text:
\begin{document}

\maketitle

\thispagestyle{empty}
\begin{abstract}
In this paper we extend the notion of specialization functor to the case of several closed submanifolds satisfying some suitable conditions. Applying this functor to the sheaf of Whitney holomorphic functions
we construct different kinds
 of sheaves of multi-asymptotically developable functions,
whose definitions are natural extensions of the definition of strongly asymptotically developable functions introduced by Majima.
\end{abstract}

\tableofcontents

\section*{Introduction}

Asymptotically developable expansions of  holomorphic functions on a sector are an
important tool to study ordinary differential equations
with irregular singularities.
When we study, in higher dimension,
a completely integrable connection with irregular singularities
along a normal crossing divisor $H = H_1 \cup \dots \cup H_\ell \subset X$,
it is known that these asymptotic expansions are too weak for this purpose.
Hence M.~Majima, in \cite{Ma84}, introduced the notion of {\it{strongly asymptotically
developable expansion}}
along $H$ for a holomorphic function defined on a poly-sector $S$, and the one of
{\it{consistent family of coefficients}} to which $f$ is
strongly asymptotically developable.

We can understand these notions
from a view point of a locally defined multi-action. For each smooth submanifold
$H_k$ ($k=1,2,\dots,\ell$), we can locally identify $X$ with the normal bundle
$T_{H_k} X$ of $H_k$ near $H_k$. A conic action on $T_{H_k} X$ by $\R^+$
induces a local action $\mu_k$ on $X$ near $H_k$. Then a poly-sector $S$
on which $f$ is defined can be regarded as {\it{a multi-cone}}
with respect to a multi-action $\mu_1$, \dots, $\mu_\ell$
in the sense that
it is an intersection of open sets $V_k$ ($k=1,2,\dots,\ell$) where each $V_k$
is a (locally) conic subset with respect to the action $\mu_k$ and its
edge is contained in $H_k$. A strongly asymptotically
developable expansion of $f$ is, roughly speaking, a formal Taylor
expansion with respect to an orbit
generated by these actions $\mu_1$, \dots, $\mu_\ell$.

Hence, from this point of view, one can expect that
strongly asymptotically developability extends to that along a more general
$H$. As a matter of fact, we have succeeded to construct
the sheaves of {\it{multi-asymptotically developable functions}}
along several kinds of $H$ by the aid of a multi-action,
whose definitions are natural extensions of the one introduced by H.~Majima in \cite{Ma84}.
These sheaves contain, as important cases, not only that
of strongly asymptotically developable functions
but also that associated with a multi-cone which appears in
{\it{a bi-normal deformation}} introduced by P.~Schapira and K.~Takeuchi in \cite{ST94}.

Now an important problem is to establish relations between these sheaves
on different spaces along different kinds of $H$,
to be more precise, we need to construct operations such as inverse images including
restrictions and direct images for these sheaves.
For that purpose, we need a uniform machinery allowing us to treat geometries
associated with these multi-actions. Namely, we need the notion of {\it{multi-normal deformation}} and the one of {\it{multi-specialization}} introduced in this paper,
which are our main subjects.

Let us briefly explain these new notions.

