1 Introduction↩︎

The study of Fourier transforms of \(\mathcal{D}\)-modules is an important subject in both algebraic analysis and algebraic geometry. Despite its long history, we do not know so far much of the global properties of the Fourier transforms, such as the monodromies and the Stokes matrices of their holomorphic solutions, even in the simplest case of dimension one (see for example, D’Agnolo-Hien-Morando-Sabbah [@DHMS20] and Hohl [@Hoh22]). For the results in higher dimensions, see Brylinski [@Bry86], Daia [@Dai00], Ito-Takeuchi [@IT20a], [@IT20b], Kashiwara-Schapira [@KS97] and Takeuchi [@Tak22]. After some pioneering works by Malgrange in [@Mal88] and [@Mal91], inspired from the theory of Fourier transforms of \(l\)-adic sheaves in positive characteristic, Bloch-Esnault [@BE04b] and García López [@Gar04] introduced independently the so-called local Fourier transforms of algebraic holonomic \(\mathcal{D}\)-modules \(\mathcal{M}\) on the complex affine line \(X= \mathbb{C}_z\). In what follows, we assume that \(\mathcal{M}\) is an algebraic meromorphic connection i.e. a localized algebraic holonomic \(\mathcal{D}\)-module on \(X= \mathbb{C}_z\). Let \(Y= \mathbb{C}_w\) be the dual of \(X= \mathbb{C}_z\) and \(\overline{X}\simeq{\mathbb{P}}^1\) (resp. \(\overline{Y}\simeq{\mathbb{P}}^1\)) the projective compactification of \(X\) (resp. \(Y\)). Then the Fourier transform \(\mathcal{M}^\wedge\) of \(\mathcal{M}\) is an algebraic holonomic \(\mathcal{D}\)-module on \(Y= \mathbb{C}_w\) and the new method of [@BE04b] and [@Gar04] enables us to describe the formal structure i.e. the exponential factors of \(\mathcal{M}^\wedge\) in terms of that of \(\mathcal{M}\). Such an explicit description was obtained by Fang [@Fan09], Graham-Squire [@Gra13] and Sabbah [@Sab08]. We thus now know that the exponential factors of \(\mathcal{M}^\wedge\) are obtained by the Legendre transform from those of \(\mathcal{M}\) (for the definition, see Section 2.6). We call it the stationary phase method. Subsequently, based on this result, Mochizuki [@Mochi10], [@Mochi18] gave also a description of the Stokes structure of \(\mathcal{M}^\wedge\) at infinity. Moreover, by using the new theories of the irregular Riemann-Hilbert correspondence established by D’Agnolo-Kashiwara [@DK16] and the enhanced Fourier-Sato transforms of Kashiwara-Schapira [@KS16a] adapted to it, in the two papers [@DK18] and [@DK23] D’Agnolo and Kashiwara reformulated and reproved the stationary phase method more elegantly. For this purpose, in [@DK18] they used some microlocal notions, such as enhanced micro-supports and multiplicity test functors, to treat the exponential factors of \(\mathcal{M}\) other than the linear ones. As we do not see any information of the linear exponential factors by the enhanced micro-support, in [@DK23] they developed a theory of nearby and vanishing cycles of enhanced ind-sheaves to treat them separately.

In this paper, we apply the Morse theoretical method in Ito-Takeuchi [@IT20a] to give a unified proof to the results in the two papers [@DK18] and [@DK23]. Moreover we obtain not only the exponential factors but also the enhanced solution complex \(Sol_{\overline{Y}}^{{\mathrm{E}}}(\widetilde{\mathcal{M}^\wedge})\) of \(\widetilde{\mathcal{M}^\wedge}\), where by the inclusion map \(i_Y : Y = \mathbb{C}_w \xhookrightarrow{\;\;\;}\overline{Y}={\mathbb{P}}^1\) we set \(\widetilde{\mathcal{M}^\wedge} := i_{Y\ast}\mathcal{M}^\wedge \simeq\mathbf{D}i_{Y\ast} \mathcal{M}^\wedge \in \mathrm{Mod}_{\mathrm{hol}}(\mathcal{D}_{\overline{Y}})\). Specifically, we describe it on a sufficiently small punctured disk centered at a singular point of \(\mathcal{M}^\wedge\) or at infinity. Note that in [@DK18] the authors determined only the multiplicities of the exponential factors of \(\mathcal{M}^\wedge\) via the “microlocal shadow" of \(\mathcal{M}^\wedge\) i.e. the enhanced micro-support associated to it. We thus have upgraded such an indirect proof of the stationary phase method to a more concrete and manipulatable one, which enables us to extract deeper informations of the Fourier transform \(\mathcal{M}^\wedge\). For its application to the monodromies at infinity of \(\mathcal{M}^\wedge\) see [@KT24]. Let us explain the outline of our proof. Recall that in [@DK18] a local description of the enhanced solution complex of the original \(\mathcal{D}\)-module \(\mathcal{M}\) was obtained. We refine this result to get also a global explicit description of it (see Section 3 for the details) and apply the Morse theoretical method in [@IT20a] to it. More precisely, by the inclusion map \(i_X : X=\mathbb{C}_z \xhookrightarrow{\;\;\;}\overline{X}={\mathbb{P}}^1\) we set \(\widetilde{\mathcal{M}} := i_{X\ast}\mathcal{M}\simeq\mathbf{D}i_{X\ast}\mathcal{M} \in\mathrm{Mod}_{\mathrm{hol}}(\mathcal{D}_{\overline{X}})\) and construct an \(\mathbb{R}\)-constructible enhanced sheaf \(G\in\mathbf{E}_{\mathbb{R}{\text{-}}\mathrm c}^0(\mathbb{C}_{\overline{X}^{{\rm an}}})\) on the underlying complex manifold \(\overline{X}^{{\rm an}}\) of \(\overline{X} = {\mathbb{P}}^1\) such that for the enhanced solution complex \(Sol_{\overline{X}}^{{\mathrm{E}}}(\widetilde{\mathcal{M}})\) of \(\widetilde{\mathcal{M}}\) we have an isomorphism \[\begin{align} Sol_{\overline{X}}^{{\mathrm{E}}}(\widetilde{\mathcal{M}}) \simeq \mathbb{C}_{\overline{X}^{{\rm an}}}^{{\mathrm{E}}}\overset{+}{\otimes}G. \end{align}\] This construction is important, because for the Fourier-Sato (Fourier-Laplace) transform \({}^\mathsf{L}G\in{\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_{\overline{Y}^{{\rm an}}})\) of \(G\) (see Definition 4) there exists an isomorphism \[\begin{align} Sol_{\overline{Y}}^{{\mathrm{E}}}(\widetilde{\mathcal{M}^\wedge}) \simeq \mathbb{C}_{\overline{Y}^{{\rm an}}}^{{\mathrm{E}}}\overset{+}{\otimes}{}^\mathsf{L}G \end{align}\] and hence all the informations of the Fourier transform \(\mathcal{M}^\wedge\) of \(\mathcal{M}\) are contained in \({}^\mathsf{L}G\). Then our remaining task is to describe \({}^\mathsf{L}G\) and it turns out that the stalk of (an \(\mathbb{R}\)-constructible sheaf on \(\overline{Y}^{{\rm an}} \times \mathbb{R}\) representing) the enhanced sheaf \({}^\mathsf{L}G\in{\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_{\overline{Y}^{{\rm an}}})\) at a point \((w,t) \in Y^{{\rm an}} \times \mathbb{R}\) is isomorphic to the complex \[\begin{align} {\mathrm{R}}\Gamma_c \left(X^{{\rm an}}; G(w,t) [1]\right) \end{align}\] associated to an \(\mathbb{R}\)-constructible sheaf \(G(w,t)\) on \(X^{{\rm an}}\) (for the definition, see Section 4.2). Next, we study very carefully how their cohomology groups \[\begin{align} H^j_c \left(X^{{\rm an}}; G(w,t) [1]\right) \simeq H^{j+1}_c \left(X^{{\rm an}}; G(w,t) \right) \qquad (j \in \mathbb{Z}) \end{align}\] change as \(t \in \mathbb{R}\) increases. This is what we call here”the Morse theoretical method" and it leads us to a very precise description of the enhanced sheaf \({}^\mathsf{L}G\) underlying the enhanced solution complex \(Sol_{\overline{Y}}^{{\mathrm{E}}}(\widetilde{\mathcal{M}^\wedge})\) of \(\widetilde{\mathcal{M}^\wedge}\). Then as in [@Tak22], by the exponential factors of \(\mathcal{M}\) we define a Lagrangian cycle \(\mathrm{CC}_{\mathrm{irr}}(\mathcal{M})\) in the cotangent bundle \(T^*X^{{\rm an}}\) of \(X^{{\rm an}}\) that we call the irregular characteristic cycle of \(\mathcal{M}\) and describe the enhanced solution complex of \(\widetilde{\mathcal{M}^\wedge}\) by it (see Figure 1 below). In this way, we obtain a new perspective of the Fourier transform, from which we can geometrically see how the exponential factors of \(\mathcal{M}\) and \(\mathcal{M}^\wedge\) are related and the criteria for \(\mathcal{M}^\wedge\) to be regular or monodromic follow immediately (see Corollaries 2 and 3). In particular, we see that the generic rank \({\rm rk} \mathcal{M}^\wedge\) of \(\mathcal{M}^\wedge\) is equal to the covering degree at infinity of the restriction of the projection \(T^*X^{{\rm an}} \simeq X^{{\rm an}} \times Y^{{\rm an}} \longrightarrow Y^{{\rm an}}\) to \(\mathrm{CC}_{\mathrm{irr}}(\mathcal{M})\) (see Figure 1 below). To the best of our knowledge, we do not find such a geometric expression of \({\rm rk} \mathcal{M}^\wedge\) in the literature. Finally, we also obtain a formula for the (classical) characteristic cycle \({\rm CC} ( \mathcal{M}^\wedge )\) of \(\mathcal{M}^\wedge\), which has never been studied successfully before.

In order to introduce our results more precisely, we prepare some notations. For the algebraic meromorphic connection \(\mathcal{M}\in\mathrm{Mod}_{\mathrm{hol}}(\mathcal{D}_X)\) on the affine line \(X=\mathbb{C}_z\) we denote its singular support \(\mathrm{sing.supp}(\mathcal{M})\) by \(D\subset X\) and set \(U:= X\left.\right\backslash D\). Then there exists an isomorphism \(\mathcal{M}\overset{\sim}{\longrightarrow}\Gamma_U(\mathcal{M})\). We denote the generic rank of \(\mathcal{M}\) by \({\rm rk} \mathcal{M}\). Define the analytification \(\widetilde{\mathcal{M}}^{{\rm an}}\in\mathrm{Mod}_{\mathrm{hol}}(\mathcal{D}_{\overline{X}^{{\rm an}}})\) of \(\widetilde{\mathcal{M}}\) by \(\widetilde{\mathcal{M}}^{{\rm an}} = \mathcal{O}_{\overline{X}^{{\rm an}}} \otimes_{\mathcal{O}_{\overline{X}}}\widetilde{\mathcal{M}}\). Then for the divisor \(\widetilde{D} :=D^{{\rm an}}\sqcup\{\infty\}\subset\overline{X}^{{\rm an}}\) in \(\overline{X}^{{\rm an}}\) we obtain an isomorphism \((\widetilde{\mathcal{M}})^{{\rm an}} \overset{\sim}{\longrightarrow}(\widetilde{\mathcal{M}})^{{\rm an}}(\ast\widetilde{D})\). We define the enhanced solution complex \(Sol_{\overline{X}}^{\mathrm{E}}(\widetilde{\mathcal{M}})\) of \(\widetilde{\mathcal{M}}\) by \[\begin{align} Sol_{\overline{X}}^{\mathrm{E}}(\widetilde{\mathcal{M}}) := Sol_{\overline{X}^{{\rm an}}}^{\mathrm{E}}(\widetilde{\mathcal{M}}^{{\rm an}}) \qquad \in{\mathbf{E}}^{\mathrm{b}}({\rm I}\mathbb{C}_{\overline{X}^{{\rm an}}}) \end{align}\] (see Section 2 for the details). Similarly, for the Fourier transform \(\mathcal{M}^\wedge\in\mathrm{Mod}_{\mathrm{hol}}(\mathcal{D}_Y)\) of \(\mathcal{M}\) we define the enhanced solution complex \(Sol_{\overline{Y}}^{\mathrm{E}}(\widetilde{\mathcal{M}^\wedge}) \in{\mathbf{E}}^{\mathrm{b}}({\rm I}\mathbb{C}_{\overline{Y}^{{\rm an}}})\) of \(\widetilde{\mathcal{M}^\wedge}\). Now our objective here is to describe \(Sol_{\overline{Y}}^{\mathrm{E}}(\widetilde{\mathcal{M}^\wedge})\) in terms of \(\mathcal{M}\). Let \(D^{{\rm an}} = \{ a_1,a_2,\dots,a_l \} \subset X^{{\rm an}}\) and set \(a_\infty\coloneq\infty\in\overline{X}^{{\rm an}}\) so what we have \[\begin{align} \widetilde{D} = \mathrm{sing.supp}(\widetilde{\mathcal{M}})^{{\rm an}} = \{a_1,a_2,\dots,a_l,a_\infty\}. \end{align}\] For a point \(a_i\in D^{{\rm an}}\) and a Puiseux germ \(f(z)\in\mathcal{P}_{S_{a_i}X^{{\rm an}}}^\prime\) (see Section 2.5 for the definition) which is holomorphic on a sector \(S \subset X^{{\rm an}}\) along \(a_i\), we define a complex Lagrangian submanifold \(\Lambda_i^f\) of \(T^\ast X^{{\rm an}} \simeq X^{{\rm an}}\times Y^{{\rm an}}=X^{{\rm an}}\times\mathbb{C}_w\) by \[\begin{align} \Lambda_i^f \coloneq \Set*{(z,f^\prime(z))}{z\in S} \;\subset T^\ast X^{{\rm an}}. \end{align}\] In this paper, we always assume that sectors are connected and open. For the point \(a_i\in D^{{\rm an}}\), let \[\begin{align} N_i =N(a_i) \colon \mathcal{P}_{S_{a_i}X^{{\rm an}}}^\prime \longrightarrow (\mathbb{Z}_{\geq 0})_{S_{a_i}X^{{\rm an}}} \end{align}\] be the multiplicity for which the enhanced solution complex \(Sol_{\overline{X}}^{\mathrm{E}}(\widetilde{\mathcal{M}})= Sol_{\overline{X}^{{\rm an}}}^{\mathrm{E}}(\widetilde{\mathcal{M}}^{{\rm an}}) \in{\mathbf{E}}^{\mathrm{b}}({\rm I}\mathbb{C}_{\overline{X}^{{\rm an}}})\) of \((\widetilde{\mathcal{M}})^{{\rm an}}\) has a normal form at it (see Definitions 1 and 3 and Proposition 5) and set \[\begin{align} r_i \coloneq N_i(0) \;\in\mathbb{Z}_{\geq0}. \end{align}\] We call \(r_i \geq 0\) the regular rank of \((\widetilde{\mathcal{M}})^{{\rm an}}\) at the point \(a_i\in D^{{\rm an}}\). The sections of \(N_i^{>0}:=N_i^{-1}( \mathbb{Z}_{>0} )\subset\mathcal{P}_{S_{a_i}X}^\prime\) are called the exponential factors of \((\widetilde{\mathcal{M}})^{{\rm an}}\) at \(a_i\). Assume that any exponential factor \(f \in N_i^{>0}\) of \((\widetilde{\mathcal{M}})^{{\rm an}}\) at \(a_i\) is a (possibly multi-valued) holomorphic function on a sufficiently small punctured disk \[\begin{align} B(a_i)^\circ \coloneq \Set*{z\in X^{{\rm an}}=\mathbb{C}}{0<\abs*{z-a_i}<\varepsilon} \quad (0< \varepsilon \ll 1) \end{align}\] centered at the point \(a_i\). Then the analytic continuation of an exponential factor along a loop in \(B(a_i)^\circ\) is again an exponential factor, as the multiplicity \(N_i \colon \mathcal{P}_{S_{a_i}X^{{\rm an}}}^\prime \longrightarrow (\mathbb{Z}_{\geq 0})_{S_{a_i}X^{{\rm an}}}\) is defined on the whole circle \(S_{a_i}X^{{\rm an}} \simeq S^1\). This implies that we can set \[\begin{align} \mathrm{CC}_{\mathrm{irr}}(\mathcal{M})_i \coloneq \left\{\sum_{f\in N_i^{>0}}N_i(f)\cdot[\Lambda_i^f]\right\} + r_i\cdot[T_{\{a_i\}}^\ast X^{{\rm an}}] \end{align}\] (see Figure 1 below), where the Lagrangian cycle \(\sum_{f\in N_i^{>0}}N_i(f)\cdot[\Lambda_i^f]\) over the punctured disk \(B(a_i)^\circ\) is defined by gluing the ones defined over some sectors along the point \(a_i\). We call \(\mathrm{CC}_{\mathrm{irr}}(\mathcal{M})_i\) the irregular characteristic cycle of \(\mathcal{M}\) at the point \(a_i\in D^{{\rm an}}\). Also for the point \(a_\infty=\infty\in\widetilde{D}\) we can define a Lagrangian cycle \(\mathrm{CC}_{\mathrm{irr}}(\mathcal{M})_\infty\) in \(T^\ast X^{{\rm an}}\) as follows. Let \[\begin{align} N_\infty = N(a_\infty) \colon \mathcal{P}_{S_{\infty}\overline{X}^{{\rm an}}}^\prime \longrightarrow (\mathbb{Z}_{\geq 0})_{S_{\infty}\overline{X}^{{\rm an}}} \end{align}\] be the multiplicity of the analytic meromorphic connection \((\widetilde{\mathcal{M}}^{{\rm an}})\) at \(a_\infty=\infty\in\overline{X}^{{\rm an}}\). Then by \(N_{\infty}^{>0}:=N_{\infty}^{-1}( \mathbb{Z}_{>0} )\subset \mathcal{P}_{S_{\infty}\overline{X}^{{\rm an}}}^\prime\) we set \[\begin{align} \mathrm{CC}_{\mathrm{irr}}(\mathcal{M})_\infty \coloneq \sum_{f\in N_\infty^{>0}}N_\infty(f)\cdot[\Lambda_\infty^f], \end{align}\] where the complex Lagrangian submanifolds \(\Lambda_\infty^f\subset X^{{\rm an}}\times\mathbb{C}\simeq T^\ast X^{{\rm an}}\) are defined similarly on a sufficiently small punctured disk \(B(a_{\infty})^{\circ}\) in \(\overline{X}^{{\rm an}}=({\mathbb{P}}^1)^{{\rm an}}\) centered at \(a_\infty=\infty\). We thus obtain a Lagrangian cycle \[\begin{align} \mathrm{CC}_{\mathrm{irr}}(\mathcal{M}) \coloneq \mathrm{CC}_{\mathrm{irr}}(\mathcal{M})_\infty + \sum_{i=1}^{l}\mathrm{CC}_{\mathrm{irr}}(\mathcal{M})_i \end{align}\] in \(T^\ast X^{{\rm an}}\) (see Figure 1). We call it the irregular characteristic cycle of the meromorphic connection \(\mathcal{M}\in\mathrm{Mod}_{\mathrm{hol}}(\mathcal{D}_X)\). Now let us consider the symplectic transform \(\chi\colon T^\ast X^{{\rm an}}\overset{\sim}{\longrightarrow}T^\ast Y^{{\rm an}}\) in [@DK18] defined by \[\begin{align} T^\ast X^{{\rm an}}=X^{{\rm an}}\times Y^{{\rm an}}\ni(z,w) \longmapsto (w,-z)\in Y^{{\rm an}}\times X^{{\rm an}}=T^\ast Y^{{\rm an}}. \end{align}\] Let \(\Lambda(\mathcal{M})\subset T^\ast X^{{\rm an}}\) be the support of the irregular characteristic cycle \(\mathrm{CC}_{\mathrm{irr}}(\mathcal{M})\). Set \(b_{\infty}:= \infty\in\overline{Y}^{{\rm an}}\). Then for a sufficiently small punctured disk \[\begin{align} B( b_\infty )^\circ \coloneq \Set*{w\in Y^{{\rm an}}=\mathbb{C}}{\abs{w}>\frac{1}{\varepsilon}} \quad(0<\varepsilon\ll1) \end{align}\] in \(\overline{Y}^{{\rm an}}=({\mathbb{P}}^1)^{{\rm an}}\) centered at \(b_{\infty}= \infty\in\overline{Y}^{{\rm an}}\) the restriction of the projection \[\begin{align} T^\ast X^{{\rm an}}=X^{{\rm an}}\times Y^{{\rm an}} \relbar\joinrel\twoheadrightarrow Y^{{\rm an}} \end{align}\] to \(\Lambda(\mathcal{M})\cap(X^{{\rm an}}\times B( b_\infty )^\circ )\) is an unramified finite covering over \(B( b_\infty )^\circ \subset Y^{{\rm an}}\).

The same is true also for the Lagrangian cycle \(\chi( \mathrm{CC}_{\mathrm{irr}}(\mathcal{M}) )\) in \(T^\ast Y^{{\rm an}} \simeq T^\ast X^{{\rm an}}\). We denote by \(d(\mathcal{M})\in\mathbb{Z}_{\geq0}\) its covering degree over \(B( b_\infty )^\circ \subset Y^{{\rm an}}\). Let \(V\subset B( b_\infty )^\circ\) be a simply connected (open) sector along the point \(\infty \in\overline{Y}^{{\rm an}}\). Since \(\chi( \mathrm{CC}_{\mathrm{irr}}(\mathcal{M}) )\) is a Lagrangian cycle, then there exist (mutually distinct) Puiseux germs \[\begin{align} g_1,g_2,\dots,g_n\in\mathcal{P}^{\prime}_{S_\infty\overline{Y}^{{\rm an}}} \end{align}\] which are holomorphic on \(V\subset B( b_\infty )^\circ\) and positive integers \(d_1, d_2,\dots, d_n>0\) such that we have \[\begin{align} \label{eq:d95ig95i} \chi(\mathrm{CC}_{\mathrm{irr}}(\mathcal{M}))\cap(V \times X^{{\rm an}} ) = \sum_{i=1}^{n}d_i\cdot[\Lambda^{g_i}], \end{align}\tag{1}\] where we set \[\begin{align} \Lambda^{g_i} \coloneq \Set*{( w, g_i^\prime(w))}{w\in V} \subset V \times X^{{\rm an}} \simeq T^\ast V \end{align}\] (see Lemmas 4 and 5). Comparing the covering degrees of the both sides over the sector \(V\subset B( b_\infty )^\circ\), we find \[\begin{align} d(\mathcal{M}) = \sum_{i=1}^{n}d_i. \end{align}\] Note that we can explicitly calculate \(g_i\in\mathcal{P}^{\prime}_{S_\infty \overline{Y}^{{\rm an}}}\) and \(d_i>0\) \((1\leq i\leq n)\) (see (the proof of) Lemma 5). Then our first main result in this paper is the following theorem.

