bullet) := \mathcal{R}\varpi_{0,*}(\tilde{\mathcal{L}}, \tilde{\mathcal{L}}_\bullet)$ is an object of PervSt$(\tilde{\mathbb{P}}_\tau^1, k, C, *0)$ and furthermore the correspondence $(\tilde{\mathcal{L}}, \tilde{\mathcal{L}}_\bullet) \mapsto (\tilde{\mathcal{G}}, \tilde{\mathcal{G}}_\bullet)$ is an equivalence. Composing it with the equivalence (8.32) yields the equivalence (7.1). In particular, $\tilde{\mathcal{G}} = \tilde{\mathcal{G}}|_{\mathbb{C}_\tau}$, satisfies $\tilde{\mathcal{G}} \simeq \mathcal{R}j_{0*}j_0^{-1}\tilde{\mathcal{G}}$, and $j_0^{-1}\tilde{\mathcal{G}}[-1]$ is a locally constant sheaf on $\mathbb{C}_\tau^*$, that can be identified with $\mathcal{L}$.

We also notice the following:

**Lemma 8.34.** For each $t \in \mathbb{C}$ and $k \in \mathbb{N}$, the following equalities hold:

$$ H^k(\tilde{\mathbb{P}}_\tau^1, \tilde{\mathcal{G}}_{<t}) = H^k(S_\tau^1, \mathcal{L}_{<t}) \quad \text{and} \quad H^k(\tilde{\mathbb{P}}_\tau^1, \tilde{\mathcal{G}}_{\le t}) = H^k(S_\tau^1, \mathcal{L}_{\le t}). $$

*Proof.* We argue for $\tilde{\mathcal{G}}_{\le t}$ for example. We have a long exact sequence

$$ \dots \to H_c^k(\mathbb{C}_\tau, \tilde{\mathcal{G}}) \to H^k(\tilde{\mathbb{P}}_\tau^1, \tilde{\mathcal{G}}_{\le t}) \to H^k(S_\tau^1, \mathcal{L}_{\le t}) \to \dots $$

and $H_c^k(\mathbb{C}_\tau, \tilde{\mathcal{G}}) = 0$ for any $k$, according to Lemma 7.4. $\square$
## 9. THE RIEMANN-HILBERT CORRESPONDENCE

Let $\mathcal{M}$ be a free $\mathbb{C}\{\tau'\}[\tau'^{-1}]$-module of finite rank with a connection $\nabla$ having an irregular singularity of exponential type, with exponential factors $c/\tau'$, with $c$ in a finite subset $C \subset \mathbb{C}$. In other words, $\mathcal{M}$ is an object of $\mathcal{Mero}(\infty, C)$. The construction of Deligne's meromorphic extension together with an algebraization procedure for meromorphic connections on $\mathbb{P}_{\tau}^1$ yields a unique free $\mathbb{C}[\tau, \tau^{-1}]$-module with connection $M$ such that, setting $\tau' = \tau^{-1}$, $\mathbb{C}\{\tau'\} \otimes_{\mathbb{C}[\tau']}$ $M \simeq \mathcal{M}$ and the connection on $M$ has a regular singularity at $\tau = 0$ and no other singularity in the complex plane $\mathbb{C}_{\tau}$. The correspondence $\mathcal{M} \mapsto M$ is a quasi-inverse functor to the functor $i_{\infty}^{-1}$ considered in the diagram (7.3**).

On the other hand, in the diagram (7.3**), we define

$$F_{\pm}^{\infty} : \mathrm{Mod}_{\mathrm{holreg}}^{0}(\mathbb{D}_{\mathbb{A}_{t}^{1}}, C) \to \mathcal{Mero}(\infty, C)$$

so that $F_{\pm}^{\infty} = i_{\infty}^{-1} \circ F_{\pm}$, and

$$F_{\mp}^{\infty} : \mathcal{Mero}(\infty, C) \to \mathrm{Mod}_{\mathrm{holreg}}^{0}(\mathbb{D}_{\mathbb{A}_{t}^{1}}, C)$$

so that $F_{\mp} \circ i_{\infty}^{-1} = F_{\mp}^{\infty}$. In [Sab13, Chap. 7], it is proved that the diagram

$$\begin{tikzcd}
\mathrm{Mod}_{\mathrm{holreg}}^{0}(\mathbb{D}_{\mathbb{A}_t^1}, C) \arrow[r, "F_{\pm}^{\infty}"] \arrow[d, "\mathrm{RH}"] & \mathrm{Mero}(\infty, C) \arrow[d, "\mathrm{RHB}"] \\
\mathrm{Perv}^{0}(\mathbb{C}_t, \mathbf{k}, C)[-1] \arrow[r, "F_{\pm}^{\infty, \mathrm{top}}"] & \mathrm{Sto}(S_{\tau}^{1}, \mathbf{k}, C)
\end{tikzcd}$$

commutes. In this section, we prove that, with Definition 8.17 of $F_{\mp}^{\infty, \mathrm{top}}$, the diagram

$$\begin{tikzcd}
\mathrm{Mod}_{\mathrm{holreg}}^{0}(\mathbb{D}_{\mathbb{A}_t^1}, C) \arrow[r, "F_{\mp}^{\infty}"] \arrow[d, "\mathrm{RH}"] & \mathrm{Mero}(\infty, C) \arrow[d, "\mathrm{RHB}"] \\
\mathrm{Perv}^{0}(\mathbb{C}_t, \mathbf{k}, C)[-1] \arrow[r, "F_{\mp}^{\infty, \mathrm{top}}"] & \mathrm{Sto}(S_{\tau}^{1}, \mathbf{k}, C)
\end{tikzcd}$$

also commutes. We will focus on $F_{\mp}^{\infty}, F_{-}^{\infty, \mathrm{top}}$. Together with the equivalences $\tilde{i}_{\infty}^{-1}$ seen in Section 8.f and $i_{\infty}^{-1}$ seen above, the proof of Theorem 7.3 is completed.