Let $X$ be a $n$-dimensional real analytic manifold with $\operatorname{dim}{X} = n$, and let
$\chi = \{M_1,\dots,M_\ell\}$ be
a family of connected closed submanifolds satisfying some suitable conditions. The multi-normal deformation of $X$ with respect to $\chi$. We first construct the normal deformation $\widetilde{X}_{M_1}$ of $X$ along $M_1$ defined by M.~Kashiwara and P.~Schapira in \cite{KS90}. Then taking the pull-back of $M_2$ we can obtain the normal deformation $\widetilde{X}_{M_1,M_2}$ of $\widetilde{X}_{M_1}$ along  to $M_2$. Then we can define recursively the normal deformation along $\chi$ as
$\widetilde{X}=\widetilde{X}_{M_1,\ldots,M_\ell}:=(\widetilde{X}_{M_1,\ldots,M_{\ell-1}})^\sim_{\widetilde{M}_\ell}.$ This manifold is of dimension $n+\ell$, it is locally isomorphic to $X \times \R^\ell$ and in the zero section $X \times \{0\}$ it is isomorphic to
$$
\underset{X,1\leq j\leq \ell}{\times}T_{M_j}\iota(M_j):=T_{M_1}\iota(M_1) \underset{X}{\times} T_{M_2}\iota(M_2) \underset{X}{\times} \cdots \underset{X}{\times} T_{M_\ell}\iota(M_\ell),
$$
where $\iota(M_j)$ denotes the intersection of the $M_k$'s strictly  containing $M_j$ (or $X$ if $M_j$ is maximal). There is also an action of $(\RP)^\ell$ on $\widetilde{X}$, which is obtained as a natural extension of the $\RP$-action of the normal deformation with respect to one submanifold. Its restriction to the zero section will be crucial for the definition of multi-asymptotic functions.
There are also natural notions of multi-cone and multi-normal cone extending the one of P.~Schapira and K.~Takeuchi, which will be the key to understand the geometry of the sections of the multi-specialization functor.
Given a morphism of real analytic manifolds $f:X \to Y$, we are also able to extend $f$ to a morphism $\widetilde{f}:\widetilde{X}\to\widetilde{Y}$. This is done by repeatedly employing the the usual construction
of a morphism between normal deformations,
i.e. we extend $f$ to  $\widetilde{f}_1:\widetilde{X}_{M_1} \to \widetilde{Y}_{N_1}$,
then we extend $\widetilde{f}_1$ to $\widetilde{f}_{1,2}:\widetilde{X}_{M_1,M_2} \to \widetilde{Y}_{N_1,N_2}$.
Then we can define recursively $\widetilde{f}:\widetilde{X} \to \widetilde{Y}$
by extending the morphism $\widetilde{f}_{1,\ldots,\ell-1}:\widetilde{X}_{M_1,\ldots,M_{\ell-1}} \to \widetilde{Y}_{N_1,\ldots,N_{\ell-1}}$
to the normal deformations with respect to $M_\ell$ and $N_\ell$ respectively. This morphism enable us to make a link between different kinds of multi-normal deformations. As a kind of example, desingularization map makes a link between the normal deformation with respect to a normal crossing divisor and the bispecialization of P.~Schapira and K.~Takeuchi.

Once we have constructed the multi-normal deformation, we are able to extend the definition of the specialization functor to the case of several submanifolds. Roughly speaking, it is a functor associating to a subanalytic sheaf on $X$ a subanalytic sheaf on $\underset{X,1\leq j\leq \ell}{\times}T_{M_j}\iota(M_j)$. We perform all this constructions in the subanalytic setting in order to treat sheaves of functions with growth conditions. Given a morphism of real analytic manifolds $f:X \to Y$ we give conditions for the commutation of the direct and inverse images with respect to the multi-specialization functor.

When we apply the multi-specialization functor to the subanalytic sheaf of Whitney holomorphic functions we obtain the sheaf of multi-asymptotically developable functions along $\chi$ and, outside the zero section in the normal crossing case, the sheaf of strongly asymptotically developable functions of H.~Majima. When we apply the multi-specialization functor to the subanalytic sheaf of Whitney holomorphic functions vanishing up to infinity on $M_1 \cup \cdots \cup M_\ell$ (resp. Whitney holomorphic functions on $M_1 \cup \cdots \cup M_\ell$) we obtain the sheaf of flat multi-asymptotically developable functions (resp. consistent families of coefficients) along $\chi$. The vanishing of the $H^1$ of flat multi-asymptotically developable functions allows us to prove a Borel-Ritt exact sequence for multi-asymptotic functions.

The paper is organized in the following way. In Section \ref{1} we introduce the notion of multi-normal deformation. In Section \ref{2} we define the multi-action of $(\RP)^\ell$ on the zero section of $\widetilde{X}$. In Sections \ref{3} and \ref{4} we give the definitions of multi-cone and multi-normal cone which are essential to understand the sections of the multi-specialization. Morphisms between multi-normal deformations and their restriction to the zero sections are studied in Section \ref{5}. The functorial construction is performed in Section \ref{6} with the definition of the multi-specialization functor and its relations with the functors of direct and inverse image. In Section \ref{7} we define
the sheaves of multi-asymptotically developable functions along $\chi$ whose functorial nature is proved in Section \ref{8}, where we apply the multi-specialization functor to the subanalytic sheaf of holomorphic functions with Whitney growth conditions.