Although as we see in [@DK18] D’Agnolo and Kashiwara avoided some sectors in the proof of their main theorem in [@DK18], one may deduce also a result similar to Theorem 1 from the results in [@DK18] and [@DK23] by relying on the general properties of the enhanced solution complexes to holonomic \(\mathcal{D}\)-modules. The point is that we do not use such an indirect argument in our proof. We have also another expression of the generic rank \({\rm rk} \mathcal{M}^\wedge\) (see Corollary 1). For a generalization of Theorem 1 to arbitrary holonomic \(\mathcal{D}\)-modules on \(X= \mathbb{C}\) see Section 4.4. If an exponential factor \(f \in N_i^{>0}\) at a point \(a_i \in \widetilde{D}\) is holomorphic on a sector \(S\) along it, by abuse of notations we write \(f \in N_i^{>0}(S)\). For such \(f\in N_i^{>0}(S)\) and \(w \in Y^{{\rm an}}= \mathbb{C}\) we define a holomorphic function \(f^w\) on \(S\) by \[\begin{align} \label{funcfw} f^w(z) \coloneq zw-f(z) \quad \left(z\in S \right). \end{align}\tag{2}\] Then to prove Theorem 1, for \(w \in V\) we consider the real-valued functions \(\operatorname{Re}f^w\colon S \longrightarrow\mathbb{R}\; (f\in N_i^{>0}(S))\) as Morse functions and apply a Morse theory associated to them. In contrast to the usual Morse theory, that we use here relies on the sublevel sets of “several" Morse functions. See Section 4.2 for the details.

We obtain similar results also for the other singular points of \(\mathcal{M}^{\wedge}\) as follows. For a point \(b\in Y^{{\rm an}}=\mathbb{C}\) we set \[N_{\infty,b}^{>0}\coloneq\{f\in N_\infty^{>0}\mid f(z)= bz+(\textit{lower order terms})\} \quad \subset N_\infty^{>0}.\] Then for the sufficiently small punctured disk \(B(a_\infty)^\circ\subset X^{{\rm an}}\) centered at the point \(a_\infty=\infty\in\overline{X}^{{\rm an}}\), we define a Lagrangian cycle \(\mathrm{CC}_{\mathrm{irr}}(\mathcal{M})^b\) in \(T^\ast B(a_\infty)^\circ\simeq B(a_\infty)^\circ\times Y^{{\rm an}}\subset T^\ast X^{{\rm an}}\) by \[\mathrm{CC}_{\mathrm{irr}}(\mathcal{M})^b\coloneq\sum_{f\in N_{\infty,b}^{>0}}N_\infty(f)\cdot[\Lambda_\infty^f].\] Note that \(\mathrm{CC}_{\mathrm{irr}}(\mathcal{M})^b\) is a part of \(\mathrm{CC}_{\mathrm{irr}}(\mathcal{M})_{\infty}\) which does not affect the covering degree \(d(\mathcal{M})\) in Theorem 1. Let \(\Lambda(\mathcal{M})^b\subset T^\ast X^{{\rm an}}\) be the support of \(\mathrm{CC}_{\mathrm{irr}}(\mathcal{M})^b\). Then for a sufficiently small punctured disk \(B(b)^\circ\in Y^{{\rm an}}\) centered at the point \(b\in Y^{{\rm an}}\) the restriction of the projection \[T^\ast X^{{\rm an}}=X^{{\rm an}}\times Y^{{\rm an}}\relbar\joinrel\twoheadrightarrow Y^{{\rm an}}\] to \(\Lambda(\mathcal{M})^b\cap(X^{{\rm an}} \times B(b)^\circ)\) is an unramified finite covering over \(B(b)^\circ\subset Y^{{\rm an}}\). The same is true also for the Lagrangian cycle \(\chi( \mathrm{CC}_{\mathrm{irr}}(\mathcal{M})^b )\) in \(T^\ast Y^{{\rm an}} \simeq T^\ast X^{{\rm an}}\). We denote by \(d(\mathcal{M})^b\in\mathbb{Z}_{\geq0}\) its covering degree over \(B(b)^\circ\subset Y^{{\rm an}}\). Let \(W\subset B(b)^\circ\) be a simply connected (open) sector along the point \(b\in Y^{{\rm an}}\). Since \(\chi( \mathrm{CC}_{\mathrm{irr}}(\mathcal{M})^b )\) is a Lagrangian cycle in \(T^\ast Y^{{\rm an}}\), then there exist (mutually distinct) Puiseux germs \[h_1,h_2,\dots,h_m\in\mathcal{P}^{\prime}_{S_bY^{{\rm an}}}\] which are holomorphic on \(W\subset B(b)^\circ\) and positive integers \(e_1,e_2,\dots,e_m >0\) such that \[\label{eq:e95ih95i} \chi(\mathrm{CC}_{\mathrm{irr}}(\mathcal{M})^b)\cap(W \times X^{{\rm an}} )=\sum_{i=1}^{m}e_i\cdot[\Lambda^{h_i}]\tag{3}\] (see Lemmas 4 and 5). Note that we have \(d(\mathcal{M})^b=\sum_{i=1}^{m}e_i\). Then, as in the proof of Theorem 1, we obtain the following result.

For a generalization of Theorem 2 to arbitrary holonomic \(\mathcal{D}\)-modules on \(X= \mathbb{C}\) see Section 4.4. In the situation of Theorem 2, note that for the irregularity \(\mathrm{irr}_b(\Gamma_{Y\setminus\{b\}}(\mathcal{M}^\wedge)) \in \mathbb{Z}_{\geq 0}\) of the meromorphic connection \(\Gamma_{Y\setminus\{b\}}(\mathcal{M}^\wedge)\) at the point \(b\in Y\) we have \[\mathrm{irr}_b(\Gamma_{Y\setminus\{b\}}(\mathcal{M}^\wedge)) =\sum_{i=1}^m e_i \cdot \operatorname{ord}_b(h_i).\] Finally, we prove the following formula from which we obtain the (classical) characteristic cycle \({\rm CC} ( \mathcal{M}^\wedge )\) of the Fourier transform \(\mathcal{M}^\wedge\). For the point \(b \in Y\) we denote the multiplicity of \(\mathcal{M}^\wedge\) along \(T_b^\ast Y\subset T^\ast Y\) by \(\mathop{\mathrm{mult}}_{T_b^\ast Y}(\mathcal{M}^\wedge) \in \mathbb{Z}_{\geq 0}\).

Since the Fourier transform is an exact functor, Theorem 3 holds true also for any holonomic \(\mathcal{D}\)-module on \(X= \mathbb{C}_z\) (see the discussions at the end of Section 4.1).

In Section 5, we will apply a similar Morse theoretical method also to obtain a natural basis of the stalk \(Sol_Y( \mathcal{M}^\wedge )_b\) of the solution complex \(Sol_Y( \mathcal{M}^\wedge )\) to \(\mathcal{M}^\wedge\) at a generic point \(b \in Y^{{\rm an}}\) of \(Y^{{\rm an}}\). Specifically, we identify \(Sol_Y( \mathcal{M}^\wedge )_b\) with some rapid decay homology group in the sense of Bloch-Esnault [@BE04a] and Hien [@Hi07], [@Hi09] and construct a basis of the latter. This construction is applicable to any algebraic meromorphic connection \(\mathcal{M}\) on \(X= \mathbb{C}\). Then for such \(\mathcal{M}\) it would be possible to calculate the monodromies and the Stokes matrices of the holomorphic solutions to \(\mathcal{M}^\wedge\) explicitly by looking at how the rapid decay \(1\)-cycles in the basis deform as the point \(b \in Y^{{\rm an}}\) moves.

2 Preliminaries↩︎

In this section, we recall some basic notions and results which will be used in this paper. We assume here that the reader is familiar with the theory of sheaves and functors in the framework of derived categories. For them we follow the terminologies in [@KS90] etc. For a topological space \(X\) denote by \({\mathbf{D}}^{\mathrm{b}}(\mathbb{C}_X)\) the derived category consisting of bounded complexes of sheaves of \(\mathbb{C}\)-vector spaces on it.

2.1 Enhanced sheaves↩︎

We refer to [@Tam18], [@KS16b], and [@DK16] for the details of this subsection. Let \(X\) be a complex manifold and we consider the maps \[\begin{align} X\times\mathbb{R}^2 \xrightarrow{p_1,p_2,\mu} X\times\mathbb{R}\overset{\pi}{\longrightarrow} X, \end{align}\] where \(p_1,p_2,\pi\) are the projections and we set \(\mu(x,t_1,t_2)\coloneq(x,t_1+t_2)\). Then we define the bounded derived category of enhanced sheaves \({\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_X)\) on \(X\) by \[\begin{align} {\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_X) \coloneq {\mathbf{D}}^{\mathrm{b}}(\mathbb{C}_{X\times\mathbb{R}})/\pi^{-1}{\mathbf{D}}^{\mathrm{b}}(\mathbb{C}_X). \end{align}\] The convolution functors \(\overset{+}{\otimes}\) and \({\rm R}{\mathcal{H}}om^+\) in \({\mathbf{D}}^{\mathrm{b}}(\mathbb{C}_{X\times\mathbb{R}})\) are defined by \[\begin{align} F\overset{+}{\otimes}G &\coloneq {\mathrm{R}}\mu_!(p_1^{-1}F\otimes p_2^{-1}G), \\ {\rm R}{\mathcal{H}}om^+(F,G) &\coloneq {\mathrm{R}}p_{1\ast}{\rm R}{\mathcal{H}}om(p_2^{-1}F,\mu^!G), \end{align}\] and they induce convolution functors in \({\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_X)\), which we denote also by \(\overset{+}{\otimes}\) and \({\rm R}{\mathcal{H}}om^+\), respectively. The quotient functor \[\begin{align} \mathbf{Q}\colon {\mathbf{D}}^{\mathrm{b}}(\mathbb{C}_{X\times\mathbb{R}})\longrightarrow{\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_X) \end{align}\] has fully faithful left and right adjoints \(\mathbf{L}^{\mathrm{E}},\mathbf{R}^{\mathrm{E}}\) defined by \[\begin{align} \mathbf{L}^{\mathrm{E}}(\mathbf{Q}F) &\coloneq (\mathbb{C}_{\{t\geq0\}}\oplus\mathbb{C}_{\{t\leq0\}})\overset{+}{\otimes}F, \\ \mathbf{R}^{\mathrm{E}}(\mathbf{Q}F) &\coloneq {\rm R}{\mathcal{H}}om^+(\mathbb{C}_{\{t\geq0\}}\oplus\mathbb{C}_{\{t\leq0\}},F), \end{align}\] where \(\{t\geq0\}\) stands for \(\Set*{\left(x,t\right)\in X\times\mathbb{R}}{t\geq0}\) and \(\{t\leq0\}\) is defined similarly. In this paper, we sometimes identify a sheaf \(F\) on \(X\times\mathbb{R}\) with the enhanced sheaf \(\mathbf{Q}F\) on \(X\) associated to it. The functor \(\mathbf{L}^{\mathrm{E}}\colon{\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_X)\longrightarrow {\mathbf{D}}^{\mathrm{b}}(\mathbb{C}_{X\times\mathbb{R}})\) induces a \(t\)-structure of \({\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_X)\) by the standard \(t\)-structure of \({\mathbf{D}}^{\mathrm{b}}(\mathbb{C}_{X\times\mathbb{R}})\). We denote by \(\mathbf{E}^0(\mathbb{C}_X)\) its heart. For a morphism of complex manifolds \(f\colon X\rightarrow Y\), we define the direct image functor \(\mathbf{E}f_\ast\colon {\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_X)\rightarrow{\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_Y)\) so that the following diagram commutes: \[\vcenter{ \xymatrix@M=7pt@C=36pt@R=24pt{ {\mathbf{D}}^{\mathrm{b}}(\mathbb{C}_{X\times\mathbb{R}}) \ar[r]^-{{\mathrm{R}}(f\times{\rm id}_{\mathbb{R}})_\ast} \ar[d]^-{\mathbf{Q}} & {\mathbf{D}}^{\mathrm{b}}(\mathbb{C}_{Y\times\mathbb{R}}) \ar[d]^-{\mathbf{Q}} & \\ {\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_X) \ar[r]^-{\mathbf{E}f_\ast} & {\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_Y). & }}\] The proper direct image and (proper) inverse image functors \[\begin{align} \mathbf{E}f_! &\colon {\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_X)\longrightarrow{\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_Y), \\ \mathbf{E}f^{-1},\mathbf{E}f^! &\colon {\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_Y)\longrightarrow{\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_X) \end{align}\] are defined similarly. We have a natural embedding \(\epsilon\colon{\mathbf{D}}^{\mathrm{b}}(\mathbb{C}_X)\rightarrow{\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_X)\) defined by \[\begin{align} \epsilon(L) \coloneq \mathbf{Q}(\mathbb{C}_{\{t\geq0\}}\otimes\pi^{-1}L), \end{align}\] and a bifunctor \(\pi^{-1}(\cdot)\otimes(\cdot)\colon {\mathbf{D}}^{\mathrm{b}}(\mathbb{C}_X)\times{\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_X)\rightarrow{\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_X)\) defined by \[\begin{align} \pi^{-1}L\otimes F \coloneq \mathbf{Q}(\pi^{-1}L\otimes\mathbf{L}^{\mathrm{E}}(F)). \end{align}\]

2.2 Enhanced ind-sheaves↩︎

We briefly recall some notions and results on enhanced ind-sheaves without giving detailed definitions. We refer to [@KS01] and [@KS06] for ind-sheaves, to [@DK16] for ind-sheaves on bordered spaces, and to [@DK16] and [@KS16b] for enhanced ind-sheaves. Let \(X\) be a complex manifold and \(\mathbb{R}_\infty\) the bordered space \((\mathbb{R}, \overline{\mathbb{R}}\coloneq\mathbb{R}\sqcup\{\pm\infty\})\). Denote by \({\mathbf{D}}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_X)\) and \({\mathbf{D}}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_{X\times\mathbb{R}_\infty})\) the bounded derived categories of ind-sheaves on \(X\) and ind-sheaves on the bordered space \(X\times\mathbb{R}_\infty\), respectively. We define the the bounded derived category of enhanced ind-sheaves \({\mathbf{E}}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_X)\) on \(X\) by \[{\mathbf{E}}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_X) \coloneq {\mathbf{D}}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_{X\times\mathbb{R}_\infty})/\pi^{-1}{\mathbf{D}}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_X),\] where \(\pi\colon X\times\mathbb{R}_\infty\rightarrow X\) is the projection of bordered spaces. The quotient functor \[\begin{align} \mathbf{Q}\colon {\mathbf{D}}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_{X\times\mathbb{R}_\infty}) \longrightarrow{\mathbf{E}}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_X) \end{align}\] has fully faithful left and right adjoints \[\begin{align} \mathbf{L}^{\mathrm{E}},\mathbf{R}^{\mathrm{E}}\colon {\mathbf{E}}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_X)\longrightarrow {\mathbf{D}}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_{X\times\mathbb{R}_\infty}). \end{align}\] A \(t\)-structure of \({\mathbf{E}}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_X)\) is induced by the standard \(t\)-structure of \({\mathbf{D}}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_{X\times\mathbb{R}_\infty})\) and the functor \(\mathbf{L}^{\mathrm{E}}\) as in \({\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_X)\). Denote by \(\mathbf{E}^0(\mathrm{I}\mathbb{C}_X)\) its heart. Furthermore, as in the case of enhanced sheaves we can define the convolution functors \[\begin{align} \overset{+}{\otimes}&\colon {\mathbf{E}}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_X)\times{\mathbf{E}}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_X) \longrightarrow{\mathbf{E}}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_X), \\ {\rm R}{\mathcal{I}}hom^+&\colon {\mathbf{E}}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_X)^{\scriptsize op}\times{\mathbf{E}}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_X) \longrightarrow{\mathbf{E}}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_X), \end{align}\] and the operations of (proper) direct and inverse images \[\begin{align} \mathbf{E}f_\ast,\mathbf{E}f_{!!} &\colon {\mathbf{E}}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_X) \longrightarrow{\mathbf{E}}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_Y), \\ \mathbf{E}f^{-1},\mathbf{E}f^! &\colon {\mathbf{E}}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_Y) \longrightarrow{\mathbf{E}}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_X), \end{align}\] for a morphism of complex manifolds \(f\colon X\rightarrow Y\). We have a natural embedding \(\varepsilon\colon{\mathbf{D}}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_X) \rightarrow{\mathbf{E}}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_X)\) and a bifunctor \(\pi^{-1}(\cdot)\otimes(\cdot) \colon{\mathbf{D}}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_X)\times{\mathbf{E}}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_X) \rightarrow{\mathbf{E}}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_X)\) defined by \[\begin{align} \varepsilon(\mathcal{L}) \coloneq \mathbf{Q}(\mathbb{C}_{\{t\geq0\}}\otimes\pi^{-1}\mathcal{L}), \\ \pi^{-1}\mathcal{L}\otimes\mathcal{F}\coloneq \mathbf{Q}(\pi^{-1}\mathcal{L}\otimes\mathbf{L}^{\mathrm{E}}(\mathcal{F})). \end{align}\] We set \(\mathbb{C}_X^{\mathrm{E}}\coloneq\mathbf{Q}\Bigl(\underset{\alpha \to+\infty} {``\varinjlim"}\mathbb{C}_{\{t\geq \alpha \}}\Bigr)\in{\mathbf{E}}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_X)\), where the symbol \(``\varinjlim"\) stands for the inductive limit in \(\mathrm{I}\mathbb{C}_{X\times\overline{\mathbb{R}}}\).

2.3 Exponential enhanced (ind-)sheaves↩︎

Let \(X\) be a complex manifold. Denote by \(\mathbf{E}_{\mathbb{R}{\text{-}}\mathrm c}^{\mathrm{b}}(\mathbb{C}_X)\) and \(\mathbf{E}_{\mathbb{R}{\text{-}}\mathrm c}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_X)\) the triangulated categories of \(\mathbb{R}\)-constructible enhanced (ind-)sheaves on \(X\) (see [@DK16]). Let \(U\subset X\) be an open subset and \(\phi,\phi^+,\phi^-\colon U\to\mathbb{R}\) continuous functions such that \(\phi^+\geq\phi^-\) on it. For a locally closed subset \(Z\subset U\), we define the exponential enhanced (ind-)sheaves \(\mathsf{E}_{Z\left.\right|X}^\phi, \mathsf{E}_{Z\left.\right|X}^{\phi^+\vartriangleright\phi^-}\in\mathbf{E}^0(\mathbb{C}_X)\) and \(\mathbb{E}_{Z\left.\right|X}^\phi,\mathbb{E}_{Z\left.\right|X} ^{\phi^+\vartriangleright\phi^-}\in\mathbf{E}^0(\mathrm{I}\mathbb{C}_X)\) by \[\begin{align} \mathsf{E}_{Z\left.\right|X}^\phi &\coloneq \mathbf{Q}(\mathbb{C}_{\{t+\phi\geq0\}}), \\ \mathsf{E}_{Z\left.\right|X}^{\phi^+\vartriangleright\phi^-} &\coloneq \mathbf{Q}(\mathbb{C}_{\{-\phi^+\leq t<-\phi^-\}}), \\ \mathbb{E}_{Z\left.\right|X}^\phi &\coloneq \mathbb{C}_X^{\mathrm{E}}\overset{+}{\otimes}\mathsf{E}_{Z\left.\right|X}^\phi, \\ \mathbb{E}_{Z\left.\right|X}^{\phi^+\vartriangleright\phi^-} &\coloneq \mathbb{C}_X^{\mathrm{E}}\overset{+}{\otimes}\mathsf{E}_{Z\left.\right|X} ^{\phi^+\vartriangleright\phi^-}, \end{align}\] where \(\{t+\phi\geq0\}\) and \(\{-\phi^+\leq t<-\phi^-\}\) stand for \(\Set*{(x,t)\in X\times\mathbb{R}}{x \in Z, t+\phi(x)\geq0}\) and \(\Set*{(x,t)\in X\times\mathbb{R}}{x \in Z, -\phi^+(x)\leq t<-\phi^-(x)}\), respectively. Note that we have exact sequences \[\begin{align} 0 \longrightarrow \mathsf{E}_{Z\left.\right|X}^{\phi^+\vartriangleright\phi^-} \longrightarrow \mathsf{E}_{Z\left.\right|X}^{\phi^+} \longrightarrow \mathsf{E}_{Z\left.\right|X}^{\phi^-} \longrightarrow 0, \\ 0 \longrightarrow \mathbb{E}_{Z\left.\right|X}^{\phi^+\vartriangleright\phi^-} \longrightarrow \mathbb{E}_{Z\left.\right|X}^{\phi^+} \longrightarrow \mathbb{E}_{Z\left.\right|X}^{\phi^-} \longrightarrow 0 \end{align}\] in \(\mathbf{E}^0(\mathbb{C}_X)\) and \(\mathbf{E}^0(\mathrm{I}\mathbb{C}_X)\), respectively. Note also that if \(Z\) is subanalytic and \(f\colon U\to\mathbb{C}\) is holomorphic then we have \(\mathsf{E}_{Z\left.\right|X}^{\operatorname{Re}f}\in\mathbf{E}_{\mathbb{R}{\text{-}}\mathrm c}^0(\mathbb{C}_X)\) and \(\mathbb{E}_{Z\left.\right|X}^{\operatorname{Re}f}\in\mathbf{E}_{\mathbb{R}{\text{-}}\mathrm c}^0(\mathrm{I}\mathbb{C}_X)\). For these exponential enhanced ind-sheaves we have the following basic property (see [@DK18]).

2.4 \(\mathcal{D}\)-modules↩︎

Let us recall some notions and results on \(\mathcal{D}\)-modules on a complex manifold \(X\) (we refer to [@Kas03] and [@HTT08] etc.). Denote by \(\mathcal{O}_X,\Omega_X\) and \(\mathcal{D}_X\) the sheaves of holomorphic functions, holomorphic differential forms of top degree and holomorphic differential operators on \(X\), respectively. Let \(\mathrm{Mod}(\mathcal{D}_X)\) be the abelian category of left \(\mathcal{D}_X\)-modules. Then we can define \(\mathrm{Mod}_{\scriptsize coh}(\mathcal{D}_X)\) (resp. \(\mathrm{Mod}_{\mathrm{hol}}(\mathcal{D}_X)\)) to be the subcategory of \(\mathrm{Mod}(\mathcal{D}_X)\) consisting of coherent (resp. holonomic) \(\mathcal{D}_X\)-modules. We write \({\mathbf{D}}^{\mathrm{b}}(\mathcal{D}_X)\) for the bounded derived category of left \(\mathcal{D}_X\)-modules and denote by \({\mathbf{D}}^{\mathrm{b}}_{\scriptsize coh}(\mathcal{D}_X)\) and \({\mathbf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathcal{D}_X)\) its full triangulated subcategories of objects which have coherent and holonomic cohomologies, respectively. The symbols \(\overset{D}{\otimes},\mathbf{D}f_\ast,\mathbf{D}f^\ast\) stand for the standard operations for \(\mathcal{D}\)-modules associated to a morphism of complex manifolds \(f\colon X\to Y\). The (classical) solution functor is defined by \[\begin{align} Sol_X \colon {\mathbf{D}}^{\mathrm{b}}_{\scriptsize coh}(\mathcal{D}_X)^{\scriptsize op}\longrightarrow {\mathbf{D}}^{\mathrm{b}}(\mathbb{C}_X), \quad \mathcal{M}\longmapsto{\rm R}{\mathcal{H}}om_{\mathcal{D}_X}(\mathcal{M},\mathcal{O}_X). \end{align}\] Let \(D\subset X\) be a closed hypersurface and denote by \(\mathcal{O}(\ast D)\) the sheaf of meromorphic functions on \(X\) with poles in \(D\). Then for \(\mathcal{M}\in{\mathbf{D}}^{\mathrm{b}}(\mathcal{D}_X)\), we set \[\begin{align} \mathcal{M}(\ast D)\coloneq\mathcal{M}\overset{D}{\otimes}\mathcal{O}_X(\ast D) \end{align}\] and for \(f\in\mathcal{O}_X(\ast D)\) and \(U\coloneq X\left.\right\backslash D\), set \[\begin{align} \mathcal{D}_X e^f &\coloneq \mathcal{D}_X / \Set{P\in\mathcal{D}_X}{P e^f\left.\right|_U=0}, \\ \mathcal{E}_{U\left.\right|X}^f &\coloneq \mathcal{D}_X e^f(\ast D). \end{align}\] Note that \(\mathcal{E}_{U\left.\right|X}^f\) is a holonomic \(\mathcal{D}_X\)-module. In [@DK16], the authors constructed the enhanced solution functor on a complex manifold \(X\) \[\begin{align} Sol_X^{\mathrm{E}}\colon{\mathbf{D}}^{\mathrm{b}}_{\mathrm{hol}}(\mathcal{D}_X)^{\scriptsize op} \longrightarrow\mathbf{E}_{\mathbb{R}{\text{-}}\mathrm c}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_X) \end{align}\] and proved that it is fully faithful. Instead of giving its definition, we recall some of its properties.

