${\rm SS}^{{\rm E}}(F)$ the enhanced micro-support of $F$. Let ${\rm
pr}_{T^{*}X_{\mathbb{R}}}:(T^{*}X_{\mathbb{R}})\times\mathbb{R}\longrightarrow
T^{*}X_{\mathbb{R}}$ be the projection and set

${\rm SS}_{{\rm irr}}(F):=\overline{{\rm pr}_{T^{*}X_{\mathbb{R}}}\bigl{(}{\rm
SS}^{{\rm E}}(F)\bigr{)}}\subset T^{*}X_{\mathbb{R}}$ (5.4)

(see [8, Section 2.6] for a different notation for it). We call it the
irregular micro-support of $F$. From now on, we shall consider the objects
$F\in{\mathbf{E}}^{\mathrm{b}}_{+}(\mathbb{C}_{X^{{\rm an}}})$ related to the
following special but basic holonomic D-modules. For a rational function
$f=\frac{P}{Q}:X\setminus Q^{-1}(0)\longrightarrow\mathbb{C}$
($P,Q\in\Gamma(X;\mathcal{O}_{X})$, $Q\not=0$) on $X$ we set $U:=X\setminus
Q^{-1}(0)$ and define an exponential $\mathcal{D}_{X}$-module
$\mathcal{E}^{f}_{U|X}\in\mathrm{Mod}_{\mbox{\rm\scriptsize
hol}}(\mathcal{D}_{X})$ as in the analytic case. Then by Theorem 4.1 (vi) we
have an isomorphism

$Sol_{X}^{\rm E}(\mathcal{E}^{f}_{U|X})\simeq\mathbb{E}_{U^{{\rm an}}|X^{{\rm
an}}}^{{\rm Re}f}.$ (5.5)

######  Definition 5.1.

We say that a holonomic $\mathcal{D}_{X}$-module
$\mathcal{M}\in\mathrm{Mod}_{\mbox{\rm\scriptsize hol}}(\mathcal{D}_{X})$ is
an exponentially twisted holonomic D-module if there exist a regular holonomic
$\mathcal{D}_{X}$-module $\mathcal{N}\in\mathrm{Mod}_{\mbox{\rm\scriptsize
rh}}(\mathcal{D}_{X})$ and a rational function $f=\frac{P}{Q}:U=X\setminus
Q^{-1}(0)\longrightarrow\mathbb{C}$
($P,Q\in\Gamma(X;\mathcal{O}_{X}),Q\not=0$) on $X$ such that we have an
isomorphism

$\mathcal{M}\simeq\mathcal{N}\overset{D}{\otimes}\mathcal{E}^{f}_{U|X}.$ (5.6)

Let $\mathcal{M}\in\mathrm{Mod}_{\mbox{\rm\scriptsize hol}}(\mathcal{D}_{X})$
be an exponentially twisted holonomic $\mathcal{D}_{X}$-module such that
$\mathcal{M}\simeq\mathcal{N}\overset{D}{\otimes}\mathcal{E}^{f}_{U|X}$ for a
regular holonomic $\mathcal{D}_{X}$-module
$\mathcal{N}\in\mathrm{Mod}_{\mbox{\rm\scriptsize rh}}(\mathcal{D}_{X})$ and a
rational function $f=\frac{P}{Q}:U=X\setminus
Q^{-1}(0)\longrightarrow\mathbb{C}$ on $X$. Let us set

$K:=Sol_{X}(\mathcal{N})\in{\bf D}_{\mathrm{c}}^{\mathrm{b}}(X^{{\rm an}}).$
(5.7)

We denote the support of $K\in{\bf D}_{\mathrm{c}}^{\mathrm{b}}(X^{{\rm an}})$
by $Z\subset X$. Then in this algebraic case, considering $f$ as a rational
function on a (possibly singular) compactification $\overline{Z}$ of $Z$, we
see that the restriction of $f$ to $Z$ is uniquely determined by $\mathcal{M}$
modulo constant functions on $Z$. Note also that the shift $K[N]\in{\bf
D}_{\mathrm{c}}^{\mathrm{b}}(X^{{\rm an}})$ of $K$ is a perverse sheaf on
$X^{{\rm an}}$. Then by Theorem 4.1 (vi) and Corollary 4.2 for the enhanced
sheaf $F:=\pi^{-1}K\otimes{\rm E}_{U^{{\rm an}}|X^{{\rm an}}}^{{\rm
Re}f}\in{\mathbf{E}}^{\mathrm{b}}_{+}(\mathbb{C}_{X^{{\rm an}}})$ on $X^{{\rm
an}}$ we have an isomorphism

$Sol_{X}^{\rm E}(\mathcal{M})\simeq\mathbb{C}^{\rm E}_{X^{{\rm
an}}}\overset{+}{\otimes}F.$ (5.8)

From now on, we shall study the enhanced micro-support ${\rm SS}^{{\rm
E}}(F)\subset(T^{*}X_{\mathbb{R}})\times\mathbb{R}$ and the irregular one
${\rm SS}_{{\rm irr}}(F)\subset T^{*}X_{\mathbb{R}}$ of the enhanced sheaf
$F=\pi^{-1}K\otimes{\rm E}_{U^{{\rm an}}|X^{{\rm an}}}^{{\rm
Re}f}\in{\mathbf{E}}^{\mathrm{b}}_{+}(\mathbb{C}_{X^{{\rm an}}})$. It turns
out that over the open subset $T^{*}U_{\mathbb{R}}\subset T^{*}X_{\mathbb{R}}$
the structures of these two micro-supports are very simple. Define a (not
necessarily $\mathbb{C}^{*}$-conic) complex Lagrangian submanifold
$\Lambda^{f}$ of $T^{*}U$ by

$\Lambda^{f}:=\\{(z,df(z))\ |\ z\in U=X\setminus Q^{-1}(0)\\}\subset T^{*}U.$
(5.9)

Then we can easily show that via the natural identification
$(T^{*}U)_{\mathbb{R}}\simeq T^{*}U_{\mathbb{R}}$ we have

${\rm SS}_{{\rm irr}}(F)\cap T^{*}U_{\mathbb{R}}=\bigl{(}{\rm SS}(K)\cap
T^{*}U_{\mathbb{R}}\bigr{)}+\Lambda^{f}$ (5.10)

and

$\displaystyle{\rm SS}^{{\rm
E}}(F)\cap\Bigl{\\{}(T^{*}U_{\mathbb{R}})\times\mathbb{R}\Bigr{\\}}$
$\displaystyle=\Bigl{\\{}(x,y,x^{*},y^{*},-{\rm Re}f(x+iy)\ |\
(x,y,x^{*},y^{*})\in\bigl{(}{\rm SS}(K)\cap
T^{*}U_{\mathbb{R}}\bigr{)}+\Lambda^{f}\Bigr{\\}}.$

Moreover we have the following results.

######  Lemma 5.2.

Assume that there exists a Whitney stratification $\mathcal{S}$ of $Z$ adapted
to $K$ such that any stratum $S\in\mathcal{S}$ in it is not contained in
$P^{-1}(0)\cap Z\subset Z$ nor $Q^{-1}(0)\cap Z\subset Z$ and the meromorphic
function $f|_{S\setminus Q^{-1}(0)}:S\setminus
Q^{-1}(0)\longrightarrow\mathbb{C}$ on $S$ satisfies the condition: For any
point of $S\cap P^{-1}(0)\cap Q^{-1}(0)$ there exists a local coordinate
$z_{1},z_{2},\ldots,z_{l}$ ($l:={\rm dim}S$) of $S$ around it such that
$(P|_{S})(z_{1},\ldots,z_{l})=z_{1},\ (Q|_{S})(z_{1},\ldots,z_{l})=z_{2}^{m}$
for some $m\geq 1$ and hence

$(f|_{S})(z_{1},\ldots,z_{l})=\frac{z_{1}}{z_{2}^{m}}.$ (5.11)

We denote the preimage of the closed subset $P^{-1}(0)\cap Q^{-1}(0)\cap
Z\subset Z\subset X=\mathbb{C}^{N}$ by the projection
$(T^{*}X_{\mathbb{R}})\times\mathbb{R}\longrightarrow X_{\mathbb{R}}$ simply
by $\\{P=Q=0\\}\subset(T^{*}X_{\mathbb{R}})\times\mathbb{R}$. Then we have

${\rm SS}^{{\rm E}}(F)\cap\\{P=Q=0\\}=\emptyset.$ (5.12)

###### Proof.

First, we consider the special case where $N=2$, $Z=X=\mathbb{C}^{2}$,
$K=\mathbb{C}_{X}$ and $P(z_{1},z_{2})=z_{1}$, $Q(z_{1},z_{2})=z_{2}$,
$f(z_{1},z_{2})=\frac{z_{1}}{z_{2}}$. We set $z=(z_{1},z_{2})=(x,y)$ so that
we have $f(x,y)=\frac{x}{y}$. We shall show ${\rm SS}^{{\rm
E}}(F)\cap\\{x=y=0\\}=\emptyset$. By an explicit calculation, we can easily
show that the closure of the set

${\rm SS}^{{\rm
E}}(F)\setminus\\{x=y=0\\}\subset(T^{*}X_{\mathbb{R}})\times\mathbb{R}\setminus\\{x=y=0\\}$
(5.13)

in $(T^{*}X_{\mathbb{R}})\times\mathbb{R}$ does not intersect $\\{x=y=0\\}$.
Let $\nu:\widetilde{X}\longrightarrow X$ be the blow-up of $X=\mathbb{C}^{2}$
along the origin $\\{(0,0)\\}\subset X=\mathbb{C}^{2}$ and
$M=\mathbb{C}^{2}_{u,y}$ the affine open subset of $\widetilde{X}$ such that
we have

$\nu|_{M}:M=\mathbb{C}^{2}_{u,y}\longrightarrow
X=\mathbb{C}^{2}\qquad((u,y)\longmapsto(uy,y)).$ (5.14)

Then for the exceptional divisor $\nu^{-1}(\\{(0,0)\\})\simeq{\mathbb{P}}$ we
have $\nu^{-1}(\\{(0,0)\\})\cap M=\mathbb{C}_{u}\times\\{0\\}\subset
M=\mathbb{C}^{2}_{u,y}$ and $(f\circ\nu)(u,y)=\frac{uy}{y}=u$. Note that the
meromorphic function $f\circ\nu$ on $\widetilde{X}$ has no point of
indeterminacy on the whole $\widetilde{X}$ and its pole is contained in
$\widetilde{X}\setminus M$. We define an enhanced sheaf $\widetilde{F}$ on
$\widetilde{X}$ by

$\widetilde{F}:={\rm E}_{M\setminus\\{y=0\\}|\widetilde{X}}^{{\rm
Re}(f\circ\nu)}\in{\mathbf{E}}^{\mathrm{b}}_{+}(\mathbb{C}_{\widetilde{X}^{{\rm
an}}})$ (5.15)

and set $\widetilde{\nu}:=\nu\times{\rm
id}_{\mathbb{R}}:\widetilde{X}\times\mathbb{R}\longrightarrow
X\times\mathbb{R}$ so that we have an isomorphism

${\rm R}\widetilde{\nu}_{*}\widetilde{F}\simeq F.$ (5.16)

Let

$T^{*}(X\times\mathbb{R})\overset{\varpi}{\longleftarrow}(\widetilde{X}\times\mathbb{R})\times_{(X\times\mathbb{R})}T^{*}(X\times\mathbb{R})\overset{\rho}{\longrightarrow}T^{*}(\widetilde{X}\times\mathbb{R})$
(5.17)

be the natural morphisms associated to $\widetilde{\nu}$. Then by [22,
Proposition 5.4.4] we have

${\rm SS}(F)\subset\varpi\rho^{-1}{\rm SS}(\widetilde{F}).$ (5.18)

Let

$(0,0,t,\xi,\eta,1)\in\\{t^{*}>0\\}\subset T^{*}(X\times\mathbb{R})$ (5.19)

be a point which corresponds to the covector $\xi dx+\eta dy+dt\in
T^{*}_{(0,0,t)}(X\times\mathbb{R})$ at the point $(0,0,t)\in
X\times\mathbb{R}$. Then its pull-back by the map

$\widetilde{\nu}|_{M\times\mathbb{R}}:M\times\mathbb{R}\longrightarrow
X\times\mathbb{R}\qquad((u,y,t)\longmapsto(uy,y,t))$ (5.20)

is calculated as

$\bigl{(}\widetilde{\nu}|_{M\times\mathbb{R}}\bigr{)}^{*}(\xi dx+\eta
dy+dt)=(\xi u+\eta)dy+dt.$ (5.21)

Since ${\rm Re}(f\circ\nu)(u,y)={\rm Re}u$ and there is no covector of the
form $(\xi u+\eta)dy+dt$ in $\\{t^{*}=1\\}\cap{\rm SS}(\widetilde{F})$, we
obtain the assertion ${\rm SS}^{{\rm E}}(F)\cap\\{x=y=0\\}=\emptyset$. We can
treat also the case where $N=2$, $Z=X=\mathbb{C}^{2}$, $K=\mathbb{C}_{X}$ and
$P(z_{1},z_{2})=z_{1}$, $Q(z_{1},z_{2})=z_{2}^{m}$,
$f(z_{1},z_{2})=\frac{z_{1}}{z_{2}^{m}}$ for some $m\geq 2$ by repeating blow-
ups. The general case can be treated similarly. This completes the proof. ∎

By the natural identification $(T^{*}X)_{\mathbb{R}}\simeq
T^{*}X_{\mathbb{R}}$ we regard $T^{*}X_{\mathbb{R}}$ as a complex symplectic
manifold endowed with the canonical holomorphic symplectic 2-form
$\sigma_{X}:=\sum_{i=1}^{N}dw_{i}\wedge dz_{i}$.

######  Lemma 5.3.

The irregular micro-support ${\rm SS}_{{\rm irr}}(F)\subset
T^{*}X_{\mathbb{R}}$ of $F=\pi^{-1}K\otimes{\rm E}_{U|Z}^{{\rm
Re}f}\in{\mathbf{E}}^{\mathrm{b}}_{+}(\mathbb{C}_{X^{{\rm an}}})$ is contained
in a complex isotoropic analytic subset of $(T^{*}X)_{\mathbb{R}}\simeq
T^{*}X_{\mathbb{R}}$. If we assume moreover that $I(f)=P^{-1}(0)\cap
Q^{-1}(0)=\emptyset$, then we have

${\rm SS}_{{\rm irr}}(F)=\bigl{(}{\rm SS}(K)\cap
T^{*}U_{\mathbb{R}}\bigr{)}+\Lambda^{f}.$ (5.22)

###### Proof.

Let us consider the exact sequence

$0\longrightarrow F_{\pi^{-1}(X\setminus P^{-1}(0))}\longrightarrow
F\longrightarrow F_{\pi^{-1}(P^{-1}(0))}\longrightarrow 0$ (5.23)

of enhanced sheaves on $X$. Since there exists an isomorphism

$F_{\pi^{-1}(P^{-1}(0))}\simeq\pi^{-1}K\otimes\mathbb{C}_{\\{(z,t)\in
U\times\mathbb{R}\ |\ z\in U\cap P^{-1}(0),\ t\geq 0\\}},$ (5.24)

we see that ${\rm SS}_{{\rm irr}}(F_{\pi^{-1}(P^{-1}(0))})={\rm SS}(K_{U\cap
P^{-1}(0)})$ is a Lagrangian analytic subset of $(T^{*}X)_{\mathbb{R}}\simeq
T^{*}X_{\mathbb{R}}$. Let $\nu:\widetilde{X}\longrightarrow X$ be a proper
morphism of complex manifolds such that the restriction
$\widetilde{X}\setminus\nu^{-1}(P^{-1}(0)\cup Q^{-1}(0))\longrightarrow
X\setminus(P^{-1}(0)\cup Q^{-1}(0))$ of $\nu$ is an isomorphism,
$\nu^{-1}(P^{-1}(0)\cup Q^{-1}(0))\subset\widetilde{X}$ is a normal crossing
divisor, and the meromorphic function $f\circ\nu:=(P\circ\nu)/(Q\circ\nu)$ on
$\widetilde{X}$ has no point of indeterminacy on the whole $\widetilde{X}$.
Moreover we define a complex Lagrangian submanifold
$\Lambda^{f\circ\nu}\subset T^{*}\widetilde{X}$ by

$\Lambda^{f\circ\nu}:=\\{(z,d(f\circ\nu)(z))\ |\ z\in\widetilde{X}\\}\subset
T^{*}\widetilde{X}$ (5.25)

and set $\widetilde{\nu}:=\nu\times{\rm
id}_{\mathbb{R}}:\widetilde{X}\times\mathbb{R}\longrightarrow
X\times\mathbb{R}$. Then for the enhanced sheaf
$\widetilde{\nu}^{-1}F_{\pi^{-1}(X\setminus P^{-1}(0))}$ on $\widetilde{X}$ we
can easily show that

${\rm SS}_{{\rm irr}}\bigl{(}\widetilde{\nu}^{-1}F_{\pi^{-1}(X\setminus
P^{-1}(0))}\bigr{)}={\rm SS}\bigl{(}\nu^{-1}K_{U\setminus
P^{-1}(0)}\bigr{)}+\Lambda^{f\circ\nu}$ (5.26)

in $T^{*}\widetilde{X}$. This implies that ${\rm SS}_{{\rm
irr}}\bigl{(}\widetilde{\nu}^{-1}F_{\pi^{-1}(X\setminus P^{-1}(0))}\bigr{)}$
is a Lagrangian analytic subset of $T^{*}\widetilde{X}$. Let

$T^{*}X\overset{\varpi}{\longleftarrow}\widetilde{X}\times_{X}T^{*}X\overset{\rho}{\longrightarrow}T^{*}\widetilde{X}$
(5.27)

be the natural morphisms associated to $\nu:\widetilde{X}\longrightarrow X$.
Since $\nu$ is proper and there exists an isomorphism

$F_{\pi^{-1}(X\setminus P^{-1}(0))}\overset{\sim}{\longrightarrow}{\rm
R}\widetilde{\nu}_{*}\widetilde{\nu}^{-1}(F_{\pi^{-1}(X\setminus
P^{-1}(0))}),$ (5.28)

by [22, Proposition 5.4.4] we have

${\rm SS}_{{\rm irr}}(F_{\pi^{-1}(X\setminus
P^{-1}(0))})\subset\varpi\rho^{-1}{\rm SS}_{{\rm
irr}}\bigl{(}\widetilde{\nu}^{-1}F_{\pi^{-1}(X\setminus P^{-1}(0))}\bigr{)}.$
(5.29)

It is clear that the right hand side is an analytic subset of $T^{*}X$.
Moreover, by [12, page 43] there exist some stratifications of it and
$\rho^{-1}{\rm SS}_{{\rm
irr}}\bigl{(}\widetilde{\nu}^{-1}F_{\pi^{-1}(X\setminus P^{-1}(0))}\bigr{)}$
such that the morphism

$\rho^{-1}{\rm SS}_{{\rm
irr}}\bigl{(}\widetilde{\nu}^{-1}F_{\pi^{-1}(X\setminus
P^{-1}(0))}\bigr{)}\longrightarrow\varpi\rho^{-1}{\rm SS}_{{\rm
irr}}\bigl{(}\widetilde{\nu}^{-1}F_{\pi^{-1}(X\setminus P^{-1}(0))}\bigr{)}$
(5.30)

induced by $\varpi$ is a stratified fiber bundle over each stratum. Then we
can slightly modify the proof of [19, Proposition A.54] to show that the
symplectic 2-form $\sigma_{X}$ vanishes on $\varpi\rho^{-1}{\rm SS}_{{\rm
irr}}\bigl{(}\widetilde{\nu}^{-1}F_{\pi^{-1}(X\setminus P^{-1}(0))}\bigr{)}$.
We thus obtain the first assertion. Let us prove the second one. Let
$\nu:\widetilde{X}\longrightarrow X$ be a proper morphism of complex manifolds
such that the restriction
$\widetilde{X}\setminus\nu^{-1}Q^{-1}(0)\longrightarrow X\setminus Q^{-1}(0)$
of $\nu$ is an isomorphism and $\nu^{-1}Q^{-1}(0)\subset\widetilde{X}$ is a
normal crossing divisor in $\widetilde{X}$. Then by our assumption
$I(f)=\emptyset$ also the meromorphic function
$f\circ\nu:=(P\circ\nu)/(Q\circ\nu)$ on $\widetilde{X}$ has no point of
indeterminacy on the whole $\widetilde{X}$. Moreover for the morphism
$\widetilde{\nu}=\nu\times{\rm id}_{\mathbb{R}}$ there exists an isomorphism

$F\overset{\sim}{\longrightarrow}{\rm
R}\widetilde{\nu}_{*}\widetilde{\nu}^{-1}F$ (5.31)

and we can easily see that there is no point of ${\rm SS}_{{\rm
irr}}(\widetilde{\nu}^{-1}F)$ over the normal crossing divisor
$\nu^{-1}Q^{-1}(0)\subset\widetilde{X}$. Then the assertion immediately
follows. ∎

In the special case where $X$ is $\mathbb{C}^{N}_{z}$, $Y$ is its dual vector
space $\mathbb{C}^{N}_{w}$ and $T^{*}X\simeq X\times Y$, for a point
$(z_{0},w_{0},t_{0})$ of
$(T^{*}X)_{\mathbb{R}}\times\mathbb{R}\simeq(T^{*}X_{\mathbb{R}})\times\mathbb{R}$
we shall introduce a sufficient condition for the inclusion
$(z_{0},w_{0},t_{0})\in{\rm SS}^{{\rm E}}(F)$ to hold by using meromorphic
vanishing cycle functors. For the point $w_{0}\in Y=\mathbb{C}^{N}$ we define
a rational function $f^{w_{0}}$ on $X=\mathbb{C}^{N}$ by

$f^{w_{0}}:U=X\setminus
Q^{-1}(0)\longrightarrow\mathbb{C}\qquad(z\longmapsto\langle
z,w_{0}\rangle-f(z)).$ (5.32)

######  Proposition 5.4.

In the case where $X$ is $\mathbb{C}^{N}_{z}$, $Y$ is its dual vector space
$\mathbb{C}^{N}_{w}$ and $T^{*}X\simeq X\times Y$, for a point $(z_{0},w_{0})$
of $T^{*}X\simeq X\times Y$ assume that there exists a complex number
$c\in\mathbb{C}$ such that

$\phi^{{\rm mero,c}}_{f^{w_{0}}-c}(K)_{z_{0}}\not=0$ (5.33)

and set $t_{0}:={\rm Re}c-{\rm Re}\langle z_{0},w_{0}\rangle\in\mathbb{R}$.
Then we have $(z_{0},w_{0},t_{0})\in{\rm SS}^{{\rm E}}(F)$.