We end this work with an Appendix in which we introduce the category of multi-conic sheaves. Using o-minimal geometry we construct suitable coverings of subanalytic open subsets which are helpful in order to study the sections of multi-conic sheaves.

\begin{section}{Multi-normal deformation}\label{1}

Let $X$ be a real analytic manifold with $\operatorname{dim}{X} = n$, and let
$\chi = \{M_1,\dots,M_\ell\}$ be
a family of $\ell$-closed submanifolds in $X$ ($\ell \ge 1$).
We set, for $N \in \chi$ and $p \in N$,
$$
	\operatorname{NR}_p(N) := \{M_j \in \chi;\,
p \in M_j,\,N \nsubseteq M_j \text{ and } M_j \nsubseteq N\}.
$$
Let us consider the following conditions for $\chi$.
\begin{itemize}
\item[H1] Each $M_j \in \chi$ is connected and the
submanifolds are mutually distinct, i.e. $M_j \neq M_{j'}$ for $j \neq {j'}$.
\item[H2] For any $N \in \chi$ and $p \in N$ with $\operatorname{NR}_p(N) \ne \emptyset$,
we have
\begin{equation}{\label{eq:geometric-condition}}
\left(
\underset{
M_j \in \operatorname{NR}_p(N)
}{\bigcap} T_pM_j\right) + T_pN = T_pX.
\end{equation}
\end{itemize}
Note that, if $\chi$ satisfies the condition \eqref{eq:geometric-condition},
the configuration of two submanifolds
must be either 1.~or 2.~below.
\begin{enumerate}
\item Both submanifolds intersect transversely.
\item One of them contains the other.
\end{enumerate}
We set, for $N \in \chi$,
$$
\iota_\chi(N) := \left\{
\begin{matrix}
&X \qquad & \text{There exists no $M_j\in \chi$ with $N \varsubsetneqq M_j$} \\
\\
&\underset{N \varsubsetneqq M_j}{\bigcap} M_j  \qquad &\text{Otherwise}.
\end{matrix}
\right.
$$
When there is no risk of confusion, we write for short $\iota(N)$ instead of $\iota_\chi(N)$.

Since $T_{M_j}\iota (M_j)$ is contained in the zero section $T_XX$ if $M_j$ satisfies
$M_j = \iota(M_j)$, we assume the condition H3 below for simplicity.

\begin{itemize}
\item[H3] $M_j \ne \iota(M_j)$ for any $j \in \{1,2,\dots, \ell\}$.
\end{itemize}

\begin{prop}{\label{prop:canonical-coordinate}}
The following conditions are equivalent.
\begin{enumerate}
\item The family $\chi$ satisfies the condition \eqref{eq:geometric-condition}.
\item For any $p \in X$, there exist a neighborhood $V$ of $p$ in $X$, a system
of local coordinates $(x_1,\dots,x_n)$ in $V$ and a family of subsets $\{I_j\}_{j=1}^\ell$
of the set $\{1,2,\dots,n\}$ for which the following conditions hold.
\begin{enumerate}
\item Either $I_k \subset I_j$, $I_j \subset I_k$ or $I_k \cap I_j = \emptyset$ holds
$($$k,j \in \{1,2,\dots,\ell\}$$)$.
\item A submanifold $M_j \in \chi$ with $p \in M_j$ $($$j = 1,2,\dots,\ell$$)$
is defined by $\{x_i = 0; i \in I_j\}$ in $V$.
\end{enumerate}
\end{enumerate}
\end{prop}
\dim\ \
Clearly 2.~ implies 1. We will show the converse implication.
We may assume, by taking $V$ sufficiently small, $p \in M_j$ for any $j$.
We set, for $N \in \chi$,
$$
d(N) := \max\, \{k;\,N = M_{j_1} \varsubsetneqq M_{j_2} \varsubsetneqq \dots \varsubsetneqq M_{j_k},\, M_{j_1},\dots,M_{j_k} \in \chi\},
$$
and
$$
d(\chi) := \max \{d(N);\, N \in \chi\}.
$$
We show the lemma by induction with respect to $d(\chi)$.
Clearly the equivalence holds for a family $\chi$ with $d(\chi) = 1$.
Let us consider $\chi$ with $d(\chi) = \kappa > 1$, and
suppose that $1. \Rightarrow 2.$ were true for any $\chi$ with $d(\chi) < \kappa$.
Let $L_1$, $\dots$, $L_m$ denote the least elements in $\chi$ with respect
to the partial order $\subset$ of sets.