Given $\mathcal{M}$ in $\mathcal{Mero}(\infty, C)$, that we express as $i_{\infty}^{-1}M$, the Fourier transform $N = F_{-}M$ is defined as $p_{+}(q^{+}M, \nabla - d(\tau t))$. We set

$$\mathcal{M} := (\mathbb{C}\{\tau'\}[t] \otimes_{\mathbb{C}\{\tau'\}} \mathcal{M}, \nabla - d(\tau t)).$$

This is the analytic germ along $\{\infty\} \times \mathbb{C}_t$ of $(q^+M, \nabla - d(\tau t))$, the latter being regarded as a holonomic $\mathbb{D}_{\mathbb{P}_t^1 \times \mathbb{C}_t}$-module.

Let $(\mathcal{L}, \mathcal{L}_{\bullet})$ be the $\mathbb{C}$-Stokes-filtered local system associated to $\mathcal{M}$ on $S_{\tau}^1$ by the Riemann-Hilbert-Birkhoff-Deligne-Malgrange (RHB) correspondence and let $\mathcal{L}_{<0}$ be the sheaf on $S_{\tau}^1 \times \mathbb{C}_t$ defined in Section 8.c. We will simply denote by $\varpi$—instead of $\varpi \times \mathrm{Id}$—the real blowing up (cf. Diagram (8.14)) $\tilde{\mathbb{P}}_{\tau}^1 \times \mathbb{C}_t \to \mathbb{P}_{\tau}^1 \times \mathbb{C}_t$ as well as its restriction $\tilde{D}_{\infty} = S_{\tau}^1 \times \mathbb{C}_t \to D_{\infty} = \{\infty\} \times \mathbb{C}_t$ above $D_{\infty}$.

**Theorem 9.1.** There exists a functorial isomorphism $\mathrm{DR}^{\mathrm{an}}(\mathcal{M}) \simeq \mathbb{R}\varpi_{*}\mathcal{L}_{<0}$.
Set $\mathcal{G} = \text{DR}^{\text{an}}(q^+M, \nabla - \text{d}(\tau t))$, which a constructible complex on $\mathbb{P}_{\tau}^1 \times \mathbb{C}_t$. Letting $j_{\infty}: \mathbb{C}_{\tau} \times \mathbb{C}_t \hookrightarrow \mathbb{P}_{\tau}^1 \times \mathbb{C}_t$ and $i_{\infty}: \{\infty\} \times \mathbb{C}_t \hookrightarrow \mathbb{P}_{\tau}^1 \times \mathbb{C}_t$ denote the complementary inclusions, we have $\text{DR}^{\text{an}}(\mathcal{M}) \simeq i_{\infty}^{-1}\mathcal{G}$ and $\mathbf{R}p_*\mathbf{R}j_{\infty, !}j_{\infty}^{-1}\mathcal{G} = 0$ (by Lemma 7.4), so that

$$ {}^{p}\text{DR}^{\text{an}}(N) \simeq \mathbf{R}p_* {}^{p}\text{DR}^{\text{an}}(q^+M, \nabla - \text{d}(\tau t)) = \mathbf{R}p_* \mathcal{G}[2] \simeq \mathbf{R}p_* \text{DR}^{\text{an}}(\mathcal{M})[2]. $$

According to the commutation of the analytic de Rham functor with proper push-forward of holonomic D-modules, we obtain:

**Corollary 9.2.** *The analytic de Rham complex ${}^{p}\text{DR}^{\text{an}}(N)$ is isomorphic to $\mathbf{R}\tilde{p}_{*}\mathcal{L}_{<0}[2]$.*

*Remark 9.3.* Since ${}^{p}\text{DR}^{\text{an}}(N)$ is known to be a perverse complex, and has zero global cohomology (cf. the introduction), we recover the properties of Proposition 8.16 for $\mathbf{R}\tilde{p}_{*}\mathcal{L}_{<0}[2]$ when $(\mathcal{L}, \mathcal{L}_{\bullet})$ is defined over any subfield of $\mathbb{C}$.

9.a. *First reduction for the proof of Theorem 9.1.* We realize the isomorphism stated in Theorem 9.1 as the pushforward by $\varpi$ of an isomorphism on $\tilde{\mathbb{P}}_{\tau}^1 \times \mathbb{C}_t$. For that purpose, let us consider the sheaf $\mathcal{A}_{\tilde{\mathbb{P}}_{\tau}^1 \times \mathbb{C}_t}^{\text{mod}}$ on $\tilde{\mathbb{P}}_{\tau}^1 \times \mathbb{C}_t$ consisting of holomorphic functions on $\mathbb{C}_{\tau} \times \mathbb{C}_t$ having moderate growth along $\tilde{D}_{\infty}$, and let $\mathcal{A}_{S_{\tau}^1 \times \mathbb{C}_t}^{\text{mod}}$ denote its sheaf-theoretical restriction to $S_{\tau}^1 \times \mathbb{C}_t$. We similarly set $\mathcal{O}_{\tilde{\mathbb{P}}_{\tau}^1 \times \mathbb{C}_t} = \varpi^{-1}\mathcal{O}_{\mathbb{P}_{\tau}^1 \times \mathbb{C}_t}$, $\mathcal{O}_{S_{\tau}^1 \times \mathbb{C}_t} = \mathcal{O}_{\tilde{\mathbb{P}}_{\tau}^1 \times \mathbb{C}_t}|_{S_{\tau}^1 \times \mathbb{C}_t}$ and $\mathcal{O}_{\{\infty\} \times \mathbb{C}_t}^{k, \text{an}} = \mathcal{O}_{\tilde{\mathbb{P}}_{\tau}^1 \times \mathbb{C}_t}|_{\{\infty\} \times \mathbb{C}_t}$ ($k \ge 0$). We then define $\text{DR}^{\text{mod}}(\mathcal{M})$ as the complex