###### Proof.

We define a closed half space $H_{+}\subset X_{\mathbb{R}}\times\mathbb{R}$ of
$X_{\mathbb{R}}\times\mathbb{R}\simeq\mathbb{R}^{2N+1}$ by

$H_{+}:=\\{(z,t)\in X_{\mathbb{R}}\times\mathbb{R}\ |\ t\geq t_{0}-{\rm
Re}\langle z-z_{0},w_{0}\rangle\\}.$ (5.34)

Note that its boundary $H_{0}\subset H_{+}$ is a real hyperplane in
$X_{\mathbb{R}}\times\mathbb{R}$ whose conormal vector is equal to
$\left(\begin{array}[]{c}w_{0}\\\ 1\end{array}\right)\in
Y_{\mathbb{R}}\times\mathbb{R}$. We shall now calculate the stalk ${\rm
R}\Gamma_{H_{+}}(F)_{(z_{0},t_{0})}$ of ${\rm R}\Gamma_{H_{+}}(F)$ at the
point $(z_{0},t_{0})\in X_{\mathbb{R}}\times\mathbb{R}$. First of all, let us
set

$G:=\pi^{-1}K\otimes\mathbb{C}_{\\{(z,t)\in U\times\mathbb{R}\ |\ t=-{\rm
Re}f(z)\\}}$ (5.35)

and

$F^{\circ}:=\pi^{-1}K\otimes\mathbb{C}_{\\{(z,t)\in U\times\mathbb{R}\ |\
t>-{\rm Re}f(z)\\}}$ (5.36)

so that we have an exact sequence

$0\longrightarrow F^{\circ}\longrightarrow F\longrightarrow G\longrightarrow
0$ (5.37)

of enhanced sheaves on $X$. Then for the open subset $\\{t^{*}>0\\}\subset
T^{*}X_{\mathbb{R}}\times T^{*}\mathbb{R}$ we can show the condition ${\rm
SS}(F^{\circ})\cap\\{t^{*}>0\\}=\emptyset$ to obtain an isomorphism

${\rm R}\Gamma_{H_{+}}(F)_{(z_{0},t_{0})}\overset{\sim}{\longrightarrow}{\rm
R}\Gamma_{H_{+}}(G)_{(z_{0},t_{0})}.$ (5.38)

Indeed, over the open subset $X\setminus Q^{-1}(0)\subset X$ of $X$ clearly
the condition ${\rm SS}(F^{\circ})\cap\\{t^{*}>0\\}=\emptyset$ holds. Moreover
in the exact sequence

$0\longrightarrow(F^{\circ})_{\pi^{-1}(X\setminus P^{-1}(0))}\longrightarrow
F^{\circ}\longrightarrow(F^{\circ})_{\pi^{-1}(P^{-1}(0))}\longrightarrow 0$
(5.39)

of enhanced sheaves on $X$, we have an isomorphism

$(F^{\circ})_{\pi^{-1}(P^{-1}(0))}\simeq\pi^{-1}K\otimes\mathbb{C}_{\\{(z,t)\in
U\times\mathbb{R}\ |\ z\in U\cap P^{-1}(0),\ t>0\\}}.$ (5.40)

Hence we obtain the condition

${\rm
SS}\bigl{(}(F^{\circ})_{\pi^{-1}(P^{-1}(0))}\bigr{)}\cap\\{t^{*}>0\\}=\emptyset.$
(5.41)

Let $\nu:\widetilde{X}\longrightarrow X$ be a proper morphism of complex
manifolds such that the restriction
$\widetilde{X}\setminus\nu^{-1}(P^{-1}(0)\cup Q^{-1}(0))\longrightarrow
X\setminus(P^{-1}(0)\cup Q^{-1}(0))$ of $\nu$ is an isomorphism,
$\nu^{-1}(P^{-1}(0)\cup Q^{-1}(0))\subset\widetilde{X}$ is a normal crossing
divisor, and the meromorphic function $f\circ\nu:=(P\circ\nu)/(Q\circ\nu)$ on
$\widetilde{X}$ has no point of indeterminacy on the whole $\widetilde{X}$.
Moreover we set $\widetilde{\nu}:=\nu\times{\rm
id}_{\mathbb{R}}:\widetilde{X}\times\mathbb{R}\longrightarrow
X\times\mathbb{R}$. Then we can easily see that the micro-support of
$\widetilde{\nu}^{-1}(F^{\circ})_{\pi^{-1}(X\setminus P^{-1}(0))}$ does not
meet the open subset $\\{t^{*}>0\\}\subset
T^{*}{\widetilde{X}}_{\mathbb{R}}\times T^{*}\mathbb{R}$. Since
$\widetilde{\nu}$ is proper and there exists an isomorphism

$(F^{\circ})_{\pi^{-1}(X\setminus
P^{-1}(0))}\overset{\sim}{\longrightarrow}{\rm
R}\widetilde{\nu}_{*}\widetilde{\nu}^{-1}(F^{\circ})_{\pi^{-1}(X\setminus
P^{-1}(0))}$ (5.42)

on $X\times\mathbb{R}$, by [22, Proposition 5.4.4] we obtain the desired
condition

${\rm SS}\bigl{(}(F^{\circ})_{\pi^{-1}(X\setminus
P^{-1}(0))}\bigr{)}\cap\\{t^{*}>0\\}=\emptyset.$ (5.43)

Next, for $\varepsilon,\delta>0$ such that $0<\delta<\varepsilon$ we define
two closed subsets $A_{\varepsilon},B_{\varepsilon,\delta}\subset
X_{\mathbb{R}}\times\mathbb{R}$ by

$A_{\varepsilon}:=\\{(z,t)\in X_{\mathbb{R}}\times\mathbb{R}\ |\
||z-z_{0}||\leq\varepsilon,\ |t-t_{0}+{\rm Re}\langle
z-z_{0},w_{0}\rangle|\leq\varepsilon\\}$ (5.44)

and

$B_{\varepsilon,\delta}:=\\{(z,t)\in A_{\varepsilon}\ |\ t-t_{0}+{\rm
Re}\langle z-z_{0},w_{0}\rangle\leq-\delta\\}$ (5.45)

respectively. Note that we have $B_{\varepsilon,\delta}\subset
A_{\varepsilon}$ and

$A_{\varepsilon}\setminus
H_{+}=\bigcup_{\delta:0<\delta<\varepsilon}B_{\varepsilon,\delta}.$ (5.46)

Then for a fixed $\varepsilon>0$, by applying [22, Proposition 8.3.12] (the
microlcal Bertini-Sard theorem) to the $\mathbb{R}$-construnctible sheaf
$G_{A_{\varepsilon}}$ and the real analytic Morse function

$\varphi(z,t):=t-t_{0}+{\rm Re}\langle z-z_{0},w_{0}\rangle,$ (5.47)

we obtain an isomorphism

$H^{j}(X_{\mathbb{R}}\times\mathbb{R};{\rm
R}\Gamma_{H_{+}}(G_{A_{\varepsilon}}))\simeq
H^{j}(X_{\mathbb{R}}\times\mathbb{R};G_{A_{\varepsilon}\setminus
B_{\varepsilon,\delta}})$ (5.48)

for $\delta>0$ such that $0<\delta\ll\varepsilon$ and any $j\in\mathbb{Z}$.
Since the subsets $A_{\varepsilon}\subset X_{\mathbb{R}}\times\mathbb{R}$
($\varepsilon>0$) form a fundamental system of neighborhoods of the point
$(z_{0},t_{0})\in X_{\mathbb{R}}\times\mathbb{R}$, for any $j\in\mathbb{Z}$ we
have an isomorphism

$H^{j}{\rm
R}\Gamma_{H_{+}}(G)_{(z_{0},t_{0})}\simeq\underset{\varepsilon,\delta\to+0}{\varinjlim}H^{j}(X_{\mathbb{R}}\times\mathbb{R};G_{A_{\varepsilon}\setminus
B_{\varepsilon,\delta}}),$ (5.49)

where in the inductive limit $\varepsilon>0$ and $\delta>0$ range over the
positive real numbers such that $0<\delta\ll\varepsilon$. Let

$\iota:U=X\setminus Q^{-1}(0)\hookrightarrow
X_{\mathbb{R}}\times\mathbb{R}\qquad(z\longmapsto(z,-{\rm Re}f(z))$ (5.50)

be the graph embedding defined by the function $-{\rm Re}f$ and ${\rm
pr}_{X_{\mathbb{R}}}:X_{\mathbb{R}}\times\mathbb{R}\longrightarrow
X_{\mathbb{R}}$ the projection. Then, the morphism ${\rm pr}_{X_{\mathbb{R}}}$
being finite on the support of $G_{A_{\varepsilon}\setminus
B_{\varepsilon,\delta}}$, we have isomorphisms

$\displaystyle{\rm R}({\rm
pr}_{X_{\mathbb{R}}})_{*}G_{A_{\varepsilon}\setminus
B_{\varepsilon,\delta}}\simeq{\rm R}({\rm
pr}_{X_{\mathbb{R}}})_{!}G_{A_{\varepsilon}\setminus B_{\varepsilon,\delta}}$
$\displaystyle\simeq\ $ $\displaystyle\ {\rm R}({\rm
pr}_{X_{\mathbb{R}}})_{!}\bigl{(}\iota_{!}(K|_{U})\otimes_{\mathbb{C}}\mathbb{C}_{A_{\varepsilon}\setminus
B_{\varepsilon,\delta}}\bigr{)}\ \simeq\ {\rm R}({\rm
pr}_{X_{\mathbb{R}}})_{!}\ \iota_{!}\
\bigl{(}(K|_{U})\otimes_{\mathbb{C}}\iota^{-1}\mathbb{C}_{A_{\varepsilon}\setminus
B_{\varepsilon,\delta}}\bigr{)}$ $\displaystyle\simeq\ $ $\displaystyle\
K_{U\cap\iota^{-1}(A_{\varepsilon}\setminus B_{\varepsilon,\delta})}.$

We thus obtain an isomorphism

$H^{j}{\rm
R}\Gamma_{H_{+}}(G)_{(z_{0},t_{0})}\simeq\underset{\varepsilon,\delta\to+0}{\varinjlim}H^{j}(X_{\mathbb{R}};K_{U\cap\iota^{-1}(A_{\varepsilon}\setminus
B_{\varepsilon,\delta})})$ (5.51)

for any $j\in\mathbb{Z}$. Moreover the set
$\iota^{-1}(A_{\varepsilon}\setminus B_{\varepsilon,\delta})\subset U\subset
X_{\mathbb{R}}$ is explicitly described as follows:

$\iota^{-1}(A_{\varepsilon}\setminus B_{\varepsilon,\delta})=\\{z\in U\ |\
||z-z_{0}||\leq\varepsilon,\ -\delta<{\rm
Re}f^{w_{0}}(z)-c_{0}\leq\varepsilon\\},$ (5.52)

where we set $c_{0}:={\rm Re}\langle z_{0},w_{0}\rangle+t_{0}$. Then by
Proposition 2.5 we obtain an isomorphism

$H^{j}{\rm R}\Gamma_{H_{+}}(G)_{(z_{0},t_{0})}\simeq\bigoplus_{c:\ {\rm
Re}c=c_{0}}H^{j}\phi^{{\rm mero,c}}_{f^{w_{0}}-c}(K)_{z_{0}}$ (5.53)

for any $j\in\mathbb{Z}$. From this the assertion immediately follows. ∎

######  Definition 5.5.

In the situation of Poposition 5.4, for the the sheaf $F=\pi^{-1}K\otimes{\rm
E}_{U^{{\rm an}}|X^{{\rm an}}}^{{\rm Re}f}$ on
$X_{\mathbb{R}}\times\mathbb{R}$ we define a subset ${\rm SS}_{{\rm
eva}}(F)\subset(T^{*}X)_{\mathbb{R}}$ of $(T^{*}X)_{\mathbb{R}}\simeq
T^{*}X_{\mathbb{R}}$ by

${\rm SS}_{{\rm eva}}(F):=\\{(z_{0},w_{0})\in(T^{*}X)_{\mathbb{R}}\ |\
\phi^{{\rm mero,c}}_{f^{w_{0}}-c}(K)_{z_{0}}\not=0\ \text{for some}\
c\in\mathbb{C}\\}.$ (5.54)

By Proposition 5.4 this set ${\rm SS}_{{\rm eva}}(F)\subset
T^{*}X_{\mathbb{R}}$ is contained in the irregular micro-support ${\rm
SS}_{{\rm irr}}(F)$ of $F$. With Lemmas 5.2 and 5.3 and Proposition 5.4 at
hands, now we have obtained a rough picture of the irregular micro-support
${\rm SS}_{{\rm irr}}(F)\subset T^{*}X_{\mathbb{R}}$. However, for the moment
it is not clear for us if it is a complex Lagrangian analytic subset of
$(T^{*}X)_{\mathbb{R}}\simeq T^{*}X_{\mathbb{R}}$ or not. For this reason, we
define a (not necessarily homogeneous) complex Lagrangian analytic subset
${\rm SS}^{\mathbb{C}}_{{\rm irr}}(\mathcal{F})$ of
$(T^{*}X)_{\mathbb{R}}\simeq T^{*}X_{\mathbb{R}}$ in a different way as
follows and use it instead of ${\rm SS}_{{\rm irr}}(F)$. First, as a complex
analogue of the $\mathbb{R}$-constructible sheaf $F=\pi^{-1}K\otimes{\rm
E}_{U^{{\rm an}}|X^{{\rm an}}}^{{\rm Re}f}$ on
$X_{\mathbb{R}}\times\mathbb{R}$, by applying the proof of Theorem 2.3 (i) to
the (not necessarily) closed embedding

$i_{-f}:U=X\setminus Q^{-1}(0)\hookrightarrow
X\times\mathbb{C},\qquad(z\longmapsto(z,-f(z)))$ (5.55)

associated to the rational function $-f:U\longrightarrow\mathbb{C}$, we obtain
a complex constructible sheaf

$\mathcal{F}:=(i_{-f})_{!}(K|_{U})\quad\in{\bf
D}_{\mathrm{c}}^{\mathrm{b}}(X\times\mathbb{C})$ (5.56)

on $X\times\mathbb{C}$. Then for the morphism ${\rm id}_{X}\times{\rm
Re}:X\times\mathbb{C}\longrightarrow X\times\mathbb{R}$ induced by the one
${\rm Re}:\mathbb{C}\longrightarrow\mathbb{R}$ ($\tau\longmapsto{\rm Re}\tau$)
we have an isomorphism

$F\simeq\mathbb{C}_{\\{t\geq 0\\}}\overset{+}{\otimes}{\rm R}({\rm
id}_{X}\times{\rm Re})_{!}\mathcal{F}.$ (5.57)

Note that $\mathcal{F}[N]$ is a perverse sheaf and its micro-support ${\rm
SS}(\mathcal{F}[N])={\rm SS}(\mathcal{F})$ is a homogeneous complex Lagrangian
analytic subset of $T^{*}(X\times\mathbb{C})$. Then as in the definitions of
${\rm SS}^{{\rm E}}(F)\subset(T^{*}X_{\mathbb{R}})\times\mathbb{R}$ and ${\rm
SS}_{{\rm irr}}(F)\subset T^{*}X_{\mathbb{R}}$, by forgetting the homogeneity
of ${\rm SS}(\mathcal{F})$ we define two subsets:

${\rm SS}^{{\rm E},\mathbb{C}}(\mathcal{F})\subset
T^{*}X\times\mathbb{C},\qquad{\rm SS}_{{\rm
irr}}^{\mathbb{C}}(\mathcal{F})\subset T^{*}X$ (5.58)

in the following way. First, by the closed embedding

$\iota:(T^{*}X)\times\mathbb{C}\hookrightarrow
T^{*}(X\times\mathbb{C})\qquad(((z,w),\tau)\longmapsto(z,\tau,w,1))$ (5.59)

we set

${\rm SS}^{{\rm E},\mathbb{C}}(\mathcal{F}):=\iota^{-1}{\rm
SS}(\mathcal{F})\subset T^{*}X\times\mathbb{C}.$ (5.60)

It is clear that ${\rm SS}^{{\rm E},\mathbb{C}}(\mathcal{F})$ is a complex
analytic subset of $T^{*}X\times\mathbb{C}$. Let us call it the enhanced
micro-support of $\mathcal{F}\in{\bf
D}_{\mathrm{c}}^{\mathrm{b}}(X\times\mathbb{C})$. Then we define ${\rm
SS}_{{\rm irr}}^{\mathbb{C}}(\mathcal{F})\subset T^{*}X$ to be the closure of
the image of ${\rm SS}^{{\rm E},\mathbb{C}}(\mathcal{F})$ by the projection
$T^{*}X\times\mathbb{C}\longrightarrow T^{*}X$. We call it the irregular
micro-support of $\mathcal{F}$. As the restriction of $f$ to $Z$ is uniquely
determined by $\mathcal{M}$ modulo constant functions on $Z$, we see that
${\rm SS}_{{\rm irr}}^{\mathbb{C}}(\mathcal{F})$ does not depend on the
expression $\mathcal{N}\overset{D}{\otimes}\mathcal{E}^{f}_{U|X}$ of
$\mathcal{M}$. It is easy to see that a result analogous to Lemma 5.2 holds
also for ${\rm SS}_{{\rm irr}}^{\mathbb{C}}(\mathcal{F})\subset T^{*}X$.

######  Lemma 5.6.

The irregular micro-support ${\rm SS}_{{\rm
irr}}^{\mathbb{C}}(\mathcal{F})\subset T^{*}X$ of $\mathcal{F}\in{\bf
D}_{\mathrm{c}}^{\mathrm{b}}(X\times\mathbb{C})$ is a complex Lagrangian
analytic subset of $(T^{*}X)_{\mathbb{R}}\simeq T^{*}X_{\mathbb{R}}$. If we
assume moreover that $I(f)=P^{-1}(0)\cap Q^{-1}(0)=\emptyset$, then we have

${\rm SS}_{{\rm irr}}^{\mathbb{C}}(\mathcal{F})=\bigl{(}{\rm SS}(K)\cap
T^{*}U_{\mathbb{R}}\bigr{)}+\Lambda^{f}.$ (5.61)

###### Proof.

By the inclusion map $j:X\times\mathbb{C}\hookrightarrow X\times{\mathbb{P}}$
we obtain an algebraic constructible sheaf

$j_{!}\mathcal{F}=j_{!}(i_{-f})_{!}(K|_{U})\quad\in{\bf
D}_{\mathrm{c}}^{\mathrm{b}}(X\times{\mathbb{P}})$ (5.62)

on $X\times{\mathbb{P}}$. Then by the theorem in [12, page 43] there exist
Whitney stratifications $\mathcal{S}$ and $\mathcal{S}_{0}$ of
$X\times{\mathbb{P}}$ and $X$ respectively such that

${\rm
SS}(j_{!}\mathcal{F})\subset\bigsqcup_{S\in\mathcal{S}}T^{*}_{S}(X\times{\mathbb{P}})$
(5.63)

and the projection $X\times{\mathbb{P}}\longrightarrow X$ is a stratified
fiber bundle as in its assertion. Note that for each stratum
$S_{0}\in\mathcal{S}_{0}$ in $\mathcal{S}_{0}$ there exist only finitely many
strata $S\in\mathcal{S}$ in $\mathcal{S}$ projecting to it such that ${\rm
dim}S={\rm dim}S_{0}$ and the enhanced micro-support ${\rm SS}^{{\rm
E},\mathbb{C}}(\mathcal{F})\subset T^{*}X\times\mathbb{C}$ of $\mathcal{F}$ is
determined only by the conormal bundles $T^{*}_{S}(X\times{\mathbb{P}})$ of
such “horizontal” strata $S\in\mathcal{S}$ such that $S\subset
X\times\mathbb{C}$. Then we obtain the first assertion by simple calculations.
We obtain also the second one as in the proof of Lemma 5.3. ∎

By the proof of Lemma 5.6, we obtain a Lagrangian cycle in $T^{*}X$ supported
by the irregular micro-support ${\rm SS}_{{\rm
irr}}^{\mathbb{C}}(\mathcal{F})\subset T^{*}X$ of $\mathcal{F}$. As it depends
only on $\mathcal{M}$, we call it the irregular characteristic cycle of
$\mathcal{M}$ and denote it by ${\rm CC}_{{\rm irr}}(\mathcal{M})$. Moreover
it satisfies also the following functorial property. Let
$\phi:X\longrightarrow Y$ be a proper morphism of smooth algebraic varieties
and
$\mathcal{M}^{\prime}:={\mathbf{D}}\phi_{\ast}\mathcal{M}\in{\mathbf{D}}^{\mathrm{b}}_{\rm
hol}(\mathcal{D}_{Y})$ the direct image of the exponentially twisted holonomic
$\mathcal{D}_{X}$-module $\mathcal{M}$ by it. Then by Theorem 4.1 (iii) for
the enhanced sheaf
$F^{\prime}:=\mathbf{E}\phi_{*}F[d_{X}-d_{Y}]\in{\mathbf{E}}^{\mathrm{b}}_{+}(\mathbb{C}_{Y^{{\rm
an}}})$ on $Y^{{\rm an}}$ we have an isomorphism

$Sol_{Y}^{\rm E}(\mathcal{M}^{\prime})\simeq\mathbb{C}^{\rm E}_{Y^{{\rm
an}}}\overset{+}{\otimes}F^{\prime}.$ (5.64)

Similarly to the case of $\mathcal{M}$, also for
$\mathcal{M}^{\prime}={\mathbf{D}}\phi_{\ast}\mathcal{M}$ we can define a
Lagrangian cycle in $T^{*}Y$ by using the complex constructible sheaf
$\mathcal{F}^{\prime}:={\rm R}(\phi\times{\rm
id}_{\mathbb{C}})_{*}\mathcal{F}[d_{X}-d_{Y}]\in{\bf
D}_{\mathrm{c}}^{\mathrm{b}}(Y\times\mathbb{C})$. Denote it by ${\rm CC}_{{\rm
irr}}(\mathcal{M}^{\prime})$ and let

$T^{*}Y\overset{\varpi}{\longleftarrow}X\times_{Y}T^{*}Y\overset{\rho}{\longrightarrow}T^{*}X$
(5.65)

be the natural morphisms associated to $\phi:X\longrightarrow Y$. Then there
exists an equality

${\rm CC}_{{\rm irr}}(\mathcal{M}^{\prime})=\varpi_{*}\rho^{*}{\rm CC}_{{\rm
irr}}(\mathcal{M})$ (5.66)

of Lagrangian cycles in $T^{*}Y$. Note also that by the commutative diagram

$\begin{CD}X\times\mathbb{C}@>{{\rm id}_{X}\times{\rm
Re}}>{}>X\times\mathbb{R}\\\ @V{\phi\times{\rm
id}_{\mathbb{C}}}V{}V@V{}V{\widetilde{\phi}}V\\\ Y\times\mathbb{C}@>{{\rm
id}_{Y}\times{\rm Re}}>{}>Y\times\mathbb{R}\end{CD}$ (5.67)

we obtain an isomorphism

$F^{\prime}\simeq\mathbb{C}_{\\{t\geq 0\\}}\overset{+}{\otimes}{\rm R}({\rm
id}_{Y}\times{\rm Re})_{!}\mathcal{F}^{\prime}.$ (5.68)

Also for a morphism $\psi:Y\longrightarrow X$ of smooth algebraic varieties
and the inverse image
$\mathcal{M}^{\prime}:={\mathbf{D}}\psi^{\ast}\mathcal{M}\in{\mathbf{D}}^{\mathrm{b}}_{\rm
hol}(\mathcal{D}_{Y})$ of $\mathcal{M}$ by it, we can define a Lagrangian
cycle ${\rm CC}_{{\rm irr}}(\mathcal{M}^{\prime})$ in $T^{*}Y$ and obtain a
formula which expresses it in terms of ${\rm CC}_{{\rm irr}}(\mathcal{M})$
under some non-characteristic condition. Note that in this case the inverse
image $\mathcal{M}^{\prime}={\mathbf{D}}\psi^{\ast}\mathcal{M}$ is also of
exponentially twisted type. The folowing result is a complex analogue of
Proposition 5.4.

######  Proposition 5.7.

In the case where $X$ is $\mathbb{C}^{N}_{z}$, $Y$ is its dual vector space
$\mathbb{C}^{N}_{w}$ and $T^{*}X\simeq X\times Y$, for a point $(z_{0},w_{0})$
of $T^{*}X\simeq X\times Y$ assume that there exists a complex number
$c\in\mathbb{C}$ such that

$\phi^{{\rm mero,c}}_{f^{w_{0}}-c}(K)_{z_{0}}\not=0$ (5.69)

and set $\tau_{0}:=c-\langle z_{0},w_{0}\rangle\in\mathbb{C}$. Then we have
$(z_{0},w_{0},\tau_{0})\in{\rm SS}^{{\rm E},\mathbb{C}}(\mathcal{F})$,
equivalently $(z_{0},\tau_{0},w_{0},1)\in{\rm SS}(\mathcal{F})$.