For each $k=1,2,\dots,m$,  we will determine defining functions $f_{k,1}$, $\dots$,
$f_{k, i_k}$ of the submanifold $L_k$ in the following way.
Set, for any $N \in \chi$,
$$
\chi_N = \{L \in \chi;\, N \varsubsetneqq L\}.
$$
Since we have $d(\chi_{L_k}) < \kappa$ ($k=1,2,\dots,m$),
and since the family $\chi_{L_k}$
also satisfies the condition 1.~ of the proposition,
by the induction hypothesis,
there exist local coordinate functions ($\varphi_{k,1}$, $\dots$, $\varphi_{k,n}$)
that satisfies the condition 2.~for the family $\chi_{L_k}$.
Then, for $k=1,2,\dots,m$, we take defining functions
$\{f_{k,1},\dots, f_{k,i_k}\}$
of $L_k$ so that
they contain all the coordinate functions
$\varphi_{k, i}$ which vanish on $L_k$.

As $L_{j'} \in \operatorname{NR}_p(L_{j})$ holds for $1 \le j \ne j' \le m$,
it follows from the condition \eqref{eq:geometric-condition}
that we have
$$
\underset{1\le k \le m,\, 1 \le i \le i_k}{\wedge} df_{k,i} \ne 0
\qquad \text{near $p$}.
$$
Therefore the family of these functions $\{f_{k,i}\}$
can be extended to a system of local coordinates near $p$, for which
we can easily verify 2.~of the lemma. This completes the proof.
\qed

Let $X$ be a $n$-dimensional real analytic manifold and let $\chi=\{M_1,\ldots,M_\ell\}$ be a family of smooth closed submanifolds of $X$ satisfying \eqref{eq:geometric-condition}. First recall the construction of the normal deformation of $X$ along $M_1$. We denote it by $\widetilde{X}_{M_1}$ and we denote by $t_1\in\R$ the deformation parameter. Let $\Omega_{M_1}=\{t_1>0\}$ and let us identify $\imin s(0)$ with $T_{M_1}X$. We have the commutative diagram
\begin{equation}\label{normal deformation}
\xymatrix{T_{M_1}X \ar[r]^{s_{M_1}} \ar[d]^{\tau_{M_1}} & \widetilde{X}_{M_1} \ar[d]^{p_{M_1}} &
\Omega_{M_1} \ar[l]_{\ \ i_{\Omega_{M_1}}} \ar[dl]^{\widetilde{p}_{M_1}} \\
M \ar[r]^{i_{M_1}} & X. & }
\end{equation}
Set $\widetilde{\Omega}_{M_1}=\{(x,t_1)\;;\;t_1\neq 0\}$ and define
$$
\widetilde{M}_2:=\overline{(p_{M_1}|_{\widetilde{\Omega}_{M_1}})^{-1}M_2}.
$$
Then $\widetilde{M}_2$ is a closed smooth submanifold of $\widetilde{X}_{M_1}$. Now we can define the normal deformation along $M_1,M_2$ as
$$
\widetilde{X}_{M_1,M_2}:=(\widetilde{X}_{M_1})^\sim_{\widetilde{M}_2}.
$$
Then we can define recursively the normal deformation along