2.5 Puiseux germs and normal forms of enhanced (ind-)sheaves↩︎

In this subsection, we recall some definitions on Puiseux germs and normal forms of enhanced (ind-)sheaves in [@DK18] to describe the Hukuhara-Levelt-Turrittin theorem in terms of enhanced ind-sheaves. Let \(X\) be a complex manifold of dimension one. For \(a\in X\), denote by \(\varpi_a\colon \widetilde{X_a}\to X\) the real oriented blow-up of \(X\) along \(a\) and consider the commutative diagram \[\vcenter{ \xymatrix@M=5pt{ S_aX \ar[r]^-{\widetilde{\imath}_a} & \widetilde{X_a} \ar[d]^-{\varpi_a}& \\ X\left.\right\backslash\{a\} \ar[r]^-{j_a} \ar[ur]^-{\widetilde{\jmath}_a} & X, }}\] where we set \(S_aX\coloneq\varpi_a^{-1}(a)\simeq S^1\) and \(\widetilde{\imath}_a,\widetilde{\jmath}_a,j_a\) are the natural embeddings. We denote by \(\mathcal{P}_{\widetilde{X_a}}\) the subsheaf of \(\widetilde{\jmath}_{a\ast} j_a^{-1}\mathcal{O}_X\) whose sections are defined by \[\begin{gather} \Gamma(\Omega;\mathcal{P}_{\widetilde{X_a}}) \coloneq \{f\in\Gamma(\Omega;\widetilde{\jmath}_{a\ast} j_a^{-1}\mathcal{O}_X)\mid \textit{For any \theta\in\Omega\cap S_aX,} \\ \textit{f admits a Puiseux expansion at \theta.}\} \end{gather}\] for open subsets \(\Omega\subset\widetilde{X_a}\). Then we define the sheaf of Puiseux germs \(\mathcal{P}_{S_aX}\) on \(S_aX\) to be \[\begin{align} \mathcal{P}_{S_aX}\coloneq\widetilde{\imath}_a^{\,-1}\mathcal{P}_{\widetilde{X_a}}. \end{align}\] For a rational number \(\lambda\in\mathbb{Q}\), denote by \(\mathcal{P}_{S_aX}^{\lambda}, \mathcal{P}_{S_aX}^{\leq\lambda}\subset\mathcal{P}_{S_aX}\) the subsheaf of \(\mathcal{P}_{S_aX}\) consisting of sections \(f\) whose pole order \(\operatorname{ord}_a(f)\) at \(a\) is \(\lambda\) and \(\leq\lambda\), respectively. For an interval \(I\subset\mathbb{R}\), we set \(\mathcal{P}_{S_aX}^I\coloneq \bigsqcup_{\lambda\in I\cap\mathbb{Q}}\mathcal{P}_{S_aX}^\lambda\). Let \(z_a\) be a local coordinate centered at \(a\) and denote by \(\mathcal{P}_{S_aX}^\prime\) the subsheaf of \(\mathcal{P}_{S_aX}\) consisting of sections locally contained in \[\begin{align} \bigcup_{p\in\mathbb{Z}_{\geq1}}z_a^ {-\frac{1}{p}}\mathbb{C}[z_a^{-\frac{1}{p}}] \end{align}\] for some (hence, any) branch of \(z_a^{1/p}\). Note that there exists a canonical isomorphism \(\mathcal{P}_{S_aX}^\prime\overset{\sim}{\longrightarrow}\mathcal{P}_{S_aX}/\mathcal{P}_{S_aX}^{\leq0}\).

Let \(\mathcal{M}\) be a holonomic \(\mathcal{D}_X\)-module. For \(a\in X\) the enhanced solution complex \(Sol_X^{{\mathrm{E}}}(\mathcal{M})\) has the following decomposition by some exponential enhanced ind-sheaves on sufficiently small open sectors along \(a\).

In [@DK18], D’Agnolo and Kashiwara introduced some notions on normal forms of enhanced sheaves to refine the above decomposition. We recall their definitions in a slightly modified form. In particular, as we see in Definition 1 below, our multiplicities are defined on \(\mathcal{P}_{S_aX}^\prime\) and not on \(\mathcal{P}_{S_aX}\) as in [@DK18]. In this way, we can eliminate the ambiguities of exponential factors.

Note that if \(F\in\mathbf{E}_{\mathbb{R}{\text{-}}\mathrm c}^{\mathrm{b}}(\mathbb{C}_X)\) (resp. \(\mathcal{F}\in\mathbf{E}_{\mathbb{R}{\text{-}}\mathrm c}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_X)\)) has a normal form at \(a\in X\), then the multiplicity \(N\) for which \(F\) (resp. \(\mathcal{F}\)) has a normal form at \(a\) is uniquely determined by the following lemma.

By Lemma 2 and Lemma 3, we obtain the following proposition.

We shall explain some classical notions on meromorphic connections in terms of enhanced solution complexes. Let \(\mathcal{M}\) be a holonomic \(\mathcal{D}_X\)-module and \(N\) the multiplicity for which \(Sol_X^{{\mathrm{E}}}(\mathcal{M})\) has a normal form at \(a\in X\). We call the sections of \(N^{>0}:=N^{-1}( \mathbb{Z}_{>0} )_{S_aX} \subset\mathcal{P}_{S_aX}^\prime\) the exponential factors of \(\mathcal{M}\) at \(a\). Then obviously the analytic continuation of an exponential factor of \(\mathcal{M}\) along the loop \(S_aX \simeq S^1\) is again an exponential factor. The regular rank \(r_a\in\mathbb{Z}_{\geq0}\) of \(\mathcal{M}\) at \(a\in X\) is defined by \[r_a\coloneq N(0) \;\in\mathbb{Z}_{\geq0}.\] For two exponential factors \(f_1,f_2\in N^{>0}\) such that \(f_1 \not= f_2\) which are holomorphic on a sector \(S\) along the point \(a \in X\), the Stokes curves of the pair \((f_1,f_2)\) (over \(S\)) are the irreducible components of the real analytic set \[\Set*{z\in S }{\operatorname{Re}f_1(z)-\operatorname{Re}f_2(z)=0} \quad \subset S\] and a ray tangent to a Stokes curve at \(a\) is called a Stokes line of \((f_1,f_2)\). Denote by \(\mathrm{St}_a(f_1,f_2)\) the set of the Stokes lines of the pair \((f_1,f_2)\) (over \(S\)). Then by taking a union over all the sectors \(S\) along \(a \in X\) and the exponential factors \(f_1 \not= f_2\) on them, we set \[\mathrm{St}_a(\mathcal{M})\coloneq\bigcup_{f_1,f_2\in N^{>0}, \;f_1 \not= f_2} \mathrm{St}_a(f_1,f_2).\] We call the elements of \(\mathrm{St}_a(\mathcal{M})\) the Stokes lines of \(\mathcal{M}\) at \(a\in X\).

For \(\mathcal{M}\in\mathrm{Mod}_{\mathrm{hol}}(\mathcal{D}_X)\) and \(a\in X\) such that \(\mathcal{M}(\ast a)\simeq\mathcal{M}\) we say that \(\mathcal{M}\) is a meromorphic connection along \(a\in X\). In this case, if \(Sol_X^{\mathrm{E}}(\mathcal{M})\) has a normal form at \(a\in X\) for a multiplicity \(N\colon\mathcal{P}_{S_aX}^\prime\to (\mathbb{Z}_{\geq0})_{S_aX}\), then for any \(\theta\in S_aX\) the irregularity \(\mathrm{irr}_a(\mathcal{M}) \in \mathbb{Z}_{\geq 0}\) of the meromorphic connection \(\mathcal{M}\) is equal to \[\sum_{f\in N^{>0}_{\theta}} N_{\theta}(f) \cdot \operatorname{ord}_a(f)\] (see [@Sab93] for an excellent review on this notion). By the proofs of Proposition 6 and [@IT20a] we obtain a purely topological proof of the following classical result (see [@IT20a]). For the meromorphic connection \(\mathcal{M}\) along \(a\in X\) we set \[\chi_a(Sol_X(\mathcal{M})) \coloneq \sum_{j\in\mathbb{Z}}(-1)^j {\rm dim}_\mathbb{C}H^j Sol_X(\mathcal{M})_a\] and call it the local Euler-Poincaré index of \(Sol_X(\mathcal{M})\) at the point \(a\in X\).

2.6 Legendre transform for Puiseux germs↩︎

Let \(X=\mathbb{C}_z\) be the complex affine line and \(Y=\mathbb{C}_w\) its dual. We denote by \(\overline{X}\simeq{\mathbb{P}}^1\) (resp. \(\overline{Y}\simeq{\mathbb{P}}^1\)) the projective compactification of \(X\) (resp. \(Y\)) and \(X^{\rm an}\) (resp. \(Y^{\rm an}\), \(\overline{X}^{\rm an}\) and \(\overline{Y}^{\rm an}\)) the underlying complex manifold of \(X\) (resp. \(Y\), \(\overline{X}\) and \(\overline{Y}\)). We define the symplectic transform \(\chi\colon T^\ast X^{\rm an} \overset{\sim}{\longrightarrow}T^\ast Y^{\rm an}\) by \[T^\ast X^{\rm an}= X^{\rm an}\times Y^{\rm an}\ni(z,w)\longmapsto (w,-z)\in Y^{\rm an}\times X^{\rm an}=T^\ast Y^{\rm an}.\] In this subsection, we recall an important correspondence between the Puiseux germs on \(X^{\rm an}\) and \(Y^{\rm an}\). Let \(a\in X^{\rm an}\) be a point in \(X^{\rm an}\), \(W\subset X^{\rm an}\setminus\{a\}\) an (open) sector along \(a\) and \(f\colon W\to\mathbb{C}\) a holomorphic function on \(W\). Then we define a Lagrangian submanifold \(\Lambda_a^f\) of \(T^\ast X^{\rm an}\) by \[\Lambda_a^f\coloneq\{(z,f^\prime(z))\mid z\in W\} \quad \subset T^\ast X^{\rm an}.\] Let \(g\colon V\rightarrow\mathbb{C}\) be a holomorphic function on an (open) sector \(V\subset Y^{\rm an}\) along the point \(\infty\in \overline{Y}^{\rm an}\) and set \(\Lambda^g\coloneq\{(w,g^\prime(w))\mid w\in V\}\subset T^\ast Y^{\rm an}\simeq Y^{\rm an}\times X^{\rm an}\). Then we have the following result due to D’Agnolo-Kashiwara (see the proof of [@DK18]). For the convenience of the readers, we recall also their proof.

We call the correspondence between the holomorphic functions \(f\) and \(g\) in Lemma 4 (ii) a Legendre transform. Also for holomorphic functions defined on sectors along the point \(\infty\in\overline{X}^{\rm an}\) we have a similar correspondence and call it a Legendre transform. In [@DK18], D’Agnolo and Kashiwara refined this result for Puiseux germs as follows. For a point \(a\in\overline{X}^{{\rm an}}\), we define the étalé space \(\mathrm{\acute{e}t}(\mathcal{P}_{S_a\overline{X}^{{\rm an}}})\) endowed with the natural topology by \[\mathrm{\acute{e}t}(\mathcal{P}_{S_a\overline{X}^{{\rm an}}})\coloneq \bigsqcup_{\theta\in S_a\overline{X}^{{\rm an}}} \mathcal{P}_{S_a\overline{X}^{{\rm an}},\theta}.\]

We call the correspondence in Lemma 5 the Legendre transform for Puiseux germs.

3 Enhanced solution complexes of holonomic \(\mathcal{D}\)-modules in dimension one↩︎

In this section, we focus our attention on holonomic \(\mathcal{D}\)-modules in dimension one and give an explicit description of their enhanced solution complexes, which will be effectively used in subsequent sections. Let \(X\) be a closed Riemann surface i.e. a compact complex manifold of dimension one and \(\mathcal{M}\) an analytic holonomic \(\mathcal{D}\)-module on it. For simplicity, we denote by \(D\subset X\) the singular support \(\mathrm{sing.supp}(\mathcal{M})\) of \(\mathcal{M}\) and set \(\mathcal{N}\coloneq\mathcal{M}(\ast D)\in\mathrm{Mod}_{\mathrm{hol}}(\mathcal{D}_X)\). The complex curve \(X\) being compact, \(D\subset X\) is a finite subset of \(X\). Our objective here is to give a global and explicit description of the enhanced solution complex \(Sol_X^{{\mathrm{E}}}(\mathcal{N})\in \mathbf{E}_{\mathbb{R}{\text{-}}\mathrm c}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_X)\) of the meromorphic connection \(\mathcal{N}\). Recall that by Proposition 6 for each point \(a\in D\) there exist a multiplicity \[\begin{align} N(a)\colon\mathcal{P}_{S_aX}^{\prime} \longrightarrow \left(\mathbb{Z}_{\geq 0}\right)_{S_aX} \end{align}\] and an enhanced sheaf \(F(a)\in \mathbf{E}_{\mathbb{R}{\text{-}}\mathrm c}^0(\mathbb{C}_X)\) with normal form at \(a\in D\) for the multiplicity \(N(a)\) such that for a sufficiently small closed disk \(D(a)\subset X\) centered at \(a\in D\) there exists an isomorphism \[\begin{align} \pi^{-1}\mathbb{C}_{D(a) \setminus \{ a \}} \otimes Sol_X^{{\mathrm{E}}}(\mathcal{N}) \simeq \mathbb{C}_X^{{\mathrm{E}}}\overset{+}{\otimes}F(a). \end{align}\] Then by Proposition 4 (i) we obtain an isomorphism \[\begin{align} \pi^{-1}\mathbb{C}_{D(a)} \otimes Sol_X^{{\mathrm{E}}}(\mathcal{N}) \simeq \mathbb{C}_X^{{\mathrm{E}}}\overset{+}{\otimes}F(a). \end{align}\] By Remark 7 we may assume that there exist open sectors \(V_1,V_2,\dots,V_d\subset X\) along \(a\in D\) placed in the counter-clockwise direction such that \[\begin{align} D(a)^{\circ}:= D(a)\left.\right\backslash\{a\} \subset V_1\cup V_2\cup\dots\cup V_d \end{align}\] and for \(1\leq j<j^\prime\leq d\) we have \(V_j\cap V_{j^\prime}\neq\emptyset\) if and only if \(j^\prime=j+1\) or \((j,j^\prime)=(1,d)\) for which we have isomorphisms \[\begin{align} \Phi_j \colon\pi^{-1}\mathbb{C}_{V_j}\otimes F(a) \overset{\sim}{\longrightarrow}\bigoplus_{f\in N(a)^{>0}(V_j)} \left(\mathsf{E}_{V_j\cap D(a)\left.\right|X}^{\operatorname{Re}f}\right)^{N(a)(f)} \quad \left(1\leq j\leq d\right) \end{align}\] of enhanced sheaves on \(X\). Note that the integer \(\sum_{f \in N(a)^{>0}(V_j)} N(a)(f) \geq 0\) does not depend on \(j\). We denote it by \(R\). It is clear that \(R\) is equal to the rank of the meromorphic connection \(\mathcal{N}\). Let us call it the generic rank of \(\mathcal{M}\). Moreover by Remark 7 we may assume also that for any \(1\leq j<j^\prime\leq d\) such that \(V_j\cap V_{j^\prime}\neq\emptyset\) after renumbering the exponential factors \(f_1,f_2,\dots,f_m\in N(a)^{>0}(V_j\cap V_{j^\prime})\) of \(\mathcal{N}\) on the open sector \(V_j \cap V_{j^\prime}\) they satisfy the condition \[\begin{align} \operatorname{Re}f_1(z)<\operatorname{Re}f_2(z)<\dots<\operatorname{Re}f_m(z) \quad \left(z\in V_j\cap V_{j^\prime}\right) \end{align}\] and the automorphism \[\begin{align} \left(\pi^{-1}\mathbb{C}_{V_j\cap V_{j^\prime}}\otimes\Phi_{j^\prime}\right) &\circ \left(\pi^{-1}\mathbb{C}_{V_j\cap V_{j^\prime}}\otimes\Phi_j\right)^{-1} \colon \\ &\; \bigoplus_{i=1}^m \left(\mathsf{E}_{V_j\cap V_{j^\prime}\cap D(a)\left.\right|X}^{\operatorname{Re}f_i}\right)^{N(a)(f_i)} \overset{\sim}{\longrightarrow} \bigoplus_{i=1}^m \left(\mathsf{E}_{V_j\cap V_{j^\prime}\cap D(a)\left.\right|X}^{\operatorname{Re}f_i}\right)^{N(a)(f_i)} \end{align}\] of the enhanced sheaf \[\begin{align} \bigoplus_{i=1}^m \left(\mathsf{E}_{V_j\cap V_{j^\prime}\cap D(a)\left.\right|X}^{\operatorname{Re}f_i}\right)^{N(a)(f_i)} \simeq \bigoplus_{i=1}^m \mathbb{C}_{\left\{ z\in V_j\cap V_{j^\prime}\cap D(a),\, t+\operatorname{Re}f_i(z)\geq 0\right\}} ^{\oplus N(a)(f_i)} \end{align}\] is induced by a block upper triangular matrix \(A_{j j^\prime}\in \mathrm{GL}_R(\mathbb{C})\) with respect to the decomposition \(R= \sum_{i=1}^m N(a)(f_i)\) of \(R\). From now on, we shall explicitly construct an enhanced sheaf \(G\in \mathbf{E}_{\mathbb{R}{\text{-}}\mathrm c}^0(\mathbb{C}_X)\) on \(X\) such that \[\begin{align} Sol_X^{{\mathrm{E}}}(\mathcal{N}) \simeq \mathbb{C}_X^{{\mathrm{E}}}\overset{+}{\otimes}G \end{align}\] and for any \(a\in D\) we have \[\begin{align} \pi^{-1}\mathbb{C}_{D(a)}\otimes G \simeq \pi^{-1}\mathbb{C}_{D(a)}\otimes F(a). \end{align}\] For this purpose, set \(U \coloneq X\left.\right\backslash D\subset X\) and let \[\begin{align} L \coloneq Sol_U(\mathcal{N}\left.\right|_U) \simeq Sol_U(\mathcal{M}\left.\right|_U) \end{align}\] be the local system associated to the integrable connection \(\mathcal{N}\left.\right|_U\simeq \mathcal{M}\left.\right|_U\) on \(U\). For each point \(a\in D\) we take a closed disk \(D(a)^{\prime}\subset X\) centered at it such that \(D(a)^{\prime}\subset \operatorname{Int}(D(a))\) and define a closed subset \(E\subset X\) by \[\begin{align} E \coloneq X\left.\right\backslash\left\{\bigcup_{a\in D} \operatorname{Int}(D(a)^{\prime})\right\}. \end{align}\] Then for \(a\in X\) the closed subset \[\begin{align} A(a) \coloneq D(a) \cap E = D(a)\left.\right\backslash\operatorname{Int}(D(a)^{\prime}) \quad \subset X \end{align}\] of \(X\) is an annulus centered at it. Since \(\mathcal{N}\) is an integrable connection and hence regular on a neighborhood of \(E\) in \(X\), by Proposition 4 (iv) we obtain an isomorphism \[\begin{align} \pi^{-1}\mathbb{C}_{E}\otimes Sol_X^{{\mathrm{E}}}(\mathcal{N}) \simeq \mathbb{C}_X^{{\mathrm{E}}}\overset{+}{\otimes}\epsilon(L_E). \end{align}\] For \(a\in D\) let \(L(a)\) be the local system on \(D(a)^{\circ}\) defined by gluing the constant sheaves \[\begin{align} \bigoplus_{f \in N(a)^{>0}(V_j)} \mathbb{C}_{V_j\cap D(a)}^{\oplus N(a)(f)} \end{align}\] on \(V_j \cap D(a) \subset D(a)^{\circ}\) \(\left(1\leq j\leq d\right)\) by the block upper triangular matrices \(A_{j j^{\prime}}\in\mathrm{GL}_R(\mathbb{C})\) (\(V_j \cap V_{j^{\prime}} \not= \emptyset\)) and \(j(a) \colon D(a)^{\circ} \hookrightarrow X\) the inclusion map. Then by our construction of the enhanced sheaf \(F(a)\in\mathbf{E}_{\mathbb{R}{\text{-}}\mathrm c}^0(\mathbb{C}_X)\) there exists a surjective morphism \[\begin{align} \pi^{-1}j(a)_!L(a) \longrightarrow F(a). \end{align}\] Let us take a sufficiently large \(c\gg0\) satisfying the condition \[\begin{align} c > \max_{f\in N(a)^{>0}(V_j)} \left\{ \max_{z\in A(a)\cap\overline{V_j}} \left(-\operatorname{Re}f(z)\right)\right\} \end{align}\] for any \(1\leq j\leq d\). Then there exists a surjective morphism \[\begin{align} \pi^{-1}\mathbb{C}_{A(a)}\otimes F(a) \longrightarrow \mathbb{C}_{\{t\geq c\}}\otimes \pi^{-1} \Bigl( j(a)_!L(a) \Bigr)_{A(a)} \end{align}\] and applying \(\pi_{\ast}\) to it we obtain an isomorphism \[\label{eq-T1} \pi_{\ast}F(a)\left.\right|_{A(a)} \overset{\sim}{\longrightarrow}L(a)\left.\right|_{A(a)}\tag{7}\] of sheaves on the annulus \(A(a)\).