$$ (\mathcal{A}_{S_{\tau}^1 \times \mathbb{C}_t}^{\text{mod}} \otimes_{\varpi^{-1}\mathcal{O}_{S_{\tau}^1 \times \mathbb{C}_t}} \varpi^{-1}(\mathcal{M} \otimes \mathcal{\Omega}_{\{\infty\} \times \mathbb{C}_t}^{\bullet, \text{an}}), \nabla - \text{d}(\tau t)(\tau t)). $$

The theorem is then a consequence of:

**Proposition 9.4.** *There exists a functorial isomorphism $\text{DR}^{\text{mod}}(\mathcal{M}) \simeq \mathcal{L}_{<0}$.*

*Proof of Theorem 9.1.* We use Proposition 9.4 and the isomorphism

$$ \mathbf{R}\varpi_* \text{DR}^{\text{mod}}(\mathcal{M}) \simeq \text{DR}^{\text{an}}(\mathcal{M}), $$

which follows from the isomorphism $\mathbf{R}\varpi_* \mathcal{A}_{S_{\tau}^1 \times \mathbb{C}_t}^{\text{mod}} \simeq \mathcal{O}_{\{\infty\} \times \mathbb{C}_t}(\ast D_{\infty})$.

□

The remaining part of this section is devoted to the proof of Proposition 9.4.

9.b. *The complex and real blowing up of $\mathbb{P}_{\tau}^1 \times \mathbb{C}_t$. The proof of Proposition 9.4 cannot be done easily on $\mathbb{P}_{\tau}^1 \times \mathbb{C}_t$ as we do not have the suitable asymptotic analysis at our disposal, due to the non-goodness of $\Phi \cup \{0\}$. The solution to this problem in the present setting consists in blowing up $\mathbb{P}_{\tau}^1 \times \mathbb{C}_t$ at each point $(\infty, c) \in D_{\infty}$. This is a birational morphism $e: X \to \mathbb{P}_{\tau}^1 \times \mathbb{C}_t$ which is an isomorphism away from the points $(\infty, c)$ with $c \in C$. In particular, it is an isomorphism above $\mathbb{C}_{\tau} \times \mathbb{C}_t$. The pullback $D = e^{-1}(D_{\infty})$ is a normal crossing divisor, which is the union*

$$ D = D'_{\infty} \cup E, \quad E = \bigsqcup_{c \in C} E_c, $$

where $D'_{\infty}$ is the strict transform of $D_{\infty}$ (so that $e: D'_{\infty} \to D_{\infty}$ is an isomorphism) and $E$ is the exceptional divisor, with each $E_c = e^{-1}(c)$ isomorphic to $\mathbb{P}^1$.
Correspondingly, we consider the real blowing up $\varpi_X : \tilde{X} \to X$ of the irreducible components of $D$. The diagram (8.14) can be completed as the diagram (9.5).

<figure><img src="image_4.png" /></figure>

We fix $c_o \in C$ and we denote by $\Delta_x = \Delta_x(c_o) \subset \mathbb{C}_t$ the open disc centered at $c_o$, with coordinate $x = t - c_o$, of radius $r > 0$ small enough such that $\Delta_x$ contains no other point of $C$. We restrict the diagram (9.5) to $\mathbb{P}_\tau^1 \times \Delta_x$, and we still denote by $X$ the open subset $e^{-1}(\mathbb{P}_\tau^1 \times \Delta_x)$, and similarly with $\tilde{X}$. Then $X$ is covered by the following charts.

(a) The chart with coordinates $(u, v)$ such that $e(u, v) = (\tau' = uv, x = u)$; in this chart, the divisor $D$ is defined by $uv = 0$, the strict transform $D'_\infty$ of $D_\infty$ is defined by $v = 0$ and has coordinate $u$, while the exceptional divisor $E$ is defined by $u = 0$ and has coordinate $v$; its pullback in $\tilde{X}$ has polar coordinates $((\rho_u, \theta_u), (\rho_v, \theta_v))$ with

$$ \rho_{\tau'} = \rho_u \rho_v, \quad \rho_x = \rho_u, \quad \theta = \theta_u + \theta_v, \quad \theta_x = \theta_u; $$

this chart is the open set of $(\mathbb{R}_+)^2 \times (S^1)^2$ defined by $\rho_u \in [0, r)$; the pullback $\tilde{D}$ of $D$ is identified with $\partial(\mathbb{R}_+)^2 \times (S^1)^2$, i.e., the subset where $\rho_u$ or $\rho_v$ is zero; we regard it as the (disjoint) union of $\tilde{D}'_\infty = \{\rho_v = 0\}$ and $S_u^1 \times \mathbb{C}_v^*$; of special interest is $\varpi_x^{-1}(0, 0) = S_u^1 \times S_v^1$.

(b) The chart with coordinates $(z, w)$ such that $e(z, w) = (\tau' = z, x = wz)$; in this chart, the divisor $D$ coincides with the exceptional divisor $E$ and is defined as $z = 0$, so that $w$ is a coordinate on it; its pullback in $\tilde{X}$ has polar coordinates $((\rho_z, \theta_z), w)$ with

$$ \rho_{\tau'} = \rho_z, \quad \theta = \theta_z; $$

the pullback $\tilde{E}$ is identified with $S_z^1 \times \mathbb{C}_w$ and its projection to $\tilde{\mathbb{P}}^1 \times \Delta_x$ is nothing but the projection of $S_z^1 \times \mathbb{C}_w$ to $S_\tau^1 \times \{0\}$ defined by $(\theta_z, w) \mapsto (\theta = \theta_z, 0)$.