###### Proof.

Let us consider the (not necessarily) closed embedding

$i_{f^{w_{0}}}:U=X\setminus Q^{-1}(0)\hookrightarrow
X\times\mathbb{C},\qquad(z\longmapsto(z,f^{w_{0}}(z)))$ (5.70)

associated to the rational function $f^{w_{0}}:U\longrightarrow\mathbb{C}$.
Then we obtain a non-vanishing

$\phi_{\tau-c}\bigl{(}(i_{f^{w_{0}}})_{!}(K|_{U})\bigr{)}_{(z_{0},c)}\simeq\phi^{{\rm
mero,c}}_{f^{w_{0}}-c}(K)_{z_{0}}\not=0.$ (5.71)

By [22, Proposition 8.6.3], this implies that the point $(z_{0},c,0,1)\in
T^{*}(X\times\mathbb{C})$ is contained in ${\rm
SS}\bigl{(}(i_{f^{w_{0}}})_{!}(K|_{U})\bigr{)}\subset
T^{*}(X\times\mathbb{C})$. On the other hand, for the automorphism $T_{w_{0}}$
of $X\times\mathbb{C}$ defined by

$T_{w_{0}}:X\times\mathbb{C}\overset{\sim}{\longrightarrow}X\times\mathbb{C},\qquad((z,\tau)\longmapsto(z,\tau+\langle
z,w_{0}\rangle)),$ (5.72)

we have an isomorphism

$(T_{w_{0}})_{*}(\mathcal{F})=(T_{w_{0}})_{*}(i_{-f})_{!}(K|_{U})\simeq(i_{f^{w_{0}}})_{!}(K|_{U}).$
(5.73)

Then the assertion $(z_{0},\tau_{0},w_{0},1)\in{\rm SS}(\mathcal{F})$
immediately follows from it. ∎

By Proposition 5.7 the subset ${\rm SS}_{{\rm
eva}}(F)\subset(T^{*}X)_{\mathbb{R}}$ is contained in the irregular micro-
support ${\rm SS}_{{\rm irr}}^{\mathbb{C}}(\mathcal{F})$ of
$\mathcal{F}\in{\bf D}_{\mathrm{c}}^{\mathrm{b}}(X\times\mathbb{C})$. It is
also clear that we have

${\rm SS}_{{\rm irr}}^{\mathbb{C}}(\mathcal{F})\cap
T^{*}U_{\mathbb{R}}=\bigl{(}{\rm SS}(K)\cap
T^{*}U_{\mathbb{R}}\bigr{)}+\Lambda^{f}.$ (5.74)

##  6 Fourier Transforms of Exponentially Twisted
Holonomic D-modules

In this section, we study the Fourier transforms of exponentially twisted
holonomic D-modules. We inherit the situation and the notations in Section 1.
Let

$X\overset{p}{\longleftarrow}X\times Y\overset{q}{\longrightarrow}Y$ (6.1)

be the projections. Then by Katz-Laumon [29], for an algebraic holonomic
$\mathcal{D}_{X}$-module $\mathcal{M}\in\mathrm{Mod}_{\mbox{\rm\scriptsize
hol}}(\mathcal{D}_{X})$ we have an isomorphism

$\mathcal{M}^{\wedge}\simeq\mathbf{D}q_{\ast}(\mathbf{D}p^{\ast}\mathcal{M}\overset{D}{\otimes}\mathcal{O}_{X\times
Y}e^{-\langle z,w\rangle}),$ (6.2)

where $\mathbf{D}p^{\ast},\mathbf{D}q_{\ast},\overset{D}{\otimes}$ are the
operations for algebraic D-modules and $\mathcal{O}_{X\times Y}e^{-\langle
z,w\rangle}$ is the integral connection of rank one on $X\times Y$ associated
to the canonical paring $\langle,\rangle:X\times Y\to\mathbb{C}$. In
particular the right hand side is concentrated in degree zero. 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
$\mathcal{M}\in\mathrm{Mod}_{\mbox{\rm\scriptsize hol}}(\mathcal{D}_{X})$ on
$X$ to the one
$\widetilde{\mathcal{M}}:=i_{X\ast}\mathcal{M}\simeq\mathbf{D}i_{X\ast}\mathcal{M}$
on $\overline{X}$. Denote by $\overline{X}^{{\rm an}}$ the underlying complex
manifold of $\overline{X}$ and define the analytification
$\widetilde{\mathcal{M}}^{{\ }{\rm an}}\in\mathrm{Mod}_{\mbox{\rm\scriptsize
hol}}(\mathcal{D}_{\overline{X}^{{\rm an}}})$ 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

$Sol_{\overline{X}}^{\rm E}(\widetilde{\mathcal{M}}):=Sol_{\overline{X}^{{\rm
an}}}^{\rm E}(\widetilde{\mathcal{M}}^{{\ }{\rm an}})\ \in\
{\mathbf{E}}^{\mathrm{b}}({\rm I}\mathbb{C}_{\overline{X}^{{\rm an}}}).$ (6.3)

Similarly for the Fourier transform
$\mathcal{M}^{\wedge}\in\mathrm{Mod}_{\mbox{\rm\scriptsize
hol}}(\mathcal{D}_{Y})$, by the inclusion map
$i_{Y}:Y=\mathbb{C}^{N}\xhookrightarrow{\ \ \ }\overline{Y}={\mathbb{P}}^{N}$
we define $Sol_{\overline{Y}}^{\rm
E}(\widetilde{\mathcal{M}^{\wedge}})\in{\mathbf{E}}^{\mathrm{b}}({\rm
I}\mathbb{C}_{\overline{Y}^{{\rm an}}})$. Let

$\overline{X}^{{\rm
an}}\overset{\overline{p}}{\longleftarrow}\overline{X}^{{\rm
an}}\times\overline{Y}^{{\rm
an}}\overset{\overline{q}}{\longrightarrow}\overline{Y}^{{\rm an}}$ (6.4)

be the projections. Then the following theorem is essentially due to
Kashiwara-Schapira [27] and D’Agnolo-Kashiwara [8]. For
$F\in{\mathbf{E}}^{\mathrm{b}}({\rm I}\mathbb{C}_{\overline{X}^{{\rm an}}})$
we set

${}^{\rm
L}F:=\mathbf{E}\overline{q}_{*}(\mathbf{E}\overline{p}^{-1}F\overset{+}{\otimes}\mathbb{E}_{X\times
Y|\overline{X}\times\overline{Y}}^{-{\rm Re}\langle z,w\rangle}[N])\ \in\
{\mathbf{E}}^{\mathrm{b}}({\rm I}\mathbb{C}_{\overline{Y}^{{\rm an}}})$ (6.5)

(here we denote $X^{{\rm an}}\times Y^{{\rm an}}$ etc. by $X\times Y$ etc. for
short) and call it the Fourier-Sato (Fourier-Laplace) transform of $F$.

######  Theorem 6.1.

For $\mathcal{M}\in\mathrm{Mod}_{\rm hol}(\mathcal{D}_{X})$ there exists an
isomorphism

$Sol_{\overline{Y}}^{\rm E}(\widetilde{\mathcal{M}^{\wedge}})\simeq{}^{\rm
L}Sol_{\overline{X}}^{\rm E}(\widetilde{\mathcal{M}}).$ (6.6)

First, let us consider the case where for an enhanced sheaf
$F\in{\mathbf{E}}^{\mathrm{b}}_{+}(\mathbb{C}_{X^{{\rm an}}})$ on $X^{{\rm
an}}$ such that $F\simeq\mathbb{C}_{\\{t\geq 0\\}}\overset{+}{\otimes}F$ we
have an isomorphism

$Sol_{\overline{X}}^{\rm E}(\widetilde{\mathcal{M}})\simeq\mathbb{C}^{\rm
E}_{\overline{X}^{{\rm an}}}\overset{+}{\otimes}\mathbf{E}i_{X!}F.$ (6.7)

For an enhanced sheaf
$G\in{\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_{\overline{X}^{{\rm an}}})$ on
$\overline{X}^{{\rm an}}$ we define its Fourier-Sato (Fourier-Laplace)
transform ${}^{\rm
L}G\in{\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_{\overline{Y}^{{\rm an}}})$ by

${}^{\rm
L}G:=\mathbf{E}\overline{q}_{*}(\mathbf{E}\overline{p}^{-1}G\overset{+}{\otimes}{\rm
E}_{X\times Y|\overline{X}\times\overline{Y}}^{-{\rm Re}\langle
z,w\rangle}[N])\ \in\ {\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_{\overline{Y}^{{\rm
an}}}).$ (6.8)

Since by [7, Proposition 4.7.17] we have

${}^{\rm L}(\mathbb{C}^{\rm E}_{\overline{X}^{{\rm
an}}}\overset{+}{\otimes}(\cdot))\simeq\mathbb{C}^{\rm E}_{\overline{Y}^{{\rm
an}}}\overset{+}{\otimes}{}^{\rm L}(\cdot),$ (6.9)

by Theorem 6.1 and (6.7) we obtain an isomorphism

$Sol_{\overline{Y}}^{\rm
E}(\widetilde{\mathcal{M}^{\wedge}})\simeq\mathbb{C}^{\rm
E}_{\overline{Y}^{{\rm an}}}\overset{+}{\otimes}{}^{\rm
L}(\mathbf{E}i_{X!}F).$ (6.10)

Hence it suffices to study the enhanced sheaf

${}^{\rm
L}(\mathbf{E}i_{X!}F)\simeq\mathbf{E}i_{Y!}\mathbf{E}q_{!}(\mathbf{E}p^{-1}F\overset{+}{\otimes}{\rm
E}_{X\times Y|X\times Y}^{-{\rm Re}\langle z,w\rangle}[N])\ \in\
{\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_{\overline{Y}^{{\rm an}}})$ (6.11)

on $\overline{Y}^{{\rm an}}$. As was explained in [8, Sections 2.6 and 7.3],
we can know the geometric structure of
$L(F):=\mathbf{E}q_{!}(\mathbf{E}p^{-1}F\overset{+}{\otimes}{\rm E}_{X\times
Y|X\times Y}^{-{\rm Re}\langle
z,w\rangle}[N])\in{\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_{Y^{{\rm an}}})$ to
some extent by Tamarkin’s microlocal thery of sheaves in [54] as follows.
Since we have $L(F)\in{\mathbf{E}}^{\mathrm{b}}_{+}(\mathbb{C}_{Y^{{\rm
an}}})$ we can define ${\rm SS}^{{\rm
E}}(L(F))\subset(T^{*}Y_{\mathbb{R}})\times\mathbb{R}$ and ${\rm SS}^{{\rm
E}}_{{\rm red}}(L(F))\subset T^{*}Y_{\mathbb{R}}$ also for $L(F)$. Then we
have the following theorem due to Tamarkin [54]. Let us consider the maps

$\widetilde{\chi}:(T^{*}X^{{\rm
an}})\times\mathbb{R}\overset{\sim}{\longrightarrow}(T^{*}Y^{{\rm
an}})\times\mathbb{R},\quad((z,w),t)\longmapsto((w,-z),t+{\rm Re}\langle
z,w\rangle)$ (6.12)

and

$\chi:T^{*}X^{{\rm an}}\overset{\sim}{\longrightarrow}T^{*}Y^{{\rm
an}},\quad(z,w)\longmapsto(w,-z)).$ (6.13)

######  Theorem 6.2 ([54, Theorem 3.6]).

For $F\in{\mathbf{E}}^{\mathrm{b}}_{+}(\mathbb{C}_{X^{{\rm an}}})$ and
$L(F)=\mathbf{E}q_{!}(\mathbf{E}p^{-1}F\overset{+}{\otimes}{\rm E}_{X\times
Y|X\times Y}^{-{\rm Re}\langle
z,w\rangle}[N])\in{\mathbf{E}}^{\mathrm{b}}_{+}(\mathbb{C}_{Y^{{\rm an}}})$ we
have

${\rm SS}^{{\rm E}}(L(F))=\widetilde{\chi}({\rm SS}^{{\rm E}}(F)),\quad{\rm
SS}_{{\rm irr}}(L(F))=\chi({\rm SS}_{{\rm irr}}(F)).$ (6.14)

For a rational function $f=\frac{P}{Q}:X\setminus
Q^{-1}(0)\longrightarrow\mathbb{C}$
($P,Q\in\Gamma(X;\mathcal{O}_{X})\simeq\mathbb{C}[z_{1},z_{2},\ldots,z_{N}]$,
$Q\not=0$) on $X=\mathbb{C}^{N}_{z}$ we set $U:=X\setminus Q^{-1}(0)$ and
define an exponential $\mathcal{D}_{X}$-module
$\mathcal{E}^{f}_{U|X}\in\mathrm{Mod}_{\mbox{\rm\scriptsize
hol}}(\mathcal{D}_{X})$ as in the analytic case. Then by Theorem 4.1 (vi) for
its extension
$(\mathcal{E}^{f}_{U|X})^{\sim}:=i_{X\ast}\mathcal{E}^{f}_{U|X}\simeq\mathbf{D}i_{X\ast}\mathcal{E}^{f}_{U|X}$
to $\overline{X}={\mathbb{P}}^{N}$ we have an isomorphism

$Sol_{\overline{X}}^{\rm
E}((\mathcal{E}^{f}_{U|X})^{\sim})\simeq\mathbb{E}_{U^{{\rm
an}}|\overline{X}^{{\rm an}}}^{{\rm Re}f}.$ (6.15)

From now on, we fix an exponentially twisted holonomic
$\mathcal{D}_{X}$-module $\mathcal{M}\in\mathrm{Mod}_{\mbox{\rm\scriptsize
hol}}(\mathcal{D}_{X})$ such that
$\mathcal{M}\simeq\mathcal{N}\overset{D}{\otimes}\mathcal{E}^{f}_{U|X}$ for a
regular holonomic $\mathcal{D}_{X}$-module
$\mathcal{N}\in\mathrm{Mod}_{\mbox{\rm\scriptsize rh}}(\mathcal{D}_{X})$ and a
rational function $f=\frac{P}{Q}:U=X\setminus
Q^{-1}(0)\longrightarrow\mathbb{C}$ on $X$ and study the basic properties of
its Fourier transform $\mathcal{M}^{\wedge}$. Let us set

$K:=Sol_{X}(\mathcal{N})\in{\bf D}_{\mathrm{c}}^{\mathrm{b}}(X^{{\rm an}}).$
(6.16)

Note that the shift $K[N]\in{\bf D}_{\mathrm{c}}^{\mathrm{b}}(X^{{\rm an}})$
of $K$ is a perverse sheaf on $X^{{\rm an}}$. We denote the support of
$K\in{\bf D}_{\mathrm{c}}^{\mathrm{b}}(X^{{\rm an}})$ by $Z\subset X$. Then by
Theorem 4.1 (vi) and Corollary 4.2 there exists an isomorphism

$Sol_{\overline{X}}^{\rm
E}(\widetilde{\mathcal{M}})\simeq\bigl{\\{}\pi^{-1}(i_{X})_{!}K\bigr{\\}}\otimes\mathbb{E}_{U^{{\rm
an}}|\overline{X}^{{\rm an}}}^{{\rm Re}f}.$ (6.17)

Moreover for the enhanced sheaf

$\bigl{\\{}\pi^{-1}(i_{X})_{!}K\bigr{\\}}\otimes{\rm E}_{U^{{\rm
an}}|\overline{X}^{{\rm an}}}^{{\rm
Re}f}\simeq\mathbf{E}i_{X!}\Bigl{\\{}\pi^{-1}K\otimes{\rm E}_{U^{{\rm
an}}|X^{{\rm an}}}^{{\rm
Re}f}\Bigr{\\}}\qquad\in{\mathbf{E}}^{\mathrm{b}}_{+}(\mathbb{C}_{\overline{X}^{{\rm
an}}})$ (6.18)

on $\overline{X}^{{\rm an}}$ we have an isomorphism

$Sol_{\overline{X}}^{\rm E}(\widetilde{\mathcal{M}})\simeq\mathbb{C}^{\rm
E}_{\overline{X}^{{\rm
an}}}\overset{+}{\otimes}\Bigl{\\{}\bigl{\\{}\pi^{-1}(i_{X})_{!}K\bigr{\\}}\otimes{\rm
E}_{U^{{\rm an}}|\overline{X}^{{\rm an}}}^{{\rm Re}f}\Bigr{\\}}.$ (6.19)

It follows from Theorem 6.1 that there exists an isomorphism

$Sol_{\overline{Y}}^{\rm
E}(\widetilde{\mathcal{M}^{\wedge}})\simeq\mathbb{C}^{\rm
E}_{\overline{Y}^{{\rm an}}}\overset{+}{\otimes}{}^{\rm
L}\Bigl{\\{}\bigl{\\{}\pi^{-1}(i_{X})_{!}K\bigr{\\}}\otimes{\rm E}_{U^{{\rm
an}}|\overline{X}^{{\rm an}}}^{{\rm Re}f}\Bigr{\\}}.$ (6.20)

Hence it suffices to study the enhanced sheaf

${}^{\rm L}\Bigl{\\{}\bigl{\\{}\pi^{-1}(i_{X})_{!}K\bigr{\\}}\otimes{\rm
E}_{U^{{\rm an}}|\overline{X}^{{\rm an}}}^{{\rm Re}f}\Bigr{\\}}\ \in\
{\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_{\overline{Y}^{{\rm an}}}).$ (6.21)

From now on, for the enhanced sheaf

$F:=\pi^{-1}K\otimes{\rm E}_{U^{{\rm an}}|X^{{\rm an}}}^{{\rm
Re}f}\in{\mathbf{E}}^{\mathrm{b}}_{+}(\mathbb{C}_{X^{{\rm an}}})$ (6.22)

on $X_{\mathbb{R}}$ we shall study
$L(F)=\mathbf{E}q_{!}(\mathbf{E}p^{-1}F\overset{+}{\otimes}{\rm E}_{X\times
Y|X\times Y}^{-{\rm Re}\langle
z,w\rangle}[N])\in{\mathbf{E}}^{\mathrm{b}}_{+}(\mathbb{C}_{Y^{{\rm an}}})$.
For this purpose, as a complex analogue of the $\mathbb{R}$-constructible
sheaf $F=\pi^{-1}K\otimes{\rm E}_{U^{{\rm an}}|X^{{\rm an}}}^{{\rm Re}f}$ on
$X_{\mathbb{R}}\times\mathbb{R}$, by applying the proof of Theorem 2.3 (i) to
the (not necessarily) closed embedding

$i_{-f}:U=X\setminus Q^{-1}(0)\hookrightarrow
X\times\mathbb{C},\qquad(z\longmapsto(z,-f(z)))$ (6.23)

associated to the rational function $-f:U\longrightarrow\mathbb{C}$, we obtain
a complex constructible sheaf

$\mathcal{F}:=(i_{-f})_{!}(K|_{U})\quad\in{\bf
D}_{\mathrm{c}}^{\mathrm{b}}(X\times\mathbb{C})$ (6.24)

on $X\times\mathbb{C}$. Note that $\mathcal{F}[N]$ is a perverse sheaf and its
micro-support ${\rm SS}(\mathcal{F}[N])={\rm SS}(\mathcal{F})$ is a
homogeneous complex Lagrangian analytic subset of $T^{*}(X\times\mathbb{C})$.
Then by forgetting the homogeneity of ${\rm SS}(\mathcal{F})$ we define two
subsets:

${\rm SS}^{{\rm E},\mathbb{C}}(\mathcal{F})\subset
T^{*}X\times\mathbb{C},\qquad{\rm SS}_{{\rm
irr}}^{\mathbb{C}}(\mathcal{F})\subset T^{*}X$ (6.25)

(see Section 5 for the details). Recall that the irregular micro-support ${\rm
SS}_{{\rm irr}}^{\mathbb{C}}(\mathcal{F})$ of $\mathcal{F}\in{\bf
D}_{\mathrm{c}}^{\mathrm{b}}(X\times\mathbb{C})$ is a (not necessarily
homogeneous) complex Lagrangian analytic subset of $T^{*}X$ and ${\rm
SS}_{{\rm eva}}(F)\subset(T^{*}X)_{\mathbb{R}}$ is contained in ${\rm
SS}_{{\rm irr}}^{\mathbb{C}}(\mathcal{F})$. In what follows, we set
$\Lambda:={\rm SS}_{{\rm
irr}}^{\mathbb{C}}(\mathcal{F})\subset(T^{*}X)_{\mathbb{R}}$. Recall that we
have

$\Lambda\cap T^{*}U_{\mathbb{R}}={\rm SS}_{{\rm
irr}}^{\mathbb{C}}(\mathcal{F})\cap T^{*}U_{\mathbb{R}}=\bigl{(}{\rm
SS}(K)\cap T^{*}U_{\mathbb{R}}\bigr{)}+\Lambda^{f}.$ (6.26)

Since $\Lambda$ is Lagrangian and ${\rm dim}\Lambda={\rm dim}X=N$, by e.g. the
theorem in [12, page 43] we can easily prove the following lemma.

######  Lemma 6.3.

There exists a non-empty Zariski open subset $\Omega\subset
Y=\mathbb{C}_{w}^{N}$ of $Y$ such that the restriction
$q^{-1}(\Omega)\cap\Lambda\longrightarrow\Omega$ of the projection $q:X\times
Y\longrightarrow Y$ is an unramified finite covering and any connected
component of the open subset $q^{-1}(\Omega)\cap\Lambda\subset\Lambda$ is a
fiber bundle over a complex submanifold of $X=\mathbb{C}^{N}_{z}$ contained in
$U=X\setminus Q^{-1}(0)$ or $I(f)=P^{-1}(0)\cap Q^{-1}(0)$.

######  Remark 6.4.

In the special case where $I(f)=P^{-1}(0)\cap Q^{-1}(0)=\emptyset$, we have
$\Lambda=\bigl{(}{\rm SS}(K)\cap T^{*}U_{\mathbb{R}}\bigr{)}+\Lambda^{f}$ by
Lemma 5.6 and hence there exists a (unique) maximal Zariski open subset
$\Omega\subset Y=\mathbb{C}^{N}$ satisfying the conditions in Lemma 6.3.

Let $\Omega\subset Y=\mathbb{C}_{w}^{N}$ be as in Lemma 6.3 and
$V\subset\Omega$ a contractible open subset of it. Then for the decomposition

$q^{-1}(V)\cap\Lambda=\Lambda_{V,1}\sqcup\Lambda_{V,2}\sqcup\cdots\cdots\sqcup\Lambda_{V,d}$
(6.27)

of $q^{-1}(V)\cap\Lambda$ into its connected components
$\Lambda_{V,i}\subset\Lambda$ ($1\leq i\leq d$) the morphism $q|_{\Lambda}$
induces an isomorphism $\Lambda_{V,i}\overset{\sim}{\longrightarrow}V$ for any
$1\leq i\leq d$. Now let us consider the symplectic transformation of [8]:

$\chi:T^{*}X\overset{\sim}{\longrightarrow}T^{*}Y\qquad((z,w)\longmapsto(w,-z)),$
(6.28)

where we used the natural identification $T^{*}Y\simeq Y\times X$. Then
$\chi(\Lambda_{V,i})\subset T^{*}Y$ is a Lagrangian submanifold of $T^{*}Y$
for which the natural projection $T^{*}Y\longrightarrow Y$ induces an
isomorphism $\chi(\Lambda_{V,i})\overset{\sim}{\longrightarrow}V$. It follows
that there exists a holomorphic function $g_{i}$ on $V$ such that

$\chi(\Lambda_{V,i})=\Lambda^{g_{i}}:=\\{(w,dg_{i}(w))\ |\ w\in V\\}\subset
T^{*}Y.$ (6.29)

Namely, if for a point $w\in V$ we denote by
$\zeta^{(i)}(w)=(\zeta^{(i)}_{1}(w),\zeta^{(i)}_{2}(w),\ldots,\zeta^{(i)}_{N}(w))\in
X=\mathbb{C}^{N}_{z}$ the unique point of $X$ such that
$(\zeta^{(i)}(w),w)\in\Lambda_{V,i}\subset\Lambda$, then we have

$\Bigl{(}\frac{\partial g_{i}}{\partial w_{1}}(w),\frac{\partial
g_{i}}{\partial w_{2}}(w),\ldots\ldots,\frac{\partial g_{i}}{\partial
w_{N}}(w)\Bigr{)}=-(\zeta^{(i)}_{1}(w),\zeta^{(i)}_{2}(w),\ldots,\zeta^{(i)}_{N}(w)).$
(6.30)

More precisely, we have the following higher-dimensional analogue of [8,
Lemma-Definition 7.4.2]. For $1\leq i\leq d$ by our choice of $\Omega\subset
Y=\mathbb{C}^{N}$ in Lemma 6.3, $Z_{i}:=p(\Lambda_{V,i})\subset
X=\mathbb{C}^{N}_{z}$ is a complex submanifold of $X$. We renumber them so
that for some $0\leq r\leq d$ we have $Z_{i}\subset U=X\setminus Q^{-1}(0)$
(resp. $Z_{i}\subset I(f)=P^{-1}(0)\cap Q^{-1}(0)$) if $1\leq i\leq r$ (resp.
if $r+1\leq i\leq d$).

######  Lemma 6.5.

For any $1\leq i\leq r$ there exists a unique holomorphic function $g_{i}$ on
$V\subset\Omega$ such that $\chi(\Lambda_{V,i})=\Lambda^{g_{i}}$ and the
equality

$g_{i}(w)=f(\zeta^{(i)}(w))-\langle\zeta^{(i)}(w),w\rangle$ (6.31)

holds on $V$.