By Lemma 6 we obtain a surjective morphism \[\pi^{-1}(L \left.\right|_{A(a)}) \simeq \pi^{-1}(L(a) \left.\right|_{A(a)}) \longrightarrow F(a) \left.\right|_{\pi^{-1} A(a)}\] of sheaves on \(\pi^{-1} A(a)=A(a) \times \mathbb{R}\) obtained by cutting the support of the local system \(\pi^{-1}(L \left.\right|_{A(a)})\). Namely \(F(a) \left.\right|_{\pi^{-1} A(a)}\) is a quotient of \(\pi^{-1}(L \left.\right|_{A(a)})\). Similarly, for sufficiently large \(c\gg0\) we can also construct a quotient \(G_0 \in\mathbf{E}_{\mathbb{R}{\text{-}}\mathrm c}^0(\mathbb{C}_E)\) of the local system \(\pi^{-1}(L \left.\right|_{E})\) satisfying the conditions \[G_0 \left.\right|_{\pi^{-1}(X \setminus \cup_{a \in D} D(a))} \simeq \mathbb{C}_{\{t\geq c\}} \otimes \pi^{-1} (L \left.\right|_{(X \setminus \cup_{a \in D} D(a))})\] and \[G_0 \left.\right|_{\pi^{-1}A(a)} \simeq F(a) \left.\right|_{\pi^{-1} A(a)} \qquad (a \in D).\] Gluing \(G_0 \in\mathbf{E}_{\mathbb{R}{\text{-}}\mathrm c}^0(\mathbb{C}_E)\) and \(F(a) \left.\right|_{\pi^{-1} D(a)} \in\mathbf{E}_{\mathbb{R}{\text{-}}\mathrm c}^0(\mathbb{C}_{D(a)})\) over the sets \(\pi^{-1} A(a)=A(a) \times \mathbb{R}\) (\(a \in D\)) we obtain an enhanced sheaf \(G\in\mathbf{E}_{\mathbb{R}{\text{-}}\mathrm c}^0(\mathbb{C}_X)\) globally defined on the whole \(X\). Now let us define a closed subset \(E^{\prime}\subset X\) by \[\begin{align} E^{\prime} \coloneq \bigsqcup_{a\in D}D(a). \end{align}\] Note that we have \(X=E^\prime\cup E\) and \(E^\prime\cap E=\displaystyle\sqcup_{a\in D}A(a)\). Then there exist distinguished triangles \[\begin{align} Sol_X^{\mathrm{E}}(\mathcal{N})\longrightarrow\bigl(\pi^{-1} \mathbb{C}_E \otimes Sol_X^{\mathrm{E}}(\mathcal{N})\bigr) &\oplus\bigl(\pi^{-1}\mathbb{C}_{E^\prime}\otimes Sol_X^{\mathrm{E}}(\mathcal{N})\bigr) \notag \\ &\longrightarrow\bigoplus_{a\in D}\left(\pi^{-1}\mathbb{C}_{A(a)} \otimes Sol_X^{\mathrm{E}}(\mathcal{N})\right)\overset{+1}{\longrightarrow} \end{align}\] and \[\begin{align} \mathbb{C}_X^{\mathrm{E}}\overset{+}{\otimes}G\longrightarrow\bigl(\pi^{-1} \mathbb{C}_E\otimes (\mathbb{C}_X^{\mathrm{E}}\overset{+}{\otimes}G)\bigr) &\oplus\bigl(\pi^{-1}\mathbb{C}_{E^\prime}\otimes (\mathbb{C}_X^{\mathrm{E}}\overset{+}{\otimes}G)\bigr) \notag \\ &\longrightarrow\bigoplus_{a\in D}\bigl(\pi^{-1}\mathbb{C}_{A(a)} \otimes (\mathbb{C}_X^{\mathrm{E}}\overset{+}{\otimes}G)\bigr)\overset{+1}{\longrightarrow} \end{align}\] associated to the exact sequence \[0\longrightarrow\mathbb{C}_X\longrightarrow\mathbb{C}_E\oplus\mathbb{C}_{E^\prime} \longrightarrow\bigoplus_{a\in D}\mathbb{C}_{A(a)}\longrightarrow0.\] Moreover, by the proof of Lemma 6 and Lemma 1 there exists a commutative diagram \[\vcenter{ \xymatrix@M=5pt@R=18pt@C=14pt{ \bigl(\pi^{-1}\mathbb{C}_E \otimes Sol_X^{\mathrm{E}}(\mathcal{N})\bigr) \oplus\bigl(\pi^{-1}\mathbb{C}_{E^\prime}\otimes Sol_X^{\mathrm{E}}(\mathcal{N})\bigr) \ar[r] \ar[d]^-[@!-90]{\sim} & \displaystyle\bigoplus^{{\phantom{A}}}_{a\in D}\bigl( \pi^{-1}\mathbb{C}_{A(a)}\otimes Sol_X^{{\mathrm{E}}}(\mathcal{N})\bigr) \ar[d]^-[@!-90]{\sim} \\ \bigl(\pi^{-1}\mathbb{C}_E\otimes (\mathbb{C}_X^{\mathrm{E}}\overset{+}{\otimes}G)\bigr) \oplus\bigl(\pi^{-1}\mathbb{C}_{E^\prime}\otimes (\mathbb{C}_X^{\mathrm{E}}\overset{+}{\otimes}G)\bigr) \ar[r] & \displaystyle\bigoplus^{{\phantom{A}}}_{a\in D}\bigl( \pi^{-1}\mathbb{C}_{A(a)}\otimes(\mathbb{C}_X^{{\mathrm{E}}}\overset{+}{\otimes}G)\bigr). }}\] Then by the axiom (TR 4) (see [@KS90]) we obtain an isomorphism \[Sol_X^{\mathrm{E}}(\mathcal{N})\simeq \mathbb{C}_X^{\mathrm{E}}\overset{+}{\otimes}G.\]

4 Fourier transforms of holonomic \(\mathcal{D}\)-modules in dimension one and their characteristic cycles↩︎

In this section, we give some refinements of the results of D’Agnolo-Kashiwara [@DK18] and [@DK23] on the exponential factors of Fourier transforms of holonomic \(\mathcal{D}\)-modules in dimension one. With our explicit description of their enhanced solution complexes obtained in Section 3 at hands, we can now apply a twisted Morse theory (with several Morse functions) to obtain more precise structures of their Fourier transforms.

4.1 Fourier transforms of holonomic \(\mathcal{D}\)-modules↩︎

First of all, we recall the definition of Fourier transforms of \(\mathcal{D}\)-modules and their basic properties and explain their relations with the irregular Riemann-Hilbert correspondence of D’Agnolo-Kashiwara [@DK16]. Let \(X=\mathbb{C}_z^N\) be the \(N\)-dimensional complex vector space and \(Y=\mathbb{C}_w^N\) its dual space. Here we first consider them as smooth algebraic varieties over \(\mathbb{C}\) endowed with the Zariski topology. We use the notations \(\mathcal{D}_X\) and \(\mathcal{D}_Y\) for the rings of “algebraic” differential operators on them. Denote by \(\mathrm{Mod}_{\scriptsize coh}(\mathcal{D}_X)\) (resp. \(\mathrm{Mod}_{\mathrm{hol}}(\mathcal{D}_X)\), \(\mathrm{Mod}_{\scriptsize rh}(\mathcal{D}_X)\)) the category of coherent (resp. holonomic, regular holonomic) \(\mathcal{D}_X\)-modules. Let \(W_N := \mathbb{C}[z, \partial_z]\simeq\Gamma(X; \mathcal{D}_X)\) and \(W^\ast_N := \mathbb{C}[w, \partial_w]\simeq\Gamma(Y; \mathcal{D}_Y)\) be the Weyl algebras over \(X\) and \(Y\), respectively. Then by the ring isomorphism \[W_N\overset{\sim}{\longrightarrow}W^\ast_N (z_i\mapsto-\partial_{w_i},\;\partial_{z_i}\mapsto w_i)\] we can endow a left \(W_N\)-module \(M\) with a structure of a left \(W_N^\ast\)-module. We call it the Fourier transform of \(M\) and denote it by \(M^\wedge\). For a ring \(R\) we denote by \(\mathrm{Mod}_f(R)\) the category of finitely generated \(R\)-modules. Recall that for the affine algebraic varieties \(X\) and \(Y\) we have the equivalences of categories \[\begin{align} \mathrm{Mod}_{\scriptsize coh}(\mathcal{D}_X) &\simeq \mathrm{Mod}_f(\Gamma(X; \mathcal{D}_X)) = \mathrm{Mod}_f(W_N),\\ \mathrm{Mod}_{\scriptsize coh}(\mathcal{D}_Y) &\simeq \mathrm{Mod}_f(\Gamma(Y; \mathcal{D}_Y)) = \mathrm{Mod}_f(W^\ast_N) \end{align}\] (see e.g. [@HTT08]). For a coherent \(\mathcal{D}_X\)-module \(\mathcal{M}\in\mathrm{Mod}_{\scriptsize coh}(\mathcal{D}_X)\) we thus can define its Fourier transform \(\mathcal{M}^\wedge\in\mathrm{Mod}_{\scriptsize coh}(\mathcal{D}_Y)\). It follows that we obtain an equivalence of categories \[( \cdot )^\wedge : \mathrm{Mod}_{\mathrm{hol}}(\mathcal{D}_X)\overset{\sim}{\longrightarrow}\mathrm{Mod}_{\mathrm{hol}}(\mathcal{D}_Y)\] between the categories of holonomic \(\mathcal{D}\)-modules (see [@HTT08]). Let \[\begin{align} X\overset{p}{\longleftarrow}X\times Y\overset{q}{\longrightarrow}Y \end{align}\] be the projections. Then by Katz-Laumon [@KL85], we have the following lemma.

Let \(\overline{X}\simeq{\mathbb{P}}^N\) (resp. \(\overline{Y}\simeq{\mathbb{P}}^N\)) be the projective compactification of \(X\) (resp. \(Y\)). By the inclusion map \(i_X : X=\mathbb{C}^N\xhookrightarrow{\;\;\;}\overline{X}={\mathbb{P}}^N\) we extend a holonomic \(\mathcal{D}_X\)-module \(\mathcal{M}\in\mathrm{Mod}_{\mathrm{hol}}(\mathcal{D}_X)\) to \(\widetilde{\mathcal{M}} := i_{X\ast}\mathcal{M}\simeq\mathbf{D}i_{X\ast}\mathcal{M} \in\mathrm{Mod}_{\mathrm{hol}}(\mathcal{D}_{\overline{X}})\). Denote by \(\overline{X}^{{\rm an}}\) the underlying complex manifold of \(\overline{X}\) and define the analytification \(\widetilde{\mathcal{M}}^{{\rm an}}\in\mathrm{Mod}_{\mathrm{hol}}(\mathcal{D}_{\overline{X}^{{\rm an}}})\) of \(\widetilde{\mathcal{M}}\) by \(\widetilde{\mathcal{M}}^{{\rm an}} = \mathcal{O}_{\overline{X}^{{\rm an}}} \otimes_{\mathcal{O}_{\overline{X}}}\widetilde{\mathcal{M}}\). Then we set \[\begin{align} Sol_{\overline{X}}^{\mathrm{E}}(\widetilde{\mathcal{M}}) := Sol_{\overline{X}^{{\rm an}}}^{\mathrm{E}}(\widetilde{\mathcal{M}}^{{\rm an}}) \qquad \in{\mathbf{E}}^{\mathrm{b}}({\rm I}\mathbb{C}_{\overline{X}^{{\rm an}}}). \end{align}\] Similarly for the Fourier transform \(\mathcal{M}^\wedge\in\mathrm{Mod}_{\mathrm{hol}}(\mathcal{D}_Y)\) we define \(Sol_{\overline{Y}}^{\mathrm{E}}(\widetilde{\mathcal{M}^\wedge}) \in{\mathbf{E}}^{\mathrm{b}}({\rm I}\mathbb{C}_{\overline{Y}^{{\rm an}}})\). Let \[\begin{align} \overline{X}^{{\rm an}}\overset{\overline{p}}{\longleftarrow} \overline{X}^{{\rm an}}\times\overline{Y}^{{\rm an}}\overset{\overline{q}}{\longrightarrow} \overline{Y}^{{\rm an}} \end{align}\] be the projections.

Note that these transforms preserve the \(\mathbb{R}\)-constructibility. Namely we obtain functors \({}^{\mathsf{L}}(\cdot)\colon\) \(\mathbf{E}_{\mathbb{R}{\text{-}}\mathrm c}^{\mathrm{b}}(\mathbb{C}_{\overline{X}^{{\rm an}}})\to \mathbf{E}_{\mathbb{R}{\text{-}}\mathrm c}^{\mathrm{b}}(\mathbb{C}_{\overline{Y}^{{\rm an}}})\) and \({}^{\mathsf{L}}(\cdot)\colon \mathbf{E}_{\mathbb{R}{\text{-}}\mathrm c}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_{\overline{X}^{{\rm an}}})\to \mathbf{E}_{\mathbb{R}{\text{-}}\mathrm c}^{\mathrm{b}}(\mathrm{I}\mathbb{C}_{\overline{Y}^{{\rm an}}})\). The following lemma is essentially due to Kashiwara-Schapira [@KS16a] and D’Agnolo-Kashiwara [@DK18] (see also [@IT20a]).

Note that by [@DK16] we have \[\begin{align} {}^{\mathsf{L}}\Bigl(\mathbb{C}_{\overline{X}^{{\rm an}}}^{{\mathrm{E}}} \overset{+}{\otimes}(\cdot)\Bigr)\simeq\mathbb{C}_{\overline{Y}^{{\rm an}}}^{{\mathrm{E}}} \overset{+}{\otimes}{}^{\mathsf{L}}(\cdot). \end{align}\] Therefore, if there exists \(G\in\mathbf{E}_{\mathbb{R}{\text{-}}\mathrm c}^{\mathrm{b}}(\mathbb{C}_{\overline{X}^{{\rm an}}})\) such that \[\begin{align} Sol_{\overline{X}}^{{\mathrm{E}}}(\widetilde{\mathcal{M}})\simeq \mathbb{C}_{\overline{X}^{{\rm an}}}^{{\mathrm{E}}}\overset{+}{\otimes}G, \end{align}\] then we obtain an isomorphism \[\begin{align} \label{undls} Sol_{\overline{Y}}^{{\mathrm{E}}}(\widetilde{\mathcal{M}^\wedge}) \simeq \mathbb{C}_{\overline{Y}^{{\rm an}}}^{{\mathrm{E}}}\overset{+}{\otimes}{}^\mathsf{L}G \end{align}\tag{8}\] and hence for the study of \(Sol_{\overline{X}}^{{\mathrm{E}}}(\widetilde{\mathcal{M}^\wedge})\) it suffices to study \({}^\mathsf{L}G\in \mathbf{E}_{\mathbb{R}{\text{-}}\mathrm c}^{\mathrm{b}}(\mathbb{C}_{\overline{Y}^{{\rm an}}})\).

From now on, we assume that the dimension \(N\) of \(X\) and \(Y\) is one. Let \(\mathcal{M}\in\mathrm{Mod}_{\mathrm{hol}}(\mathcal{D}_X)\) be an algebraic holonomic \(\mathcal{D}\)-module on the affine line \(X=\mathbb{C}_z\) and denote by \(D\subset X\) its singular support \(\mathrm{sing.supp}(\mathcal{M})\). Then there exists a distinguished triangle \[{\mathrm{R}}\Gamma_D(\mathcal{M}) \longrightarrow \mathcal{M}\longrightarrow {\mathrm{R}}\Gamma_{X\left.\right\backslash D}(\mathcal{M}) \overset{+1}{\longrightarrow}.\] Since \(U:= X\left.\right\backslash D\) is an affine open subset of \(X\), we have \[\begin{align} H^j{\mathrm{R}}\Gamma_U(\mathcal{M}) \simeq \begin{cases} \;\Gamma_U(\mathcal{M}) & (j=0), \\ \;0 & (j\neq0) \end{cases} \end{align}\] and \(\Gamma_U(\mathcal{M})\) is nothing but the localization of \(\mathcal{M}\) along the divisor \(D\subset X\). We thus obtain an exact sequence \[0 \longrightarrow \Gamma_D(\mathcal{M}) \longrightarrow \mathcal{M}\longrightarrow \Gamma_U(\mathcal{M}) \longrightarrow H^1{\mathrm{R}}\Gamma_D(\mathcal{M}) \longrightarrow 0.\] Note that \(\Gamma_D(\mathcal{M})\) and \(H^1{\mathrm{R}}\Gamma_D(\mathcal{M})\) are supported in the finite set \(D\subset X\) and hence their Fourier transforms are integrable connections on \(X= \mathbb{C}\). Since the Fourier transform \((\cdot)^\wedge\) is an exact functor, this implies that for the initial study of \(\mathcal{M}^\wedge\) it suffices to study the Fourier transform of \(\Gamma_U(\mathcal{M})\). For this reason, in this paper we first assume that \[\begin{align} \label{eq:localize} \mathcal{M}\overset{\sim}{\longrightarrow}\Gamma_U(\mathcal{M}) \end{align}\tag{9}\] and discuss the general case in Section 4.4. We call \(\mathcal{M}\in \mathrm{Mod}_{\mathrm{hol}}(\mathcal{D}_X)\) satisfying (9 ) a localized holonomic \(\mathcal{D}\)-module or a meromorphic connection on \(X=\mathbb{C}_z\). We denote its generic rank by \({\rm rk} \mathcal{M}\).

4.2 The proof of Theorem 1 and related results↩︎

For the proof of Theorem 1, recall the notation in Section 1. If an exponential factor \(f \in N_i^{>0}\) at a point \(a_i \in \widetilde{D}\) is holomorphic on a sector \(S\) along it, by abuse of notations we write \(f \in N_i^{>0}(S)\). For such \(f\in N_i^{>0}(S)\) and \(w \in Y^{{\rm an}}= \mathbb{C}\) we define a holomorphic function \(f^w\) on \(S\) by \[\begin{align} f^w(z) \coloneq zw-f(z) \quad \left(z\in S \right). \end{align}\] For \(a_i\in D^{{\rm an}}\) let \(D(a_i)\) be a closed disk centered at it such that \(D(a_i)^\circ := D(a_i) \setminus \{ a_i \} \subset B(a_i)^\circ\). Shrinking \(B(a_i)^\circ\) if necessary, we may assume that for any non-zero exponential factor \(f \in N_i^{>0}\) of \(\mathcal{M}^{{\rm an}}\) at \(a_i\) the morphism \(f^{\prime}: B(a_i)^{\circ} \longrightarrow Y^{{\rm an}}= \mathbb{C}\) is an unramified finite covering over the punctured disk \(B(b_{\infty})^{\circ}\). Recall that we have \[(f^w)^\prime(z)=0 \quad \iff \quad f^\prime(z)=w.\] Then we can take the disk \(D(a_i)\) so that for any non-zero \(f \in N_i^{>0}\) and \(w \in V \subset B(b_{\infty})^{\circ}\) all the critical points of the (possibly multi-valued) holomorphic function \(f^w\) on \(B(a_i)^\circ\) are contained in \({\rm Int} D(a_i)^{\circ}\).

Then as in Section 3 we can explicitly construct an enhanced sheaf \(F_i=F(a_i)\in\mathbf{E}_{\mathbb{R}{\text{-}}\mathrm c}^0(\mathbb{C}_{\overline{X}^{{\rm an}}})\) on \(\overline{X}^{{\rm an}}\) such that \[\begin{align} \pi^{-1}\mathbb{C}_{D(a_i)^\circ}\otimes Sol_{\overline{X}}^{{\mathrm{E}}}(\widetilde{\mathcal{M}}) \simeq \mathbb{C}_{\overline{X}^{{\rm an}}}^{{\mathrm{E}}}\overset{+}{\otimes}F_i. \end{align}\] Similarly, for the point \(a_\infty=\infty\in\widetilde{D}\) we take a closed disk \(D(a_\infty)\) centered at it such that \(D(a_{\infty})^\circ := D(a_{\infty}) \setminus \{ a_{\infty} \} \subset B(a_{\infty})^\circ\), and construct \(F_\infty=F(a_\infty)\in\mathbf{E}_{\mathbb{R}{\text{-}}\mathrm c}^0(\mathbb{C}_{\overline{X}^{{\rm an}}})\) for which we have an isomorphism \[\begin{align} \pi^{-1}\mathbb{C}_{D(a_{\infty})^\circ}\otimes Sol_{\overline{X}}^{{\mathrm{E}}}(\widetilde{\mathcal{M}}) \simeq \mathbb{C}_{\overline{X}^{{\rm an}}}^{{\mathrm{E}}}\overset{+}{\otimes}F_\infty. \end{align}\] We set \(L\coloneq Sol_U(\mathcal{M}\left.\right|_U)\). As in Section 3, then by gluing \(F_i\) \((1\leq i\leq l)\), \(F_\infty\) and \(\mathbb{C}_{\{t\geq c\}}\otimes\pi^{-1}L\) (\(c\gg0\)), we obtain an enhanced sheaf \(G\in\mathbf{E}_{\mathbb{R}{\text{-}}\mathrm c}^0(\mathbb{C}_{\overline{X}^{{\rm an}}})\) such that \[\begin{align} Sol_{\overline{X}}^{{\mathrm{E}}}(\widetilde{\mathcal{M}}) \simeq \mathbb{C}_{\overline{X}^{{\rm an}}}^{{\mathrm{E}}}\overset{+}{\otimes}G. \end{align}\] It follows from the results in Section 4.1 also that there exists an isomorphism \[\begin{align} \label{sbehdwfz} Sol_{\overline{Y}}^{{\mathrm{E}}}(\widetilde{\mathcal{M}^\wedge}) \simeq \mathbb{C}_{\overline{Y}^{{\rm an}}}^{{\mathrm{E}}}\overset{+}{\otimes}{}^\mathsf{L}G. \end{align}\tag{10}\] Let \[\begin{align} \left(X^{{\rm an}} \times\mathbb{R}_s\right) \overset{\;p_1}{\longleftarrow} \left(X^{{\rm an}} \times\mathbb{R}_s\right)\times\left(Y^{{\rm an}} \times\mathbb{R}_t\right) \overset{p_2}{\longrightarrow} \left(Y^{{\rm an}} \times\mathbb{R}_t\right) \end{align}\] be the projections and set \(G^{\circ}:= G|_{X^{{\rm an}} \times\mathbb{R}_s}\). Then by [@DK18] on \(Y^{{\rm an}}\subset\overline{Y}^{{\rm an}}\) there exists an isomorphism \[\begin{align} \label{qisoms} {}^\mathsf{L}G \simeq \mathbf{Q}\left({\mathrm{R}}p_{2!}(p_1^{-1} G^{\circ} \otimes\mathbb{C}_{\{t-s-\operatorname{Re}zw\geq0\}}[1])\right), \end{align}\tag{11}\] where \(\mathbf{Q}\colon{\mathbf{D}}^{\mathrm{b}}(\mathbb{C}_{Y^{{\rm an}}\times\mathbb{R}_t}) \rightarrow{\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_{Y^{{\rm an}}})\) is the quotient functor. Moreover, for a point \((w,t)\in V\times\mathbb{R}\subset Y^{{\rm an}}\times\mathbb{R}\) we have an isomorphism \[\begin{align} \label{stalkfs} \left({\mathrm{R}}p_{2!}(p_1^{-1}G^{\circ} \otimes \mathbb{C}_{\{t-s-\operatorname{Re}zw\geq0\}}[1])\right)_{(w,t)} \simeq {\mathrm{R}}\Gamma_c \left(X^{{\rm an}}; {\mathrm{R}}\pi_!(G^{\circ} \otimes\mathbb{C}_{\{t-s-\operatorname{Re}zw\geq0\}})[1]\right). \end{align}\tag{12}\] In view of 8 , 11 and 12 , for the proof of Theorem 1 it suffices to calculate the right hand side of 12 for each point \((w,t)\in V\times\mathbb{R}\).