Let us make explicit the relations between the two charts on their intersection $\mathbb{C}_v^* \times \Delta_u$, defined respectively by $v \neq 0$ and $w \neq 0$. It is also identified with the open subset $\{z \neq 0, |w| < r/|z|\}$ of $\mathbb{C}_w^* \times \mathbb{C}_z$. We have $w = 1/v$ (this is the change of chart on $E \simeq \mathbb{P}^1$) and $z = uv$. Concerning polar coordinates, we have $\rho_v \neq 0$ and $w \neq 0$ on the intersection, and $\theta_z = \theta_u + \theta_v$.

9.c. The set of exponential factors $\Phi$. We consider the set $\Phi = \{\varphi_c \mid c \in C\}$ of exponential factors of $(\mathcal{L}, \mathcal{L}_\bullet)$ and its pullback $\Phi_X = \{\psi_c := \varphi_c \circ e \mid c \in C\}$. In the local setting of Section 9.b, $\Phi_X$ is as follows:

* in the chart $(u, v)$, $\psi_c = (c - c_o)/uv - 1/v$,

* in the chart $(z, w)$, $\psi_c = (c - c_o)/z$ modulo holomorphic functions.
Then $\Phi_X$ is a finite set of global sections of $\mathcal{O}_X(*D)/\mathcal{O}_X$, that we regard as defining a stratified covering $\Sigma \to D$ with $\Sigma$ contained in the sheaf space of $\mathcal{O}_X(*D)/\mathcal{O}_X$ (cf. [Sab13, Chap. 1] for this notion). The stratified property comes from the fact that the pole divisor of $\psi_{c_o}$ is equal to the strict transform $D'_\infty$ of $D_\infty$, and not to $D'_\infty \cup E_{c_o}$. Then $\Phi_X$ is good at each point of $D$ since $\Phi$ as good at each point of $D_\infty$. Even better, $\Phi_X$ is very good since for $c \neq c' \in C$, $\psi_c - \psi_{c'}$ has a pole along each component of $D$.

**Lemma 9.6.** Let $\tilde{x} \in \tilde{D}$. For $c \neq c' \in C$, we have

$$ \psi_c \leq_{\tilde{x}} \psi_{c'} \iff \varphi_c \leq_{\tilde{e}(\tilde{x})} \varphi_{c'} $$

and

$$ \psi_c <_{\tilde{x}} \psi_{c'} \iff \varphi_c <_{\tilde{e}(\tilde{x})} \varphi_{c'} $$

*Proof.* The second equivalence follows from the first one, since for $c \neq c'$, $\psi_c - \psi_{c'}$ is nowhere holomorphic:

* in the chart $(u, v)$, $\psi_c - \psi_c' = (c - c')/uv$,

* in the chart $(z, w)$, $\psi_c - \psi_c' = (c - c')/z$ modulo holomorphic functions. $\square$

9.d. The Stokes-filtered local system $(\mathcal{L}_X, \mathcal{L}_{X,\bullet})$ on $\tilde{D}$. Let $\mathcal{L}_X$ be the local system $\tilde{e}^{-1}\mathcal{L} = (\tilde{e} \circ \tilde{q})^{-1}\mathcal{L}$ and let us set $\mathcal{L}_{X,\leq \psi_c} = \tilde{e}^{-1}\mathcal{L}_{\leq \varphi_c} = (\tilde{e} \circ \tilde{q})^{-1}\mathcal{L}_{\leq c}$, and similarly with $\leq < \psi_c$. Then $\mathrm{gr}_{\psi_c}\mathcal{L}_X = (\tilde{e} \circ \tilde{q})^{-1}\mathrm{gr}_c\mathcal{L}$ is a local system and, according to the lemma above, the filtration condition for $\mathcal{L}_{X,\bullet}$ holds at $\tilde{x}$ iff it holds at $\tilde{e}(\tilde{x})$ for $\mathcal{L}_{\bullet}$, i.e., iff it holds at $(\tilde{e} \circ \tilde{q})(\tilde{x}) \in S_\tau^1$ for $\mathcal{L}_{\bullet}$. It follows that $(\mathcal{L}_X, \mathcal{L}_{X,\bullet})$ is a Stokes-filtered local system on $\tilde{D}$ indexed by $\Phi_X$.

We now consider the set $\Phi_X \cup \{0\}$. We note that it is good at each point of $D$: for $c \in C$, $\psi_c - 0 = \psi_c$ is purely monomial both if $c \neq c_o$ and $c = c_o$. (However, it is not very good any more, since $\psi_{c_o}$ does not have a pole along $E_{c_o}$.) We can then extend $(\mathcal{L}_X, \mathcal{L}_{X,\bullet})$ as a Stokes-filtered local system indexed by $\Phi_X \cup \{0\}$ by setting, for each $\tilde{x} \in \tilde{D}$:

$$ \mathcal{L}_{X, \leq 0, \tilde{x}} = \sum_{c | \psi_c \leq \tilde{x} 0} \mathcal{L}_{X, \leq \psi_c, \tilde{x}}, \quad \mathcal{L}_{X, < 0, \tilde{x}} = \sum_{c | \psi_c < \tilde{x} 0} \mathcal{L}_{X, \leq \psi_c, \tilde{x}} $$