$\chi$ as
$$
\widetilde{X}=\widetilde{X}_{M_1,\ldots,M_\ell}:=(\widetilde{X}_{M_1,\ldots,M_{\ell-1}})^\sim_{\widetilde{M}_\ell}.
$$
Set $S=\{t_1,\ldots,t_\ell=0\}$, $M=\bigcap_{i=1}^\ell M_i$ and $\Omega=\{t_1,\ldots,t_\ell>0\}$. Then we have the commutative diagram
\begin{equation}\label{multi normal deformation}
\xymatrix{S \ar[r]^s \ar[d]^\tau & \widetilde{X} \ar[d]^p &
\Omega \ar[l]_{\ \ i_\Omega} \ar[dl]^{\widetilde{p}} \\
M \ar[r]^{i_M} & X. & }
\end{equation}
In local coordinate let $I_1,\ldots I_\ell \subseteq \{1,\ldots,n\}$ such that $M_i=\{x_k=0\;;\;k\in I_i\}$. Set
$$
J_i=\{k\in\{1,\ldots,\ell\}\;;\;i\in I_k\}, \ \ \ \ t_{J_i}=\prod_{k\in J_i}t_k,
$$
where $t_1,\ldots,t_\ell\in\R$ and $t_{J_i}=1$ if $J_i=\emptyset$.
Then $p:\widetilde{X}\to X$ is defined by
$$
(x_1,\ldots,x_n,t_1,\ldots,t_\ell) \mapsto (t_{J_1}x_1,\ldots,t_{J_n}x_n).
$$
We are interested in the bundle structure of the zero section
$$
S :=\{t_1=\ldots=t_\ell=0\} \subset \widetilde{X}.
$$
Let us consider the canonical map $T_{M_j}\iota(M_j) \overset{}{\to} M_j \hookrightarrow X$, $j=1,\dots,\ell$,
we write for short $$\underset{X,1\leq j\leq \ell}{\times}T_{M_j}\iota(M_j)$$ instead of
$$
T_{M_1}\iota(M_1) \underset{X}{\times} T_{M_2}\iota(M_2) \underset{X}{\times} \cdots \underset{X}{\times} T_{M_\ell}\iota(M_\ell).
$$
\begin{prop}\label{prop:zero section} Assume that $\chi$ satisfies the conditions H1-H2-H3.

Then we have
\begin{equation}

S \simeq \underset{X,1\leq j\leq \ell}{\times}T_{M_j}\iota(M_j).
\end{equation}
\end{prop}
\dim\ \
Let $\hat{X}$ be a copy of $X$, and $\varphi: X \to \hat{X}$
a local coordinate transformation near $p \in X$. We set $\hat{M}_j = \varphi(M_j)$.
We may assume that $X = {\mathbb R}^n$
(resp. $\hat{X} = {\mathbb R}^n$)
with coordinates  $(x_1,\dots,x_n)$
(resp. $(\hat{x}_1, \dots, \hat{x}_n)$), and $\varphi$ is given by
$\hat{x}_i = \varphi_i(x_1,\dots,x_n)$ ($i=1,2,\dots,n$).
Moreover, by Proposition \ref{prop:canonical-coordinate}, we may also suppose
that there exist a family of subsets $\{I_j\}_{j=1}^\ell$ of $\{1,2,\dots,n\}$
that satisfies 2.~(a) and 2.~(b) of Proposition \ref{prop:canonical-coordinate}
for the both coordinate systems.

Let $J_k \subset \{1,2,\dots,\ell\}$ ($k=1,2,\dots,n$) denote the set
\begin{equation}\label{eq:J_k}
\left\{j \in \{1,2,\dots,\ell\};\, k \in I_j\right\}.
\end{equation}
For a subset $J$ of $\{1,2,\dots,\ell\}$, we set
$t_{J} := \underset{j \in J}{\prod} t_j$ if $J$ is non-empty and
$t_{J} := 1$ if $J = \emptyset$.