Let us fix a point \((w,t)\in V\times\mathbb{R}\) and describe the structure of \[\begin{align} G(w,t) \coloneq {\mathrm{R}}\pi_!(G^{\circ} \otimes\mathbb{C}_{\{t-s-\operatorname{Re}zw\geq0\}} [1])\; \in{\mathbf{D}}^{\mathrm{b}}(\mathbb{C}_{X^{{\rm an}}}). \end{align}\] By our construction of \(G\), we can easily see that the restriction of \(G(w,t)\) to \(D^{{\rm an}}=\{a_1,\dots,a_l\}\subset X^{{\rm an}}\) is zero and the restriction of \(\pi\) to the support of \(G^{\circ} \otimes\mathbb{C}_{\{t-s-\operatorname{Re}zw\geq0\}}\) is proper on \(U^{{\rm an}}=X^{{\rm an}} \setminus D^{{\rm an}}\). Then we obtain an isomorphism \[\begin{align} {\mathrm{R}}\pi_!(G^{\circ} \otimes\mathbb{C}_{\{t-s-\operatorname{Re}zw\geq0\}}) \overset{\sim}{\longrightarrow} {\mathrm{R}}\pi_*(G^{\circ} \otimes\mathbb{C}_{\{t-s-\operatorname{Re}zw\geq0\}}) \end{align}\] on \(U^{{\rm an}} \subset X^{{\rm an}}\) and hence a morphism \[\begin{align} \label{surmorp} L \simeq H^0( {\mathrm{R}}\pi_* G^{\circ})|_{U^{{\rm an}}} \longrightarrow H^{-1}G(w,t)|_{U^{{\rm an}}} \end{align}\tag{13}\] of \(\mathbb{R}\)-constructible sheaves on \(U^{{\rm an}}\) is induced by the canonical morphism \(G^{\circ} \longrightarrow G^{\circ} \otimes\mathbb{C}_{\{t-s-\operatorname{Re}zw\geq0\}}\). We will see below that it is surjective. Namely we show that \(H^{-1}G(w,t)|_{U^{{\rm an}}}\) is a quotient sheaf of the local system \(L\). For a point \(a_i\in D^{{\rm an}}\) by our construction of the enhanced sheaf \(F_i=F(a_i)\in\mathbf{E}_{\mathbb{R}{\text{-}}\mathrm c}^0(\mathbb{C}_{X^{{\rm an}}})\) there exist open sectors \(V_1,V_2,\dots,V_d\) along \(a_i\in D^{{\rm an}}\) placed in the counter-clockwise direction such that \[\begin{align} D(a_i)^\circ \subset V_1\cup V_2\cup\dots\cup V_d \subset B(a_i)^\circ \end{align}\] and for \(1\leq j< j^{\prime} \leq d\) we have \(V_j\cap V_{j^{\prime}} \not= \emptyset\) if and only if \(j^{\prime} =j+1\) or \((j, j^{\prime})=(1,d)\) for which we have isomorphisms \[\begin{align} \Phi_j \colon \pi^{-1}\mathbb{C}_{V_j}\otimes F_i \overset{\sim}{\longrightarrow}\bigoplus_{f\in N_i^{>0}(V_j)} \left( \mathbb{C}_{ \{ (z,s) | z \in V_j \cap D(a_i)^\circ, \; s+ \operatorname{Re}f(z) \geq 0 \} } \right)^{N_i(f)} \quad \left(1\leq j\leq d\right). \end{align}\] We also impose the condition (iv) in Remark 7 on the covering \(D(a_i)^\circ \subset V_1\cup\dots\cup V_d\). Let \(f_1,f_2, \ldots, f_{m_i} \in N_i^{>0}(V_j)\) be the exponential factors of \(\mathcal{M}^{{\rm an}}\) on the sector \(V_j\). Then we can easily show that the restriction of \(G(w,t)\) to \(V_j \cap D(a_i)^\circ \subset D(a_i)^\circ\) is isomorphic to \[\begin{align} \label{locstru} \bigoplus_{k=1}^{m_i} \Bigl(\mathbb{C}_{\left\{z\in V_j \cap D(a_i)^\circ \mid \operatorname{Re}(f_k^w)(z) \leq t\right\}} [1] \Bigr)^{N_i(f_k)} \end{align}\tag{14}\] (see 2 ). Moreover the restriction of \(G(w,t)\) to the punctured disk \(D(a_i)^\circ\) is obtained by gluing these \(\mathbb{R}\)-constructible sheaves on \(V_j \cap D(a_i)^\circ\) (\(1 \leq j \leq d\)) by the transition matrices \(A_{j j^{\prime}} \in {\rm GL}_{R}( \mathbb{C})\) (\(V_j \cap V_{j^{\prime}} \not= \emptyset\)), where we set \(R:= {\rm rk} \mathcal{M}\). Recall that if \(V_j \cap V_{j^{\prime}} \not= \emptyset\) after renumbering the exponential factors on \(V_j \cap V_{j^{\prime}}\) the matrix \(A_{j j^{\prime}}\) becomes block upper triangular. For \(j \not= j^{\prime}\) such that \(V_j \cap V_{j^{\prime}} \not= \emptyset\) and \(V_j \cup V_{j^{\prime}}\) is simply connected, we can easily show that there exists an isomorphism \[\begin{align} \label{twolocdesc} G(w,t)|_{V_j \cup V_{j^{\prime}}} \simeq \bigoplus_{f \in N_i^{>0}(V_j \cup V_{j^{\prime}})} \Bigl(\mathbb{C}_{\left\{z\in V_j \cup V_{j^{\prime}} \mid \operatorname{Re}f^w(z) \leq t\right\}} [1] \Bigr)^{N_i(f)}. \end{align}\tag{15}\] Let \(S \subset D(a_i)^\circ\) be a simply connected sector in \(D(a_i)^\circ\). Then, the restriction of \(L\) to \(S\) being isomorphic to the constant sheaf \(\mathbb{C}_S^R\), similarly we obtain an isomorphism \[\begin{align} \label{locdesc} G(w,t)|_S \simeq \bigoplus_{f \in N_i^{>0}(S)} \Bigl(\mathbb{C}_{\left\{z\in S \mid \operatorname{Re}f^w(z) \leq t\right\}} [1] \Bigr)^{N_i(f)}. \end{align}\tag{16}\] In the following example, for the reader’s understanding of the proofs of Lemma 9 and Theorem 1, we illustrate how the sublevel sets \(\{\operatorname{Re}f^w(z)\leq t\}\) of \(\operatorname{Re}f^w\) change as \(t\in\mathbb{R}\) increases.

Also for the point \(a_\infty=\infty\in\widetilde{D}\) we have a similar description of \(G(w,t)\) on a neighborhood of it and can define holomorphic functions \(f^w\;(f\in N_\infty^{>0})\) defined on some open sectors along it. For \(w\in V\) we define a real-valued function \(\xi^w\) on the open subset \[\begin{align} \Omega \coloneq X^{{\rm an}}\left.\right\backslash\bigl(D(a_i)\cup\dots\cup D(a_l)\cup D(a_\infty)\bigr)\;\subset U^{{\rm an}} \end{align}\] of \(U^{{\rm an}}\) by \[\begin{align} \xi^w(z)\coloneq\operatorname{Re}(zw)+c \quad (z\in\Omega), \end{align}\] where \(c\in\mathbb{R}\) is the sufficiently large number used in the construction of the enhanced sheaf \(G\) (see Section 3). Then it is easy to see that for any \(w\in V\) and \(t\in\mathbb{R}\) the restriction of \(G(w,t)\) to \(\Omega\subset U^{{\rm an}}\) is isomorphic to \[\begin{align} (L\left.\right|_\Omega)\otimes \mathbb{C}_{\{ z\in \Omega | \xi^w(z)\leq t \}} [1]. \end{align}\] From these local descriptions of \(G(w,t)\), we see that the morphism in 13 is surjective.

To prove Theorem 1, for \(w \in V\) we consider the real-valued functions \(\operatorname{Re}f^w\colon V_j \cap D(a_i)^\circ \rightarrow\mathbb{R}\; (f\in N_i^{>0}(V_j))\) on the open subsets \(V_j \cap D(a_i)^\circ \subset D(a_i)^\circ\) as Morse functions and apply a Morse theory associated to them. In contrast to the usual Morse theory, that we use here relies on the sublevel sets of “several" Morse functions. Note that the critical points of \(\operatorname{Re}f^w\colon V_j\rightarrow\mathbb{R}\) are those of the holomorphic function \(f^w\colon V_j\rightarrow\mathbb{C}\). For \(w \in V\) and \(a_i\in D^{{\rm an}}\) let \(\gamma_{i j}(w)\in {\rm Int}D (a_i)^{\circ}\) \((1\leq j\leq n_i)\) be the points in the punctured disk \({\rm Int}D(a_i)^{\circ}\) such that \[\begin{align} (f^w)^\prime( \gamma_{i j}(w)) = 0 \quad \Longleftrightarrow \quad f^\prime ( \gamma_{i j}(w)) = w \end{align}\] for some non-zero exponential factor \(f\in N_i^{>0}\) of \(\mathcal{M}^{{\rm an}}\) at \(a_i\). Note that for each point \(\gamma_{i j}(w)\) the non-zero exponential factor \(f \in N_i^{>0}\) satisfying the condition \(f^\prime ( \gamma_{i j}(w)) = w\) is unique. Then by using such \(f\in N_i^{>0}\) we set \[\begin{align} c_{i j}(w) \coloneq \operatorname{Re}(f^w)(\gamma_{i j}(w)) \;\in\mathbb{R}\quad (1\leq j\leq n_i) \end{align}\] and \[\begin{align} N_{i j} \coloneq N_i(f) \;>0 \quad (1\leq j\leq n_i). \end{align}\] Namely \(c_{i j}(w)\) \((1\leq j\leq n_i)\) are the critical values of the (possibly multi-valued) functions \(\operatorname{Re}(f^w)\) \((f\in N_i^{>0}, f\neq0)\). Also for \(w \in V\) and \(a_\infty=\infty\in\widetilde{D}\) we can define points \(\gamma_{\infty j}(w)\in {\rm Int}D(a_{\infty})^{\circ} \subset X^{{\rm an}}=\mathbb{C}_z\) \((1\leq j\leq n_\infty)\) and \(c_{\infty j}(w)\in\mathbb{R}\), \(N_{\infty j}\) \((1\leq j\leq n_\infty)\) similarly.

For \(w\in V\) and \(a_i\in D^{{\rm an}}\) we define a subset \(Q(a_i)_+^w\) (resp. \(Q(a_i)_-^w\)) of \(\partial D(a_i)\simeq S^1\) to be the set of the points \(z\in\partial D(a_i)\) such that for some \(f\in N_i^{>0}\) the (possibly multi-valued) real-valued function \(\operatorname{Re}(f^w\left.\right|_{\partial D(a_i)})\) on \(\partial D(a_i)\simeq S^1\) has a local maximum (resp. local minimum) at \(z\in\partial D(a_i)\). Note that by our definition \(f^w(z)=zw-f(z)\) in 2 if \(w \rightarrow \beta \infty\) in the sector \(V\) for some non-zero \(\beta \in \mathbb{C}\), then the points in \(Q(a_i)_+^w\) (resp. \(Q(a_i)_-^w\)) converge to a point in \(\partial D(a_i)\simeq S^1\). Hence by shrinking the punctured disk \(B(b_{\infty})^{\circ}\) centered at \(b_{\infty} = \infty\in\overline{Y}^{{\rm an}}\) if necessary, we may assume that there exists a simply connected sector \(V_+\) (resp. \(V_-\)) along the point \(a_i\) such that \(Q(a_i)_+^w\subset V_+\) (resp. \(Q(a_i)_-^w\subset V_-\)) for any \(w\in V\). We may assume also that for any \(w\in V\) and \(f\in N_i^{>0}\) the (possibly multi-valued) real-valued function \(\operatorname{Re}(f^w\left.\right|_{\partial D(a_i)})\) on \(\partial D(a_i)\) has no critical point on \(\partial D(a_i)\left.\right\backslash(Q(a_i)_+^w\sqcup Q(a_i)_-^w)\). Specifically, by the exponential factors \(f_1,\dots,f_{m_i}\in N_i^{>0}(V_+)\) of \(\mathcal{M}^{{\rm an}}\) on the sector \(V_+\) we define real-valued functions \(\phi_{+,1}^w,\dots,\phi_{+,m_i}^w\) on \(\partial D(a_i)\cap V_+\) by \[\begin{align} \phi_{+,k}^w(z)\coloneq\operatorname{Re}f_k^w(z)=\operatorname{Re}(zw)-\operatorname{Re}f_k(z) \quad (z\in\partial D(a_i)\cap V_+, 1 \leq k \leq m_i) \end{align}\] and set \[\begin{align} N_i^+(k)\coloneq N_i(f_k) \, \in\mathbb{Z}_{>0} \quad (1 \leq k \leq m_i). \end{align}\] Similarly we define real-valued functions \(\phi_{-,1}^w,\dots,\phi_{-,m_i}^w\) on \(\partial D(a_i)\cap V_-\) and \(N_i^-(k)\in\mathbb{Z}_{>0}\) \((1\leq k \leq m_i)\). For \(1\leq k\leq m_i\) and \(w\in V\) let \(a_{ik}^+(w)\in\partial D(a_i)\cap V_+\) (resp. \(a_{ik}^-(w)\in\partial D(a_i)\cap V_-\)) be the (unique) point where the function \(\phi_{+,k}^w\) (resp. \(\phi_{-,k}^w\)) takes its maximum (resp. minimum) and set \[\begin{align} A_{ik}^+(w)\coloneq\phi_{+,k}^w(a_{ik}^+(w)) \quad \textrm{(resp. A_{ik}^-(w)\coloneq\phi_{-,k}^w(a_{ik}^-(w))}. \end{align}\] Also for the point \(a_\infty=\infty\in\widetilde{D}\) we define real-valued functions \(A_{\infty k}^+(w)\), \(A_{\infty k}^-(w)\) (\(1 \leq k \leq m_{\infty}\)) on the sector \(V\subset Y^{{\rm an}}\) along the point \(b_\infty=\infty\in\overline{Y}^{{\rm an}}\) and \(N_\infty^+(k),N_\infty^-(k)\in\mathbb{Z}_{>0}\) (\(1 \leq k \leq m_{\infty}\)) similarly.

In what follows, we use also \(\xi^w\colon \Omega = X^{{\rm an}}\left.\right\backslash\bigl(D(a_i)\cup \dots \cup D(a_l)\cup D(a_\infty)\bigr) \rightarrow \mathbb{R}\) as a Morse function to calculate \({}^{\mathsf{L}}G\). Let \(\widetilde{\xi^w}\colon\overline{\Omega} \rightarrow \mathbb{R}\) be the (unique) continuous extension of \(\xi^w\colon\Omega\to\mathbb{R}\) to \(\overline{\Omega}\) and for \(a_i\in\widetilde{D}=D^{{\rm an}}\sqcup\{\infty\}\) set \[\begin{align} L_i^+(w)&\coloneq\max_{z\in\partial D(a_i)}\widetilde{\xi^w}(z), \\ L_i^-(w)&\coloneq\min_{z\in\partial D(a_i)}\widetilde{\xi^w}(z). \end{align}\] Then by our choice of \(c\in\mathbb{R}\) in the construction of \(G\), we have the following result.

By applying the Morse theoretical method in the proof of [@IT20a] to the Morse functions \({\rm Re} f^w\) (\(f \in N_i^{>0}\)) and \(\xi^w\) we obtain the following result.

Now we apply Lemma 1 and Lemma 11 to Proposition 10. Then we obtain an isomorphism \[\begin{align} \pi^{-1}\mathbb{C}_V\otimes Sol_{\overline{Y}}^{{\mathrm{E}}}(\widetilde{\mathcal{M}^\wedge}) \quad &\simeq \quad \mathbb{C}_{\overline{Y}^{{\rm an}}}^{{\mathrm{E}}} \overset{+}{\otimes}\left( \pi^{-1}\mathbb{C}_V\otimes{}^{\mathsf{L}}G \right) \\ &\simeq \quad \bigoplus_{i=1}^l\biggl\{ \bigoplus_{j=1}^{n_i} \bigl(\mathbb{E}_{V\left.\right|\overline{Y}^{{\rm an}}}^{-c_{ij}}\bigr)^{N_{ij}} \oplus\bigl(\mathbb{E}_{V\left.\right|\overline{Y}^{{\rm an}}}^{-c_i}\bigr)^{r_i}\biggr\} \oplus \bigoplus_{j=1}^{n_\infty} \bigl(\mathbb{E}_{V\left.\right|\overline{Y}^{{\rm an}}}^{-c_{\infty j}}\bigr)^{N_{\infty j}}. \end{align}\] We can also rewrite the right hand side to \[\begin{align} \bigoplus_{i=1}^n(\mathbb{E}_{V\left.\right|\overline{Y}^{{\rm an}}}^{\operatorname{Re}g_i})^{\oplus d_i} \end{align}\] by Lemma 4. This completes the proof of Theorem 1. 0◻
We can describe the generic rank \(d(\mathcal{M})\) of the Fourier transform \(\mathcal{M}^\wedge\) in terms of the irregularities of \(\mathcal{M}\) as follows. First, note that for \(\theta\in S_{a_i}X^{{\rm an}}\) \((1\leq i\leq l)\) the non-negative number \[\begin{align} \sum_{f\in(N_i^{>0})_\theta}N_i(f)\cdot\mathrm{ord}_{a_i}(f)\,\geq0 \end{align}\] does not depend on the choice of \(\theta\in S_{a_i}X^{{\rm an}}\). We know moreover that it is an integer and call it the irregularity of the meromorphic connection \(\mathcal{M}^{{\rm an}}\) at \(a_i\in D^{{\rm an}}\) (see e.g. Sabbah [@Sab93]). We denote it by \(\mathrm{irr}_{a_i}(\mathcal{M})\). Note also that for \(\theta\in S_\infty\overline{X}^{{\rm an}}\) the non-negative number \[\begin{align} \sum_{f\in(N_\infty^{>0})_\theta}N_\infty(f)\cdot \min\left\{\mathrm{ord}_\infty(f)-1,0\right\}\,\geq0 \end{align}\] does not depend on \(\theta\in S_\infty\overline{X}^{{\rm an}}\). We can easily show that it is an integer (see e.g. [@Sab08]). Then we denote it by \(e_\infty(\mathcal{M})\in\mathbb{Z}_{\geq0}\) and obtain the following corollary of Theorem 1.

4.3 The proofs of Theorems 2 and 3 and related results↩︎

First, let us prove Theorem 2. Since the sector \(W\) is simply connected, in view of the proof of Proposition 6 it suffices to treat the case where it is sufficiently narrow. For \(w\in W\subset B(b)^\circ\) let \(\gamma_{\infty j}^b(w)\in {\rm Int}D (a_\infty)^\circ\) \((1\leq j\leq n_\infty(b))\) be the points in the punctured disk \({\rm Int}D (a_\infty)^\circ\) such that \[(f^w)^\prime(\gamma_{\infty j}^b(w))=0\] for some (non-zero) exponential factor \(f\in N_{\infty, b}^{>0}\) such that \(f\neq bz\) and set \[c_{\infty j}^b\coloneq \operatorname{Re}(f^w)(\gamma_{\infty j}^b(w))\in\mathbb{R} \quad (1\leq j\leq n_\infty(b))\] and \[N_{\infty j}^b\coloneq N_\infty(f)>0 \quad (1\leq j\leq n_\infty(b)).\] Then we have \[\sum_{j=1}^{n_\infty(b)} N_{\infty j}^b =\sum_{i=1}^m e_i =d(\mathcal{M})^b.\] For a point \(a_i\in D^{{\rm an}}\) and \(w\in W\subset B(b)^\circ\) let us count the total number of the critical points of the real-valued functions \(\operatorname{Re}f^w\vert_{\partial D(a_i)}\colon \partial D(a_i)\to\mathbb{R}\) \((f\in N_i^{>0},f\neq0)\). Namely we count them with the multiplicities \(N_i(f)\). Since \(\operatorname{Re}f^w\vert_{\partial D(a_i)}\) is a linear perturbation of \(- \operatorname{Re}f \vert_{\partial D(a_i)}\), after shrinking the disk \(D(a_i)\) if necessary, it suffices to count the total number of the critical points of the real-valued functions \(\operatorname{Re}f\vert_{\partial D(a_i)}\colon \partial D(a_i)\to\mathbb{R}\) \((f\in N_i^{>0},f\neq0)\). For a point \(\theta\in S_{a_i}X^{{\rm an}}\) we say that \(f_1,f_2\in (N_i^{>0})_\theta\) are equivalent if \(f_1\) is analytically continued to \(f_2\) along some path in the punctured disk \(B(a_i)^\circ\). Let \((N_i^{>0})_\theta^\sim\) be the quotient set of \((N_i^{>0})_\theta\) obtained by this equivalence relation. By analytic continuations, for \(\theta_1,\theta_2\in S_{a_i}X^{{\rm an}}\) there exists a (unique) bijection between \((N_i^{>0})_{\theta_1}^\sim\) and \((N_i^{>0})_{\theta_2}^\sim\). Hence by fixing a point \(\theta\in S_{a_i}X^{{\rm an}}\) we set \((N_i^{>0})^\sim\coloneq(N_i^{>0})_\theta^\sim\) for short. For \([f]\in(N_i^{>0})^\sim\) \((f\in(N_i^{>0})_\theta)\) denote by \(v([f])\) the number of the elements of \((N_i^{>0})_\theta\) equivalent to \(f\). Note that if \(f_1,f_2\in(N_i^{>0})_\theta\) are equivalent then \(N_i(f_1)=N_i(f_2)\) and \(\operatorname{ord}_{a_i}(f_1)=\operatorname{ord}_{a_i}(f_2)\). We thus obtain morphisms \[N_i\colon(N_i^{>0})^\sim \longrightarrow\mathbb{Z}_{>0},\quad \operatorname{ord}_{a_i}(\cdot)\colon(N_i^{>0})^\sim\left.\right\backslash\{0\} \longrightarrow\mathbb{Q}_{>0}.\] Then we can easily show that for \(w\in W\subset B(b)^\circ\) the total number of the critical points of the functions \(\operatorname{Re}f^w\vert_{\partial D(a_i)}\colon \partial D(a_i)\to\mathbb{R}\) \((f\in N_i^{>0},f\neq0)\) is equal to \[2\times\sum_{[f]\in (N_i^{>0})^\sim\left.\right\backslash\{0\}} N_i([f])\cdot v([f])\cdot\operatorname{ord}_{a_i}([f]).\] The number of the critical points with local maximal value is the half of it i.e. \[m_i(b)\coloneq\sum_{[f]\in (N_i^{>0})^\sim\left.\right\backslash\{0\}} N_i([f])\cdot v([f])\cdot\operatorname{ord}_{a_i}([f]) =\mathrm{irr}_{a_i}(\mathcal{M})\] and we denote by \(B_{ik}^+(w)\) \((1\leq k\leq m_i(b))\) the corresponding local maximal values. Note that they are bounded continuous functions on \(W\).

Similarly, also for the point \(a_\infty=\infty\in\widetilde{D}\), we define \((N_\infty^{>0})^\sim\) and the morphisms \(v\colon(N_\infty^{>0})^\sim\to\mathbb{Z}_{>0}\), \(N_{\infty}\colon(N_\infty^{>0})^\sim\to\mathbb{Z}_{>0}\), \(\operatorname{ord}_{a_\infty}(\cdot)\colon (N_\infty^{>0})^\sim\left.\right\backslash\{0\}\to\mathbb{Q}_{>0}\). Let \((N_{\infty,b}^{>0})^\sim\) be the image of \(N_{\infty,b}^{>0}\) in \((N_\infty^{>0})^\sim\). Then we can easily show that the total number of the critical points with local maximal value of the functions \(\operatorname{Re}f^w\vert_{\partial D(a_\infty)}\colon \partial D(a_\infty)\to\mathbb{R}\) \((f\in N_\infty^{>0})\) is equal to \[\begin{align} m_\infty(b)&\coloneq \sum_{[f]\in(N_\infty^{>0})^\sim,\;\operatorname{ord}_{\infty}(f)>1} N_\infty([f])\cdot v([f])\cdot\operatorname{ord}_\infty([f]) \notag \\ &+ \sum_{[f]\in(N_\infty^{>0})^\sim\left.\right\backslash(N_{\infty,b}^{>0})^\sim, \;\operatorname{ord}_\infty([f])\leq1} N_\infty([f])\cdot v([f]) \notag \\ &+ \sum_{[f]\in(N_{\infty,b}^{>0})^\sim,\;f\neq bz} N_\infty([f])\cdot v([f])\cdot\operatorname{ord}_\infty([f-bz]) \notag \\ &+N_\infty(bz). \end{align}\] Note that we have \(m_\infty(b)\geq N_\infty(bz)\). We denote by \(B_{\infty k}^+(w)\) \((1\leq k\leq m_\infty(b))\) the corresponding local maximal values. As in the proof of Theorem 1, by the Morse function \(\xi^w\colon\Omega\to\mathbb{R}\) \((w\in W)\) we define also real-valued functions \(c_i\colon W\to\mathbb{R}\), \(L_i^\pm\colon W\to\mathbb{R}\) \((1\leq i\leq l)\) and \(L_\infty^\pm\colon W\to\mathbb{R}\). Also these functions are bounded and continuous on \(W\).