As the subsets $\{\tilde{x} | \psi_c \leq_{\tilde{x}} \psi_{c'}\}$ and $\{\tilde{x} | \psi_c \leq_{\tilde{x}} 0\}$ are open, $\mathcal{L}_{X, \leq 0}$ is a subsheaf of $\mathcal{L}_X$. On the other hand, the subsets $\{\tilde{x} | \psi_c <_{\tilde{x}} 0\}$ are not open, as for example in the local setting of Section 9.b, for $\tilde{x} \in S_u^1 \times S_v^1$, we have $\psi_{c_o} \leq_{\tilde{x}} 0$ iff $\theta_v \in (-\pi/2, \pi/2)$ (with $\tilde{x} = (\theta_u, \theta_v)$), and this is equivalent to $\psi_{c_o} <_{\tilde{x}} 0$, but the latter property does not hold at any nearby point $\tilde{y}$ above a point of the exceptional divisor $E_{c_o} \setminus \{(\infty, c_o)\}$. We thus find:

* on $\varpi_X^{-1}(D'_\infty)$, $\mathcal{L}_{X, < 0}$ is a subsheaf of $\mathcal{L}_X$ which coincides with $\mathcal{L}_{X, \leq 0}$,

* on $\varpi_X^{-1}(E_{c_o} \setminus \{(\infty, c_o)\})$ where $\psi_{c_o}$ has no pole, we have $\mathcal{L}_{X, \leq 0} = (\tilde{q} \circ \tilde{e})^{-1}\mathcal{L}_{\leq c_o}$ and $\mathcal{L}_{X, < 0} = (\tilde{q} \circ \tilde{e})^{-1}\mathcal{L}_{< c_o}$.

This illustrates the “stratified” definition of a Stokes-filtered local system along a normal crossing divisor (cf. [Sab13, §1.9]).

9.e. *Proof of Proposition 9.4.* We consider the sheaf $A_{\tilde{X}}^{\mathrm{mod}}$ on $\tilde{X}$ of holomorphic functions on $\mathbb{C}_\tau \times \mathbb{C}_t$ having moderate growth along $\tilde{D}$ and its sheaf-theoretical restriction $A_{\tilde{D}}^{\mathrm{mod}}$ to $\tilde{D}$. We denote by $\mathcal{M}_X$ the pullback meromorphic bundle with flat connection $e^+\mathcal{M}$. We can thus define the moderate de Rham complex $\mathrm{DR}_{\tilde{D}}^{\mathrm{mod}}(\mathcal{M}_X)$, which is a complex on $\tilde{D}$. The proof of Proposition 9.4 is done in three steps.
(1) The Hukuhara theorem in dimension two yields:

**Theorem 9.7.** We have $\mathcal{H}^k(\mathrm{DR}_{\tilde{D}}^{\mathrm{mod}}(\mathcal{M}_X)) = 0$ if $k \neq 0$ and a natural isomorphism $\mathcal{H}^0 \mathrm{DR}_{\tilde{D}}^{\mathrm{mod}}(\mathcal{M}_X) \simeq \mathcal{L}_{X, \le 0}$.

(2) The moderate de Rham complex behaves well by blowing down:

**Proposition 9.8.** There exists a natural isomorphism

$$ \mathcal{R} \tilde{e}_* \mathrm{DR}_{\tilde{D}}^{\mathrm{mod}} (\mathcal{M}_X) \simeq \mathrm{DR}_{S_{\tau}^1 \times \mathbb{C}_t}^{\mathrm{mod}} (\mathcal{M}). $$

*Proof.* See e.g. [Sab13, Prop. 8.9]. □

<figure><img src="image_5.png" /></figure>

(3) Similarly, the pushforward of $\mathcal{L}_{X, \le 0}$ by $\tilde{e}$ is isomorphic to $\mathcal{L}_{<0}$:

**Proposition 9.9.** We have $\mathcal{R}^k \tilde{e}_* \mathcal{L}_{X, \le 0} = 0$ for $k \neq 0$ and the natural morphism $\tilde{e}_* \mathcal{L}_{X, \le 0} \to \tilde{q}^{-1} \mathcal{L}$ induces an isomorphism $\tilde{e}_* \mathcal{L}_{X, \le 0} \simeq \mathcal{L}_{<0}$, and so, $\mathcal{R} \tilde{e}_* \mathcal{L}_{X, \le 0} \simeq \mathcal{L}_{<0}$.

We obtain Proposition 9.4 by applying $\mathcal{R} \tilde{e}_*$ to the isomorphism of Theorem 9.7 and then Propositions 9.8 and 9.9 on each side. □

<figure><img src="image_6.png" /></figure>

9.f. *Proof of Theorem 9.7.* For the first part, cf. e.g. [Sab13, Cor. 12.7]. For the second part, we first exhibit a morphism $\mathcal{H}^0 \mathrm{DR}_{\tilde{D}}^{\mathrm{mod}}(\mathcal{M}) \to \mathcal{L}_X$ and then show that it is an isomorphism onto $\mathcal{L}_{X, \le 0}$. For that purpose, let us set $\mathcal{L}^* = j_\infty^{-1} \mathrm{DR}^{\mathrm{an}}(M)$, which is a local system on $\mathbb{C}_\tau^*$. Then, $j_\infty^{-1} \mathrm{DR}^{\mathrm{an}}(q^+M, q^+\nabla - d(\tau t))$ is a local system on $\mathbb{C}_\tau^* \times \mathbb{C}_t$, canonically isomorphic to $\mathcal{L}^* := q^{-1}\mathcal{L}^*$.

Let us denote now by $\tilde{\mathcal{J}}_\infty, \tilde{\iota}_\infty$ the pair of complementary inclusions $\mathbb{C}_\tau^* \times \mathbb{C}_t \hookrightarrow \tilde{X}$ and $\tilde{D} \hookrightarrow X$.