Then, outside of $t_1t_2\ldots t_\ell = 0$, the coordinate transformation
between multi-normal deformations
$$
(x_1,\dots,x_n;\,t_1,\dots,t_\ell) \to
(\hat{x}_1,\dots,\hat{x}_n;\,\hat{t}_1,\dots,\hat{t}_\ell)
$$
is given by
$$
\left\{
\begin{aligned}
&\hat{t}_{J_k}\hat{x}_k = \varphi_k(t_{J_1}x_1,\, t_{J_2}x_2,\,\dots,\, t_{J_n}x_n)
\qquad (k = 1,2,\dots,n),\\
&\hat{t}_j = t_j \qquad (j = 1,2,\dots,\ell).
\end{aligned}
\right.
$$
For any $k\in \underset{1 \le j \le \ell}{\bigcup} I_j$, we set
$$
I(k) := \underset{k \in I_j}{\bigcap} I_j = \underset{j \in J_k}{\bigcap} I_j.
$$
As the condition 2.~(a) of Proposition \ref{prop:canonical-coordinate} is equivalently
saying that
$$
I_{j} \cap I_{j'} \ne \emptyset \Rightarrow
I_{j} \subset I_{j'} \text{ or } I_{j'} \subset I_{j},
$$
the set $\{I_j;\, k \in I_j\}$
is totally ordered with respect to ``$\subset$``,
and $I(k)$ is its minimal element.
Hence, for any $k \in \underset{1 \le j \le \ell}{\bigcup} I_j$,
there exists $j(k) \in \{1,2,\dots,\ell\}$ such that $I(k) = I_{j(k)}$.
By expanding $\varphi_k(x_1,\dots,x_n)$ along the submanifold $M_{j(k)}$,
we obtain
$$
\varphi_k(x_1,\dots,x_n) =
\sum_{i \in I_{j(k)}}\left.\frac{\partial \varphi_k}{\partial x_i}\right|_{M_{j(k)}} x_i
+ {1\over 2}\sum_{i_1,i_2 \in I_{j(k)}}
\left.
\frac{\partial^2 \varphi_k}{\partial x_{i_1}\partial x_{i_2}}\right|_{M_{j(k)}}
x_{i_1}x_{i_2}
+ \dots,
$$
as $\varphi_k\vert_{M_{j(k)}} = 0$ holds.
Then we get
$$
t_{J_k}\hat{x}_k =
\sum_{i \in I_{j(k)}}\left.\frac{\partial \varphi_k}{\partial x_i}\right|_{M_{j(k)}}
t_{J_i}x_i
+ {1\over 2}\sum_{i_1,i_2 \in I_{j(k)}}
\left.
\frac{\partial^2 \varphi_k}{\partial x_{i_1}\partial x_{i_2}}\right|_{M_{j(k)}}
t_{J_{i_1}}t_{J_{i_2}}x_{i_1}x_{i_2}
+ \dots.
$$
We can easily see $J_k \subseteq J_i$ for any $i \in I_{j(k)}$. Indeed,
this follows from the facts
$$
\begin{aligned}
&\text{$j \in J_k$ and $i \in I_{j(k)}$} \Longrightarrow
\text{$k \in I_j$ and $i \in I(k) = \underset{k \in I_\beta}{\bigcap}I_\beta$} \\
&\Longrightarrow i \in I_j \Longrightarrow j \in J_i.
\end{aligned}
$$
Therefore, by letting $t \to 0$,  we have
$$
\hat{x}_k =
\sum_{i \in I_{j(k)},\, J_k = J_i}
\left.\frac{\partial \varphi_k}{\partial x_i}\right|_M
x_i.
$$
where $M := \underset{1 \le j \le l}{\bigcap} M_j$.
Now we claim, for $k \in \underset{1 \le j \le \ell}{\bigcup} I_j$,
$$
\left\{i \in I_{j(k)};\, J_k = J_i\right\} =
\left\{i \in \{1,2,\dots,n\};\, j(k) = j(i)\right\},
$$
which is proved in the following way:
We first prove the implication ($\Leftarrow$).
Assume that $i$ satisfies $j(k) = j(i)$, which implies
$i \in I_{j(i)} = I_{j(k)}$. We have already proved the fact
$J_k \subseteq J_i$ for $i \in I_{j(k)}$.
Therefore it suffices to show $J_i \subseteq J_k$.
Let $\beta \in J_i$. Then, as $i \in I_\beta$, we have
$k \in I_{j(k)} = I_{j(i)} \subset I_\beta$, from which $\beta \in J_k$ follows.