###### Proof.

By the condition $Z_{i}=p(\Lambda_{V,i})\subset U=X\setminus Q^{-1}(0)$ and
(6.26), the subset $\Lambda_{V,i}-\Lambda^{f}\subset(T^{*}U)_{\mathbb{R}}$ of
$(T^{*}U)_{\mathbb{R}}\simeq T^{*}U_{\mathbb{R}}$ is contained in the
$\mathbb{C}^{*}$-conic Lagrangian analytic set ${\rm
SS}(K)\cap(T^{*}U)_{\mathbb{R}}$. By [19, Lemma A.52] this implies that we
have $\Lambda_{V,i}\subset T^{*}_{Z_{i}}X+\Lambda^{f}$. Choose a holomorphic
function $g_{i}$ on $V\subset\Omega$ such that
$\chi(\Lambda_{V,i})=\Lambda^{g_{i}}$ and set

$k_{i}(w):=g_{i}(w)-f(\zeta^{(i)}(w))+\langle\zeta^{(i)}(w),w\rangle\qquad(w\in
V).$ (6.32)

Then by using the condition $(\zeta^{(i)}(w),w)\in\Lambda_{V,i}\subset
T^{*}_{Z_{i}}X+\Lambda^{f}$ ($w\in V$) we can easily show that the condition
(6.30) is equivalent to the one

$\frac{\partial k_{i}}{\partial w_{j}}(w)=0\qquad(1\leq j\leq N,w\in V).$
(6.33)

This implies that $k_{i}$ is constant on $V$. Then by substracting its value
from $g_{i}$ we obtain the assertion. ∎

Recall that for the closed embedding

$\iota:(T^{*}X)\times\mathbb{C}\hookrightarrow
T^{*}(X\times\mathbb{C})\qquad(((z,w),\tau)\longmapsto(z,\tau,w,1))$ (6.34)

we have

${\rm SS}^{{\rm E},\mathbb{C}}(\mathcal{F})=\iota^{-1}{\rm SS}(\mathcal{F}).$
(6.35)

Then for $r+1\leq i\leq d$, by the proof of Lemma 5.6 we obtain the following
description of ${\rm SS}^{{\rm
E},\mathbb{C}}(\mathcal{F})\cap(\Lambda_{V,i}\times\mathbb{C})$. For $r+1\leq
i\leq d$ let $\rho_{i}:Z_{i}\times_{X}T^{*}X\longrightarrow T^{*}Z_{i}$ be the
natural morphism associated to the inclusion map $Z_{i}\hookrightarrow
X=\mathbb{C}^{N}$.

######  Lemma 6.6.

For any $r+1\leq i\leq d$ there exist holomorphic functions
$h_{ik}:Z_{i}\longrightarrow\mathbb{C}$ ($1\leq k\leq n_{i}$) on $Z_{i}$ such
that for the complex submanifolds

$\Gamma_{k}:=\\{(z,-h_{ik}(z))\in X\times\mathbb{C}\ |\ z\in Z_{i}\\}\subset
X\times\mathbb{C}\qquad(1\leq k\leq n_{i})$ (6.36)

of $X\times\mathbb{C}$ we have

${\rm SS}^{{\rm
E},\mathbb{C}}(\mathcal{F})\cap\Bigl{\\{}\rho_{i}^{-1}\rho_{i}(\Lambda_{V,i})\times\mathbb{C}\Bigr{\\}}=\bigcup_{k=1}^{n_{i}}\iota^{-1}\Bigl{\\{}T^{*}_{\Gamma_{k}}(X\times\mathbb{C})\Bigr{\\}}$
(6.37)

and

$\rho_{i}(\Lambda_{V,i})=\Lambda_{h_{ik}}\qquad(1\leq k\leq n_{i}),$ (6.38)

where we set

$\Lambda_{h_{ik}}:=\\{(z,dh_{ik}(z))\ |\ z\in Z_{i}\\}\subset T^{*}Z_{i}.$
(6.39)

###### Proof.

Note that as $\Lambda_{V,i}\simeq V$ the complex manifold
$Z_{i}=p(\Lambda_{V,i})\subset I(f)$ is contractible. Let
$\Lambda_{i}^{\circ}\subset q^{-1}(\Omega)\cap\Lambda$ be the unique connected
component of $\Lambda^{\circ}:=q^{-1}(\Omega)\cap\Lambda$ containing the
connected set $\Lambda_{V,i}\simeq V$. Then by our choice of $\Omega\subset
Y=\mathbb{C}^{N}$ in Lemma 6.3, the smooth variety $\Lambda_{i}^{\circ}$ is a
fiber bundle over a complex submanifold $S_{i}$ of $X=\mathbb{C}^{N}$
contained in $I(f)$ such that $Z_{i}\subset S_{i}$. Note that even if
$i\not=i^{\prime}$ we may have
$\Lambda_{i}^{\circ}=\Lambda_{i^{\prime}}^{\circ}$. As the natural morphism
$\rho_{i}:S_{i}\times_{X}T^{*}X\longrightarrow T^{*}S_{i}$ associated to the
inclusion map $S_{i}\hookrightarrow X=\mathbb{C}^{N}$ is an extension of
$\rho_{i}$, we denote it also by $\rho_{i}$. Note that the closure
$\overline{\Lambda_{i}^{\circ}}$ of $\Lambda_{i}^{\circ}$ in $T^{*}X$ is an
irreducible component of $\Lambda$. Moreover by the proof of Lemma 5.6, it is
the closure of the projection of the conormal bundle of a “horizontal” stratum
over an open dense subset of $S_{i}$. This implies that the subset
$\overline{\Lambda_{i}^{\circ}}\cap(S_{i}\times_{X}T^{*}X)$ of
$S_{i}\times_{X}T^{*}X$ is a union of some fibers of $\rho_{i}$. Namely there
exists a Lagrangian subvariety $\Lambda_{S_{i}}\subset T^{*}S_{i}$ such that

$\overline{\Lambda_{i}^{\circ}}\cap(S_{i}\times_{X}T^{*}X)=\rho_{i}^{-1}\Lambda_{S_{i}}.$
(6.40)

By the proof of Lemma 5.6 there exists also a complex hypersurface
$D_{i}\subset S_{i}$ of $S_{i}$ such that for any point $z_{0}\in
S_{i}\setminus D_{i}$ of $S_{i}^{\circ}:=S_{i}\setminus D_{i}$ we have
holomorphic functions $h_{ik}$ ($1\leq k\leq m_{i}$) defined on a neighborhood
$W$ of it in $S_{i}^{\circ}$ and satisfying the condition

${\rm SS}^{{\rm
E},\mathbb{C}}(\mathcal{F})\cap(\overline{\Lambda_{i}^{\circ}}\times\mathbb{C})=\bigcup_{k=1}^{m_{i}}\iota^{-1}\Bigl{\\{}T^{*}_{\Gamma_{k}}(X\times\mathbb{C})\Bigr{\\}}$
(6.41)

over $W$. It follows that also the condition

$\overline{\Lambda_{i}^{\circ}}=\rho_{i}^{-1}\Bigl{(}\bigcup_{k=1}^{m_{i}}\Lambda_{h_{ik}}\Bigr{)}=\rho_{i}^{-1}\Lambda_{S_{i}}$
(6.42)

is satisfied over $W$. By taking the closure of the union of the smooth
Lagrangian subvarieties $\cup_{k=1}^{m_{i}}\Lambda_{h_{ik}}\subset T^{*}W$ for
various $W\subset S_{i}^{\circ}$ in $T^{*}S_{i}$ we obtain the possibly
singular one $\Lambda_{S_{i}}\subset T^{*}S_{i}$. Since $\Lambda_{i}^{\circ}$
is smooth and a fiber bundle over $S_{i}$, by (6.40) for any point $z_{0}\in
D_{i}\subset S_{i}$ there exist a point $(z_{0},w_{0})\in\Lambda_{i}^{\circ}$
over it and its neighborhood $W$ in $\Lambda_{i}^{\circ}$ whose image
$\rho_{i}(W)\subset T^{*}S_{i}$ by $\rho_{i}$ is a submanifold of $T^{*}S_{i}$
such that the restriction of the projection $T^{*}S_{i}\longrightarrow S_{i}$
to it is a submersion to its image. Comparing this with (6.42) and using again
the fact that $\Lambda_{i}^{\circ}$ is a fiber bundle over $S_{i}$, one can
show that there exists a positive integer $e_{i}>0$ such that for any point
$z_{0}\in S_{i}$ of $S_{i}$ its fiber of the morphism
$\rho_{i}(\Lambda_{i}^{\circ})\longrightarrow S_{i}$ induced by the projection
$T^{*}S_{i}\longrightarrow S_{i}$ consists of $e_{i}$ points. In particular,
we have $\rho_{i}(\Lambda_{i}^{\circ})=\Lambda_{S_{i}}$. We thus find that
$\Lambda_{S_{i}}\subset T^{*}S_{i}$ is a smooth Lagrangian subvariety of
$T^{*}S_{i}$ and the morphism $\Lambda_{S_{i}}\longrightarrow S_{i}$ is an
unramified covering of degree $e_{i}$. Restricting it to the contractible open
subset $Z_{i}\subset S_{i}$, we obtain a trivial covering
$\Lambda_{Z_{i}}:=Z_{i}\times_{S_{i}}\Lambda_{S_{i}}\longrightarrow Z_{i}$
over $Z_{i}$ of degree $e_{i}$. Since
$\rho_{i}(\Lambda_{V,i})\subset\Lambda_{Z_{i}}$ is connected, it is contained
in only one connected component of $\Lambda_{Z_{i}}$. Together with the
condition $p(\Lambda_{V,i})=Z_{i}$ we obtain $\rho_{i}(\Lambda_{V,i})\simeq
Z_{i}$. This implies that some of the holomorphic functions
$h_{ik}:W\longrightarrow\mathbb{C}$ on $W\subset S_{i}^{\circ}$ ($1\leq k\leq
m_{i}$) extend to $Z_{i}$ to satisfy the conditions (6.37) and (6.38) there.
By renumbering $h_{ik}$ ($1\leq k\leq m_{i}$) we may assume that for some
$1\leq n_{i}\leq m_{i}$ the first $n_{i}$ functions $h_{ik}$ ($1\leq k\leq
n_{i}$) have this property. This completes the proof. ∎

Moreover by the condition (6.38) we see also that for any $1\leq
k,k^{\prime}\leq n_{i}$ such that $k\not=k^{\prime}$ we have
$\Lambda_{h_{ik}}=\Lambda_{h_{ik^{\prime}}}$ and hence
$h_{ik}-h_{ik^{\prime}}:S_{i}\longrightarrow\mathbb{C}$ is a non-zero constant
function on $S_{i}$. Then as in the proof of Lemma 6.5 we obtain the following
result.

######  Lemma 6.7.

For any $r+1\leq i\leq d$ and $1\leq k\leq n_{i}$ there exists a unique
holomorphic function $g_{ik}$ on $V\subset\Omega$ such that
$\chi(\Lambda_{V,i})=\Lambda^{g_{ik}}$ and the equality

$g_{ik}(w)=h_{ik}(\zeta^{(i)}(w))-\langle\zeta^{(i)}(w),w\rangle$ (6.43)

holds on $V$.

######  Example 6.8.

Consider the case where $N=2$ and set $z=(z_{1},z_{2})=(x,y)$ and
$w=(w_{1},w_{2})=(\xi,\eta)$. Assume that $U=\\{(x,y)\in X=\mathbb{C}^{2}\ |\
x\not=0\\}\subset X$, $K=\mathbb{C}_{X}$ and the function
$f:U\longrightarrow\mathbb{C}$ is defined to be the ratio of $P(x,y)=y$ and
$Q(x,y)=x$ as

$f(x,y)=\frac{P(x,y)}{Q(x,y)}=\frac{y}{x}.$ (6.44)

Then by Lemma 5.2 we have

$\Lambda={\rm SS}_{\rm
irr}(F)=\Bigl{\\{}\bigl{(}x,y,-\frac{y}{x^{2}},\frac{1}{x}\bigr{)}\ |\
(x,y)\in U\Bigr{\\}}\subset T^{*}X\simeq X\times Y.$ (6.45)

By solving the equation $\xi=-\frac{y}{x^{2}}$, $\eta=\frac{1}{x}$, we find
that $d=1$, $\Omega=\\{(\xi,\eta)\in Y=\mathbb{C}^{2}\ |\ \eta\not=0\\}\subset
Y$ and for $(\xi,\eta)\in\Omega$ we have

$x=\zeta_{1}(\xi,\eta):=\frac{1}{\eta},\quad
y=\zeta_{2}(\xi,\eta):=-\frac{\xi}{\eta^{2}}.$ (6.46)

We can also verify that the function

$g(\xi,\eta):=f\bigl{(}\frac{1}{\eta},-\frac{\xi}{\eta^{2}}\bigr{)}-\Bigl{\langle}\bigl{(}\frac{1}{\eta},-\frac{\xi}{\eta^{2}}\bigr{)},(\xi,\eta)\Bigr{\rangle}=-\frac{\xi}{\eta}$
(6.47)

satisfies the condition $\chi(\Lambda)=\Lambda^{g}$.

######  Example 6.9.

Let $N=2$ and $z=(z_{1},z_{2})=(x,y)$, $w=(w_{1},w_{2})=(\xi,\eta)$ be as in
Example 6.8. Assume that $U=\\{(x,y)\in X=\mathbb{C}^{2}\ |\ x\not=0\\}\subset
X$, $K=\mathbb{C}_{X}$ and the function $f:U\longrightarrow\mathbb{C}$ is
defined to be the ratio of $P(x,y)=x-y^{3}$ and $Q(x,y)=x$ as

$f(x,y)=\frac{P(x,y)}{Q(x,y)}=\frac{x-y^{3}}{x}=1-\frac{y^{3}}{x}.$ (6.48)

Then we have

$\Lambda_{0}:={\rm SS}_{\rm
irr}(F)\cap(T^{*}U)_{\mathbb{R}}=\Bigl{\\{}\bigl{(}x,y,\frac{y^{3}}{x^{2}},-\frac{3y^{2}}{x}\bigr{)}\
|\ (x,y)\in U\Bigr{\\}}\subset T^{*}X\simeq X\times Y.$ (6.49)

By solving the equation $\xi=\frac{y^{3}}{x^{2}}$, $\eta=-\frac{3y^{2}}{x}$,
we find that $\Omega=\\{(\xi,\eta)\in Y=\mathbb{C}^{2}\ |\
\xi\not=0,\eta\not=0\\}\subset Y$ and for $(\xi,\eta)\in\Omega$ we have

$x=\zeta_{1}(\xi,\eta):=-\frac{\eta^{3}}{27\xi^{2}},\quad
y=\zeta_{2}(\xi,\eta):=\frac{\eta^{2}}{9\xi}.$ (6.50)

We can also verify that the function

$g(\xi,\eta):=f\bigl{(}-\frac{\eta^{3}}{27\xi^{2}},\frac{\eta^{2}}{9\xi}\bigr{)}-\Bigl{\langle}\bigl{(}-\frac{\eta^{3}}{27\xi^{2}},\frac{\eta^{2}}{9\xi}\bigr{)},(\xi,\eta)\Bigr{\rangle}=1-\frac{\eta^{3}}{27\xi}$
(6.51)

satisfies the condition $\chi(\Lambda_{0})=\Lambda^{g}$.

For a point $w\in V\subset\Omega$ we define two regular functions
$\ell^{w}:X=\mathbb{C}^{N}\longrightarrow\mathbb{C}$ and $f^{w}:U=X\setminus
Q^{-1}(0)\longrightarrow\mathbb{C}$ by

$\ell^{w}:X\longrightarrow\mathbb{C}\qquad(z\longmapsto\langle z,w\rangle)$
(6.52)

and

$f^{w}=\ell^{w}|_{U}-f:U=X\setminus
Q^{-1}(0)\longrightarrow\mathbb{C}\qquad(z\longmapsto\langle z,w\rangle-f(z))$
(6.53)

respectively.

######  Lemma 6.10.

For any $1\leq i\leq r$ and $w\in V\subset\Omega$ the holomorphic function
$f^{w}|_{Z_{i}}:Z_{i}\longrightarrow\mathbb{C}$ on $Z_{i}$ has a non-
degenerate (complex Morse) critical point at $\zeta^{(i)}(w)\in Z_{i}$.

###### Proof.

In the proof of Lemma 6.5, for $1\leq i\leq r$ we saw that $\Lambda_{V,i}$ is
an open subset of $T^{*}_{Z_{i}}X+\Lambda^{f}\subset T^{*}U$. In other words,
for the diffeomorphism

$\Phi:T^{*}U\overset{\sim}{\longrightarrow}T^{*}U\qquad((z,w)\longmapsto(z,w+df(z))$
(6.54)

of $T^{*}U$ we have the inclusion

$\Lambda_{V,i}\subset\Phi(T^{*}_{Z_{i}}X).$ (6.55)

Moreover, for $w\in V$ the condition $(\zeta^{(i)}(w),w)\in\Lambda_{V,i}$
implies

$(\zeta^{(i)}(w),df^{w}(\zeta^{(i)}(w)))=(\zeta^{(i)}(w),w-df(\zeta^{(i)}(w)))\in
T^{*}_{Z_{i}}X$ (6.56)

and hence the holomorphic function
$f^{w}|_{Z_{i}}:Z_{i}\longrightarrow\mathbb{C}$ has a critical point at
$\zeta^{(i)}(w)\in Z_{i}$. By the choice of $\Omega$, for any $w\in
V\subset\Omega_{0}\subset\Omega$ the fiber $q^{-1}(w)\simeq X=\mathbb{C}^{N}$
of $q:X\times Y\longrightarrow Y$ intersects
$\Lambda_{V,i}\subset\Phi(T^{*}_{Z_{i}}X)$ transversally. Moreover, we have
$q^{-1}(w)=\Lambda^{\ell^{w}}$. This implies that
$\Phi^{-1}(\Lambda^{\ell^{w}})=\Lambda^{\ell^{w}-f}=\Lambda^{f^{w}}$
intersects $\Phi^{-1}(\Lambda_{V,i})\subset T^{*}_{Z_{i}}X$ transversally.
Then by Kashiwara-Schapira [21, Lemma 7.2.2], the holomorphic function
$f^{w}|_{Z_{i}}=(\ell^{w}-f)|_{Z_{i}}:Z_{i}\longrightarrow\mathbb{C}$ on
$Z_{i}$ has a non-degenerate (complex Morse) critical point at
$\zeta^{(i)}(w)\in Z_{i}$. ∎

As in the proof of Lemma 6.10, we obtain also the following result. For
$r+1\leq i\leq d$, $1\leq k\leq n_{i}$ and $w\in V\subset\Omega$ we define a
holomorphic function $h_{ik}^{w}:Z_{i}\longrightarrow\mathbb{C}$ on $Z_{i}$ by

$h_{ik}^{w}:=\ell^{w}|_{Z_{i}}-h_{ik}:Z_{i}\longrightarrow\mathbb{C},\qquad(z\longmapsto\langle
z,w\rangle-h_{ik}(z))$ (6.57)

so that we have

$g_{ik}(w)=h_{ik}(\zeta^{(i)}(w))-\langle\zeta^{(i)}(w),w\rangle=-h_{ik}^{w}(\zeta^{(i)}(w))\qquad(w\in
V).$ (6.58)

######  Lemma 6.11.

For any $r+1\leq i\leq d$, $1\leq k\leq n_{i}$ and $w\in V\subset\Omega$ the
function $h_{ik}^{w}:Z_{i}\longrightarrow\mathbb{C}$ on $Z_{i}$ has a non-
degenerate (complex Morse) critical point at $\zeta^{(i)}(w)\in Z_{i}$.

Note that for $1\leq i\leq r$ and $w\in V\subset\Omega$ the point

$(\zeta^{(i)}(w),df^{w}(\zeta^{(i)}(w)))=(\zeta^{(i)}(w),w-df(\zeta^{(i)}(w)))\in(T^{*}X)_{\mathbb{R}}$
(6.59)

is contained in the intersection of $T^{*}_{Z_{i}}X$ and the smooth part of
${\rm SS}(K|_{U})\subset(T^{*}U)_{\mathbb{R}}$. We denote by $m(i)\geq 1$ the
multiplicity of the regular holonomic $\mathcal{D}_{X}$-module
$\mathcal{N}\in\mathrm{Mod}_{\mbox{\rm\scriptsize rh}}(\mathcal{D}_{X})$ (or
of the perverse sheaf $K[N]$) there.

######  Proposition 6.12.

For $1\leq i\leq r$ and $w\in V\subset\Omega$ we have isomorphisms

$H^{j}\phi^{{\rm mero,c}}_{f^{w}-c}(K[N])_{\zeta^{(i)}(w)}\simeq
H^{j}\phi_{f^{w}-c}(K[N])_{\zeta^{(i)}(w)}\simeq\begin{cases}\mathbb{C}^{m(i)}&(j=-1)\\\
\\\ 0&(\mbox{otherwise}),\end{cases}$ (6.60)

where we set
$c:=f^{w}(\zeta^{(i)}(w))=\langle\zeta^{(i)}(w),w\rangle-f(\zeta^{(i)}(w))\in\mathbb{C}$.

###### Proof.

Since for $1\leq i\leq r$ we have $\zeta^{(i)}(w)\in Z_{i}\subset U=X\setminus
Q^{-1}(0)$ and hence the function $f^{w}-c$ is holomorphic on a neighborhood
of $\zeta^{(i)}(w)$, there exists an isomorphism

$\phi^{{\rm
mero,c}}_{f^{w}-c}(K[N])_{\zeta^{(i)}(w)}\simeq\phi_{f^{w}-c}(K[N])_{\zeta^{(i)}(w)}.$
(6.61)

By Lemma 6.10 the holomorphic function
$f^{w}|_{Z_{i}}:Z_{i}\longrightarrow\mathbb{C}$ on $Z_{i}$ has a non-
degenerate (complex Morse) critical point at $\zeta^{(i)}(w)\in Z_{i}$.
Moreover by our definition of $m(i)\geq 1$ the perverse sheaf $K[N]\in{\bf
D}_{\mathrm{c}}^{\mathrm{b}}(X^{{\rm an}})$ is isomorphic to
$\mathbb{C}_{Z_{i}}^{\oplus m(i)}[{\rm dim}Z_{i}]\in{\bf
D}_{\mathrm{c}}^{\mathrm{b}}(X^{{\rm an}})$ in the localized category ${\bf
D}^{\mathrm{b}}(X^{{\rm an}},\\{(\zeta^{(i)}(w),df^{w}(\zeta^{(i)}(w)))\\})$
(see [22, Definition 6.1.1] for the definition). Then by [22, Proposition
8.6.3] we obtain the desired isomorphisms

$H^{j}\phi_{f^{w}-c}(K[N])_{\zeta^{(i)}(w)}\simeq
H^{j}\phi_{f^{w}-c}(\mathbb{C}_{Z_{i}}^{\oplus m(i)}[{\rm
dim}Z_{i}])_{\zeta^{(i)}(w)}\simeq\begin{cases}\mathbb{C}^{m(i)}&(j=-1)\\\ \\\
0&(\mbox{otherwise}),\end{cases}$ (6.62)

∎

Note that by (6.37) for $r+1\leq i\leq d$, $1\leq k\leq n_{i}$ and $w\in
V\subset\Omega$ the point

$(\zeta^{(i)}(w),-h_{ik}(\zeta^{(i)}(w)),w,1)\in T^{*}(X\times\mathbb{C})$
(6.63)

is contained in the smooth part of ${\rm SS}(\mathcal{F}[N])={\rm
SS}(\mathcal{F})\subset T^{*}(X\times\mathbb{C})$. We denote by $m(i,k)\geq 1$
the multiplicity of of the perverse sheaf
$\mathcal{F}[N]=(i_{-f})_{!}(K[N]|_{U})\in{\bf
D}_{\mathrm{c}}^{\mathrm{b}}(X^{{\rm an}}\times\mathbb{C})$ there.

######  Proposition 6.13.

For any $r+1\leq i\leq d$, $1\leq k\leq n_{i}$ and $w\in V\subset\Omega$ we
have isomorphisms

$H^{j}\phi^{{\rm
mero,c}}_{f^{w}-c_{k}}(K[N])_{\zeta^{(i)}(w)}\simeq\begin{cases}\mathbb{C}^{m(i,k)}&(j=-1)\\\
\\\ 0&(\mbox{otherwise}),\end{cases}$ (6.64)

where we set
$c_{k}:=h_{ik}^{w}(\zeta^{(i)}(w))=\langle\zeta^{(i)}(w),w\rangle-
h_{ik}(\zeta^{(i)}(w))\in\mathbb{C}$.