Let \(\beta\in\mathbb{C}^\ast=\mathbb{C}\left.\right\backslash\{0\}\) be a non-zero complex number such that the sector \(W\subset B(b)^\circ\) is a sectorial neighborhood of the point \(b+\beta\cdot0\in S_bY^{{\rm an}}\) and fix a point \(w_0\in(b+\mathbb{R}_{>0}\beta)\cap W\). For \(w\in W\) let \(P(a_\infty)_+^w\in\partial D(a_\infty)\) be the point where the real-valued function \(\widetilde{\xi^w}\vert_{\partial D(a_\infty)}\colon \partial D(a_\infty)\to\mathbb{R}\) takes its maximal value \(L_\infty^+(w)\). Then for a sufficiently small \(\varepsilon_0>0\) we define an open subset \(U(W)\subset\Omega=X^{{\rm an}}\left.\right\backslash(D(a_1)\cup \dots\cup D(a_l)\cup D(a_\infty))\subset U^{{\rm an}}\) by \[U(W)\coloneq\Set*{z\in\Omega} {\widetilde{\xi^{w_0}}(z)>L_\infty^+(w_0)-\varepsilon_0}.\] It is clear that \(U(W)\) is convex and \(P(a_\infty)_+^{w_0}\in\overline{U(W)} \cap\partial D(a_\infty)\) (see Figure 6). Moreover, shrinking the sector \(W\subset B(b)^\circ\) if necessary, we may assume that \(W\) is a sectorial open neighborhood of the point \(b+\beta\cdot0\in S_bY^{{\rm an}}\) and for any \(w\in W\) we have \(P(a_\infty)_+^w\in\overline{U(W)} \cap\partial D(a_\infty)\). In order to describe the restriction \(\mathsf{E}_W\) of the enhanced sheaf \(\pi^{-1}\mathbb{C}_W\otimes{}^\mathsf{L}G\in{\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_{\overline{Y}^{{\rm an}}})\) to \(Y^{{\rm an}}\subset\overline{Y}^{{\rm an}}\) we set \[\begin{align} \begin{cases} \;\mathsf{E}_W^\prime\coloneq\pi^{-1}\mathbb{C}_W\otimes {\mathrm{R}}p_{2!}(p_1^{-1}G_{(X^{{\rm an}}\left.\right\backslash U(W))\times\mathbb{R}}^\circ \otimes\mathbb{C}_{\{t-s-\operatorname{Re}zw\geq0\}}[1]), \\ \;\mathsf{E}_W^{\prime\prime}\coloneq\pi^{-1}\mathbb{C}_W\otimes {\mathrm{R}}p_{2!}(p_1^{-1}G_{U(W)\times\mathbb{R}}^\circ \otimes\mathbb{C}_{\{t-s-\operatorname{Re}zw\geq0\}}[1]), \\ \end{cases} \end{align}\] and consider the distinguished triangle \[\mathsf{E}_W^{\prime\prime}\longrightarrow\mathsf{E}_W \longrightarrow\mathsf{E}_W^\prime\overset{+1}{\longrightarrow}.\] We need this decomposition of \(\mathsf{E}_W\), because we do not know so far how to calculate it directly. Then by applying the Morse theoretical method in the proof of Theorem 1 to our situation, we obtain an isomorphism \[\begin{align} \label{eq:sfEW} \mathsf{E}_W^\prime&\simeq \bigoplus_{i=1}^l\Biggl\{\bigoplus_{k=1}^{m_i(b)} \mathsf{E}_{W\left.\right|Y^{{\rm an}}}^{-B_{ik}^+} \oplus\bigl(\mathsf{E}_{W\left.\right|Y^{{\rm an}}} ^{-L_i^-+c\,\vartriangleright -L_i^-}[1] \bigr)^{r_i} \notag \\ &\oplus\bigl(\mathsf{E}_{W\left.\right|Y^{{\rm an}}}^{-L_i^-}\bigr)^{\mathop{\mathrm{rk}}(\mathcal{M})-r_i} \oplus\bigl(\mathsf{E}_{W\left.\right|Y^{{\rm an}}}^{-c_i}\bigr)^{r_i}\Biggr\} \oplus\Biggl(\bigoplus_{k=1}^{m_\infty(b)} \mathsf{E}_{W\left.\right|Y^{{\rm an}}}^{-B_{\infty k}^+}\Biggr) \notag \\ &\oplus \Biggl\{\bigoplus_{j=1}^{n_\infty(b)} \bigl(\mathsf{E}_{W\left.\right|Y^{{\rm an}}}^{-c_{\infty j}^b}\bigr) ^{N_{\infty j}^b}\Biggr\}. \end{align}\tag{18}\] Let us set \[\begin{align} \begin{cases} (\mathsf{E}_W^\prime)_\mathrm{reg}\coloneq &\displaystyle\bigoplus_{i=1}^l\Biggl\{ \bigoplus_{k=1}^{m_i(b)}\mathsf{E}_{W\left.\right|Y^{{\rm an}}}^{-B_{ik}^+} \oplus\bigl(\mathsf{E}_{W\left.\right|Y^{{\rm an}}} ^{-L_i^-+c\,\vartriangleright -L_i^-}[1]\bigr)^{r_i} \\ &\oplus\bigl(\mathsf{E}_{W\left.\right|Y^{{\rm an}}}^{-L_i^-}\bigr)^{\mathop{\mathrm{rk}}(\mathcal{M})-r_i} \oplus\bigl(\mathsf{E}_{W\left.\right|Y^{{\rm an}}}^{-c_i}\bigr) ^{r_i}\Biggr\} \oplus\Biggl( \displaystyle\bigoplus_{k=1}^{m_\infty(b)} \mathsf{E}_{W\left.\right|Y^{{\rm an}}}^{-B_{\infty k}^+}\Biggr), \\ (\mathsf{E}_W^\prime)_\mathrm{irr}\coloneq &\Biggl\{\displaystyle\bigoplus_{j=1}^{n_\infty(b)} \bigl(\mathsf{E}_{W\left.\right|Y^{{\rm an}}}^{-c_{\infty j}^b}\bigr) ^{N_{\infty j}^b}\Biggr\} \end{cases} \end{align}\] so that we have an isomorphism \(\mathsf{E}_W^\prime\simeq(\mathsf{E}_W^\prime)_\mathrm{reg} \oplus(\mathsf{E}_W^\prime)_\mathrm{irr}\). Then by Lemma 1 it is easy to see that the enhanced ind-sheaf \(\mathbb{C}_{Y^{\rm an}}^{\mathrm{E}}\overset{+}{\otimes}(\mathsf{E}_W^\prime)_\mathrm{reg}\) is isomorphic to \(( \mathbb{E}_{W\left.\right|Y^{{\rm an}}}^0)^{\nu_b}\), where we set \[\label{eq:natural} \nu_b\coloneq \Biggl\{\sum_{i=1}^l(\mathrm{irr}_{a_i}(\mathcal{M})+\mathop{\mathrm{rk}}(\mathcal{M}))\Biggr\} +m_\infty(b).\tag{19}\] We can also show that \[\label{eq:star} m_\infty(b)+ \biggl(\sum_{j=1}^{n_\infty(b)}N_{\infty j}^b\biggr) =e_\infty(b)+\mathop{\mathrm{rk}}(\mathcal{M}).\tag{20}\] Indeed, for \(f\in N_{\infty,b}^{>0}\) such that \(f\neq bz\) we set \[\lambda\coloneq\operatorname{ord}_\infty(f-bz) \quad \in(0,1).\] Then there exists a non-zero complex number \(\alpha\in\mathbb{C}\) such that \[f(z)=bz+\alpha z^\lambda+ (\textit{lower order terms})\] and hence for \(w\in W\) and \(z\in B(a_\infty)^\circ\) the condition \((f^w)^\prime(z)=0\) is equivalent to \[w-b=\alpha\lambda z^{\lambda-1}+(\textit{lower order terms}).\] This implies that we have \[\sum_{j=1}^{n_\infty(b)}N_{\infty j}^b =\sum_{[f]\in(N_{\infty,b}^{>0})^\sim,\;f\neq bz} N_\infty([f])\cdot v([f])\cdot(1-\operatorname{ord}_\infty([f-bz]))\] from which (20 ) immediately follows. Similarly, for \(\mathsf{E}_W^{\prime\prime}\) we obtain an isomorphism \[\label{eq:Ewprpr} \mathsf{E}_W^{\prime\prime}[1]\simeq (\mathsf{E}_{W\left.\right|Y^{{\rm an}}}^{-L_\infty^+})^{\mathop{\mathrm{rk}}(\mathcal{M})}.\tag{21}\] Then it follows from the distinguished triangle \[\mathsf{E}_W\longrightarrow\mathsf{E}_W^\prime\longrightarrow \mathsf{E}_W^{\prime\prime} [1] \overset{+1}{\longrightarrow}\] that there exists an exact sequence \[0\longrightarrow H^0\mathsf{E}_W\simeq\mathsf{E}_W\longrightarrow H^0\mathsf{E}_W^\prime\overset{\Phi_W}{\longrightarrow} H^0\mathsf{E}_W^{\prime\prime}[1]\simeq\mathsf{E}_W^{\prime\prime}[1] \longrightarrow0\] of enhanced sheaves on \(Y^{{\rm an}}\). Note that by Lemma 4 (ii) we have \[\mathbb{C}_{Y^{\rm an}}^{\mathrm{E}}\overset{+}{\otimes}H^0(\mathsf{E}_W^\prime)_\mathrm{irr}\simeq \bigoplus_{j=1}^{n_\infty(b)} (\mathbb{E}_{W\left.\right|Y^{\rm an}}^{-c_{\infty j}^b})^{N_{\infty j}^b} \simeq\bigoplus_{i=1}^m(\mathbb{E}_{W\left.\right|Y^{\rm an}}^{\operatorname{Re}h_i})^{e_i}\] and \[\label{eq:dMb} \sum_{j=1}^{n_{\infty}(b)}N_{\infty j}^b= \sum_{i=1}^m e_i=d(\mathcal{M})^b.\tag{22}\] Moreover there exist isomorphisms \[\mathbb{C}_{Y^{\rm an}}^{\mathrm{E}}\overset{+}{\otimes} H^0(\mathsf{E}_W)\simeq\pi^{-1}\mathbb{C}_W\otimes Sol_Y^{\mathrm{E}}(\mathcal{M}^\wedge)\] and \[\mathbb{C}_{Y^{\rm an}}^{\mathrm{E}}\overset{+}{\otimes}H^0(\mathsf{E}_W^{\prime\prime}[1]) \simeq(\mathbb{E}_{W\left.\right|Y^{\rm an}}^0)^{\mathop{\mathrm{rk}}(\mathcal{M})}.\] We thus obtain an exact sequence \[\begin{align} 0\longrightarrow\pi^{-1}\mathbb{C}_W\otimes Sol_Y^{\mathrm{E}}(\mathcal{M}^\wedge)\longrightarrow \biggl\{\bigoplus_{i=1}^m (\mathbb{E}_{W\left.\right|Y^{\rm an}}^{\operatorname{Re}h_i})^{e_i}\biggr\} \oplus(\mathbb{E}_{W\left.\right|Y^{\rm an}}^0)^{\nu_b} \\ \longrightarrow (\mathbb{E}_{W\left.\right|Y^{\rm an}}^0)^{\mathop{\mathrm{rk}}(\mathcal{M})}\longrightarrow0 \end{align}\] of enhanced ind-sheaves on \(Y^{\rm an}\). Then in view of (19 ) and (20 ) and Lemma 2, we can apply the multiplicity test functor in [@DK18] to obtain the first assertion of Theorem 2. If \(b\neq0\) and \(N_{\infty,b}^{>0}=\emptyset\), then we can replace \(W\subset B(b)\) by the open disk \(B(b)\) to obtain the last assertion of the theorem. This completes the proof of Theorem 2. 0◻
Next, let us prove Theorem 3. Recall that we have \[H^j_{\{b\}}(\mathcal{M}^\wedge)\simeq0 \quad (j\neq0,1)\] and for \(j=0,1\) the holonomic \(\mathcal{D}_Y\)-modules \(H^j_{\{b\}}(\mathcal{M}^\wedge)\) are direct sums of some copies of the standard one \[\mathcal{B}_{\{b\}\left.\right|Y}\coloneq H^1_{\{b\}}(\mathcal{O}_Y)\simeq\mathcal{D}_Y / \mathcal{D}_Y(w-b).\] Let \(k_0,k_1\in\mathbb{Z}_{\geq0}\) be the non-negative integers such that \[H_{\{b\}}^j(\mathcal{M}^\wedge)\simeq \mathcal{B}_{\{b\}\left.\right|Y}^{\oplus k_j}\quad (j=0,1).\] Then by applying the functor \(Sol_Y(\cdot)\) to the distinguished triangle \[\label{eq:dt-rsectb1} \tau^{\leq0}{\mathrm{R}}\Gamma_{\{b\}}(\mathcal{M}^\wedge) \simeq\Gamma_{\{b\}}(\mathcal{M}^\wedge)\longrightarrow {\mathrm{R}}\Gamma_{\{b\}}(\mathcal{M}^\wedge)\longrightarrow \tau^{\geq1}{\mathrm{R}}\Gamma_{\{b\}}(\mathcal{M}^\wedge) \simeq H_{\{b\}}^1(\mathcal{M}^\wedge)[-1] \overset{+1}{\longrightarrow},\tag{23}\] we obtain \[\chi_b(Sol_Y({\mathrm{R}}\Gamma_{\{b\}}(\mathcal{M}^\wedge)))=k_1-k_0\] where \(\chi_b(Sol_Y({\mathrm{R}}\Gamma_{\{b\}}(\mathcal{M}^\wedge)))\) is the local Euler-Poincaré index \[\chi_b(Sol_Y({\mathrm{R}}\Gamma_{\{b\}}(\mathcal{M}^\wedge)))\coloneq \sum_{j\in\mathbb{Z}}(-1)^j{\rm dim}_\mathbb{C}H^j Sol_Y({\mathrm{R}}\Gamma_{\{b\}}(\mathcal{M}^\wedge))_b\] of \(Sol_Y({\mathrm{R}}\Gamma_{\{b\}}(\mathcal{M}^\wedge))\) at \(b\in Y^{\rm an}\). On the other hand, it follows from the distinguished triangle \[{\mathrm{R}}\Gamma_{\{b\}}(\mathcal{M}^\wedge)\longrightarrow \mathcal{M}^\wedge\longrightarrow {\mathrm{R}}\Gamma_{Y\setminus\{b\}}(\mathcal{M}^\wedge) \simeq\Gamma_{Y\setminus\{b\}} (\mathcal{M}^\wedge)\overset{+1}{\longrightarrow}\] that we have \[\chi_b(Sol_Y(\mathcal{M}^\wedge)) =\chi_b(Sol_Y(\Gamma_{Y\setminus\{b\}}(\mathcal{M}^\wedge))) +\chi_b(Sol_Y({\mathrm{R}}\Gamma_{\{b\}}(\mathcal{M}^\wedge))).\] Moreover by Kashiwara’s index theorem for holonomic \(\mathcal{D}\)-modules (see [@Kas83]) we have \[\begin{align} \chi_b(Sol_Y(\mathcal{M}^\wedge)) &= \mathop{\mathrm{mult}}_{T_Y^\ast Y}(\mathcal{M}^\wedge) -\mathop{\mathrm{mult}}_{T_b^\ast Y}(\mathcal{M}^\wedge) \notag \\ &=\mathop{\mathrm{rk}}(\mathcal{M}^\wedge)-\mathop{\mathrm{mult}}_{T_b^\ast Y}(\mathcal{M}^\wedge) \notag \\ &=d(\mathcal{M})-\mathop{\mathrm{mult}}_{T_b^\ast Y}(\mathcal{M}^\wedge) \end{align}\] and by applying Proposition 9 to the meromorphic connection \(\Gamma_{Y\setminus\{b\}}(\mathcal{M}^\wedge)\) along \(b\in Y\) we obtain \[\chi_b(Sol_Y(\Gamma_{Y\setminus\{b\}}(\mathcal{M}^\wedge))) =-\mathrm{irr}_b(\Gamma_{Y\setminus\{b\}}(\mathcal{M}^\wedge)).\] Combining these results together, we have \[\label{eq:indexthm} \mathop{\mathrm{mult}}_{T_b^\ast Y}(\mathcal{M}^\wedge) =d(\mathcal{M})+\mathrm{irr}_b(\Gamma_{Y\setminus\{b\}}(\mathcal{M}^\wedge)) -(k_1-k_0).\tag{24}\] Therefore, for the proof of the first assertion, it suffices to compute the local Euler-Poincaré index \[\chi_b(Sol_Y({\mathrm{R}}\Gamma_{\{b\}}(\mathcal{M}^\wedge)))=k_1-k_0\] of \(Sol_Y({\mathrm{R}}\Gamma_{\{b\}}(\mathcal{M}^\wedge))\) at \(b\in Y^{\rm an}\). Moreover by [@IT20b], we may assume that \(b\neq0\). As in the proof of Theorem 2, for a sufficiently small \(\varepsilon_0>0\) we define an open subset \(U(b)\subset\Omega\) by \[U(b)\coloneq\Set*{z\in\Omega} {\widetilde{\xi^b}(z)>L_\infty^+(b)-\varepsilon_0}\] and set \[\begin{align} \begin{cases} \;\mathsf{E}_b\coloneq\pi^{-1}\mathbb{C}_{\{b\}}\otimes{}^\mathsf{L}G, \\ \;\mathsf{E}_b^\prime\coloneq\pi^{-1}\mathbb{C}_{\{b\}}\otimes {\mathrm{R}}p_{2!}(p_1^{-1}G_{(X^{{\rm an}}\left.\right\backslash U(b))\times\mathbb{R}}^\circ \otimes\mathbb{C}_{\{t-s-\operatorname{Re}zw\geq0\}}[1]), \\ \;\mathsf{E}_b^{\prime\prime}\coloneq\pi^{-1}\mathbb{C}_{\{b\}}\otimes {\mathrm{R}}p_{2!}(p_1^{-1}G_{U(b)\times\mathbb{R}}^\circ \otimes\mathbb{C}_{\{t-s-\operatorname{Re}zw\geq0\}}[1]). \end{cases} \end{align}\] Note that we have \[\begin{align} \mathbb{C}_{Y^{\rm an}}^{\mathrm{E}}\overset{+}{\otimes}\mathsf{E}_b&\simeq\pi^{-1}\mathbb{C}_{\{b\}}\otimes Sol_Y^{\mathrm{E}}(\mathcal{M}^\wedge) \notag \\ &\simeq Sol_Y^{\mathrm{E}}({\mathrm{R}}\Gamma_{\{b\}}(\mathcal{M}^\wedge)). \end{align}\] Let us consider the distinguished triangle \[\label{eq:sfEb} \mathsf{E}_b^{\prime\prime}\longrightarrow\mathsf{E}_b \longrightarrow\mathsf{E}_b^\prime\overset{+1}{\longrightarrow}.\tag{25}\] By applying the Morse theoretical method in the proof of Theorem 1 to our situation, we can show that there exist isomorphisms \[\label{eq:sfEb1} \mathbb{C}_{Y^{\rm an}}^{\mathrm{E}}\overset{+}{\otimes}\mathsf{E}_b^\prime\simeq \pi^{-1}\mathbb{C}_{\{b\}}\otimes(\mathbb{C}_{Y^{\rm an}}^{\mathrm{E}})^{\nu_b-N_\infty(bz)}\tag{26}\] and \[\label{eq:sfEb2} \mathbb{C}_{Y^{\rm an}}^{\mathrm{E}}\overset{+}{\otimes}(\mathsf{E}_b^{\prime\prime}[1])\simeq \pi^{-1}\mathbb{C}_{\{b\}}\otimes(\mathbb{C}_{Y^{\rm an}}^{\mathrm{E}})^{\mathop{\mathrm{rk}}(\mathcal{M})}.\tag{27}\] Indeed, for \(f\in N_{\infty,b}^{>0}\) such that \(f\neq bz\) there exists a non-zero Puiseux germ \(f_\mathrm{red}\in\mathcal{P}_{S_\infty\overline{Y}^{\rm an}}^\prime\) such that \(0<\operatorname{ord}_\infty(f_\mathrm{red})<1\) and \[f(z)=bz+f_\mathrm{red}(z).\] This implies that for a sufficiently small punctured disk \(D(a_\infty)^\circ\) centered at the point \(a_\infty=\infty\in\overline{Y}^{\rm an}\), the Morse function \[\operatorname{Re}f^b(z)=-\operatorname{Re}f_\mathrm{red}(z)\] has no critical point in \(B(a_\infty)^\circ=\operatorname{Int}D(a_\infty)^\circ\). Moreover, for the linear factor \(f=bz\in N_{\infty,b}^{>0}\) the Morse function \(\operatorname{Re}f^b(z)\) is identically zero on \(D(a_\infty)^\circ\) and hence for any \(t\in\mathbb{R}\) its level set at \(t\) is equal to \[\Set*{z\in D(a_\infty)^\circ}{\operatorname{Re}f^b(z)\leq t}= \begin{cases} \;D(a_\infty)^\circ & (t\geq0), \\ \;\emptyset & (t<0). \end{cases}\] As in the proof of Lemma 9, for any local system \(L_0\) on \(D(a_\infty)^\circ\), we can show the vanishing \[{\mathrm{R}}\Gamma_c(D(a_\infty)^\circ; L_0)\simeq0.\] Therefore, the isomorphism (26 ) is obtained in the same way as (18 ). We can show the isomorphism (27 ) as we showed (21 ). By applying the functor \(Sol_Y^{\mathrm{E}}(\cdot)\) to the distinguished triangle (23 ) we obtain a distinguished triangle \[\label{eq:dt-rsectb2} Sol_Y^{\mathrm{E}}(\mathcal{B}_{\{b\}\left.\right|Y}^{\oplus k_1})[1]\longrightarrow Sol_Y^{\mathrm{E}}({\mathrm{R}}\Gamma_{\{b\}}(\mathcal{M}^\wedge))\longrightarrow Sol_Y^{\mathrm{E}}(\mathcal{B}_{\{b\}\left.\right|Y}^{\oplus k_0}) \overset{+1}{\longrightarrow}.\tag{28}\] Note that for the regular holonomic \(\mathcal{D}_Y\)-modules \(\mathcal{B}_{\{b\}\left.\right|Y}\) by Proposition 4 (iv) there exists an isomorphism \[Sol_Y^{\mathrm{E}}(\mathcal{B}_{\{b\}\left.\right|Y}) \simeq \pi^{-1}\mathbb{C}_{\{b\}}\otimes \mathbb{C}_{Y^{\rm an}}^{\mathrm{E}}[-1].\] Then we obtain \[\begin{align} H^j(\mathbb{C}_{Y^{\rm an}}^{\mathrm{E}}\overset{+}{\otimes}\mathsf{E}_b) &\simeq H^j Sol_Y^{\mathrm{E}}({\mathrm{R}}\Gamma_{\{b\}}(\mathcal{M}^\wedge)) \notag \\ &\simeq \begin{cases} \;\pi^{-1}\mathbb{C}_{\{b\}}\otimes(\mathbb{C}_{Y^{\rm an}}^{\mathrm{E}})^{k_{1-j}} & (j=0,1), \\ \;0 & (otherwise). \end{cases} \end{align}\] Therefore, it follows from the distinguished triangle (25 ) that we have \[k_1-k_0=\nu_b-N_\infty(bz)-\mathop{\mathrm{rk}}(\mathcal{M}).\] Combining this equation with Corollary 1, (19 ), (20 ) and (22 ), we obtain the first assertion of Theorem 2. Now let us consider the case where \(N_{\infty,b}^{>0}\) does not contain the linear factor \(bz\). In this case, as in the proof of (26 ), we can directly show \[\begin{align} \label{eq:nolinear} Sol_Y^{\mathrm{E}}({\mathrm{R}}\Gamma_{\{b\}}(\mathcal{M}^\wedge)) &\simeq \mathbb{C}_{Y^{\rm an}}^{\mathrm{E}}\overset{+}{\otimes}\mathsf{E}_b \notag \\ &\simeq \pi^{-1}\mathbb{C}_{\{b\}}\otimes(\mathbb{C}_{Y^{\rm an}}^{\mathrm{E}})^{d(\mathcal{M})-d(\mathcal{M})^b}. \end{align}\tag{29}\] Then by comparing (29 ) with (28 ) we obtain \(k_0=0\), \(k_1=d(\mathcal{M})-d(\mathcal{M})^b\) and hence the second assertion. This completes the proof of Theorem 3. 0◻
We recall the following definition of Verdier [@Ver83] in the simplest case of \(N=1\).