The converse implication comes from
$$
I_{j(k)} = \underset{\beta \in J_k}{\bigcap}I_\beta
= \underset{\beta \in J_i}{\bigcap}I_\beta = I_{j(i)}.
$$

We divide the set $\underset{1 \le j \le \ell}{\bigcup} I_j \subset \{1,2,\dots,n\}$
into equivalent classes $\{B_\alpha\}$
by the equivalence relation ``$i \sim k \iff j(i) = j(k)$''.
Then, for an equivalent class $B$, we obtain
$$
\hat{x}_i =
\sum_{k \in B}
\left.\frac{\partial \varphi_i}{\partial x_k}\right|_M
x_k \qquad \text{for $i \in B$}.
$$
We denote by $E_B$ the vector bundle over $M$ defined by
the above equations where its fiber coordinates are given by $x_k$'s ($k \in B$).
Then, by the above observation,
$S$ is a direct sum of bundles $E_B$'s over $M$.
Note that, since each equivalent class can be written in the form
$$
I_{j(k)} \setminus \left(\underset{I_{j(i)} \varsubsetneqq I_{j(k)}}{\bigcup} I_{j(i)}\right)
=
I_{j(k)} \setminus \left(\underset{I_j \varsubsetneqq I_{j(k)}}{\bigcup} I_j\right)
$$
for some $j(k) \in \{1,2,\dots,l\}$ with $k \in \underset{1 \le j \le \ell}{\bigcup} I_j$, the bundle $E_B$ is isomorphic to
$T_{M_{j(k)}} \iota(M_{j(k)}) \underset{X}{\times} M$.

For any $j \in \{1,2,\dots,\ell\}$, the set
$$
\hat{I}_j := I_j \setminus \left(\underset{I_k \varsubsetneqq I_j}{\bigcup} I_k\right)
$$
is not empty by the condition H3, and it gives an equivalent class
of $\underset{1 \le j \le \ell}{\bigcup} I_j / \sim$. Further it follows from the
condition H1 and H2 that we get
$$
\underset{1 \le j \le \ell}{\bigcup} I_j = \hat{I}_1 \sqcup \dots \sqcup \hat{I}_l.
$$
Hence we have obtained that $S$ is a direct sum of bundles
$T_{M_j} \iota(M_j) \underset{X}{\times} M$
($j=1,2,\dots.\ell$). This completes the proof.
\qed

\begin{es} Let us see three typical examples of multi-normal deformations.
\begin{enumerate}
\item (Majima) Let $X={\mathbb C}^2$ ($\simeq\R^4$ as a real manifold)
with coordinates $(z_1, z_2)$ and let $\chi=\{M_1,M_2\}$ with $M_1 = \{z_1 = 0\}$ and $M_2 = \{z_2 = 0\}$. Then $\chi$ satisfies H1-H2-H3.
We have $I_1=\{1\}$, $I_2=\{2\}$, $J_1=\{1\}$, $J_2=\{2\}$ (in $\R^4$, if $z_1=(x_1,x_2)$ and $z_2=(x_3,x_4)$ we have $I_1=\{1,2\}$, $I_2=\{3,4\}$, $J_1=J_2=\{1\}$, $J_3=J_4=\{2\}$).
The map $p:\widetilde{X} \to X$ is defined by
$$
(z_1,z_2,t_1,t_2) \mapsto (t_1z_1,t_2z_2).
$$
We have $\iota(M_1)=\iota(M_2)=X$ and then the zero section of $\widetilde{X}$ is isomorphic to $T_{M_1}X \underset{X}{\times} T_{M_2}X$.

\item (Takeuchi)
Let $X={\mathbb R}^3$ with coordinates $(x_1, x_2, x_3)$ and let $\chi=\{M_1,M_2,M_3\}$ with
$M_1 = \{0\}$, $M_2 = \{x_2 = x_3 = 0\}$ and $M_3 = \{x_3 = 0\}$. Then $\chi$ satisfies H1-H2-H3. We have $I_1=\{1,2,3\}$, $I_2=\{2,3\}$, $I_3=\{3\}$, $J_1=\{1\}$, $J_2=\{1,2\}$, $J_3=\{1,2,3\}$.
The map $p:\widetilde{X} \to X$ is defined by
$$
(x_1,x_2,x_3,t_1,t_2,t_3) \mapsto (t_1x_1,t_1t_2x_2,t_1t_2t_3x_3).
$$
We have $\iota(M_1)=M_2$, $\iota(M_2)=M_3$, $\iota(M_3)=X$ and then the zero section of $\widetilde{X}$ is isomorphic to $T_{M_1}M_2 \underset{X}{\times} T_{M_2}M_3 \underset{X}{\times} T_{M_3}X$.