###### Proof.

Note the for the (not necessarily closed) embedding
$i_{f^{w}}:U\hookrightarrow X\times\mathbb{C}$ ($z\longmapsto(z,f^{w}(z))$)
and any $c\in\mathbb{C}$ we have an isomorphism

$\phi^{{\rm
mero,c}}_{f^{w}-c}(K[N])_{\zeta^{(i)}(w)}\simeq\phi_{\tau-c}\bigl{(}(i_{f^{w}})_{!}(K[N]|_{U})\bigr{)}_{(\zeta^{(i)}(w),c)}.$
(6.65)

By Lemma 6.11 the holomorphic function
$h_{ik}^{w}|_{Z_{i}}:Z_{i}\longrightarrow\mathbb{C}$ on $Z_{i}$ has a non-
degenerate (complex Morse) critical point at $\zeta^{(i)}(w)\in Z_{i}$. Then
by the proof of Proposition 5.7 and [22, Proposition 8.6.3] we obtain
isomorphisms

$H^{j}\phi_{\tau-
c_{k}}\bigl{(}(i_{f^{w}})_{!}(K[N]|_{U})\bigr{)}_{(\zeta^{(i)}(w),c_{k})}\simeq\begin{cases}\mathbb{C}^{m(i,k)}&(j=-1)\\\
\\\ 0&(\mbox{otherwise})\end{cases}$ (6.66)

from which the assertion immediately follows. ∎

By Proposition 5.7 and (6.37), for any $r+1\leq i\leq d$, $w\in
V\subset\Omega$ and $c\in\mathbb{C}$ such that
$c\not=h_{ik}^{w}(\zeta^{(i)}(w))$ ($1\leq k\leq n_{i}$) we have also a
vanishing

$\phi^{{\rm mero,c}}_{f^{w}-c}(K[N])_{\zeta^{(i)}(w)}\simeq 0.$ (6.67)

Together with Proposition 6.13 this implies that for $r+1\leq i\leq d$ the
positive integer

$m(i):=\displaystyle\sum_{k=1}^{n_{i}}m(i,k)\geq 1$ (6.68)

satisfies the condition

$m(i)=\displaystyle\sum_{c\in\mathbb{C}}{\rm dim}H^{N-1}\phi^{{\rm
mero,c}}_{f^{w}-c}(K)_{\zeta^{(i)}(w)}\qquad(w\in V).$ (6.69)

######  Corollary 6.14.

For the open subset $q^{-1}(\Omega)=X\times\Omega\subset X\times Y$ of
$X\times Y\simeq T^{*}X$ we have

${\rm SS}_{{\rm irr}}^{\mathbb{C}}(\mathcal{F})\cap q^{-1}(\Omega)={\rm
SS}_{{\rm eva}}(F)\cap q^{-1}(\Omega).$ (6.70)

###### Proof.

By Proposition 5.7 it suffices to prove only the inclusion

$\Lambda\cap q^{-1}(\Omega)={\rm SS}_{{\rm irr}}^{\mathbb{C}}(\mathcal{F})\cap
q^{-1}(\Omega)\subset{\rm SS}_{{\rm eva}}(F)\cap q^{-1}(\Omega).$ (6.71)

For a point $(z_{0},w_{0})\in\Lambda\cap q^{-1}(\Omega)$ we take a
contractible open subset $V\subset\Omega$ such that $q((z_{0},w_{0}))=w_{0}\in
V$ and use the several notations that we introduced for it above. Then by
Propositions 6.12 and 6.13 we obtain the desired condition
$(z_{0},w_{0})\in{\rm SS}_{{\rm eva}}(F)\cap q^{-1}(\Omega)$. ∎

######  Lemma 6.15.

Let $\mathcal{S}$ be a Whitney stratification of $Z\cap U\subset U=X\setminus
Q^{-1}(0)$ adapted to $K|_{U}\in{\bf D}_{\mathrm{c}}^{\mathrm{b}}(U^{{\rm
an}})$ such that

${\rm SS}(K|_{U})\subset\bigcup_{S\in\mathcal{S}}T^{*}_{S}X$ (6.72)

and $\mathcal{S}_{{\rm red}}\subset\mathcal{S}$ its subset consisting of
strata $S\in\mathcal{S}$ satisfying the condition $T^{*}_{S}X\subset{\rm
SS}(K|_{U})$ so that we have

${\rm SS}(K|_{U})=\bigcup_{S\in\mathcal{S}_{{\rm red}}}\overline{T^{*}_{S}X}.$
(6.73)

Then for any $w\in V\subset\Omega$ and $S\in\mathcal{S}_{{\rm red}}$ the
holomorphic function $f^{w}|_{S}:S\longrightarrow\mathbb{C}$ on $S$ is tame at
infinity.

###### Proof.

Since by definition the morphisms $\zeta^{(i)}:\Omega\longrightarrow
U=X\setminus Q^{-1}(0)$ ($1\leq i\leq r$) are holomorphic, there exists $R\gg
0$ and $0<\varepsilon\ll 1$ such that for the open ball
$B_{\varepsilon}(w)\subset V\subset\Omega$ with radius $\varepsilon>0$
centered at $w\in V$ we have the inclusion

$\zeta^{(i)}(B_{\varepsilon}(w))\subset U\cap B_{R}(0)\qquad(1\leq i\leq r).$
(6.74)

This implies that for the stratum $S\in\mathcal{S}_{{\rm red}}$ and any $u\in
B_{\varepsilon}(0)\subset Y=\mathbb{C}^{N}$ the linear perturbation of
$f^{w}|_{S}$:

$(f^{w+u}|_{S})(z)=(f^{w}|_{S})(z)+\langle z,u\rangle\qquad(z\in S)$ (6.75)

has no critical point in the set

$S\setminus B_{R}(0)=\\{z\in S\ |\ ||z||\geq R\\}.$ (6.76)

Suppose now that there exists a point $z\in S\setminus B_{R}(0)$ such that
$||d{\rm Re}(f^{w}|_{S})(z)||<\varepsilon$. Then, the image of
$B_{\varepsilon}(0)\subset\mathbb{C}^{N}$ by the surjective linear map
$\Psi_{z}:T^{*}_{z}X_{\mathbb{R}}\simeq\mathbb{C}^{N}\longrightarrow
T^{*}_{z}S_{\mathbb{R}}$ being also an open ball with radius $\varepsilon>0$,
we have the equality $d{\rm Re}(f^{w}|_{S})(z)=\Psi_{z}(u)$ for some $u\in
B_{\varepsilon}(0)$. Moreover, for the holomorphic function
$\ell^{\overline{u}}:S\longrightarrow\mathbb{C}$ on $S$ defined by

$\ell^{\overline{u}}(z):=\langle z,\overline{u}\rangle=(z,u)\qquad(z\in S)$
(6.77)

we have $d{\rm Re}\ell^{\overline{u}}(z)=\Psi_{z}(u)$ by (3.22). This implies
that the holomorphic function $(f^{w-\overline{u}}|_{S})(z)$ on $S$ satisfies
the condition $d{\rm Re}(f^{w-\overline{u}}|_{S})(z)=0$ at the point $z\in
S\setminus B_{R}(0)$. Then by the Cauchy-Riemann equation, we obtain also
$d(f^{w-\overline{u}}|_{S})(z)=0$. Since we have $-\overline{u}\in
B_{\varepsilon}(0)$, we get a contradiction. ∎

Similarly we obtain the following result. For a point $w\in\Omega$ by the (not
necessarily closed) embedding $i_{f^{w}}:U\hookrightarrow X\times\mathbb{C}$
($z\longmapsto(z,f^{w}(z))$) we set

$L^{w}:=(i_{f^{w}})_{!}(K|_{U})\quad\in{\bf
D}_{\mathrm{c}}^{\mathrm{b}}(X\times\mathbb{C}).$ (6.78)

######  Lemma 6.16.

For a point $w\in\Omega$ let $\mathcal{S}$ and $\mathcal{S}_{0}$ be (complex
analytic) Whitney stratifications of $X\times\mathbb{C}$ and $X$ respectively
such that

${\rm SS}(L^{w})\subset\bigcup_{S\in\mathcal{S}}T^{*}_{S}(X\times\mathbb{C})$
(6.79)

and the projection $X\times\mathbb{C}\longrightarrow X$ is a stratified fiber
bundle as in the assertion of the theorem in [12, page 43]. Then for any
stratum $S\in\mathcal{S}$ in $\mathcal{S}$ such that
$T^{*}_{S}(X\times\mathbb{C})\subset{\rm SS}(L^{w})$ the restriction
$h|_{S}:S\longrightarrow\mathbb{C}$ of the function
$h:X\times\mathbb{C}\longrightarrow\mathbb{C}$ ($(z,\tau)\longmapsto\tau$) to
$S\subset X\times\mathbb{C}$ is relatively tame at infinity for the projection
$X\times\mathbb{C}\longrightarrow X$ in the sense of Definition 3.8.

With Theorem 3.7, Lemmas 6.5, 6.10 and 6.15 and Proposition 6.12 at hands, in
the special case where $I(f)=P^{-1}(0)\cap Q^{-1}(0)=\emptyset$
($\Longrightarrow r=d$) we obtain the following result as in the proof of [17,
Theorem 4.4] (see the proof of Theorem 6.19 for the details).

######  Theorem 6.17.

In the situation as above, assume also that $I(f)=P^{-1}(0)\cap
Q^{-1}(0)=\emptyset$ so that we have $r=d$. Then we have an isomorphism

$\pi^{-1}\mathbb{C}_{V}\otimes\big{(}Sol_{\overline{Y}}^{\rm
E}(\widetilde{\mathcal{M}^{\wedge}})\big{)}\simeq\bigoplus_{i=1}^{d}\bigl{(}\mathbb{E}_{V^{{\rm
an}}|\overline{Y}^{{\rm an}}}^{{\rm Re}g_{i}}\bigr{)}^{\oplus m(i)}$ (6.80)

of enhanced ind-sheaves on $\overline{Y}^{{\rm an}}$.

######  Example 6.18.

We consider the situation in Example 6.9 and inherit the notations there.
Moreover we assume that $Z=X=\mathbb{C}^{2}$ and $K=\mathbb{C}_{X}$. For
$w=(\xi,\eta)\in\Omega$ the meromorphic function $f^{w}:X\setminus
Q^{-1}(0)=\mathbb{C}^{2}\setminus\\{x=0\\}\longrightarrow\mathbb{C}$ is
written as

$f^{w}(x,y)=(\xi x+\eta y)-f(x,y)=\frac{\xi x^{2}+\eta xy+y^{3}-x}{x}.$ (6.81)

For $t\in\mathbb{C}$ we set also

$R_{t}^{w}(x,y):=(\xi x^{2}+\eta xy+y^{3}-x)-tx$ (6.82)

so that we have

$f^{w}(x,y)-t:=\frac{R_{t}^{w}(x,y)}{Q(x,y)}\qquad(z=(x,y)\in Z\setminus
Q^{-1}(0))$ (6.83)

and

$\overline{(f^{w})^{-1}(t)}=(R_{t}^{w})^{-1}(0)\subset Z=X=\mathbb{C}^{2}.$
(6.84)

Then for $t\not=-1$ the complex curve $(R_{t}^{w})^{-1}(0)$ in
$X=\mathbb{C}^{2}$ is smooth on a neighborhood of the origin
$\\{(0,0)\\}=P^{-1}(0)\cap Q^{-1}(0)$. If $t=-1$, for any
$w=(\xi,\eta)\in\Omega$ the defining polynomial

$R_{-1}^{w}(x,y):=\xi x^{2}+\eta xy+y^{3}$ (6.85)

of the curve $(R_{t}^{w})^{-1}(0)$ is Newton non-degenerate at the origin
$\\{(0,0)\\}$. Then by applying the classical Kouchnirenko’s theorem (see
[33]) to it, we see that for any $w=(\xi,\eta)\in\Omega$ its Milnor number at
the origin $\\{(0,0)\\}$ is equal to $1$. Then [45, Lemma 2.4] implies that we
can take $\Omega_{0}$ to be $\Omega$ in this case.

In the general case i.e. if we do not assume the condition $I(f)=\emptyset$,
we have the following result. Note that by Lemma 6.7 for any $r+1\leq i\leq d$
and $1\leq k,k^{\prime}\leq n_{i}$ such that $k\not=k^{\prime}$ the difference
$g_{ik}-g_{ik^{\prime}}:Z_{i}\longrightarrow\mathbb{C}$ is a non-zero constant
function on $Z_{i}$. Then for any $r+1\leq i\leq d$ and $1\leq
k,k^{\prime}\leq n_{i}$ we have an isomorphism

$\mathbb{E}_{V^{{\rm an}}|\overline{Y}^{{\rm an}}}^{{\rm
Re}g_{ik}}\simeq\mathbb{E}_{V^{{\rm an}}|\overline{Y}^{{\rm an}}}^{{\rm
Re}g_{ik^{\prime}}}$ (6.86)

of enhanced ind-sheaves on $\overline{Y}^{{\rm an}}$. For this reason, for
each $r+1\leq i\leq d$ by choosing an index $1\leq k\leq n_{i}$ we set
$g_{i}(w):=g_{ik}(w)$ ($w\in V$) and consider the enhanced ind-sheaf
$\mathbb{E}_{V^{{\rm an}}|\overline{Y}^{{\rm an}}}^{{\rm Re}g_{i}}$ in what
follows.

######  Theorem 6.19.

In the situation as above, we have an isomorphism

$\pi^{-1}\mathbb{C}_{V}\otimes\big{(}Sol_{\overline{Y}}^{\rm
E}(\widetilde{\mathcal{M}^{\wedge}})\big{)}\simeq\bigoplus_{i=1}^{d}\bigl{(}\mathbb{E}_{V^{{\rm
an}}|\overline{Y}^{{\rm an}}}^{{\rm Re}g_{i}}\bigr{)}^{\oplus m(i)}$ (6.87)

of enhanced ind-sheaves on $\overline{Y}^{{\rm an}}$.

###### Proof.

By (6.20) and (6.86), it suffices to prove that there exists an isomorphism

$\displaystyle\pi^{-1}\mathbb{C}_{V}\otimes{}^{\rm
L}\Bigl{\\{}\bigl{\\{}\pi^{-1}(i_{X^{{\rm an}}})_{!}K\bigr{\\}}\otimes{\rm
E}_{U^{{\rm an}}|\overline{X}^{{\rm an}}}^{{\rm Re}f}\Bigr{\\}}$
$\displaystyle\simeq\Bigl{(}\bigoplus_{i=1}^{r}\bigl{(}{\rm E}_{V^{{\rm
an}}|\overline{Y}^{{\rm an}}}^{{\rm Re}g_{i}}\bigr{)}^{\oplus
m(i)}\Bigr{)}\bigoplus\Bigl{\\{}\bigoplus_{i=r+1}^{d}\Bigl{(}\bigoplus_{k=1}^{n_{i}}\bigl{(}{\rm
E}_{V^{{\rm an}}|\overline{Y}^{{\rm an}}}^{{\rm Re}g_{ik}}\bigr{)}^{\oplus
m(i,k)}\Bigr{)}\Bigr{\\}}.$

of enhanced sheaves on $V\subset\Omega\subset Y$. Let

$X\times\mathbb{R}_{s}\overset{p_{1}}{\longleftarrow}(X\times\mathbb{R}_{s})\times(Y\times\mathbb{R}_{t})\overset{p_{2}}{\longrightarrow}Y\times\mathbb{R}_{t}$
(6.88)

be the projections. Then by D’Agnolo-Kashiwara [8, Lemma 7.2.1] on
$Y\subset\overline{Y}$ we have an isomorphism

$\displaystyle{}^{\rm L}\Bigl{\\{}\bigl{\\{}\pi^{-1}(i_{X^{{\rm
an}}})_{!}K\bigr{\\}}\otimes{\rm E}_{U^{{\rm an}}|\overline{X}^{{\rm
an}}}^{{\rm Re}f}\Bigr{\\}}$ $\displaystyle\simeq\mathbf{Q}\Bigl{(}{\rm
R}p_{2!}\Bigl{(}p_{1}^{-1}\Bigl{\\{}\bigl{\\{}\pi^{-1}(i_{X^{{\rm
an}}})_{!}K\bigr{\\}}\otimes{\rm E}_{U^{{\rm an}}|\overline{X}^{{\rm
an}}}^{{\rm Re}f}\Bigr{\\}}\otimes\mathbb{C}_{\\{t-s-{\rm Re}\langle
z,w\rangle\geq 0\\}}[N]\Bigl{)}\Bigr{)},$

where $\mathbf{Q}:{\mathbf{D}}^{\mathrm{b}}(\mathbb{C}_{\overline{Y}^{{\rm
an}}\times\mathbb{R}})\to{\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_{\overline{Y}^{{\rm
an}}})$ is the quotient functor. For a point $(w,t)\in Y^{{\rm
an}}\times\mathbb{R}$ we have also isomorphisms

$\displaystyle\Bigl{(}{\rm
R}p_{2!}\Bigl{(}p_{1}^{-1}\Bigl{\\{}\bigl{\\{}\pi^{-1}(i_{X^{{\rm
an}}})_{!}K\bigr{\\}}\otimes{\rm E}_{U^{{\rm an}}|\overline{X}^{{\rm
an}}}^{{\rm Re}f}\Bigr{\\}}\otimes\mathbb{C}_{\\{t-s-{\rm Re}\langle
z,w\rangle\geq 0\\}}[N]\Bigl{)}\Bigr{)}_{(w,t)}$ $\displaystyle\simeq{\rm
R}\Gamma_{c}(\\{(z,s)\in U^{{\rm an}}\times\mathbb{R}\ |\ t-s-{\rm Re}\langle
z,w\rangle\geq 0,s+{\rm Re}f(z)\geq 0\\};\pi^{-1}K[N])$
$\displaystyle\simeq{\rm R}\Gamma_{c}(U^{{\rm an}};{\rm
R}\pi_{!}(\mathbb{C}_{\\{(z,s)\in U^{{\rm an}}\times\mathbb{R}\ |\ t-s-{\rm
Re}\langle z,w\rangle\geq 0,\ s+{\rm Re}f(z)\geq 0\\}}\otimes\pi^{-1}K[N]))$
$\displaystyle\simeq{\rm R}\Gamma_{c}(U^{{\rm an}};({\rm
R}\pi_{!}\mathbb{C}_{\\{(z,s)\in U^{{\rm an}}\times\mathbb{R}\ |\ t-s-{\rm
Re}\langle z,w\rangle\geq 0,\ s+{\rm Re}f(z)\geq 0\\}})\otimes K[N])$
$\displaystyle\simeq{\rm R}\Gamma_{c}(\\{z\in U^{{\rm an}}\ |\ {\rm
Re}f^{w}(z)\leq t\\};K[N]),$

where we used

${\rm R}\pi_{!}\mathbb{C}_{\\{(z,s)\in U^{{\rm an}}\times\mathbb{R}\ |\
t-s-{\rm Re}\langle z,w\rangle\geq 0,\ s+{\rm Re}f(z)\geq
0\\}}\simeq\mathbb{C}_{\\{z\in U^{{\rm an}}\ |\ {\rm Re}\langle
z,w\rangle-{\rm Re}f(z)\leq t\\}}$ (6.89)

in the last isomorphism. Fix $w\in V\subset\Omega\subset Y=\mathbb{C}^{N}$.
Then as in the proof of [17, Theorem 4.4] we can prove the vanishing

${\rm R}\Gamma_{c}(\\{z\in U^{{\rm an}}\ |\ {\rm Re}f^{w}(z)\leq
t\\};K[N])\simeq 0$ (6.90)

for $t\ll 0$ as follows. First of all, we set

$\displaystyle L(Sol_{X}(\mathcal{M})):=$ $\displaystyle\Bigl{(}{\rm
R}p_{2!}\Bigl{(}p_{1}^{-1}\Bigl{\\{}\bigl{\\{}\pi^{-1}(i_{X^{{\rm
an}}})_{!}K\bigr{\\}}\otimes{\rm E}_{U^{{\rm an}}|\overline{X}^{{\rm
an}}}^{{\rm Re}f}\Bigr{\\}}\otimes\mathbb{C}_{\\{t-s-{\rm Re}\langle
z,w\rangle\geq 0\\}}[N]\Bigl{)}\Bigr{)}.$

Then for $t\in\mathbb{R}$ its stalk at $(w,t)\in Y\times\mathbb{R}$:

$\big{(}L(Sol_{X}(\mathcal{M}))\big{)}_{(w,t)}\simeq{\rm
R}\Gamma_{c}\big{(}\\{z\in U^{{\rm an}}\ |\ {\rm Re}f^{w}\leq
t\\};K[N]\big{)}$ (6.91)

is calculated as follows:

$\displaystyle{\rm R}\Gamma_{c}\big{(}\\{z\in U^{{\rm an}}\ |\ {\rm
Re}f^{w}\leq t\\};K[N]\big{)}$ $\displaystyle\simeq{\rm
R}\Gamma_{c}\big{(}\\{\tau\in\mathbb{C}\ |\ {\rm Re}\tau\leq t\\};{\rm
R}f^{w}_{!}(K|_{U}[N])\big{)}.$

Since the morphism $f^{w}:U=X\setminus Q^{-1}(0)\longrightarrow\mathbb{C}$ is
algebraic, the proper direct image ${\rm R}f^{w}_{!}(K|_{U}[N])$ of the
perverse sheaf $(K|_{U}[N])\in{\bf D}_{\mathrm{c}}^{\mathrm{b}}(U^{{\rm an}})$
is constructible. Then in particular, for $t\ll 0$ the restrictions of its
cohomology sheaves to the closed half space $\\{\tau\in\mathbb{C}\ |\ {\rm
Re}\tau\leq t\\}\subset\mathbb{C}$ of $\mathbb{C}$ are locally constant. Thus
for $t\ll 0$ we obtain the vanishing

${\rm R}\Gamma_{c}\bigl{(}\\{\tau\in\mathbb{C}\ |\ {\rm Re}\tau\leq t\\};{\rm
R}f^{w}_{!}(K|_{U}[N])\bigr{)}\simeq 0.$ (6.92)

For $1\leq i\leq r$ we set

$c(i):=f^{w}(\zeta^{(i)}(w))=\langle\zeta^{(i)}(w),w\rangle-f(\zeta^{(i)}(w))\in\mathbb{C}.$
(6.93)

Moreover for $r+1\leq i\leq d$ and $1\leq k\leq n_{i}$ we set

$c(i,k):=h_{ik}^{w}(\zeta^{(i)}(w))=\langle\zeta^{(i)}(w),w\rangle-
h_{ik}(\zeta^{(i)}(w))\in\mathbb{C}.$ (6.94)

Let $B\subset\mathbb{C}$ be the union of these points in $\mathbb{C}$ and
write it as

$B=\\{c_{1},c_{2},\ldots,c_{l}\\}=\\{c_{j}\in\mathbb{C}\ |\ 1\leq j\leq
l\\}\subset\mathbb{C}$ (6.95)

so that for any $j\not=j^{\prime}$ we have $c_{j}\not=c_{j^{\prime}}$. Then
for any $1\leq i\leq r$ there exists a unique index $1\leq j\leq l$ such that
$c_{j}=c(i)$ and we denote it by $j(i)$. Similarly, for any $r+1\leq i\leq d$
and $1\leq k\leq n_{i}$ exists a unique index $1\leq j\leq l$ such that
$c_{j}=c(i,k)$ and we denote it by $j(i,k)$. Then by Theorem 3.10 and Lemma
6.16 for any $1\leq j\leq l$ we obtain an isomorphism

$\displaystyle\phi_{\tau-c_{j}}({\rm R}f^{w}_{!}(K|_{U}[N]))$
$\displaystyle\simeq\Bigl{(}\bigoplus_{i:\
j(i)=j}\phi_{f^{w}-c_{j}}(K[N])_{\zeta^{(i)}(w)}\Bigr{)}\bigoplus\Bigl{(}\bigoplus_{(i,k):\
j(i,k)=j}\phi^{{\rm mero,c}}_{f^{w}-c_{j}}(K[N])_{\zeta^{(i)}(w)}\Bigr{)},$

where in the first (resp. the last) direct sum $\oplus$ the index $1\leq i\leq
r$ (resp. the pair $(i,k)$ of $r+1\leq i\leq d$ and $1\leq k\leq n_{i}$)
ranges over the ones satisfying the condition $j(i)=j$ (resp. $j(i,k)=j$).
Recall that for any $r+1\leq i\leq d$ there exists at most one index $1\leq
k\leq n_{i}$ such that $j(i,k)=j$. By Proposition 6.12 for any $1\leq i\leq r$
such that $j(i)=j$ we have isomorphisms