**Lemma 9.10.** We have $\mathcal{L}_X = \tilde{\iota}_\infty^{-1} \tilde{\mathcal{J}}_{\infty, *} \mathcal{L}^* \simeq \tilde{\iota}_\infty^{-1} \mathcal{R} \tilde{\mathcal{J}}_{\infty, *} \mathcal{L}^*$.

*Proof.* The analysis of neighborhoods made in Section 9.b, (a) and (b) shows that there exists a basis of open neighborhoods $U$ of a point of $S_\tau^1 \times \mathbb{C}_t$ in $\tilde{\mathbb{P}}_\tau^1 \times \mathbb{C}_t$, or of a point of $\tilde{D}$ in $\tilde{X}$ which is not above a crossing point of $D$, of the form $I \times [0, \varepsilon) \times \Delta$, where $I$ is an open interval in $S^1$ and $\Delta$ is an open disc in $\mathbb{C}$. Above a crossing point of $D$, the neighborhoods take the form $I^2 \times [0, \varepsilon)^2$. The trace of $\mathbb{C}_\tau^* \times \mathbb{C}_t$, or $X \setminus D$, on such an open neighborhood is then $I \times (0, \varepsilon) \times \Delta$ or $I^2 \times (0, \varepsilon)^2$. In any case, this is a contractible open set. It follows that

$$ i_{\infty}^{-1} \mathcal{R} j_{\infty, *} \mathcal{L}^* \simeq i_{\infty}^{-1} j_{\infty, *} \mathcal{L}^* \quad \text{and} \quad \tilde{i}_{\infty}^{-1} \mathcal{R} \tilde{j}_{\infty, *} \mathcal{L}^* \simeq \tilde{i}_{\infty}^{-1} \tilde{j}_{\infty, *} \mathcal{L}^*. $$

On the other hand, we have a natural morphism $\tilde{e}^{-1} j_{\infty, *} \mathcal{L}^* \to \tilde{j}_{\infty, *} \mathcal{L}^*$, and thus, by applying $\tilde{i}_\infty^{-1}$, a morphism $\mathcal{L}_X = \tilde{e}^{-1} \mathcal{L} \to \tilde{i}_\infty^{-1} \tilde{j}_{\infty, *} \mathcal{L}^*$. That it is an isomorphism is thus a local question at points of $\tilde{E} := \varpi_{\tilde{X}}^{-1}(\bigcup_c E_c)$. As we have seen above, any $\tilde{x} \in \tilde{E}$ has a basis of open neighborhoods $\tilde{U}$ in $\tilde{X}$ such that $\tilde{U}^* = \tilde{U} \setminus \tilde{D}$ is contractible. A similar property holds for its image $\tilde{e}(\tilde{x}) = (\theta_{\tau'}, c) \in \tilde{D}_\infty = S_\tau^1 \times \mathbb{C}_t$ with $\tilde{V} \subset \tilde{\mathbb{P}}_\tau^1 \times \mathbb{C}_t$ and $\tilde{V}^* = \tilde{V} \setminus (S_\tau^1 \times \mathbb{C}_t)$. On the one hand, $\mathcal{L}_{X, \tilde{x}} = \lim_{\to \tilde{V}^*} \Gamma(\tilde{V}^*, \mathcal{L}^*)$, and $(\tilde{i}_\infty^{-1} \tilde{j}_{\infty, *} \mathcal{L}^*)_{\tilde{x}} = \lim_{\to \tilde{U}^*} \Gamma(\tilde{U}^*, \mathcal{L}^*)$. If $\tilde{e}(\tilde{U}^*) \subset \tilde{V}^*$, then the restriction morphism $\Gamma(\tilde{V}^*, \mathcal{L}^*) \to \Gamma(\tilde{U}^*, \mathcal{L}^*)$ is an isomorphism since $\mathcal{L}^*$ is locally constant and both $\tilde{U}^*$ and $\tilde{V}^*$ are contractible. This concludes the proof. □

<figure><img src="image_7.png" /></figure>
According to this lemma, from the natural morphism

$$ \mathcal{H}^0 \mathrm{DR}_{\tilde{D}}^{\mathrm{mod}}(\mathcal{M}_X) \to \tilde{\mathcal{J}}_{\infty, *} \tilde{\mathcal{J}}_{\infty}^{-1} \mathcal{H}^0 \mathrm{DR}_{\tilde{D}}^{\mathrm{mod}}(\mathcal{M}_X) = \tilde{\mathcal{J}}_{\infty, *} \mathcal{L}^* $$

we obtain, by applying $\tilde{\mathcal{I}}_{\infty}^{-1}$, the desired morphism

$$ \mathcal{H}^0 \mathrm{DR}_{\tilde{D}}^{\mathrm{mod}}(\mathcal{M}_X) \to \tilde{\mathcal{I}}_{\infty}^{-1} \tilde{\mathcal{J}}_{\infty, *} \mathcal{L}^* = \mathcal{L}_X. $$

We now show that the image of the latter morphism is $\mathcal{L}_{X, \le 0}$. This is clear away from the pullback of the points $(\infty, c)$. Since the question is local, we fix $\tilde{x}$ projecting to $(\infty, c_o) \in D_\infty$ with $c_o \in C$. Up to tensoring all data, we can, and will, assume that $c_o = 0$, so that the coordinate $x = t - c_o$ is equal to $t$ (cf. Section 9.b).

In the chart $(z, w)$, we have $e^* d(\tau t) = e^* d(t/\tau') = dw$, so that $\mathcal{M}_X$ is locally isomorphic to $e^+q^+M$, and $\psi_c = c/z = c/\tau'$, so that the Stokes-filtered local system attached to $\mathcal{M}_X$, which is indexed by $\Phi_X$, is the pullback by $\tilde{q} \circ \tilde{e}$ of that of $M$, which is indexed by $C/\tau'$.