Then we obtain the following very simple consequence of Theorem 3 (see [@IT20b] for a related result in higher dimensions).

We also obtain the following consequence of the results of this section.

4.4 Generalizations of Theorems 1 and 2 to arbitrary holonomic \(\mathcal{D}\)-modules↩︎

In this subsection, we prove results similar to Theorems 1 and 2 in the case where the algebraic holonomic \(\mathcal{D}\)-module \(\mathcal{M}\in\mathrm{Mod}_{\mathrm{hol}}(\mathcal{D}_X)\) on \(X=\mathbb{C}_z\) is not necessarily a meromorphic connection. Let \(D\subset X\) be its singular support \(\mathrm{sing.supp}(\mathcal{M})\) and \(a_1,\dots a_l\in D^{\rm an}\) the points in \(D^{\rm an}\subset X^{\rm an}\) and set \(U\coloneq X\left.\right\backslash D\). Recall that there exists a distinguished triangle \[\label{eq:locM} {\mathrm{R}}\Gamma_D(\mathcal{M})\longrightarrow \mathcal{M}\longrightarrow \Gamma_U(\mathcal{M})\overset{+1}{\longrightarrow}.\tag{30}\] For \(1\leq i\leq l\), let \(k_0(a_i)\), \(k_1(a_i)\geq0\) be the non-negative integers such that \[H_D^j(\mathcal{M})\simeq \bigoplus_{i=1}^l \mathcal{B}_{\{a_i\}\left.\right|X}^{\oplus k_j(a_i)} \quad (j=0,1).\] Then it follows from the distinguished triangle \[\Gamma_D(\mathcal{M})\longrightarrow{\mathrm{R}}\Gamma_D(\mathcal{M}) \longrightarrow H_D^1(\mathcal{M})[-1]\overset{+1}{\longrightarrow},\] the isomorphisms \[\left(H_D^j(\mathcal{M})\right)^\wedge\simeq \bigoplus_{i=1}^l \left(\mathcal{O}_Y e^{-a_i w}\right)^{\oplus k_j(a_i)} \quad (j=0,1)\] and Proposition 4 (iii) that there exists a distinguished triangle \[\label{eq:holD1} \bigoplus_{i=1}^l\bigl(\mathbb{E}_{Y^{\rm an}\left.\right|\overline{Y}^{\rm an}}^{-\operatorname{Re}a_i w}\bigr)^{\oplus k_1(a_i)}[1] \longrightarrow Sol_{\overline{Y}}^{\mathrm{E}}\bigl(\widetilde{{\mathrm{R}}\Gamma_D(\mathcal{M})^\wedge}\bigr) \longrightarrow \bigoplus_{i=1}^l\bigl(\mathbb{E}_{Y^{\rm an}\left.\right|\overline{Y}^{\rm an}}^{-\operatorname{Re}a_i w}\bigr)^{\oplus k_0(a_i)} \overset{+1}{\longrightarrow}.\tag{31}\] Applying the Fourier transform and the functor \({\rm Sol}_{\overline{Y}}^{\mathrm{E}}(\widetilde{(\cdot)})\) to (30 ), we also obtain \[\label{eq:holD2} Sol_{\overline{Y}}^{\mathrm{E}}\bigl(\widetilde{\Gamma_U(\mathcal{M})^\wedge}\bigr) \longrightarrow Sol_{\overline{Y}}^{\mathrm{E}}(\widetilde{\mathcal{M}^\wedge}) \longrightarrow Sol_{\overline{Y}}^{\mathrm{E}}\bigl(\widetilde{{\mathrm{R}}\Gamma_D(\mathcal{M})^\wedge}\bigr) \overset{+1}{\longrightarrow}.\tag{32}\] Note that \(\Gamma_U(\mathcal{M})\) is an algebraic meromorphic connection along the divisor \(D\subset X\). Then we obtain the following lemma.

By the multiplicities \(N_i\) (\(1\leq i\leq l\)) and \(N_{\infty}\) of the meromorphic connection \(\Gamma_U(\mathcal{M})\) we set \[\begin{align} \mathrm{CC}_{\mathrm{irr}}(\mathcal{M})_i &\coloneq \sum_{f\in N_i^{>0}}N_i(f)\cdot[\Lambda_i^f] \\ &+ \Bigl\{ \mathop{\mathrm{mult}}_{T_{a_i}^\ast X}(\mathcal{M}) + r_i - \mathop{\mathrm{rk}}(\mathcal{M}) - \mathrm{irr}_{a_i}(\Gamma_U(\mathcal{M})) \Bigr\} \cdot[T_{\{a_i\}}^\ast X^{{\rm an}}] \quad (1\leq i\leq l) \end{align}\] and \[\mathrm{CC}_{\mathrm{irr}}(\mathcal{M})_\infty \coloneq \sum_{f\in N_\infty^{>0}}N_\infty(f)\cdot[\Lambda_\infty^f] =\mathrm{CC}_{\mathrm{irr}}(\Gamma_U(\mathcal{M}))_\infty.\] Then we define the irregular characteristic cycle \(\mathrm{CC}_{\mathrm{irr}}(\mathcal{M})\) of the holonomic \(\mathcal{D}\)-module \(\mathcal{M}\in\mathrm{Mod}_{\mathrm{hol}}(\mathcal{D}_X)\) by \[\mathrm{CC}_{\mathrm{irr}}(\mathcal{M})\coloneq\mathrm{CC}_{\mathrm{irr}}(\mathcal{M})_\infty + \sum_{i=1}^l\mathrm{CC}_{\mathrm{irr}}(\mathcal{M})_i.\] Define \(g_i, d_i\) (\(1\leq i\leq n\)) and \(d(\mathcal{M})= \sum_{i=1}^n d_i\) by \(\mathrm{CC}_{\mathrm{irr}}(\mathcal{M})\) as in the case of meromorphic connections. Then, as in the proof of Lemma 12, we obtain the following generalization of Theorem 1 to holonomic \(\mathcal{D}\)-modules.

For a point \(b\in Y^{\rm an}\) we also set \[\mathrm{CC}_{\mathrm{irr}}(\mathcal{M})^b\coloneq\sum_{f\in N_{\infty,b}^{>0}}N_\infty(f)\cdot[\Lambda_\infty^f] =\mathrm{CC}_{\mathrm{irr}}(\Gamma_U(\mathcal{M}))^b.\] Then by defining \(h_i, e_i\) (\(1\leq i\leq m\)) and \(d(\mathcal{M})^b= \sum_{i=1}^m e_i\) by \(\mathrm{CC}_{\mathrm{irr}}(\mathcal{M})^b\), Theorem 2 is also extended to the case of holonomic \(\mathcal{D}\)-modules as follows.

5 Fourier transforms of \(\mathcal{D}\)-modules and rapid decay homology cycles↩︎

In this section, we first describe the stalks of the solution complexes of the Fourier transforms of holonomic \(\mathcal{D}\)-modules at generic points by the theory of rapid decay homology groups developed by Bloch-Esnault [@BE04a] and Hien [@Hi07; @Hi09]. Then we construct their natural bases via a twisted Morse theory similar to the one that we used in the proof of Theorem 1. For the basic properties of twisted homology groups, we refer to Aomoto-Kita [@AK11] and Pajitnov [@Paj06].

5.1 Rapid decay homology groups↩︎

In this subsection, we recall the theory of rapid decay homology groups developed by [@BE04a], [@Hi07] and [@Hi09] in the simplest case of dimension one. Let \(U\) be a smooth algebraic curve over \(\mathbb{C}\) and \((\mathcal{E},\nabla)\) \((\nabla\colon\mathcal{E}\to\Omega_U^1\otimes_{\mathcal{O}_U}\mathcal{E})\) an algebraic integrable connection on it. Let \(i\colon U\hookrightarrow Z\) be a smooth compactification of \(U\) and set \(D\coloneq Z\left.\right\backslash U\). Then for the underlying complex manifold \(Z^{{\rm an}}\) of \(Z\) and the subset \(D^{{\rm an}}\subset Z^{{\rm an}}\) let \(\varpi\colon\widetilde{Z}\to Z^{{\rm an}}\) be the real oriented blow-up of \(Z^{{\rm an}}\) along \(D^{{\rm an}}\). Recall that for a point \(a\in D^{{\rm an}}\) the subset \(\varpi^{-1}(a)\subset\widetilde{Z}\) of \(\widetilde{Z}\) is isomorphic to a circle \(S^1\). For \(p\geq0\) and a subset \(B\subset\widetilde{Z}\) denote by \(S_p(B)\) the \(\mathbb{C}\)-vector space generated by the piecewise smooth maps \(C\colon\Delta^p\to B\) from the \(p\)-dimensional simplex \(\Delta^p\). We denote by \(\mathcal{C}_{\widetilde{Z},\varpi^{-1}(D^{{\rm an}})}^{-p}\) the sheaf on \(\widetilde{Z}\) associated to the presheaf \[\begin{align} V \longmapsto S_p\bigl(\widetilde{Z},(\widetilde{Z}\left.\right\backslash V) \cup\varpi^{-1}(D^{{\rm an}})\bigr) = S_p\bigl(\widetilde{Z}\bigr)/S_p\bigl((\widetilde{Z}\left.\right\backslash V) \cup\varpi^{-1}(D^{{\rm an}})\bigr). \end{align}\] Now let \[\begin{align} L\coloneq H^{-1}DR_U(\mathcal{E})= \operatorname{Ker}\Bigl\{\nabla^{{\rm an}}\colon\mathcal{E}^{{\rm an}}\rightarrow \Omega_{U^{{\rm an}}}^1\otimes_{\mathcal{O}_{U^{{\rm an}}}}\mathcal{E}^{{\rm an}}\Bigr\} \end{align}\] be the sheaf of the horizontal sections of the analytic connection \((\mathcal{E}^{{\rm an}},\nabla^{{\rm an}})\) associated to \((\mathcal{E},\nabla)\) and \(\iota\colon U^{{\rm an}}\hookrightarrow\widetilde{Z}\) the inclusion map. Then \(\widetilde{L}\coloneq\iota_\ast L\) is a local system on \(\widetilde{Z}\). We define the sheaf \(\mathcal{C}_{\widetilde{Z}, \varpi^{-1}(D^{{\rm an}})}^{-p}(\widetilde{L})\) of relative twisted \(p\)-chains on the pair \(\bigl(\widetilde{Z},\varpi^{-1}(D^{{\rm an}})\bigr)\) with value in \(\widetilde{L}\) by \[\begin{align} \mathcal{C}_{\widetilde{Z},\varpi^{-1}(D^{{\rm an}})}^{-p}(\widetilde{L}) \coloneq \mathcal{C}_{\widetilde{Z},\varpi^{-1}(D^{{\rm an}})}^{-p} \otimes_{\mathbb{C}_{\widetilde{Z}}}\widetilde{L}. \end{align}\]

Note that this definition does not depend on the local coordinate \(x\) nor the local trivialization of \((i_\ast\mathcal{E})^{{\rm an}}\). We denote by \(\mathcal{C}_{\widetilde{Z},\varpi^{-1}(D^{{\rm an}})}^{{\rm rd},-p}(\widetilde{L})\) the subsheaf of \(\mathcal{C}_{\widetilde{Z},\varpi^{-1}(D^{{\rm an}})}^{-p}(\widetilde{L})\) consisting of rapid decay \(p\)-chains and set \[\begin{align} S_p^{{\rm rd}}(U^{{\rm an}};\mathcal{E},\nabla) \coloneq \Gamma(\widetilde{Z};\mathcal{C}_{\widetilde{Z},\varpi^{-1}(D^{{\rm an}})}^{{\rm rd},-p}(\widetilde{L})). \end{align}\] We thus obtain a complex \[\begin{align} S_{\bullet}^{{\rm rd}}(U^{{\rm an}};\mathcal{E},\nabla) := \Bigl[ 0 \longleftarrow S_{0}^{{\rm rd}}(U^{{\rm an}};\mathcal{E},\nabla) \longleftarrow S_{1}^{{\rm rd}}(U^{{\rm an}};\mathcal{E},\nabla) \longleftarrow \cdots \cdots \Bigr]. \end{align}\] We call \[\begin{align} H_p^{{\rm rd}}(U^{{\rm an}};\mathcal{E},\nabla) \coloneq H_p\left[S_{\bullet}^{{\rm rd}}(U^{{\rm an}};\mathcal{E},\nabla)\right] \quad (p\in\mathbb{Z}) \end{align}\] the rapid decay homology groups of the integrable connection \((\mathcal{E},\nabla)\) on \(U\). For \((\mathcal{E},\nabla)\) we denote by \(H_{\mathrm{dR}}^p(U;\mathcal{E},\nabla)\) \((p\in\mathbb{Z})\) the algebraic de Rham cohomology groups associated to it. Then we have the following celebrated theorem of [@BE04a], [@Hi07] and [@Hi09].

By (the proof of) [@ET15], we have also a purely topological reinterpretation of \(H_p(U^{{\rm an}};\mathcal{E},\nabla)\) \((p\in\mathbb{Z})\) in terms of relative twisted homology groups, which will be used in Section 5.3.

5.2 Fourier transforms of \(\mathcal{D}\)-modules and rapid decay homologies↩︎

In this subsection, we describe the stalks of the solution complexes of the Fourier transforms of holonomic \(\mathcal{D}\)-modules at generic points. Our results below are inspired by those in Hien-Roucairol [@HR08]. Assume that \(X=\mathbb{C}_z^N\) and \(Y=\mathbb{C}_w^N\) and regard them algebraic varieties over \(\mathbb{C}\) endowed with the Zariski topology. Let \(U\subset X\) be an affine open subset and \(j\colon U\hookrightarrow X\) the inclusion map. Then for an integrable connection \(\mathcal{N}\in \mathrm{Mod}_{\mathrm{hol}}(\mathcal{D}_U)\) on \(U\) we set \(\mathcal{M}\coloneq\mathbf{D}j_\ast(\mathcal{N})\simeq j_\ast\mathcal{N}\in\mathrm{Mod}_{\mathrm{hol}}(\mathcal{D}_X)\) and consider its Fourier transform \(\mathcal{M}^\wedge\in\mathrm{Mod}_{\mathrm{hol}}(\mathcal{D}_Y)\). Let \[\begin{align} X \overset{\,p}{\longleftarrow} X\times Y \overset{q}{\longrightarrow} Y \end{align}\] be the projections. Then by Lemma 7 we have an isomorphism \[\begin{align} \mathcal{M}^\wedge \simeq \mathbf{D}q_\ast (\mathbf{D}p^\ast\mathcal{M}\overset{D}{\otimes}\mathcal{O}_{X\times Y}e^{-\inprod{z,w}}). \end{align}\] Let \(b\in Y\) be a point such that \(b\notin\mathrm{sing.supp}(\mathcal{M}^\wedge)\). Since \(\mathcal{M}^\wedge\) is an integrable connection and hence regular on a Zariski open neighborhood of \(b\), by [@HTT08] there exist isomorphisms \[\begin{align} \label{eq-T9} Sol_Y(\mathcal{M}^\wedge)_b &\simeq \Bigl\{DR_Y(\mathbb{D}_Y(\mathcal{M}^\wedge))\Bigr\}_b[-N] \notag \\ &\simeq DR_{\{b\}}(\mathbb{D}_{\{b\}}\circ\mathbf{D}i_b^\ast\circ\mathbb{D}_Y) (\mathbb{D}_Y(\mathcal{M}^\wedge)) \notag \\ &\simeq \bigl[\mathbf{D}i_b^\ast(\mathcal{M}^\wedge)\bigr]^\ast. \end{align}\tag{33}\] Let \(\widetilde{i_b}\colon X\times\{b\}\hookrightarrow X\times Y\) be the inclusion map and consider the Cartesian diagram \[\vcenter{ \xymatrix@M=7pt{ X\times\{b\} \ar@{^{(}->}[r]^-{\widetilde{i_b}} \ar@{->}[d]_-{q_b} \ar@{}[dr]|\square & X\times Y \ar@{->}[d]^-{q} \\ \{b\}\ar@{^{(}->}[r]_-{i_b} & Y. }}\] Let us identity the projection \(q_b\colon X\times\{b\}\rightarrow\{b\}\) with the map \(a_X\colon X\rightarrow \{\mathrm{pt}\}\) to a point. Then for the map \(a_U\colon U\rightarrow\{\mathrm{pt}\}\) to a point by [@HTT08] we obtain isomorphisms \[\begin{align} \mathbf{D}i_b^\ast(\mathcal{M}^\wedge) &\simeq \mathbf{D}i_b^\ast\mathbf{D}q_\ast (\mathbf{D}p^\ast\mathcal{M}\overset{D}{\otimes}\mathcal{O}_{X\times Y}e^{-\inprod{z,w}}) \notag \\ &\simeq \mathbf{D}a_{X\ast}\mathbf{D}\widetilde{i_b}^\ast (\mathbf{D}p^\ast\mathcal{M}\overset{D}{\otimes}\mathcal{O}_{X\times Y}e^{-\inprod{z,w}}) \notag \\ &\simeq \mathbf{D}a_{X\ast}(\mathcal{M}\overset{D}{\otimes}\mathcal{O}_Xe^{-\inprod{z,b}}) \notag \\ &\simeq \mathbf{D}a_{X\ast} (\mathbf{D}j_\ast(\mathcal{N})\overset{D}{\otimes}\mathcal{O}_Xe^{-\inprod{z,b}}) \notag \\ &\simeq \mathbf{D}a_{X\ast}\mathbf{D}j_\ast (\mathcal{N}\overset{D}{\otimes}\mathcal{O}_Ue^{-\inprod{z,b}}) \notag \\ &\simeq \mathbf{D}a_{U\ast}(\mathcal{N}\overset{D}{\otimes}\mathcal{O}_Ue^{-\inprod{z,b}}). \end{align}\] Combining this with (33 ) and Theorem 14 we finally get \[\begin{align} Sol_Y(\mathcal{M}^\wedge)_b &\simeq H^0Sol_Y(\mathcal{M}^\wedge)_b \notag \\ &\simeq \Bigl[H^0\mathbf{D}a_{U\ast} (\mathcal{N}\overset{D}{\otimes}\mathcal{O}_Ue^{-\inprod{z,b}})\Bigr]^\ast \notag \\ &\simeq H_N^{{\rm rd}}(U^{{\rm an}};\mathcal{E}_b^\ast,\nabla_b^\ast), \end{align}\] where we used the standard fact that \[\begin{align} H^j\mathbf{D}a_{U\ast}(\mathcal{N}\overset{D}{\otimes}\mathcal{O}_Ue^{-\inprod{z,b}}) \simeq H^j{\mathrm{R}}\Gamma(U;\mathcal{D}_{\{\mathrm{pt}\}\leftarrow U} \overset{L}{\otimes}_{\mathcal{D}_U} (\mathcal{N}\overset{D}{\otimes}\mathcal{O}_Ue^{-\inprod{z,b}})) \quad (j\in\mathbb{Z}) \end{align}\] are the algebraic de Rham cohomology groups of the integrable connection \((\mathcal{E}_b,\nabla_b)\) on \(U\) defined by \[\begin{align} \mathcal{E}_b \coloneq \mathcal{N}\overset{D}{\otimes}\mathcal{O}_Ue^{-\inprod{z,b}} \simeq \mathcal{N}\otimes_{\mathcal{O}_U}\mathcal{O}_Ue^{-\inprod{z,b}} \end{align}\] and denoted \((\mathcal{E}_b^\ast,\nabla_b^\ast)\) its dual connection. See [@HTT08] for the details.