$H^{n}\phi_{f^{w}-c_{j}}(K[N])_{\zeta^{(i)}(w)}\simeq\begin{cases}\mathbb{C}^{m(i)}&(n=-1)\\\
\\\ 0&(\mbox{otherwise}).\end{cases}$ (6.96)

Moreover by Proposition 6.13 for any pair $(i,k)$ of $r+1\leq i\leq d$ and
$1\leq k\leq n_{i}$ such that $j(i,k)=j$ we have isomorphisms

$H^{n}\phi^{{\rm
mero,c}}_{f^{w}-c_{j}}(K[N])_{\zeta^{(i)}(w)}\simeq\begin{cases}\mathbb{C}^{m(i,k)}&(n=-1)\\\
\\\ 0&(\mbox{otherwise}).\end{cases}$ (6.97)

For $1\leq j\leq l$ let us set

$d_{j}:=\Bigl{(}\displaystyle\sum_{i:\
j(i)=j}m(i)\Bigr{)}+\Bigl{(}\displaystyle\sum_{(i,k):\
j(i,k)=j}m(i,k)\Bigr{)}.$ (6.98)

Then for any $1\leq j\leq l$ we thus obtain isomorphisms

$H^{n}\phi_{\tau-c_{j}}({\rm
R}f^{w}_{!}(K|_{U}[N]))\simeq\begin{cases}\mathbb{C}^{d_{j}}&(n=-1)\\\ \\\
0&(\mbox{otherwise}).\end{cases}$ (6.99)

Again by Theorem 3.10 and Lemma 6.16, for any $c\in\mathbb{C}$ such that
$c\notin B=\\{c_{1},c_{2},\ldots,c_{l}\\}$ we obtain a vanishing

$\phi_{\tau-c}({\rm R}f^{w}_{!}(K|_{U}[N]))\simeq 0.$ (6.100)

By the proof of [18, Lemma 2.1] this implies that the constructible sheaf
${\rm
R}f^{w}_{!}(K|_{U}[N])\in{\mathbf{D}}^{\mathrm{b}}_{\mathbb{C}-c}(\mathbb{C})$
on $\mathbb{C}$ is smooth outside the finite subset
$B=\\{c_{1},c_{2},\ldots,c_{l}\\}\subset\mathbb{C}$. For $1\leq j\leq l$ set
$\gamma_{j}:={\rm Re}c_{j}\in\mathbb{R}$. For the fixed point $w\in
V\subset\Omega$, after reordering
$\gamma_{1},\gamma_{2},\ldots,\gamma_{l}\in\mathbb{R}$ we may assume that

$\gamma_{1}\ \leq\gamma_{2}\ \leq\cdots\cdots\ \leq\gamma_{l}.$

If for some $1\leq i<j\leq l$ such that $\gamma_{i}<\gamma_{j}$ the open
interval $(\gamma_{i},\gamma_{j})\subset\mathbb{R}$ does not intersect the set
$\\{\gamma_{1},\gamma_{2},\ldots,\gamma_{l}\\}$, then for any
$t_{1},t_{2}\in\mathbb{R}$ such that $\gamma_{i}<t_{1}<t_{2}<\gamma_{j}$ we
obtain an isomorphism

$\displaystyle{\rm R}\Gamma_{c}(\\{z\in U^{{\rm an}}\ |\ {\rm Re}f^{w}(z)\leq
t_{2}\\};K[N])$ (6.101) $\displaystyle\overset{\sim}{\longrightarrow}{\rm
R}\Gamma_{c}(\\{z\in U^{{\rm an}}\ |\ {\rm Re}f^{w}(z)\leq t_{1}\\};K[N]).$
(6.102)

Equivalently, we have an isomorphism

$\displaystyle{\rm R}\Gamma_{c}(\\{\tau\in\mathbb{C}\ |\ {\rm Re}\tau\leq
t_{2}\\};{\rm R}f^{w}_{!}(K|_{U}[N]))$ (6.103)
$\displaystyle\overset{\sim}{\longrightarrow}{\rm
R}\Gamma_{c}(\\{\tau\in\mathbb{C}\ |\ {\rm Re}\tau\leq t_{1}\\};{\rm
R}f^{w}_{!}(K|_{U}[N])).$ (6.104)

Indeed, this follows from Kashiwara’s non-characteristic deformation lemma
(see [22, Proposition 2.7.2]) as follows. Let $\widetilde{{\mathbb{P}}}$ be
the real oriented blow-up of ${\mathbb{P}}:={\mathbb{P}}^{1}$ along
$\\{\infty\\}\subset{\mathbb{P}}$. Recall that the fiber of
$\infty\in{\mathbb{P}}$ in $\widetilde{{\mathbb{P}}}$ is isomorphic to the
circle $S^{1}$ and denote it by $C$. Let
$\iota:\mathbb{C}\hookrightarrow\widetilde{{\mathbb{P}}}$ be the inclusion map
and $a_{+}\in C$ (resp. $a_{-}\in C$) the unique point of $C$ contained in the
closure of the positive (resp. negative) real axis of $\mathbb{C}$ in
$\widetilde{{\mathbb{P}}}$. We set
$\widetilde{{\mathbb{P}}}^{\circ}:=\widetilde{{\mathbb{P}}}\setminus\\{a_{+},a_{-}\\}$.
Then there exists a continuous map
$\ell:\widetilde{{\mathbb{P}}}^{\circ}\longrightarrow\mathbb{R}$ which extends
the one ${\rm Re}:\mathbb{C}\longrightarrow\mathbb{R}$. Now, by applying
Kashiwara’s non-characteristic deformation lemma to the Morse function
$\ell:\widetilde{{\mathbb{P}}}^{\circ}\longrightarrow\mathbb{R}$, we obtain an
isomorphism

$\displaystyle{\rm R}\Gamma_{c}(\\{\tau\in\widetilde{{\mathbb{P}}}^{\circ}\ |\
\ell(\tau)\leq t_{2}\\}\sqcup\\{a_{-}\\};\iota_{!}{\rm
R}f^{w}_{!}(K|_{U}[N]))$ (6.105)
$\displaystyle\overset{\sim}{\longrightarrow}{\rm
R}\Gamma_{c}(\\{\tau\in\widetilde{{\mathbb{P}}}^{\circ}\ |\ \ell(\tau)\leq
t_{1}\\}\sqcup\\{a_{-}\\};\iota_{!}{\rm R}f^{w}_{!}(K|_{U}[N])),$ (6.106)

which is equivalent to the desired one. For $1\leq j\leq l$ we define a closed
half space $G_{j}\subset\mathbb{C}$ of $\mathbb{C}$ by

$G_{j}:=\\{\tau\in\mathbb{C}\ |\ {\rm
Re}\tau\geq\gamma_{j}\\}\subset\mathbb{C}.$ (6.107)

Then for any $1\leq j\leq l$ we have isomorphisms

$\displaystyle H^{n}{\rm R}\Gamma_{G_{j}}({\rm
R}f^{w}_{!}(K|_{U}[N]))_{c_{j}}$ $\displaystyle\simeq H^{n}\phi_{\tau-
c_{j}}({\rm R}f^{w}_{!}(K|_{U}[N]))[-1]$ (6.108)
$\displaystyle\simeq\begin{cases}\mathbb{C}^{d_{j}}&(n=0)\\\ \\\
0&(\mbox{otherwise}).\end{cases}$ (6.109)

Hence for any $t=\gamma_{j}={\rm Re}c_{j}\in\mathbb{R}$ ($1\leq j\leq l$)
there exists $0<\varepsilon\ll 1$ such that

$\displaystyle{\rm R}\Gamma_{c}(\\{\tau\in\mathbb{C}\ |\ {\rm Re}\tau\leq
t+\varepsilon\\};{\rm R}f^{w}_{!}(K|_{U}[N]))$ $\displaystyle\simeq{\rm
R}\Gamma_{c}(\\{\tau\in\mathbb{C}\ |\ {\rm Re}\tau\leq t\\};{\rm
R}f^{w}_{!}(K|_{U}[N]))$ $\displaystyle\simeq{\rm
R}\Gamma_{c}(\\{\tau\in\mathbb{C}\ |\ {\rm Re}\tau\leq t-\varepsilon\\};{\rm
R}f^{w}_{!}(K|_{U}[N]))\oplus\mathbb{C}^{d_{j}}.$

This implies that the restriction of $L(Sol_{X}(\mathcal{M}))$ to the fiber
$\pi^{-1}(w)\simeq\mathbb{R}$ of the point $w\in V\subset\Omega$ is isomorphic
to that of the sheaf

$\displaystyle\Bigl{(}\bigoplus_{i=1}^{r}\bigl{(}\mathbb{C}_{\\{t+{\rm
Re}g_{i}(w)\geq 0\\}}\bigr{)}^{\oplus
m(i)}\Bigr{)}\bigoplus\Bigl{\\{}\bigoplus_{i=r+1}^{d}\Bigl{(}\bigoplus_{k=1}^{n_{i}}\bigl{(}\mathbb{C}_{\\{t+{\rm
Re}g_{ik}(w)\geq 0\\}}\bigr{)}^{\oplus m(i,k)}\Bigr{)}\Bigr{\\}}.$
$\displaystyle\simeq\Bigl{(}\bigoplus_{i=1}^{r}\bigl{(}{\rm E}_{V^{{\rm
an}}|\overline{Y}^{{\rm an}}}^{{\rm Re}g_{i}}\bigr{)}^{\oplus
m(i)}\Bigr{)}\bigoplus\Bigl{\\{}\bigoplus_{i=r+1}^{d}\Bigl{(}\bigoplus_{k=1}^{n_{i}}\bigl{(}{\rm
E}_{V^{{\rm an}}|\overline{Y}^{{\rm an}}}^{{\rm Re}g_{ik}}\bigr{)}^{\oplus
m(i,k)}\Bigr{)}\Bigr{\\}}.$

Since the subsets $(V\times\mathbb{R})\cap\\{t+{\rm Re}g_{i}(w)\geq 0\\}\simeq
V\times\mathbb{R}_{\geq 0}$ and $(V\times\mathbb{R})\cap\\{t+{\rm
Re}g_{ik}(w)\geq 0\\}\simeq V\times\mathbb{R}_{\geq 0}$ of $V\times\mathbb{R}$
are connected and simply connected, we can extend this isomorphism to the
whole $V\times\mathbb{R}\subset Y^{{\rm an}}\times\mathbb{R}$. This completes
the proof. ∎

######  Corollary 6.20.

In the situation of Theorem 6.19, the restriction of the Fourier transform
$\mathcal{M}^{\wedge}\in\mathrm{Mod}_{\rm hol}(\mathcal{D}_{Y})$ of
$\mathcal{M}$ to $\Omega\subset Y=\mathbb{C}^{N}_{w}$ is an integrable
connection. Moreover its rank is equal to

$\displaystyle\sum_{i=1}^{d}m(i)=\Bigl{(}\displaystyle\sum_{i=1}^{r}m(i)\Bigr{)}+\displaystyle\sum_{i=r+1}^{d}\Bigl{(}\displaystyle\sum_{k=1}^{n_{i}}m(i,k)\Bigr{)}.$
(6.110)

Since the proof of this corollary is completely the same as that of [17,
Corollary 4.5], we omit it here. Next fix a point $w\in Y=\mathbb{C}^{N}$ such
that $w\neq 0$ and set

${\mathbb{L}}:=\mathbb{C}w=\\{\lambda w\ |\ \lambda\in\mathbb{C}\\}\subset
Y=\mathbb{C}^{N}.$

Then ${\mathbb{L}}$ is a complex line isomorphic to $\mathbb{C}_{\lambda}$.
Assume that ${\mathbb{L}}$ is not contained in $D=Y\setminus\Omega$. Let
${\mathbb{P}}:={\mathbb{L}}\sqcup\\{\infty\\}\subset\overline{Y}={\mathbb{P}}^{N}$
be the projective compactification of ${\mathbb{L}}$ and
$i_{{\mathbb{P}}}:{\mathbb{P}}\hookrightarrow\overline{Y}={\mathbb{P}}^{N}$
the inclusion map. Then by Corollary 6.20 the holonomic D-module
$\mathcal{L}:=H^{0}\mathbf{D}(i_{{\mathbb{P}}})^{\ast}\mathcal{M}^{\wedge}\in\mathrm{Mod}_{\mbox{\rm\scriptsize
hol}}(\mathcal{D}_{\mathbb{P}})$ on ${\mathbb{P}}$ is a meromorphic connection
on a neighborhood of the point $\infty\in{\mathbb{P}}$. The following result
is a generalization of [11, Theorem 5.6] and [17, Theorem 4.6]. Let
$\varpi_{\mathbb{P}}:\widetilde{{\mathbb{P}}}\to{\mathbb{P}}$ be the real
blow-up of ${\mathbb{P}}$ along the divisor $\\{\infty\\}\subset{\mathbb{P}}$.

######  Theorem 6.21.

In the situation of Theorem 6.19, for any point
$\theta\in\varpi_{{\mathbb{P}}}^{-1}(\\{\infty\\})\simeq S^{1}$ there exists
its open neighborhood $W$ in $\widetilde{{\mathbb{P}}}$ such that we have an
isomorphism

$\mathcal{L}^{\mathcal{A}}|_{W}\simeq\Bigr{(}\bigoplus_{i=1}^{d}\bigl{(}(\mathcal{E}_{{\mathbb{L}}|{\mathbb{P}}}^{g_{i}(\lambda
w)})^{\mathcal{A}}\bigr{)}^{\oplus m(i)}\Bigl{)}|_{W}$

of $\mathcal{D}_{\mathbb{P}}^{\mathcal{A}}$-modules (see Section 7 for the
definition) on $W$. In particular, the functions $g_{i}(\lambda w)$ of
$\lambda$ are the exponential factors of the meromorphic connection
$\mathcal{L}$ at the point $\infty\in{\mathbb{P}}$. Moreover the multiplicity
of $g_{i}(\lambda w)$ is equal to $m(i)$.

###### Proof.

By [7, Corollary 9.4.12] we have isomorphisms

$\displaystyle Sol_{\mathbb{P}}^{\rm
E}(\mathcal{E}_{{\mathbb{L}}|{\mathbb{P}}}^{g_{i}(\lambda
w)})\simeq\mathbb{E}_{{\mathbb{L}}|{\mathbb{P}}}^{{\rm Re}g_{i}(\lambda
w)}\simeq\underset{a\to+\infty}{``\varinjlim"}\ \mathbb{C}_{\\{t\geq-{\rm
Re}g_{i}(\lambda w)+a\\}}.$

On the other hand, by Theorem 6.19 for any
$\theta\in\varpi_{{\mathbb{P}}}^{-1}(\\{\infty\\})\simeq S^{1}$ there exists
its sectorial neighborhood
$V_{\theta}\subset{\mathbb{P}}\backslash\\{\infty\\}$ such that we have an
isomorphism

$\pi^{-1}\mathbb{C}_{V_{\theta}}\otimes Sol_{\mathbb{P}}^{\rm
E}(\mathcal{L})\simeq\bigoplus_{i=1}^{d}\pi^{-1}\mathbb{C}_{V_{\theta}}\otimes\Big{(}\underset{a\to+\infty}{``\varinjlim"}\
\mathbb{C}_{\\{t\geq-{\rm Re}g_{i}(\lambda w)+a\\}}^{\oplus m(i)}\Big{)}.$

We thus obtain an isomorphism

$\pi^{-1}\mathbb{C}_{V_{\theta}}\otimes Sol_{\mathbb{P}}^{\rm
E}(\mathcal{L})\simeq\pi^{-1}\mathbb{C}_{V_{\theta}}\otimes
Sol_{\mathbb{P}}^{\rm
E}\Bigl{(}\bigoplus_{i=1}^{d}\big{(}\mathcal{E}_{{\mathbb{L}}|{\mathbb{P}}}^{g_{i}(\lambda
w)}\big{)}^{\oplus m(i)}\Bigr{)}.$

Then the assertions follow from [17, Corollary 3.11 and Theorem 3.18]. ∎

As in Esterov-Takeuchi [11, Remark 5.7], by Theorems 6.19 and 6.21 we easily
obtain the Stokes lines of the meromorphic connection
$\mathcal{L}\in\mathrm{Mod}_{\mbox{\rm\scriptsize
hol}}(\mathcal{D}_{\mathbb{P}})$ at $\infty\in{\mathbb{P}}$. We leave the
precise formulation to the readers.

By Theorem 6.19, at generic points $v\in Y\setminus\Omega$ where
$D:=Y\setminus\Omega$ is a smooth hypersurface we obtain also the irregularity
and the exponential factors of $\mathcal{M}^{\wedge}$ along it. More
precisely, let $D_{\rm reg}\subset D$ be the smooth part of $D$ and $v\in
D_{\rm reg}$ such a generic point. Take a subvariety $M\subset Y$ of
$Y=\mathbb{C}^{N}$ which intersects $D_{\rm reg}$ at $v$ transversally. We
call it a normal slice of $D$ at $v$. By definition $M$ is smooth and of
dimension $1$ on a neighborhood of $v$. Let $i_{M}:M\xhookrightarrow{\ \ \
}Y=\mathbb{C}^{N}$ be the inclusion map and set
$\mathcal{K}={\mathbf{D}}i_{M}^{\ast}\mathcal{M}^{\wedge}\in\mathrm{Mod}_{\mbox{\rm\scriptsize
hol}}(\mathcal{D}_{M})$. Then we can describe the irregularity ${\rm
irr}(\mathcal{K}(\ast\\{v\\}))$ of the meromorphic connection
$\mathcal{K}(\ast\\{v\\})$ on $M$ along $\\{v\\}\subset M$ as follows.
Shrinking the normal slice $M$ if necessary we may assume that
$M=\\{u\in\mathbb{C}\ |\ |u|<\varepsilon\\}$ for some $\varepsilon>0$,
$\\{v\\}=\\{u=0\\}$ and $M\setminus\\{v\\}\subset\Omega$. Let
$i_{0}:M\setminus\\{v\\}\hookrightarrow\Omega$ be the inclusion map and define
(possibly multi-valued) holomorphic functions
$\varphi_{i}:M\setminus\\{v\\}\rightarrow\mathbb{C}$ ($1\leq i\leq d$) by

$\varphi_{i}(u)=g_{i}(i_{0}(u)).$

Then it is easy to see that $\varphi_{i}(u)$ are Laurent Puiseux series of $u$
(see Kirwan [32, Section 7.2] etc.). For each Laurent Puiseux series

$\varphi_{i}(u)=\sum_{a\in\mathbb{Q}}c_{i,a}u^{a}\qquad(c_{i,a}\in\mathbb{C})$

set $r_{i}=\min\\{a\in\mathbb{Q}\ |\ c_{i,a}\not=0\\}$ and define its pole
order ${\rm ord}_{\\{v\\}}(\varphi_{i})\geq 0$ by

${\rm ord}_{\\{v\\}}(\varphi_{i})=\begin{cases}-r_{i}&(r_{i}<0)\\\ \\\
0&(\mbox{\rm otherwise}).\end{cases}$

Then we obtain the following theorem.

######  Theorem 6.22.

The exponential factors appearing in the Hukuhara-Levelt-Turrittin
decomposition of the meromorphic connection $\mathcal{K}(\ast\\{v\\})$ at
$v\in M$ are the pole parts of $\varphi_{i}\ (1\leq i\leq d)$. Moreover for
any $1\leq i\leq d$ the multiplicity of the pole part of $\varphi_{i}$ is
equal to $m(i)$. In particular, the irregularity of the meromorphic connection
$\mathcal{K}(\ast\\{v\\})$ along $v\in M$ is given by

${\rm irr}(\mathcal{K}(\ast\\{v\\}))=\sum_{i=1}^{d}m(i)\cdot{\rm
ord}_{\\{v\\}}(\varphi_{i}).$

##  7 Toward the Study of Fourier Transforms of General Holonomic D-modules

In this section, we prove some new formulas which might be useful to extend
our results in Section 6 to arbitrary holonomic D-modules. Recall that by
Theorem 4.1 (vi) for an exponential D-module $\mathscr{E}_{U|X}^{f}$ on $X$
associated to a meromorphic function $f\in\mathcal{O}_{X}(\ast D)$ along
$D\subset X$ and $U=X\setminus D$ we have an isomorphism in
${\mathbf{E}}^{\mathrm{b}}({\rm I}\mathbb{C}_{X})$

$Sol_{X}^{\rm E}\big{(}\mathscr{E}_{U|X}^{f}\big{)}\simeq\mathbb{E}_{U|X}^{\rm
Ref}.$ (7.1)

This formula for the enhanced solution complex $Sol_{X}^{\rm
E}\big{(}\mathscr{E}_{U|X}^{f}\big{)}$ played a central role in the proof of
the main results in [7]. Aiming at the study of the Fourier transforms of
general holonomic D-modules, we shall extend it to more general meromorphic
connections. Our argument below is a higher-dimensional analogue of those of
Kashiwara-Schapira [25, Section 7] and Morando [42, Section 2.1]. Assuming
that $M$ is a real analytic manifold we recall some basic definitions and
results.

######  Definition 7.1.

Let $U\subset M$ be an open subset and $f:U\longrightarrow\mathbb{C}$ a
$\mathbb{C}$-valued $C^{\infty}$-function on it. Then for a point $p\in M$ we
say that $f$ has polynomial growth at $p$ if for a local coordinate system
$x=(x_{1},x_{2}.\ldots,x_{n})$ of $M$ around $p$ there exists a compact
neighborhood $K$ of $p$ in $M$ and $N>0$ such that

$\sup_{x\in K\setminus U}{\rm dist}(x,K\setminus U)^{N}\cdot|f(x)|<+\infty.$
(7.2)

Obviously $f$ has polynomial growth at any point of $U$. We say that $f$ is
tempered at $p\in M$ if all its derivatives have polynomial growth at $p$.
Moreover we say that $f$ is tempered on $U$ if it is tempered at any point
$p\in M$.

For an open subset $U\subset M$ we denote by $C^{\infty,t}_{M}(U)$ the
$\mathbb{C}$-vector space consisting of $\mathbb{C}$-valued
$C^{\infty}$-functions on $U$ that are tempered on $U$. Then for any pair
$(U_{1},U_{2})$ of relatively compact subanalytic open subsets
$U_{1},U_{2}\subset M$ we have an exact sequence

$0\longrightarrow C^{\infty,t}_{M}(U_{1}\cup U_{2})\longrightarrow
C^{\infty,t}_{M}(U_{1})\oplus C^{\infty,t}_{M}(U_{2})\longrightarrow
C^{\infty,t}_{M}(U_{1}\cap U_{2})\longrightarrow 0$ (7.3)

and hence we get a sheaf $C^{\infty,t}_{M}$ on the subanalytic site of $M$ in
the sense of [24]. Note that this important result is an immediate consequence
of the following Lojasiewicz’s inequality (see Bierstone-Milman [2] and
Lojasiewicz [34] for the details).

######  Proposition 7.2 (Lojasiewicz’s inequality).

Let $U\subset\mathbb{R}^{n}_{x}$ be an open subset of $\mathbb{R}^{n}_{x}$ and
$Z_{1},Z_{2}\subset U$ its closed subanalytic subsets. Then for any point
$p\in Z_{1}\cap Z_{2}$ there exist an open neighborhood $V\subset U$ of $p$ in
$U$ and positive real numbers $C,r>0$ such that we have

${\rm dist}(x,Z_{1})+{\rm dist}(x,Z_{2})\geq C\cdot{\rm dist}(x,Z_{1}\cap
Z_{2})^{r}$ (7.4)

for any $x\in V$.

Following [42, Definition 1.1.4], for a real analytic curve
$\gamma(t):[0,\varepsilon)\longrightarrow M$ ($\varepsilon>0$) on $M$ (defined
on a neighborhood of $0\in\mathbb{R}$) we call the subset
$\Gamma:=\gamma((0,\varepsilon))\subset M$ of its image a semi-analytic arc
with an endpoint $\gamma(0)\in\overline{\Gamma}$. In what follows, let $X$ be
a complex manifold and consider $\mathbb{C}$-valued $C^{\infty}$-functions on
its underlying real analytic manifold $X_{\mathbb{R}}$. Then we have the
following higher-dimensional analogue of Kashiwara-Schapira [25, Lemma 7.2]
and Morando [42, Proposition 2.1.1].