In the chart $(u, v)$, we have $e^* d(\tau t) = e^* d(x\tau) d(1/v)$, and $\psi_c = c/\tau' - 1/v$ if $c \ne 0$ and $\psi_0 = -1/v$. Hence both $\mathcal{M}_X$ and $\Phi_X$ are obtained from $e^+q^+M$ by twisting by $-d(1/v)$ or by adding $-1/v$ to $\Phi$. We conclude that the Stokes-filtered local system of $\mathcal{M}_X$ on $S_u^1 \times S_v^1$ is simply the pullback of that of $M$ by the map $(\theta_u, \theta_v) \mapsto \theta_u + \theta_v$, and this concludes the proof of Theorem 9.7. $\square$

9.g. *Proof of Proposition 9.9.* Away from $S_\tau^1 \times C \subset S_\tau^1 \times \mathbb{C}_t$, the map $\tilde{e} : \tilde{D} \to S_\tau^1 \times \mathbb{C}_t$ is an isomorphism, and the assertion amounts to $\mathcal{L}_{\le 0} = \mathcal{L}_{< 0}$, which follows from the definition of $\mathcal{L}_{< 0}$ in Section 8.c.

Let us fix $c_o \in C$. The inverse image $\tilde{e}^{-1}(S_\tau^1 \times \{c_o\})$ is isomorphic to $\tilde{E}_{c_o} \times S_z^1$, where $\tilde{E}_{c_o}$ is the real blow-up of the exceptional divisor $E_{c_o}$ at $w = \infty$, so that $\tilde{E}_{c_o}$ is diffeomorphic to a closed disc, with boundary $S_w^1$. The map $\tilde{e} : \tilde{E}_{c_o} \times S_z^1 \to S_\tau^1 \times \{c_o\}$ is induced by the projection $\tilde{E}_{c_o} \times S_z^1 \to S_z^1$ composed with the isomorphism $S_z^1 \xrightarrow{\sim} S_\tau^1$ induced by $\theta_z \mapsto \theta$. We will describe the restriction to $\tilde{e}^{-1}(S_\tau^1 \times \{c_o\})$ of the sheaves $\mathcal{L}_{X, \le 0}$ (and we now omit to mention this restriction functor).

* Let us set $E_{c_o}^\circ = E_{c_o} \setminus \infty = \tilde{E}_{c_o} \setminus S_w^1$. On $E_{c_o}^\circ \times S_z^1$, $\mathcal{L}_{X, \le 0}$ coincides with $\tilde{e}^{-1}(\mathcal{L}_{\le c_o})$.

* On $S_w^1 \times S_z^1$, we have inclusions

$$ \tilde{e}^{-1}(\mathcal{L}_{< c_o}) \subset \mathcal{L}_{X, \le 0} \subset \tilde{e}^{-1}(\mathcal{L}_{\le c_o}). $$

More precisely, locally with respect to $S_z^1$ that we identify with $S_\tau^1$, we have decompositions

$$ \mathcal{L}_{\le c_o, \theta_z} \simeq \bigoplus_{c \le \theta_z} \mathrm{gr}_c \mathcal{L}_{\theta_z} \quad \text{and} \quad \mathcal{L}_{< c_o, \theta_z} \simeq \bigoplus_{\substack{c \ne c_o \\ c \le \theta_z}} \mathrm{gr}_c \mathcal{L}_{\theta_z}. $$

Then at a point $(\theta_w, \theta_z) \in S_w^1 \times S_z^1$, we have

$$ \mathcal{L}_{X, \le 0, (\theta_w, \theta_z)} = \left\{ \bigoplus_{c \le \theta_z} \mathrm{gr}_c \mathcal{L}_{\theta_z} \quad \text{if } \theta_w \in (-\pi/2, \pi/2), \right. \\ \left. \mathcal{L}_{< c_o, \theta_z} \simeq \bigoplus_{\substack{c \ne c_o \\ c \le \theta_z}} \mathrm{gr}_c \mathcal{L}_{\theta_z} \quad \text{otherwise.} \right. $$

In conclusion, let $U_{c_o}$ be the open subset of $\tilde{E}_{c_o}$ which is the complement of the closed subset $[\pi/2, 3\pi/2]$ of $S_w^1$. Then, on $\tilde{E}_{c_o} \times S_z^1$, we have an exact sequence

$$ 0 \to \tilde{e}^{-1}(\mathcal{L}_{< c_o}) \to \mathcal{L}_{X, \le 0} \to \mathcal{D} \to 0, $$
where the quotient sheaf $\mathcal{Q}$ is isomorphic to the sheaf

$$(\tilde{e}^{-1} \mathrm{gr}_{c_o} \mathcal{L})_{U_{c_o} \times S_z^1} \simeq \mathbb{C}_{U_{c_o}} \boxtimes \mathrm{gr}_{c_o} \mathcal{L}$$

(i.e., the extension by zero of its restriction to $U_{c_o} \times S_z^1$). We have

$$R\tilde{e}_*\tilde{e}^{-1}(\mathcal{L}_{<c_o}) \simeq R\tilde{e}_*\mathbb{C}_{\tilde{E}_{c_o}} \otimes \mathcal{L}_{<c_o} \simeq \mathcal{L}_{<c_o} (\tilde{E}_{c_o} \text{ is a closed disc}),$$

$$R\tilde{e}_*(\mathbb{C}_{U_{c_o}} \boxtimes \mathrm{gr}_{c_o} \mathcal{L}) \simeq R\Gamma_c(U_{c_o}) \otimes \mathrm{gr}_{c_o} \mathcal{L} = 0 \quad (\text{Lemma 7.4}).$$

We conclude that the natural morphism $\mathcal{L}_{<0}|_{S_+^1 \times \{c_o\}} \simeq \mathcal{L}_{<c_o} \hookrightarrow \tilde{e}_*\mathcal{L}_{X, \le 0} \simeq R\tilde{e}_*\mathcal{L}_{X, \le 0}$ is an isomorphism. $\square$

## References

[BG12] A. Beilinson and D. Gaitsgory, *A corollary of the b-function lemma*, Selecta Math. (N.S.) **18** (2012), no. 2, 319–327.