5.3 A Morse theoretical construction of rapid decay cycles in dimension one↩︎

In this subsection, inheriting the notations in Subsection 5.2 we assume also that the dimension \(N\) of \(X=\mathbb{C}^N\) is one and construct a natural basis of the rapid decay homology groups \(H_1^{{\rm rd}}(U^{{\rm an}};\mathcal{E}_b^\ast,\nabla_b^\ast)\) by using a twisted Morse theory similar to the one that we used in the proof of Theorem 1. For the meromorphic connection \(\mathcal{M}=\mathbf{D}j_\ast\mathcal{N}\simeq j_\ast\mathcal{N}\in\mathrm{Mod}_{\mathrm{hol}}(\mathcal{D}_X)\) set \(D\coloneq {\rm sing.supp} ( \mathcal{M}) \subset X\) and let \(a_1,a_2, \dots,a_l\) be the points in \(D^{{\rm an}}\subset X^{{\rm an}}\). Moreover we set \(a_\infty\coloneq\infty\in\overline{X}^{{\rm an}}\) and \(\widetilde{D}\coloneq D^{{\rm an}}\sqcup\{a_\infty\}\) as in Section 4 and inherit the notations related to \(\mathcal{M}\) there. Set \(Z\coloneq\overline{X}^{{\rm an}}\) and let \(\varpi\colon\widetilde{Z}\rightarrow Z\) be the real oriented blow-up of \(Z\) along the divisor \(\widetilde{D}=\{a_1,a_2,\dots,a_l,a_\infty\}\). Let \(B(b_{\infty})^\circ \subset Y^{{\rm an}}=\mathbb{C}\) be a sufficiently small punctured disk in \(\overline{Y}^{{\rm an}}\) centered at the point \(b_\infty=\infty\in\overline{Y}^{{\rm an}}\) and assume that \(b\in B(b_{\infty})^\circ\). Then in order to construct a basis of the rapid decay homology group \(H_1^{{\rm rd}}(U^{{\rm an}};\mathcal{E}_b^\ast,\nabla_b^\ast)\) for the integrable connection \[\begin{align} \mathcal{E}_b = \mathcal{N}\otimes_{\mathcal{O}_U}\mathcal{O}_Ue^{-bz} \end{align}\] on \(U=X \setminus D\) we first consider the problem on a neighborhood of each point \(a_i\in D^{{\rm an}}\). Let \(D(a_{i})^\circ \subset V_1\cup\dots\cup V_d\) be the open covering of the punctured disk \(D(a_i)^\circ\) centered at \(a_i\) in the proof of Theorem 1 and for the inclusion map \(\iota \colon U^{{\rm an}} \hookrightarrow\widetilde{Z}\) consider the local system \(\widetilde{L}\coloneq \iota_\ast Sol_U(\mathcal{N})\) on \(\widetilde{Z}\). Here we assume also that \[\begin{align} B(a_i)^\circ = V_1 \cup V_2 \cup \dots \cup V_d \end{align}\] and set \(B(a_i):= B(a_i)^\circ \sqcup \{ b \}\). For the sector \(V_j\) set \(\widetilde{V_j}\coloneq\operatorname{Int}\bigl(\overline{i(V_j)}\bigr)\subset\widetilde{Z}\) and let \(f_1, f_2, \dots,f_{m_i} \in N_i^{>0}(V_j)\) be the exponential factors of \(\mathcal{M}\) on \(V_j\). Moreover for each \(f_k\) \((1\leq k\leq m_i)\) we set \[\begin{align} P_k \coloneq \varpi^{-1}(a_i)\cap \overline{ \iota \Set*{z\in V_j}{\operatorname{Re}(f_k(z)-bz)\geq 0}} \qquad \subset \varpi^{-1}(a_i) \simeq S^1 \end{align}\] and \[\begin{align} Q_k\coloneq ( \varpi^{-1} (a_i) \cap \widetilde{V_j} ) \setminus P_k \quad \subset \varpi^{-1} (a_i) \cap \widetilde{V_j} \quad \subset \varpi^{-1}(a_i)\simeq S^1. \end{align}\] Then in view of (the proof of) [@ET15], for the reinterpretation of \(H_1^{{\rm rd}}(U^{{\rm an}};\mathcal{E}_b^\ast,\nabla_b^\ast)\) in terms of relative twisted homology groups as in it on the open subset \(\widetilde{V_j}\subset\widetilde{Z}\) of \(\widetilde{Z}\), it suffices to consider the relative twisted chains \[\begin{align} \bigoplus_{k=1}^{m_i}S_p(V_j\cup Q_k ,Q_k; \mathbb{C}_{\widetilde{Z}}^{N_i(f_k)} ) \qquad (p \in \mathbb{Z}). \end{align}\] Namely we consider the singular 1-chains on \(V_j\cup Q_k\) modulo those on \(Q_k\). Set \(R_i:= \sum_{k=1}^{m_i} N_i(f_k)\). Then we recall that for two adjacent sectors \(V_j\) and \(V_{j^\prime}\) such that \(V_j\cap V_{j^\prime}\neq\emptyset\) after renumbering the exponential factors \(f_1, f_2, \dots,f_{m_i} \in N_i^{>0}(V_j\cap V_{j^\prime})\) we have the condition \[\begin{align} \operatorname{Re}f_1(z)<\operatorname{Re}f_2(z)<\dots<\operatorname{Re}f_{m_i}(z) \quad (z\in V_j\cap V_{j^\prime}) \end{align}\] and the transition matrix \(A_{jj^\prime}\in\mathrm{GL}_{R_i}(\mathbb{C})\) is block upper triangular with respect to the decomposition \(R_i= \sum_{k=1}^{m_i} N_i(f_k)\) of \(R_i\). In particular, for \(1\leq k_1<k_2\leq m_i\) we have \[\begin{align} Q_{k_1}\cap( \widetilde{V_j} \cap \widetilde{V}_{j^\prime}) \supset Q_{k_2}\cap( \widetilde{V_j} \cap \widetilde{V}_{j^\prime}). \end{align}\] Since the local system \({\widetilde{L} \left.\right|_{\varpi^{-1}B(a_i)}}\) on \(\varpi^{-1} B(a_i) \subset \widetilde{Z}\) is obtained by gluing the constant sheaves \(\mathbb{C}_{\widetilde{V}_j}^{R_i}\) on \(\widetilde{V_j}\) by the transition matrices \(A_{jj^\prime}\in \mathrm{GL}_{R_i}(\mathbb{C})\) and the condition on singular chains for \(Q=(Q_1,Q_2, \ldots, Q_{m_i})\) is preserved by it, we can thus define relative twisted chains modulo \(Q\) that we denote by \[\begin{align} S_p(B(a_i)^{\circ} \cup Q,Q ;\widetilde{L}) \qquad (p \in \mathbb{Z}) \end{align}\] for short. The same is true also over the disk \(B(a_{\infty})= B(a_{\infty})^{\circ} \sqcup \{ a_{\infty} \} \subset \overline{X}^{{\rm an}}\) in \(\overline{X}^{{\rm an}}\) centered at the point \(a_\infty=\infty\in\overline{X}^{{\rm an}}\). As we do not impose any condition on twisted chains outside \(D= {\rm sing.supp} ( \mathcal{M}) \subset X\), similarly we obtain relative twisted chains modulo \(Q\) \[\begin{align} S_p(U^{{\rm an}} \cup Q,Q ;\widetilde{L}) \qquad (p \in \mathbb{Z}) \end{align}\] globally defined over \(\widetilde{Z}\). We denote by \(H_p(U^{{\rm an}}\cup Q,Q;\widetilde{L})\) (\(p \in \mathbb{Z}\)) the homology groups associated to them. Then by a Mayer-Vietoris exact sequence and the proof of [@ET15] we obtain isomorphisms \[\begin{align} H_p^{{\rm rd}}(U^{{\rm an}};\mathcal{E}_b^\ast,\nabla_b^\ast) \simeq H_p(U^{{\rm an}}\cup Q,Q;\widetilde{L}) \qquad (p \in \mathbb{Z}). \end{align}\] Now our basic idea for the construction of a basis of \(H_1(U^{{\rm an}}\cup Q,Q;\widetilde{L})\) is to use a Morse theory (with several Morse functions) as in the proof of Theorem 1. But this time, for \(i=1,2, \ldots,l\) or \(\infty\) and the exponential factors \(f_1,f_2, \ldots, f_m \in N_i^{>0}(V_j)\) on a sector \(V_j\) along \(a_i\) we consider the Morse functions \(\psi_k(z):= -\operatorname{Re}f_k^b(z)\) instead of \(\phi_k^b(z)= \operatorname{Re}f_k^b(z)\) on the open sector \(V_j \subset B(a_i)^{\circ}\) along \(a_i\). For \(t \in \mathbb{R}\) we set \(W_t^{\psi_k} := \{ z \in V_j \;| \;\psi_{k}(z)<t \} \subset V_j\). We define also a function \(\psi: B(a_1)^{\circ} \cup \cdots \cup B(a_l)^{\circ} \longrightarrow \mathbb{R}\) by \[\begin{align} \psi (z):= -\operatorname{Re}(bz) \qquad (z \in B(a_1)^{\circ} \cup \cdots \cup B(a_l)^{\circ}). \end{align}\] Note that if \(1 \leq i \leq l\) and \(f_k=0\) for some \(1 \leq k \leq m_i\), we have \(r_i>0\) and \(\psi_k(z)= \psi (z)\) on \(V_j\). Moreover, on the closed subset \[\begin{align} K \coloneq X^{{\rm an}}\left.\right\backslash\left\{ B(a_1) \cup\dots\cup B(a_l) \cup B(a_{\infty}) \right\} \end{align}\] of \(X^{{\rm an}}\) we use the Morse function \[\begin{align} \eta (z) \coloneq - \operatorname{Re}(bz)+c \quad (z\in K), \end{align}\] where \(c>0\) is a sufficient large real number. For \(t \in \mathbb{R}\) we set \(W_t^{\eta} := \{ z \in K \;| \;\eta (z)<t \} \subset K\). Then for \(i=1,2, \ldots,l\) or \(\infty\) and \(t \in \mathbb{R}\) on the open subset \(\widetilde{V_j}\subset\widetilde{Z}\) of \(\widetilde{Z}\), we consider the relative twisted chains \[\begin{align} \bigoplus_{k=1}^{m_i}S_p( W_t^{\psi_k} \cup Q_k ,Q_k; \mathbb{C}_{\widetilde{Z}}^{N_i(f_k)}) \qquad (p \in \mathbb{Z}). \end{align}\] It also follows from the condition \[\begin{align} \operatorname{Re}f_1(z)<\operatorname{Re}f_2(z)<\dots<\operatorname{Re}f_{m_i}(z) \quad (z\in V_j\cap V_{j^\prime}) \end{align}\] that for \(1\leq k_1<k_2\leq m_i\) we have \[\begin{align} W_t^{\psi_{k_1}} \cap( \widetilde{V_j} \cap \widetilde{V}_{j^\prime}) \supset W_t^{\psi_{k_2}} \cap( \widetilde{V_j} \cap \widetilde{V}_{j^\prime}). \end{align}\] We thus can define relative twisted chains with value in the local system \(\widetilde{L}\) contained in the level set \(W_t^i= ( W_t^{\psi_1}, W_t^{\psi_2}, \ldots, W_t^{\psi_{m_i}} )\) modulo \(Q=(Q_1,Q_2, \ldots, Q_{m_i})\) that we denote by \[\begin{align} S_p( W_t^i \cup Q,Q ;\widetilde{L}) \qquad (p \in \mathbb{Z}) \end{align}\] for short. We can regard them also as relative twisted chains with value in an \(\mathbb{R}\)-constructible subsheaf of \(\widetilde{L}\). For any \(t \in \mathbb{R}\) we thus can define relative twisted chains with value in the local system \(\widetilde{L}\) contained in the level set \(W_t= (W_t^1, \ldots, W_t^l, W_t^{\infty}, W_t^{\eta} )\) modulo \(Q\) \[\begin{align} S_p( W_t \cup Q,Q ;\widetilde{L}) \qquad (p \in \mathbb{Z}). \end{align}\] We denote by \(H_p( W_t \cup Q,Q;\widetilde{L})\) (\(p \in \mathbb{Z}\)) the homology groups associated to them. Then as in the proof of Lemma 9 for any \(p \in \mathbb{Z}\) and \(t \ll 0\) we can easily show the vanishing \[\begin{align} H_p(W_t \cup Q,Q;\widetilde{L}) \simeq 0 \end{align}\] Moreover for any \(p \in \mathbb{Z}\) and \(t \gg 0\) there exists an isomorphism \[\begin{align} H_p(W_t \cup Q,Q;\widetilde{L}) \simeq H_p(U^{{\rm an}}\cup Q,Q;\widetilde{L}). \end{align}\] Hence we can now apply the arguments in [@ET15] to construct a basis of \(H_1(U^{{\rm an}}\cup Q,Q;\widetilde{L})\) indexed by the critical points of the Morse functions \(\psi_k\) as follows. As in the proof of Theorem 1, for a point \(a_i\in D^{{\rm an}}\) let \(\gamma_{i j}(b)\in B(a_i)^{\circ}\) \((1\leq j\leq n_i)\) be the (non-degenerate) critical points of the (possibly multi-valued) functions \(\operatorname{Re}(f^b)\) \((f\in N_i^{>0},f\neq0)\) and set \[\begin{align} c_{ij}(b) \coloneq \operatorname{Re}(f^b)(\gamma_{i j}(b)) \;\in\mathbb{R}\quad (1\leq j\leq n_i). \end{align}\] Then for the point \(\gamma_{i j}(b)\in B(a_i)^{\circ}\) and the (non-zero) exponential factor \(f\in N_i^{>0}\) such that \((f^b)^\prime(\gamma_{i j}(b))=0\) there exists a holomorphic Morse coordinate \(\zeta =x +\sqrt{-1} y\) \((x,y \in\mathbb{R})\) on a neighborhood \(\Omega_{i j}\) of \(\gamma_{i j}(b)\) such that \(\gamma_{i j}(b)=\{\zeta=0\}\) and \[\begin{align} f^b(\zeta)=f^b(\gamma_{i j}(b))+\zeta^2. \end{align}\] This implies that we have \[\begin{align} -\operatorname{Re}(f^b)(\zeta)=-c_{i j}(b)+y^2-x^2 \end{align}\] on \(\Omega_{i j}\). Hence we can regard the 1-dimensional smooth submanifold \(S_{i j}\coloneq\{ y=0\}\subset \Omega_{i j}\) of \(\Omega_{i j}\) as the stable submanifold of the gradient flow of our Morse function \[\begin{align} -\operatorname{Re}(f^b) \colon \Omega_{i j}\longrightarrow\mathbb{R}. \end{align}\] Shrinking it if necessary we may assume that it is homeomorphic to an open interval \((-\varepsilon_{i j},\varepsilon_{i j}) \subset\mathbb{R}\) \((\varepsilon_{i j}>0)\). Also for the point \(a_\infty=\infty\in \overline{X}^{{\rm an}}\) we define points \(\gamma_{\infty j}(b)\in B(a_{\infty})^\circ \subset X^{{\rm an}}\) \((1\leq j\leq n_\infty)\), \(N_{\infty j}>0\) and \(c_{\infty j}(b)\in\mathbb{R}\) \((1\leq j\leq n_{\infty})\) and obtain 1-dimensional submanifolds \(S_{\infty j}\subset B(a_{\infty})^\circ\) similarly. For \(1 \leq i \leq l\), let \(D_i^\prime\subset X^{{\rm an}}\) be a sufficiently small closed disk in \(X^{{\rm an}}\) centered at \(a_i\) such that \(D_i^\prime\subset B(a_i)\) and set \(c_i(b)\coloneq\psi(a_i)\in\mathbb{R}\) and \[\begin{align} S_i \coloneq \partial D_i^\prime\cap \psi^{-1}\bigl((c_i(b),+\infty)\bigr)\; \subset \partial D_i^\prime\simeq S^1. \end{align}\] Then as in the proof of [@ET15] for any \(t\in\mathbb{R}\) and \(0<\varepsilon\ll1\) there exist sequence \[\begin{align} 0 \longrightarrow &H_1(W_{t-\varepsilon}\cup Q,Q;\widetilde{L}) \longrightarrow H_1(W_{t+\varepsilon}\cup Q,Q;\widetilde{L}) \longrightarrow \\ &\Biggl\{\bigoplus^{{\phantom{A}}}_{(i,j)\colon-c_{i j}(b)=t} H_1(\overline{S_{i j}},\partial S_{i j}; \mathbb{C}_{X^{{\rm an}}}^{N_{i j}})\Biggr\} \oplus \Biggl\{\bigoplus^{{\phantom{A}}}_{i\colon c_i(b)=t} H_1(\overline{S_i},\partial S_i; \mathbb{C}_{X^{{\rm an}}}^{r_i})\Biggr\} \longrightarrow 0, \end{align}\] where in the first sum \(\oplus\) the pair \((i,j)\) ranges through the set \[\begin{align} \Set*{(i,j)}{1\leq i\leq l,1\leq j\leq n_i, -c_{i j}(b)=t} \cup \Set*{(\infty,j)}{1\leq j\leq n_\infty, -c_{\infty j}(b)=t}. \end{align}\] Moreover, for each such pair \((i,j)\) we have an isomorphism \[\begin{align} H_1(\overline{S_{i j}},\partial S_{i j}; \mathbb{C}_{X^{{\rm an}}}^{N_{i j}}) \simeq \mathbb{C}^{N_{i j}} \end{align}\] and can take a basis \[\begin{align} \sigma_{i j k}^0 \in H_1(\overline{S_{i j}},\partial S_{i j};\mathbb{C}_{X^{{\rm an}}}^{N_{i j}}) \quad (1\leq k\leq N_{i j}) \end{align}\] of \(H_1(\overline{S_{i j}},\partial S_{i j}; \mathbb{C}_{X^{{\rm an}}}^{N_{i j}})\). Similarly, for each \(1\leq i\leq l\) such that \(c_i(b)=t\) we can take a basis \[\begin{align} \sigma_{i j}^0 \in H_1(\overline{S_i},\partial S_i;\mathbb{C}_{X^{{\rm an}}}^{r_i}) \simeq \mathbb{C}^{r_i} \quad (1\leq j\leq r_i) \end{align}\] of \(H_1(\overline{S_i},\partial S_i;\mathbb{C}_{X^{{\rm an}}}^{r_i})\). From now, we will show that \(\sigma_{i j k}^0\) and \(\sigma_{i j}^0\) can be lifted to some elements of the rapid decay homology group \(H_1(W_{t+\varepsilon}\cup Q,Q;\widetilde{L})\). To explain our idea better, first we consider the case where we have the condition \[\begin{align} \mathrm{ord}_\infty(f)\leq1 \end{align}\] for any exponential factor \(f\in N_\infty^{>0}\) of \(\mathcal{M}^{{\rm an}}\) at \(a_\infty=\infty\). In this case, by our assumption \(\abs{b}\gg1\) we have \(n_\infty=0\). For \(1\leq i\leq l\), \(1\leq j\leq n_i\) and \(f \in N_i^{>0}\) such that \((f^b)^{\prime}( \gamma_{ij}(b))=0\), let \(C_{i j}^\circ\subset B(a_i)^\circ\) be the maximal integral curve of the gradient flow of the function \(\operatorname{Re}(f^b)\) on \(B(a_i)^\circ\) such that \(C_{i j}^\circ\supset S_{i j}\). Then each boundary point of \(C_{i j}^\circ\) is either \(a_i\) or contained in \(\partial B(a_i) \simeq S^1\) and we can naturally extend \(\sigma_{i j k}^0\) \((1\leq k\leq N_{i j})\) to some twisted 1-chains \[\begin{align} \sigma_{i j k}^\circ = \overline{C_{i j}^\circ}\otimes s_{i j k} \quad (s_{i j k}\in\widetilde{L}) \end{align}\] with value in the local system \(\widetilde{L}\) where \(\overline{C_{i j}^\circ}\) is the closure in the real oriented blow-up \(\widetilde{Z}\). Since we assume here that \(\abs{b}\gg 0\), if a boundary point of \(\overline{C_{i j}^\circ}\) is contained in \(\partial B(a_i)\) the tangent vector of \(\overline{C_{i j}^\circ}\) at it is almost parallel to the vector \(\operatorname{grad}\operatorname{Re}(bz)\) and we can add a half line \((\simeq\mathbb{R}_{>0})\) in \(U^{{\rm an}}\) emanating from it to \(\overline{C_{i j}^\circ}\) to extend \(\sigma_{i j k}^\circ\) to rapid decay 1-chains \(\sigma_{i j k}\) such that \[\begin{align} [\sigma_{i j k}] \in H_1(W_{t+\varepsilon}\cup Q,Q;\widetilde{L}) \quad (1\leq k\leq N_{i j}). \end{align}\] Similarly for \(1\leq i\leq l\), at the two boundary points of the curve \(S_i\subset\partial D_i^\prime\simeq S^1\) the tangent vectors of \(\overline{S_i}\) are parallel to the vector \(\operatorname{grad}\operatorname{Re}(bz)\) and we can add two half lines \((\simeq\mathbb{R}_{>0})\) in \(U^{{\rm an}}\) emanating from them to \(\overline{S_i}\) to extend \(\sigma_{i j}^0\) \((1\leq j\leq r_i)\) to rapid decay 1-chains \(\sigma_{i j}\) such that \[\begin{align} [\sigma_{i j}] \in H_1(W_{t+\varepsilon}\cup Q,Q;\widetilde{L}) \quad (1\leq j\leq r_i). \end{align}\] We can remove the condition \[\begin{align} \mathrm{ord}_\infty(f) \leq 1 \quad (f\in N_\infty^{>0}) \end{align}\] as follows. Assume that it does not hold. Then for \(1\leq i\leq l\) and \(1\leq j\leq n_i\) if a boundary point of \(C_{i j}^\circ\) is contained in \(\partial B(a_i) \simeq S^1\) we first extend \(\sigma_{i j k}^\circ\) \((1\leq k\leq N_{i j})\) to some twisted 1-chains \[\begin{align} \sigma_{i j k}^\prime = C_{i j}^\prime\otimes s_{i j k} \quad (s_{i j k}\in\widetilde{L}), \end{align}\] where \(C_{i j}^\prime\) is a curve in \(\widetilde{Z}\) such that \(C_{i j}^\circ\subset C_{i j}^\prime\) and one of the boundary points of it is in \(\partial B(a_{\infty}) \subset X^{{\rm an}}\). Fix \(1\leq i\leq l\), \(1\leq j\leq n_i\) and \(1\leq k\leq N_{i j}\) and set \[\begin{align} C^\prime\coloneq C_{i j}^\prime, \quad \sigma^\prime\coloneq \sigma_{i j k}^\prime = C^\prime\otimes s \quad (s\in\widetilde{L}) \end{align}\] for short. Then at the boundary point \(C^\prime\cap\partial B(a_{\infty})\) of \(C^\prime=C_{i j}^\prime\) we have a decomposition \[\begin{align} s = \sum_{j=1}^{q}s_j \quad (s_j\in\widetilde{L}) \end{align}\] of the section \(s\in\widetilde{L}\) such that there exist curves \(\Gamma_j\subset B(a_{\infty})^\circ\) \((1\leq j\leq q)\) in \(B(a_{\infty})^\circ\) starting from it and ending at a point in \(\varpi^{-1}(\infty)\simeq S^1\) along which the extensions \(\widetilde{s_j} \in\widetilde{L}\) of the sections \(s_j \in\widetilde{L}\) satisfy the rapid decay condition. Then the twisted 1-chain \[\begin{align} \sigma \coloneq \sigma^\prime+ \sum_{j=1}^{q}\Gamma_j\otimes \widetilde{s_j} \end{align}\] with value in the local system \(\widetilde{L}\) satisfies the desired condition \[\begin{align} [\sigma] \in H_1(W_{t+\varepsilon}\cup Q,Q;\widetilde{L}). \end{align}\] The same arguments can be applied to extend \(\sigma_{\infty j k}^0\) \((resp.\;\sigma_{i j}^0)\) to rapid decay 1-chains \(\sigma_{\infty j k}\) \((resp.\;\sigma_{i j})\) such that \[\begin{align} [\sigma_{\infty j k}], [\sigma_{i j}] \in H_1(W_{t+\varepsilon}\cup Q,Q;\widetilde{L}). \end{align}\] Considering all \(t\in\mathbb{R}\), we thus obtain elements \([\sigma_{i j k}]\) \((1\leq i\leq l, 1\leq j\leq n_i, 1\leq k\leq N_{i j})\), \([\sigma_{\infty j k}]\) \((1\leq j\leq n_\infty, 1\leq k\leq N_{\infty j})\) and \([\sigma_{i j}]\) \((1\leq i\leq l, 1\leq j \leq r_i)\) of the rapid decay homology group \[\begin{align} H_1^{{\rm rd}}(U^{{\rm an}};\mathcal{E}_b^\ast,\nabla_b^\ast) \simeq H_1(U^{{\rm an}}\cup,Q, Q;\widetilde{L}). \end{align}\]

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1. Mathematical Institute, Tohoku University, Aramaki Aza-Aoba 6-3, Aobaku, Sendai, 980-8578, Japan. E-mail: kazuki.kudomi.q3@dc.tohoku.ac.jp↩︎

2. Mathematical Institute, Tohoku University, Aramaki Aza-Aoba 6-3, Aobaku, Sendai, 980-8578, Japan. E-mail: takemicro@nifty.com↩︎

3. 2010 Mathematics Subject Classification: 32C38, 32S40, 34M35, 34M40, 35A27.↩︎

4. Keywords: Characteristic cycles, D-modules, Fourier transforms, Irregularity, Riemann-Hilbert correspondence.↩︎