######  Proposition 7.3.

Let $X$ be a compact complex manifold, $D\subset X$ a normal crossing divisor
on it, and $f\in\mathcal{O}_{X}(\ast D)$ a meromorphic function along $D$. Set
$U:=X\setminus D$. Assume that $f$ does not have any point of indeterminacy on
the whole $X$ and has a pole along each irreducible component of $D$. Then for
any relatively compact subanalytic open subset $W\subset U=X\setminus D$ the
following conditions are equivalent.

  1. (i)

The function $\rm Ref|_{W}:W\longrightarrow\mathbb{R}$ is bounded from above.

  2. (ii)

The $\mathbb{C}$-valued $C^{\infty}$-function
$\exp(f)|_{W}:W\longrightarrow\mathbb{C}$ is is tempered on $W\subset
X_{\mathbb{R}}$.

###### Proof.

Since $f$ has no point of indeterminacy on the whole $X$, it defines a
holomorphic map from $X$ to ${\mathbb{P}}={\mathbb{P}}^{1}$. We denote it also
by $f$. This non-constant map $f:X\longrightarrow{\mathbb{P}}$ being proper
and open, we obtain a relatively compact subanalytic open subset
$f(W)\subset{\mathbb{P}}$ of ${\mathbb{P}}$ such that
$f(W)\subset{\mathbb{P}}\setminus\\{\infty\\}=\mathbb{C}$. Since the proof of
(i) $\Longrightarrow$ (ii) is trivial, we shall prove only (ii)
$\Longrightarrow$ (i). We prove it by showing a contradiction. Suppose that
the function $\rm Ref|_{W}:W\longrightarrow\mathbb{R}$ is not bounded from
above. Then there exists a sequence $p_{k}\in W$ ($k=1,2,3,\ldots$) such that

$\lim_{k\to+\infty}{\rm Re}f(p_{k})=+\infty.$ (7.5)

By taking a subsequence of it, we may assume also that it converges to a point
$p\in X$. Then by the condition $f\in\mathcal{O}_{X}(\ast D)$, we have $p\in
D\cap\overline{W}$. By the curve selection lemma, there exists a real analytic
curve $\gamma(t):[0,\varepsilon)\longrightarrow\overline{W}$ ($\varepsilon>0$)
(defined on a neighborhood of $0\in\mathbb{R}$) such that $\gamma(0)=p\in
D\cap\overline{W}$, $\gamma(t)\in W$ for any $t\in(0,\varepsilon)$, and

$\lim_{t\to+0}\rm Ref(\gamma(t))=+\infty.$ (7.6)

Consider the semi-analytic arc $\Gamma:=\gamma((0,\varepsilon))\subset
W\subset X_{\mathbb{R}}$ with an endpoint $p=\gamma(0)\in D\cap\overline{W}$.
Let $z=(z_{1},z_{2},\ldots,z_{n})$ be a holomorphic coordinate system of $X$
on a neighborhood $V$ of the point $p\in D$ such that we have
$\\{p\\}=\\{z=0\\}$ and

$D=\\{z_{1}z_{2}\cdots z_{l}=0\\},\quad
f(z)=\frac{1}{z_{1}^{m_{1}}z_{2}^{m_{2}}\cdots z_{l}^{m_{l}}}$ (7.7)

for some $1\leq l\leq n$ and $m_{i}\in\mathbb{Z}_{>0}$ ($1\leq i\leq l$).
Shrinking the semi-analytic arc $\Gamma\subset W$ if necessary, we may assume
that $\overline{\Gamma}\subset V\subset\mathbb{C}^{n}$. Then we can apply
Proposition 7.2 the pair $(Z_{1},Z_{2})$ of the closed subanalytic subsets
$Z_{1}=\overline{\Gamma}$ and $Z_{2}=\partial W$ of $V\subset\mathbb{C}^{n}$
(satisfying the condition $Z_{1}\cap Z_{2}=\\{p\\}=\\{z=0\\}$) to show that
there exist an open neighborhood $V_{1}\subset V$ of $p$ in $V$ and positive
real numbers $C_{1},r_{1}>0$ such that we have

$||z||={\rm dist}(z,\\{p\\})\leq C_{1}\cdot{\rm dist}(z,\partial W)^{r_{1}}$
(7.8)

for any $z\in\overline{\Gamma}\cap V_{1}\subset V\subset\mathbb{C}^{n}$. On
the other hand, by the condition $\\{p\\}=\\{z=0\\}\subset D$ we have also

${\rm dist}(z,D)\leq{\rm dist}(z,\\{p\\})=||z||$ (7.9)

for any $z\in\overline{\Gamma}\cap V_{1}$. Set

$g(z):=\frac{1}{f(z)}=z_{1}^{m_{1}}z_{2}^{m_{2}}\cdots z_{l}^{m_{l}}.$ (7.10)

Then we can easily show that there exist an open neighborhood $V_{2}\subset V$
of $p$ in $V$ and positive real numbers $C_{2},r_{2}>0$ such that we have

$|g(z)|=\frac{1}{|f(z)|}\leq C_{2}\cdot{\rm dist}(z,D)^{r_{2}}$ (7.11)

for any $z\in V_{2}\setminus D=V_{2}\cap U$. Now let us set $V_{0}:=V_{1}\cap
V_{2}\subset V$. Then there exist positive real numbers $C,r>0$ such that we
have

$|g(z)|\leq C\cdot{\rm dist}(z,\partial W)^{r}$ (7.12)

for any $z\in(V_{0}\cap\overline{\Gamma})\setminus
D=(V_{0}\cap\overline{\Gamma})\cap U$. Shrinking the semi-analytic arc
$\Gamma$ once again, we may assume that $\Gamma\subset V_{0}\cap W$. Note that
the image $f(\Gamma)\subset{\mathbb{P}}\setminus\\{\infty\\}=\mathbb{C}^{n}$
of $\Gamma$ by $f:X\longrightarrow{\mathbb{P}}$ is a semi-analytic arc in
${\mathbb{P}}$ with an endpoint $\infty\in{\mathbb{P}}$. Indeed, for the real
analytic curve $(f\circ\gamma)(t):[0,\varepsilon)\longrightarrow{\mathbb{P}}$
($\varepsilon>0$) we have $(f\circ\gamma)(0)=\infty\in{\mathbb{P}}$. It
satisfies also the condition

$\lim_{t\to+0}\rm Re(f\circ\gamma)(t)=+\infty.$ (7.13)

Then by the proofs of [25, Lemma 7.2] and [42, Proposition 2.1.2], we see that
for any $M>0$ and $N>0$ there exists a point $\tau\in
f(\Gamma)\subset\mathbb{C}$ such that

$|\exp(\tau)|>M|\tau|^{N}.$ (7.14)

In particular, if we take increasing sequences $M_{k}>0$, $N_{k}>0$
($k=1,2,3,\ldots$) satisfying the condition

$\lim_{k\to+\infty}N_{k}=+\infty,\quad\lim_{k\to+\infty}\frac{M_{k}}{C^{N_{k}}}=+\infty,$
(7.15)

there exist points $\tau_{k}\in f(\Gamma)\subset\mathbb{C}$ ($k=1,2,3,\ldots$)
such that

$|\exp(\tau_{k})|>M_{k}|\tau_{k}|^{N_{k}}.$ (7.16)

We may assume also that $\lim_{k\to+\infty}\tau_{k}=\infty\in{\mathbb{P}}$.
For each $k\geq 1$ we choose a point $z_{k}\in\Gamma$ such that
$f(z_{k})=\tau_{k}$. Then we obtain

$|\exp(f(z_{k}))|>M_{k}|f(z_{k})|^{N_{k}}=M_{k}|g(z_{k})|^{-N_{k}}$ (7.17)

for any $k\geq 1$. By the condition

$\lim_{k\to+\infty}f(z_{k})=\lim_{k\to+\infty}\tau_{k}=\infty,$ (7.18)

the sequence $z_{k}\in\Gamma$ ($k=1,2,3,\ldots$) satisfies the condition

$\lim_{k\to+\infty}|g(z_{k})|=0.$ (7.19)

Then it follows also from Proposition 7.2 that we have

$\lim_{k\to+\infty}{\rm dist}(z_{k},D)=\lim_{k\to+\infty}||z_{k}||=0.$ (7.20)

Hence we get

$\lim_{k\to+\infty}z_{k}=p\in D.$ (7.21)

On the other hand, by (7.12) and (7.17) we have

$|\exp(f(z_{k}))|>\frac{M_{k}}{C^{N_{k}}}\cdot{\rm dist}(z_{k},\partial
W)^{-rN_{k}}$ (7.22)

for any $k\geq 1$. By (7.15) and (7.21) this implies that the
$\mathbb{C}$-valued $C^{\infty}$-function
$\exp(f)|_{W}:W\longrightarrow\mathbb{C}$ does not have polynomial growth at
$p\in D\cap\overline{W}$. This completes the proof. ∎

As in the the proofs of [25, Proposition 7.3] and [42, Lemma 3.1.1], by
Proposition 7.3 we obtain the following result.

######  Theorem 7.4.

In the situation of Proposition 7.3, for $R>0$ we set

$W_{R}:=\\{z\in U\ |\ {\rm Re}f(z)<R\\}\subset X.$ (7.23)

Then we have an isomorphism

$H^{0}Sol_{X}^{t}\big{(}\mathscr{E}_{U|X}^{f}\big{)}\simeq\underset{R\to+\infty}{``\varinjlim"}\
(\mathbb{C}_{X})_{W_{R}}.$ (7.24)

Now let us recall some notions and results in [7, §7]. Let $X$ be a complex
manifold and $D\subset X$ a normal crossing divisor in it. Denote by
$\varpi_{X}:\widetilde{X}\to X$ the real oriented blow-up of $X$ along $D$
(sometimes we denote it simply by $\varpi$). Then we set

$\displaystyle\mathcal{O}_{\widetilde{X}}^{\rm t}$ $\displaystyle:={\rm
R}{\mathcal{H}}om_{\varpi^{-1}\mathcal{D}_{\overline{X}}}(\varpi^{-1}\mathcal{O}_{\overline{X}},\mathcal{D}b_{\widetilde{X_{\mathbb{R}}}}^{\rm
t}),$ $\displaystyle\mathcal{A}_{\widetilde{X}}$
$\displaystyle:=\alpha_{\widetilde{X}}\mathcal{O}_{\widetilde{X}}^{\rm t},$
$\displaystyle\mathcal{D}_{\widetilde{X}}^{\mathcal{A}}$
$\displaystyle:=\mathcal{A}_{\widetilde{X}}\otimes_{\varpi^{-1}\mathcal{O}_{X}}\varpi^{-1}\mathcal{D}_{X},$

where $\mathcal{D}b_{\widetilde{X}}^{\rm t}$ stands for the ind-sheaf of
tempered distributions on $\widetilde{X}$ (for the definition see [7, Notation
7.2.4]). Recall that a section of $\mathcal{A}_{\widetilde{X}}$ is a
holomorphic function having moderate growth at $\varpi_{X}^{-1}(D)$. Note that
$\mathcal{A}_{\widetilde{X}}$ and $\mathcal{D}_{\widetilde{X}}^{\mathcal{A}}$
are sheaves of rings on $\widetilde{X}$. For
$\mathcal{M}\in{\mathbf{D}}^{\mathrm{b}}(\mathcal{D}_{X})$ we define an object
$\mathcal{M}^{\mathcal{A}}\in{\mathbf{D}}^{\mathrm{b}}(\mathcal{D}_{\widetilde{X}}^{\mathcal{A}})$
by

$\displaystyle\mathcal{M}^{\mathcal{A}}$
$\displaystyle:=\mathcal{D}_{\widetilde{X}}^{\mathcal{A}}\overset{L}{\otimes}_{\varpi^{-1}\mathcal{D}_{X}}\varpi^{-1}\mathcal{M}\simeq\mathcal{A}_{\widetilde{X}}\overset{L}{\otimes}_{\varpi^{-1}\mathcal{O}_{X}}\varpi^{-1}\mathcal{M}.$

Note that if $\mathcal{M}$ is a holonomic $\mathcal{D}_{X}$-module such that
$\mathcal{M}\overset{\sim}{\longrightarrow}\mathcal{M}(\ast D)$ and ${\rm
sing.supp}(\mathcal{M})\subset D$, then one has
$\mathcal{M}^{\mathcal{A}}\simeq\mathcal{D}_{\widetilde{X}}^{\mathcal{A}}\otimes_{\varpi^{-1}\mathcal{D}_{X}}\varpi^{-1}\mathcal{M}$
(see [7, Lemma 7.3.2]). Moreover we have an isomorphism
$\mathcal{M}^{\mathcal{A}}\overset{\sim}{\longrightarrow}\mathcal{M}(\ast
D)^{\mathcal{A}}$ for any holonomic $\mathcal{D}_{X}$-module $\mathcal{M}$
(see [7, Lemma 7.2.2]). Let us take local coordinates
$(u,v)=(u_{1},\ldots,u_{l},v_{1},\ldots,v_{d_{X}-l})$ of $X$ such that
$D=\\{u_{1}u_{2}\cdots u_{l}=0\\}$. We define a partial order $\leq$ on the
set $\mathbb{Z}^{l}$ by

$a\leq a^{\prime}\ \Longleftrightarrow\ a_{i}\leq a_{i}^{\prime}\ (1\leq i\leq
l).$

Then for a meromorphic function $\varphi\in\mathcal{O}_{X}(\ast D)$ on $X$
along $D$ by using its Laurent expansion

$\varphi=\sum_{a\in\mathbb{Z}^{l}}c_{a}(\varphi)(v)\cdot u^{a}\
\in\mathcal{O}_{X}(\ast D)$

with respect to $u_{1},\ldots,u_{l}$ we define its order ${\rm
ord}(\varphi)\in\mathbb{Z}^{l}$ to be the minimum

$\min\Big{(}\\{a\in\mathbb{Z}^{l}\ |\
c_{a}(\varphi)\not=0\\}\cup\\{0\\}\Big{)}$

if it exists. In [40, Chapter 5] Mochizuki defined the notion of good sets of
irregular values on $(X,D)$ to be finite subsets $S\subset\mathcal{O}_{X}(\ast
D)/\mathcal{O}_{X}$ satisfying some properties. We do not recall here its
precise definition. Just recall that for any $\varphi\not=\psi$ in such a set
$S$ the order ${\rm ord}(\varphi-\psi)\in\mathbb{Z}^{l}$ is defined and the
leading term $c_{{\rm ord}(\varphi-\psi)}(\varphi-\psi)(v)$ of its Laurent
expansion does not vanish at any point $v\in
Y=\\{u_{1}=u_{2}=\cdots=u_{l}=0\\}\subset D$.

######  Definition 7.5.

Let $X$ be a complex manifold and $D\subset X$ a normal crossing divisor in
it. Then we say that a holonomic $\mathcal{D}_{X}$-module $\mathcal{M}$ has a
normal form along $D$ if
(i) $\mathcal{M}\overset{\sim}{\longrightarrow}\mathcal{M}(\ast D)$
(ii) $\rm{sing.supp}(\mathcal{M})\subset D$
(iii) for any $\theta\in\varpi^{-1}(D)\subset\widetilde{X}$, there exist an
open neighborhood $U\subset X$ of $\varpi({\theta})\in D$ in $X$, a good set
$S=\\{[\varphi_{1}],[\varphi_{2}],\ldots,[\varphi_{k}]\\}\subset\mathcal{O}_{X}(\ast
D)/\mathcal{O}_{X}$ ($\varphi_{i}\in\mathcal{O}_{X}(\ast D)$) of irregular
values on $(U,D\cap U)$, positive integers $m_{i}>0$ ($1\leq i\leq k$) and an
open neighborhood $W$ of $\theta$ with $W\subset\varpi^{-1}(U)$ such that

$\mathcal{M}^{\mathcal{A}}|_{W}\simeq\bigoplus_{i=1}^{k}\Bigl{(}\bigl{(}\mathcal{E}_{U\backslash
D|U}^{\varphi_{i}}\bigr{)}^{\mathcal{A}}|_{W}\Bigr{)}^{\oplus m_{i}}.$

By [17, Proposition 3.19] the good set $S\subset\mathcal{O}_{X}(\ast
D)/\mathcal{O}_{X}$ of irregular values for $\mathcal{M}$ in this definition
does not depend on the point $\theta\in\in\varpi^{-1}(D)$. Moreover by [17,
Proposition 3.5] for any $\theta\in\varpi^{-1}(D\cap U)$ there exists its
sectorial open neighborhood $V\subset U\setminus D$ such that

$\pi^{-1}\mathbb{C}_{V}\otimes Sol_{X}^{\rm
E}(\mathcal{M})\simeq\bigoplus_{i=1}^{k}\Bigl{(}\bigl{(}\mathbb{E}_{V|X}^{\rm
Re\varphi_{i}}\Bigr{)}^{\oplus m_{i}}.$ (7.25)

######  Lemma 7.6.

In the situation as above, there exists a sectorial open neighborhood
$V\subset U\setminus D$ of $\theta\in\varpi^{-1}(D\cap U)$ such that for any
$1\leq i,j\leq k$ the natural morphism

${\rm Hom}^{\rm E}({\rm E}_{V|M}^{\rm Re\varphi_{i}},{\rm E}_{V|M}^{\rm
Re\varphi_{j}})\longrightarrow{\rm Hom}^{\rm E}(\mathbb{E}_{V|X}^{\rm
Re\varphi_{i}},\mathbb{E}_{V|X}^{\rm Re\varphi_{j}})$ (7.26)

is an isomorphism.

###### Proof.

The proof is similar to that of [8, Lemma 5.2.1 (ii)]. It suffices to consider
only pairs $(i,j)$ such that $i\not=j$. Then by
$[\varphi_{i}]\not=[\varphi_{j}]$ the function
$\varphi_{j}-\varphi_{i}\in\mathcal{O}_{X}(\ast D)$ has a pole along the
normal crossing divisor $D\subset X$. For a local coordinate system
$(u,v)=(u_{1},\ldots,u_{l},v_{1},\ldots,v_{d_{X}-l})$ of $X$ such that
$D=\\{u_{1}u_{2}\cdots u_{l}=0\\}$ and $\varpi(\theta)\in
Y:=\\{u_{1}=u_{2}=\cdots=u_{l}=0\\}\subset D$ let

$(\varphi_{j}-\varphi_{i})(u,v)=\sum_{a\in\mathbb{Z}^{l}}c_{a}(\varphi_{j}-\varphi_{i})(v)\cdot
u^{a}\ \in\mathcal{O}_{X}(\ast D)$ (7.27)

be the Laurent expansion of $\varphi_{j}-\varphi_{i}$ with respect to
$u_{1},u_{2},\ldots,u_{l}$. Then by the goodness of the set $S$ the order
$\alpha:={\rm ord}(\varphi_{j}-\varphi_{i})\in\mathbb{Z}^{l}$ is defined and
satisfies the condition

$\alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{l})\quad\in(\mathbb{Z}_{\leq
0})^{l}\setminus\\{0\\}.$ (7.28)

If there exists a sectorial open neighborhood $V\subset X\setminus D$ of
$\theta\in\varpi^{-1}(D)$ such that $\rm Re(\varphi_{j}-\varphi_{i})\leq 0$ on
$V$ , then by the proof of [8, Lemmas 3.1.1 and 3.2.2] we have isomorphisms

${\rm Hom}^{\rm E}({\rm E}_{V|M}^{\rm Re\varphi_{i}},{\rm E}_{V|M}^{\rm
Re\varphi_{j}})\simeq\mathbb{C},\quad{\rm Hom}^{\rm E}(\mathbb{E}_{V|X}^{\rm
Re\varphi_{i}},\mathbb{E}_{V|X}^{\rm Re\varphi_{j}})\simeq\mathbb{C}.$ (7.29)

Otherwise, the point $\theta\in\varpi^{-1}(D)$ is contained in the closure
$\overline{R}$ of the open subset

$R:=\\{(u,v)\in X\setminus D\ |\ \rm Re(\varphi_{j}-\varphi_{i})(u,v)>0\\}$
(7.30)

of $\widetilde{X}$. As the meromorphic function
$\varphi_{j}-\varphi_{i}\in\mathcal{O}_{X}(\ast D)$ has no point of
indeterminacy by the goodness of the set $S$, we can easily show that $R$ is a
subanalytic open subset of $\widetilde{X}$. Then by the curve selection lemma
there exists a real analytic curve
$\gamma(t):[0,\varepsilon)\longrightarrow\overline{R}$ ($\varepsilon>0$)
(defined on a neighborhood of $0\in\mathbb{R}$) such that
$\gamma(0)=\theta\in\varpi^{-1}(D)\cap\overline{R}$ and $\gamma(t)\in R$ for
any $t\in(0,\varepsilon)$. By the leading term
$c_{\alpha}(\varphi_{j}-\varphi_{i})(v)\cdot u^{\alpha}$ of the Laurent
expansion of $\varphi_{j}-\varphi_{i}$, we define a complex valued real
analytic function $\psi:\varpi^{-1}(Y)\longrightarrow\mathbb{C}$ on the real
analytic manifold $\varpi^{-1}(Y)\simeq(S^{1})^{l}\times Y$ (defined on a
neighborhood of the point $\theta\in\varpi^{-1}(Y)$) by

$\psi(e^{i\delta_{1}},e^{i\delta_{2}},\ldots,e^{i\delta_{l}},v):=c_{\alpha}(\varphi_{j}-\varphi_{i})(v)\cdot
e^{i\alpha_{1}\delta_{1}}e^{i\alpha_{2}\delta_{2}}\cdots
e^{i\alpha_{l}\delta_{l}}.$ (7.31)

Recall that the holomorphic function $c_{\alpha}(\varphi_{j}-\varphi_{i})$ on
$Y$ does not vanish at any point $v\in Y$. Suppose that $\rm
Re\psi(\theta)<0$. Then for the real analytic curve $\gamma$ we have

$\lim_{t\to+0}\rm Re(\varphi_{j}-\varphi_{i})(\gamma(t))=-\infty.$ (7.32)

Then contradicts to the condition $\gamma(t)\in R$ ($t\in(0,\varepsilon)$). We
thus obtain $\rm Re\psi(\theta)\geq 0$. Then, by the above definition of
$\psi$, for any sectorial open neighborhood $V\subset X\setminus D$ of
$\theta\in\varpi^{-1}(Y)$ there exists a point
$\theta^{\prime}=(e^{i\delta_{1}},\ldots,e^{i\delta_{l}},v)\in\varpi^{-1}(Y)\cap{\rm
Int}(\overline{V})$ such that $\rm Re\psi(\theta^{\prime})>0$. Define a line
$\gamma^{\prime}(t):(0,\varepsilon^{\prime})\longrightarrow V$
($\varepsilon^{\prime}>0$) in $V$ by

$\gamma^{\prime}(t)=(te^{i\delta_{1}},te^{i\delta_{2}},\ldots,te^{i\delta_{l}},v)\in
V\qquad(0<t<\varepsilon^{\prime})$ (7.33)

so that we have

$\lim_{t\to+0}\gamma^{\prime}(t)=\theta^{\prime}=(e^{i\delta_{1}},\ldots,e^{i\delta_{l}},v).$
(7.34)

Then by $\rm Re\psi(\theta^{\prime})>0$ we obtain

$\lim_{t\to+0}\rm Re(\varphi_{j}-\varphi_{i})(\gamma^{\prime}(t))=+\infty,$
(7.35)

which implies also that $\theta^{\prime}\in\overline{R}$. By the proof of [8,
Lemma 3.1.1 (i)] there exist isomorphisms

${\rm Hom}^{\rm E}({\rm E}_{V|M}^{\rm Re\varphi_{i}},{\rm E}_{V|M}^{\rm
Re\varphi_{j}})\simeq{\rm Hom}_{\mathbb{C}_{X}}(\mathbb{C}_{V\cap\\{\rm
Re(\varphi_{j}-\varphi_{i})\leq 0\\}},\mathbb{C}_{X})\simeq 0.$ (7.36)

Moreover by the proof of [8, Lemma 3.2.2 (i)] we have

${\rm Hom}^{\rm E}(\mathbb{E}_{V|X}^{\rm Re\varphi_{i}},\mathbb{E}_{V|X}^{\rm
Re\varphi_{j}})\simeq\varinjlim_{c\to+\infty}{\rm
Hom}_{\mathbb{C}_{X}}(\mathbb{C}_{V\cap\\{\rm Re(\varphi_{j}-\varphi_{i})\leq
c\\}},\mathbb{C}_{X})\simeq 0.$ (7.37)

We thus obtain the desired isomorphism

${\rm Hom}^{\rm E}({\rm E}_{V|M}^{\rm Re\varphi_{i}},{\rm E}_{V|M}^{\rm
Re\varphi_{j}})\overset{\sim}{\longrightarrow}{\rm Hom}^{\rm
E}(\mathbb{E}_{V|X}^{\rm Re\varphi_{i}},\mathbb{E}_{V|X}^{\rm Re\varphi_{j}})$
(7.38)

for the pair $(i.j)$. Clearly we can take a sectorial open neighborhood
$V\subset X\setminus D$ of $\theta\in\varpi^{-1}(D)$ so that this isomorphism
holds for any pair $(i.j)$. ∎

A ramification of $X$ along $D$ on a neighborhood $U$ of $x\in D$ is a finite
map $\rho:X^{\prime}\to U$ of complex manifolds of the form $z^{\prime}\mapsto
z=(z_{1},z_{2},\ldots,z_{n})=\rho(z^{\prime})=(z^{\prime
m_{1}}_{1},\ldots,z^{\prime
m_{l}}_{l},z^{\prime}_{l+1},\ldots,z^{\prime}_{n})$ for some
$(m_{1},\ldots,m_{l})\in(\mathbb{Z}_{>0})^{l}$, where
$(z^{\prime}_{1},\ldots,z^{\prime}_{n})$ is a local coordinate system of
$X^{\prime}$ and $(z_{1},\ldots,z_{n})$ is that of $U$ such that $D\cap
U=\\{z_{1}\cdots z_{l}=0\\}$.