[DK16] A. D'Agnolo and M. Kashiwara, *Riemann-Hilbert correspondence for holonomic D-modules*, Publ. Math. Inst. Hautes Études Sci. **123** (2016), 69–197.

[HTT08] R. Hotta, K. Takeuchi, and T. Tanisaki, *D-Modules, perverse sheaves, and representation theory*, Progress in Math., vol. 236, Birkhäuser, Boston, Basel, Berlin, 2008, in Japanese: 1995.

[Ked10] K. Kedlaya, *Good formal structures for flat meromorphic connections, I: surfaces*, Duke Math. J. **154** (2010), no. 2, 343–418.

[Ked11] —, *Good formal structures for flat meromorphic connections, II: excellent schemes*, J. Amer. Math. Soc. **24** (2011), no. 1, 183–229.

[KKP08] L. Katzarkov, M. Kontsevich, and T. Pantev, *Hodge theoretic aspects of mirror symmetry*, From Hodge theory to integrability and TQFT: tt*-geometry (R. Donagi and K. Wendland, eds.), Proc. Symposia in Pure Math., vol. 78, American Mathematical Society, Providence, RI, 2008, pp. 87–174.

[Mac74] R.D. MacPherson, *Chern classes for singular varieties*, Ann. of Math. **100** (1974), 423–432.

[Mai23] Ph. Maisonobe, *Filtration relative, l'idéal de Bernstein et ses pentes*, Rend. Sem. Mat. Univ. Padova **150** (2023), 81–125.

[Mal91] B. Malgrange, *Équations différentielles à coefficients polynomiaux*, Progress in Math., vol. 96, Birkhäuser, Basel, Boston, 1991.

[Meb90] Z. Mebkhout, *Le théorème de positivité de l'irrégularité pour les $\mathbb{D}_X$-modules*, The Grothendieck Festschrift, Progress in Math., vol. 88, no. 3, Birkhäuser, Basel, Boston, 1990, pp. 83–132.

[Meb04] —, *Le théorème de positivité, le théorème de comparaison et le théorème d'existence de Riemann*, Éléments de la théorie des systèmes différentiels géométriques, Séminaires & Congrès, vol. 8, Société Mathématique de France, Paris, 2004, pp. 165–310.

[Moc09a] T. Mochizuki, *Good formal structure for meromorphic flat connections on smooth projective surfaces*, Algebraic Analysis and Around (Kyoto, June 2007), Advanced Studies in Pure Math., vol. 54, Math. Soc. Japan, Tokyo, 2009, pp. 223–253.

[Moc09b] —, *On Deligne-Malgrange lattices, resolution of turning points and harmonic bundles*, Ann. Inst. Fourier (Grenoble) **59** (2009), no. 7, 2819–2837.

[Moc11a] —, *Stokes structure of a good meromorphic flat bundle*, J. Inst. Math. Jussieu **10** (2011), no. 3, 675–712.

[Moc11b] —, *Wild harmonic bundles and wild pure twistor D-modules*, Astérisque, vol. 340, Société Mathématique de France, Paris, 2011.

[Moc14] —, *Holonomic D-modules with Betti structure*, Mém. Soc. Math. France (N.S.), vol. 138–139, Société Mathématique de France, Paris, 2014.

[Moc15] —, *Mixed twistor D-modules*, Lect. Notes in Math., vol. 2125, Springer, Heidelberg, New York, 2015.

[Moc18] —, *Stokes shells and Fourier transforms*, 2018, arXiv:1808.01037.

[Rib24] G. Ribeiro, *Generic vanishing for holonomic D-modules: a study via Cartier duality*, Ph.D. thesis, Institut polytechnique de Paris, 2024.
[Sab93] C. Sabbah, *Équations différentielles à points singuliers irréguliers en dimension 2*, Ann. Inst. Fourier (Grenoble) **43** (1993), 1619–1688.

[Sab00] ——, *Équations différentielles à points singuliers irréguliers et phénomène de Stokes en dimension 2*, Astérisque, vol. 263, Société Mathématique de France, Paris, 2000.

[Sab13] ——, *Introduction to Stokes structures*, Lect. Notes in Math., vol. 2060, Springer-Verlag, 2013.

[Sab17] ——, *A remark on the irregularity complex*, J. Singul. **16** (2017), 100–114, Erratum: *Ibid.*, p. 194.

[Tey23] J.-B. Teyssier, *Moduli of Stokes torsors and singularities of differential equations*, J. Eur. Math. Soc. (JEMS) **25** (2023), 369–411.

[Wu21] L. Wu, *Riemann-Hilbert correspondence for Alexander complexes*, arXiv:2104.06941, 2021.

[YZ24] T. Y. Yu and S. Zhang, *Topological Laplace transform and decomposition of nc-Hodge structures*, 2024, arXiv:2405.19549.

CMLS, CNRS, ÉCOLE POLYTECHNIQUE, INSTITUT POLYTECHNIQUE DE PARIS, 91128 PALAISEAU CEDEX, FRANCE

Email address: Claude.Sabbah@polytechnique.edu

URL: https://perso.pages.math.cnrs.fr/users/claudesabbah