######  Definition 7.7.

Let $X$ be a complex manifold and $D\subset X$ a normal crossing divisor in
it. Then we say that a holonomic $\mathcal{D}_{X}$-module $\mathcal{M}$ has a
quasi-normal form along $D$ if it satisfies the conditions (i) and (ii) of
Definition 7.5, and if for any point $x\in D$ there exists a ramification
$\rho:X^{\prime}\to U$ on a neighborhood $U$ of it such that
$\mathbf{D}\rho^{\ast}(\mathcal{M}|_{U})$ has a normal form along the normal
crossing divisor $\rho^{-1}(D\cap U)$.

Note that $\mathbf{D}\rho^{\ast}(\mathcal{M}|_{U})$ as well as
$\mathbf{D}\rho_{\ast}\mathbf{D}\rho^{\ast}(\mathcal{M}|_{U})$ is concentrated
in degree zero and $\mathcal{M}|_{U}$ is a direct summand of
$\mathbf{D}\rho_{\ast}\mathbf{D}\rho^{\ast}(\mathcal{M}|_{U})$. Now let
$\mathcal{M}$ be a holonomic $\mathcal{D}_{X}$-module having a quasi-normal
form along the normal crossing divisor $D\subset X$. Then for any point $x\in
D$ there exists a ramification $\rho:X^{\prime}\to U$ on a neighborhood $U$ of
it such that $\mathbf{D}\rho^{\ast}(\mathcal{M}|_{U})$ has a normal form along
the normal crossing divisor $D^{\prime}:=\rho^{-1}(D\cap U)\subset
X^{\prime}$. Note that $\rho^{-1}(x)\subset D^{\prime}$ is a point and denote
it by $x^{\prime}$. Let $\varpi^{\prime}:\widetilde{X^{\prime}}\to X^{\prime}$
be the real oriented blow-up of $X^{\prime}$ along $D^{\prime}$ and
$\widetilde{\rho}:\widetilde{X^{\prime}}\to\widetilde{X}$ the morphism induced
by $\rho$. Then by [17, Proposition 3.19] there exist a unique good set
$S=\\{[\varphi_{1}],[\varphi_{2}],\ldots,[\varphi_{k}]\\}\subset\mathcal{O}_{X^{\prime}}(\ast
D^{\prime})/\mathcal{O}_{X^{\prime}}$
($\varphi_{i}\in\mathcal{O}_{X^{\prime}}(\ast D^{\prime})$) of irregular
values on a neighborhood of $x^{\prime}\in D^{\prime}$ in $X^{\prime}$ and
positive integers $m_{i}>0$ ($1\leq i\leq k$) such that for any
$\theta^{\prime}\in(\varpi^{\prime})^{-1}(D^{\prime})$ and its sufficiently
small sectorial open neighborhood $V^{\prime}\subset X^{\prime}\setminus
D^{\prime}$ we have an isomorphism

$\pi^{-1}\mathbb{C}_{V^{\prime}}\otimes Sol_{X^{\prime}}^{\rm
E}(\mathbf{D}\rho^{\ast}(\mathcal{M}|_{U}))\simeq\bigoplus_{i=1}^{k}\Bigl{(}\mathbb{E}_{V^{\prime}|X^{\prime}}^{\rm
Re\varphi_{i}}\Bigr{)}^{\oplus m_{i}}.$ (7.39)

For a point $\theta\in\varpi^{-1}(D\cap U)$ and its sufficiently small
sectorial open neighborhood $V\subset U\setminus D$ we take a point
$\theta^{\prime}\in(\varpi^{\prime})^{-1}(D^{\prime})$ such that
$\widetilde{\rho}(\theta^{\prime})=\theta$ and its sectorial open neighborhood
$V^{\prime}\subset X^{\prime}\setminus D^{\prime}$ such that
$\rho|_{V^{\prime}}:V^{\prime}\overset{\sim}{\longrightarrow}V$. Define
holomorphic functions $f_{i}:V\to\mathbb{C}$ ($1\leq i\leq k$) by
$f_{i}:=\varphi_{i}\circ(\rho|_{V^{\prime}})^{-1}$. Then by [17, Proposition
3.5] we obtain an isomorphism

$\pi^{-1}\mathbb{C}_{V}\otimes Sol_{X}^{\rm
E}(\mathcal{M})\simeq\bigoplus_{i=1}^{k}\Bigl{(}\mathbb{E}_{V|X}^{\rm
Ref_{i}}\Bigr{)}^{\oplus m_{i}}.$ (7.40)

Moreover by the proof of Lemma 7.6 we obtain the following result.

######  Lemma 7.8.

In the situation as above, there exists a sectorial open neighborhood
$V\subset U\setminus D$ of $\theta\in\varpi^{-1}(D\cap U)$ such that for any
$1\leq i,j\leq k$ the natural morphism

${\rm Hom}^{\rm E}({\rm E}_{V|M}^{\rm Ref_{i}},{\rm E}_{V|M}^{\rm
Ref_{j}})\longrightarrow{\rm Hom}^{\rm E}(\mathbb{E}_{V|X}^{\rm
Ref_{i}},\mathbb{E}_{V|X}^{\rm Ref_{j}})$ (7.41)

is an isomorphism.

######  Definition 7.9.

In the situation as above, we say that an $\mathbb{R}$-constructible enhanced
sheaf $F\in{\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_{X})$ on $X$ has a quasi-
normal form along the normal crossing divisor $D$ at $x\in D$ if for a
ramification $\rho:X^{\prime}\to U$ on a neighborhood $U$ of $x$ and the
normal crossing divisor $D^{\prime}=\rho^{-1}(D\cap U)\subset X^{\prime}$
there exist a good set
$S=\\{[\varphi_{1}],[\varphi_{2}],\ldots,[\varphi_{k}]\\}\subset\subset\mathcal{O}_{X^{\prime}}(\ast
D^{\prime})/\mathcal{O}_{X^{\prime}}$
($\varphi_{i}\in\mathcal{O}_{X^{\prime}}(\ast D^{\prime})$) of irregular
values on a neighborhood of $x^{\prime}=\rho^{\prime}(x)\in D^{\prime}$ in
$X^{\prime}$ and positive integers $m_{i}>0$ ($1\leq i\leq k$) such that for
any $\theta\in\varpi^{-1}(D\cap U)$ and its sufficiently small sectorial open
neighborhood $V\subset U\setminus D$ defining holomorphic functions
$f_{i}:V\to\mathbb{C}$ ($1\leq i\leq k$) by $\varphi_{i}$ ($1\leq i\leq k$) as
above we have an isomorphism

$\pi^{-1}\mathbb{C}_{V}\otimes F\simeq\bigoplus_{i=1}^{k}\Bigl{(}{\rm
E}_{V|X}^{\rm Ref_{i}}\Bigr{)}^{\oplus m_{i}}.$ (7.42)

By using Lemma 7.8 we obtain the following higher-dimensional analogue of [8,
Proposition 5.4.5]. Since the proof is similar to that of [8, Proposition
5.4.5], we omit it here.

######  Proposition 7.10.

Assume that a holonomic $\mathcal{D}_{X}$-module $\mathcal{M}$ has a quasi-
normal form along the normal crossing divisor $D\subset X$. Then for any point
$x\in D$ there exist an open neighborhood $U$ of $x$ in $X$ and an
$\mathbb{R}$-constructible enhanced sheaf
$F\in{\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_{X})$ on $X$ having a quasi-normal
form along the normal crossing divisor $D$ at $x\in D$ such that

$\pi^{-1}\mathbb{C}_{U\setminus\\{x\\}}\otimes Sol_{X}^{\rm
E}(\mathcal{M})\simeq\mathbb{C}_{X}^{\rm E}\overset{+}{\otimes}F.$ (7.43)

The following fundamental result is due to Kedlaya and Mochizuki.

######  Theorem 7.11 ([30, 31, 40]).

For a holonomic $\mathcal{D}_{X}$-module $\mathcal{M}$ and $x\in X$, there
exist an open neighborhood $U$ of $x$, a closed hypersurface $Y\subset U$, a
complex manifold $X^{\prime}$ and a projective morphism $\nu:X^{\prime}\to U$
such that
(i) $\rm{sing.supp}(\mathcal{M})\cap U\subset Y$,
(ii) $D:=\nu^{-1}(Y)$ is a normal crossing divisor in $X^{\prime}$,
(iii) $\nu$ induces an isomorphism $X^{\prime}\backslash
D\overset{\sim}{\longrightarrow}U\backslash Y$,
(iv) $(\mathbf{D}\nu^{\ast}\mathcal{M})(\ast D)$ has a quasi-normal form along
$D$.

This is a generalization of the classical Hukuhara-Levelt-Turrittin theorem to
higher dimensions. Now let $X$ be a compact complex manifold, $Y\subset X$ a
closed hypersurface and $\mathcal{M}$ a holonomic $\mathcal{D}_{X}$-module
such that $\rm{sing.supp}(\mathcal{M})\subset Y$ and
$\mathcal{M}(*Y)\simeq\mathcal{M}$. Assume that there exist a projective
morphism $\nu:Z\to X$ of a compact complex manifold $Z$ such that
$D:=\nu^{-1}(Y)\subset Z$ is a normal crossing divisor and $\nu|_{Z\setminus
D}:Z\setminus D\to X\setminus Y$ is an isomorphism. Assume also that there
exist a ramification $\rho:Z^{\prime}\to U$ on a neighborhood $U$ of $D$ in
$Z$, meromorphic functions

$\varphi_{1},\varphi_{2},\ldots,\varphi_{k}\in\Gamma\bigl{(}Z^{\prime}\ ;\
\mathcal{O}_{Z^{\prime}}(\ast\rho^{-1}(D))\bigr{)}$ (7.44)

and positive integers $m_{i}>0$ ($1\leq i\leq k$) such that
$\mathbf{D}\rho^{\ast}(\mathbf{D}\nu^{\ast}\mathcal{M}|_{U})$ has a normal
form along the normal crossing divisor $D^{\prime}:=\rho^{-1}(D)\subset
Z^{\prime}$ for the set

$S=\\{[\varphi_{1}],[\varphi_{2}],\ldots,[\varphi_{k}]\\}\subset\mathcal{O}_{Z^{\prime}}(\ast
D^{\prime})/\mathcal{O}_{Z^{\prime}}$ (7.45)

which is good at each point of $D^{\prime}$ and $m_{i}>0$ ($1\leq i\leq k$).
Then by the proof of Proposition 7.10 we can also show that there exist an
open neighborhood $V\subset U$ of $D$ in $Z$ and an $\mathbb{R}$-constructible
enhanced sheaf $F\in{\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_{Z})$ on $Z$ having a
quasi-normal form along the normal crossing divisor $D$ at each point of $D$
such that

$\pi^{-1}\mathbb{C}_{V\setminus D}\otimes Sol_{Z}^{\rm
E}(\mathbf{D}\nu^{\ast}\mathcal{M})\simeq\mathbb{C}_{Z}^{\rm
E}\overset{+}{\otimes}F.$ (7.46)

Since in this situation there exists an isomorphism

$\mathcal{M}\overset{\sim}{\longrightarrow}\mathbf{D}\nu_{\ast}(\mathbf{D}\nu^{\ast}\mathcal{M}),$
(7.47)

for the $\mathbb{R}$-constructible enhanced sheaf
$G:=\mathbf{E}\nu_{\ast}F\in{\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_{X})$ on $X$
and the open neighborhood $W:=\nu(V)$ of $Y$ in $X$ we obtain an isomorphism

$\pi^{-1}\mathbb{C}_{W\setminus Y}\otimes Sol_{X}^{\rm
E}(\mathcal{M})\simeq\mathbb{C}_{X}^{\rm E}\overset{+}{\otimes}G.$ (7.48)

One can also slightly modify the $\mathbb{R}$-constructible enhanced sheaf
$G\in{\mathbf{E}}^{\mathrm{b}}(\mathbb{C}_{X})$ so that we have isomorphisms

$Sol_{X}^{\rm E}(\mathcal{M}(*Y))\simeq\pi^{-1}\mathbb{C}_{X\setminus
Y}\otimes Sol_{X}^{\rm E}(\mathcal{M})\simeq\mathbb{C}_{X}^{\rm
E}\overset{+}{\otimes}G.$ (7.49)

By this very explicit description of the enhanced solution complex
$Sol_{X}^{\rm E}(\mathcal{M}(*Y))$, in the one dimensional case $d_{X}={\rm
dim}X=1$ we can apply the Morse theoretical argument in [17] to reprove the
main results in [8]. See [53] for the details.

######  Proposition 7.12.

Let $D\subset X$ be a normal crossing divisor in $X$ and assume that a
holonomic $\mathcal{D}_{X}$-module $\mathcal{M}$ has a quasi-normal form along
$D$. Let $i_{D}:D\hookrightarrow X$ be the inclusion map. Then for any $j\geq
1$ there exists a complex constructible “sheaf” $F_{j}$ on $D$ such that we
have

$H^{j}Sol_{X}^{t}\big{(}\mathcal{M}\big{)}\simeq(i_{D})_{*}F_{j}.$ (7.50)

###### Proof.

For a point $x\in D$, by our assumption there exists a ramification map
$\rho:X^{\prime}\to U$ on a neighborhood $U$ of $x$ such that
$\mathbf{D}\rho^{\ast}(\mathcal{M}|_{U})$ has a normal form along the normal
crossing divisor $D^{\prime}:=\rho^{-1}(D\cap U)\simeq D\cap U$ in
$X^{\prime}$. Let $i_{D^{\prime}}:D^{\prime}\hookrightarrow X^{\prime}$ be the
inclusion map. Then by the proof of [17, Proposition 3.14] for any $j\geq 1$
there exists a complex constructible “sheaf” $G_{j}$ on $D^{\prime}$ such that
we have

$H^{j}Sol_{X^{\prime}}^{t}\big{(}\mathbf{D}\rho^{\ast}(\mathcal{M}|_{U})\big{)}\simeq(i_{D^{\prime}})_{*}G_{j}.$
(7.51)

Moreover, by [24, Theorem 7.4.6] and [19, Propositon 4.39] we have an
isomorphism

$Sol_{U}^{t}\big{(}\mathbf{D}\rho_{\ast}\mathbf{D}\rho^{\ast}(\mathcal{M}|_{U})\big{)}\simeq{\rm
R}\rho_{*}Sol_{X^{\prime}}^{t}\big{(}\mathbf{D}\rho^{\ast}(\mathcal{M}|_{U})\big{)}.$
(7.52)

The ramification map $\rho$ being finite, for any $j\geq 1$ we obtain an
isomorphism

$H^{j}Sol_{U}^{t}\big{(}\mathbf{D}\rho_{\ast}\mathbf{D}\rho^{\ast}(\mathcal{M}|_{U})\big{)}\simeq(\rho\circ
i_{D^{\prime}})_{*}G_{j}.$ (7.53)

Since $\mathcal{M}|_{U}$ is a direct summand of
$\mathbf{D}\rho_{\ast}\mathbf{D}\rho^{\ast}(\mathcal{M}|_{U})$, the same is
true also for $Sol_{U}^{t}\big{(}\mathcal{M}|_{U}\big{)}$ and
$Sol_{U}^{t}\big{(}\mathbf{D}\rho_{\ast}\mathbf{D}\rho^{\ast}(\mathcal{M}|_{U})\big{)}$.
Hence we verified the assertion locally on $U\subset X$. We can easily check
that it holds also globally on $X$. ∎

From now on, let $\mathcal{M}$ be a holonomic D-module on $X$ such that for a
closed hypersurface $D\subset X$ of $X$ we have
(i) $\mathcal{M}\overset{\sim}{\longrightarrow}\mathcal{M}(\ast D)$
(ii) $\rm{sing.supp}(\mathcal{M})\subset D$.
Set $U:=X\setminus D$ and let $j:U\hookrightarrow X$ be the inclusion map.
Then $L:=\mathcal{H}om_{\mathcal{D}_{U}}(\mathcal{M}|_{U},\mathcal{O}_{U})$ is
a local system on $U$. We denote its rank by $r\geq 0$. Let
$\mathcal{R}\in\mathrm{Mod}_{\mbox{\rm\scriptsize hol}}(\mathcal{D}_{X})$ be
the regular meromorphic connection on $X$ along $D$ such that we have an
isomorphism

$Sol_{X}(\mathcal{R})\simeq j_{!}L$ (7.54)

(see e.g. [16, Theorem 5.3.8]). We thus obtain an isomorphism

$\Phi:L=\mathcal{H}om_{\mathcal{D}_{U}}(\mathcal{M}|_{U},\mathcal{O}_{U})\simeq\mathcal{H}om_{\mathcal{D}_{U}}(\mathcal{R}|_{U},\mathcal{O}_{U})$
(7.55)

of local systems on $U$. Fix a point $p\in U$ and let
$\phi_{1},\ldots,\phi_{r}\in\mathcal{H}om_{\mathcal{D}_{U}}(\mathcal{R}|_{U},\mathcal{O}_{U})$
(resp.
$\psi_{1},\ldots,\psi_{r}\in\mathcal{H}om_{\mathcal{D}_{U}}(\mathcal{M}|_{U},\mathcal{O}_{U})$)
be the $\mathbb{C}$-basis of the local system
$\mathcal{H}om_{\mathcal{D}_{U}}(\mathcal{R}|_{U},\mathcal{O}_{U})$ (resp.
$L=\mathcal{H}om_{\mathcal{D}_{U}}(\mathcal{M}|_{U},\mathcal{O}_{U})$) on a
neighborhood of $p$ such that $\Phi(\psi_{i})=\phi_{i}$ for any $1\leq i\leq
r$. Assume that there exist holomorphic functions $f_{1}.\ldots,f_{r}$ defined
on a neighborhood of $p$ such that

$\psi_{i}=\exp(f_{i})\cdot\psi_{i}\quad(1\leq i\leq r).$ (7.56)

As $\phi_{i}$ (resp. $\psi_{i}$) ($1\leq i\leq r$) extend to multi-valued
global sections of the local system
$\mathcal{H}om_{\mathcal{D}_{U}}(\mathcal{R}|_{U},\mathcal{O}_{U})$ (resp.
$\mathcal{H}om_{\mathcal{D}_{U}}(\mathcal{M}|_{U},\mathcal{O}_{U})$), the same
is true also for the holomorphic functions $f_{i}$ ($1\leq i\leq r$). Since we
have $\Phi(\psi_{i})=\phi_{i}$ even after the extensions, their real parts
${\rm Re}f_{i}:U\longrightarrow\mathbb{R}$ are single-valued. Moreover, the
isomorphism $\Phi$ of sheaves being compatible with restrictions, if the
section $\phi_{i}$ is continued to $\sum_{i=1}^{r}c_{i}\phi_{i}$
($c_{i}\in\mathbb{C}$) along a closed continuous curve
$\gamma(t):[0,1]\longrightarrow U$ in $U$ such that $\gamma(0)=\gamma(1)=p$,
the section $\psi_{i}$ is continued to $\sum_{i=1}^{r}c_{i}\psi_{i}$ along it.
By this observation, we see that the enhanced sheaf

$\bigoplus_{i=1}^{r}\mathbb{C}_{\\{(z,t)\in X\times\mathbb{R}\ |\ z\in U,\
t+{\rm Re}f_{i}(z)\geq 0\\}}$ (7.57)

defined on a neighborhood of $p$ can be naturally extended to the one $F$ on
the whole $X$ such that there exists a surjective morphism
$\pi^{-1}j_{!}L\longrightarrow F$ and we have $\pi^{-1}\mathbb{C}_{U}\otimes
F\simeq F$.

######  Theorem 7.13.

In the situation as above, we have an isomorphism

$Sol_{X}^{\rm E}(\mathcal{M})\simeq\mathbb{C}_{X}^{\rm
E}\overset{+}{\otimes}F.$ (7.58)

###### Proof.

Let $\nu:Y\longrightarrow X$ be a projective morphism of complex manifolds
such that $E:=\nu^{-1}D\subset Y$ is a normal crossing divisor in $Y$, the
restriction $\nu|_{Y\setminus E}:Y\setminus E\longrightarrow U=X\setminus D$
of $\nu$ is an isomorphism and the holonomic D-module
$\mathcal{N}:={\mathbf{D}}\nu^{\ast}\mathcal{M}$ on $Y$ has a quasi-normal
form along $E$. Set $V:=Y\setminus E=\nu^{-1}U$. By Theorem 4.1 (ii) and (v),
we have

$\pi^{-1}\mathbb{C}_{U}\otimes Sol_{X}^{\rm E}(\mathcal{M})\simeq Sol_{X}^{\rm
E}(\mathcal{M})$ (7.59)

and

$\pi^{-1}\mathbb{C}_{V}\otimes Sol_{Y}^{\rm E}(\mathcal{N})\simeq Sol_{Y}^{\rm
E}(\mathcal{N})\simeq\mathbf{E}\nu^{-1}Sol_{X}^{\rm E}(\mathcal{M}).$ (7.60)

Since $\nu|_{Y\setminus E}:V\longrightarrow U$ is an isomorphism, there exist
also isomorphisms

$\displaystyle\mathbf{E}\nu_{*}Sol_{Y}^{\rm
E}(\mathcal{N})\simeq\mathbf{E}\nu_{*}\Bigl{(}\pi^{-1}\mathbb{C}_{V}\otimes\mathbf{E}\nu^{-1}Sol_{X}^{\rm
E}(\mathcal{M})\Bigr{)}$ $\displaystyle\simeq\pi^{-1}\mathbb{C}_{U}\otimes
Sol_{X}^{\rm E}(\mathcal{M})\simeq Sol_{X}^{\rm E}(\mathcal{M}).$

Then it suffices to calculate $Sol_{Y}^{\rm E}(\mathcal{N})$ for $\mathcal{N}$
having a quasi-normal form along $E\subset Y$. Let

$i:Y\times\mathbb{R}_{\infty}\longrightarrow Y\times{\mathbb{P}}$ (7.61)

be the morphism of bordered spaces obtained by composing the natural ones
$Y\times\mathbb{R}_{\infty}\longrightarrow Y\times P$ and $Y\times
P\hookrightarrow Y\times{\mathbb{P}}$ for the real projective line
$P=\mathbb{R}\sqcup\\{\infty\\}$. Then we have an isomorphism

$Sol_{Y}^{\rm E}(\mathcal{N})\simeq
i^{!}Sol^{t}_{Y\times{\mathbb{P}}}\Bigl{(}\mathcal{N}\boxtimes\mathscr{E}_{\mathbb{C}|{\mathbb{P}}}^{\tau}\Bigr{)}[2]$
(7.62)

for the holonomic D-module
$\mathcal{N}\boxtimes\mathscr{E}_{\mathbb{C}|{\mathbb{P}}}^{\tau}$ on
$Y\times{\mathbb{P}}$ having a quasi-normal form along the normal crossing
divisor