\item (Mixed)
Let $X={\mathbb R}^3$ with coordinates $(x_1, x_2, x_3)$ and let $\chi=\{M_1,M_2,M_3\}$ with
$M_1 = \{0\}$, $M_2 = \{x_2 = 0\}$ and $M_3 = \{x_3 = 0\}$. Then $\chi$ satisfies H1-H2-H3. We have $I_1=\{1,2,3\}$, $I_2=\{2\}$, $I_3=\{3\}$, $J_1=\{1\}$, $J_2=\{1,2\}$, $J_3=\{1,3\}$.
The map $p:\widetilde{X} \to X$ is defined by
$$
(x_1,x_2,x_3,t_1,t_2,t_3) \mapsto (t_1x_1,t_1t_2x_2,t_1t_3x_3).
$$
We have $\iota(M_1)=M_2 \cap M_3$, $\iota(M_2)=\iota(M_3)=X$ and then the zero section is isomorphic to $T_{M_1}(M_2 \cap M_3) \underset{X}{\times} T_{M_2}X \underset{X}{\times} T_{M_3}X$.
\end{enumerate}
\end{es}

\end{section}

\begin{section}{Multi-action}{\label{2}}

Let $X$ be a real analytic manifold and let $\chi = \{M_1, M_2, \dots, M_\ell\}$ be closed submanifolds. In what follows, we always assume that $\chi$ satisfies the conditions H1-H2-H3.
Consider the diagram \eqref{multi normal deformation}.

There is a $(\RP)^\ell$ action
$$
\mu:\widetilde{X} \times (\RP)^\ell \to \widetilde{X}
$$
which is described in local coordinate system by
$$
((x_1,\ldots,x_n,t_1,\ldots,t_\ell),(c_1,\ldots,c_\ell)) \mapsto \left(c_{J_1}x_1,\ldots,c_{J_n}x_n,{t_1\over c_1},\ldots,{t_\ell\over c_\ell}\right).
$$
More precisely, the $j$-th component of the action is given by
$$
\mu_j:((x_1,\ldots,x_n,t_1,\ldots,t_\ell),c_j) \mapsto \left(c_{1j}x_1,\ldots,c_{nj}x_n,t_1,\ldots,{t_j\over c_j},\ldots,t_\ell\right),
$$
where $c_{ij}=c_j$ if $i\in I_j$ and $c_{ij}=1$ otherwise.

First we consider the actions on $\widetilde{X}$ which are compatible
with those on

$\underset{X,1\leq \alpha\leq \ell}{\times}T_{M_\alpha}\iota(M_\alpha)$.
Set
$$
\begin{aligned}
J_{\supsetneq M_j} &:= \{\alpha \in \{1,2,\dots,\ell\};\,  M_\alpha \supsetneq M_j,\,
\text{ there is no $\beta$ with $M_\alpha \supsetneq M_\beta \supsetneq M_j$ } \}\\
J_{\subsetneq M_j} &:= \{\alpha \in \{1,2,\dots,\ell\};\,  M_\alpha \subsetneq M_j,\,
\text{ there is no $\beta$ with $M_\alpha \subsetneq M_\beta \subsetneq M_j$ } \}
\end{aligned}
$$
Note that, by the conditions, the set $J_{\subsetneq M_j}$ either is empty or consists of only 1 index.

Using the actions $\mu_j$
for $j = 1,2, \dots, \ell$, we define the action
$\tau_j(\lambda): \widetilde{X} \to \widetilde{X}$
by
$$
\tau_j(\lambda) :=
\mu_j(\lambda) \underset{\beta \in J_{\supsetneq M_j}}{\prod} \mu_\beta(\lambda^{-1})
\qquad (j = 1,2,\dots,\ell).
$$
On the zero section
$\underset{X,1\leq \alpha\leq \ell}{\times}T_{M_\alpha}\iota(M_\alpha)$
of $\widetilde{X}$,
the action $\tau_j(\lambda)$ coincides with
$$
\left.\tau_j(\lambda)\right|_{T_{M_{\alpha}}\iota (M_{\alpha})} =
\left\{
\begin{matrix}
&T_{M_{\alpha}}\iota (M_{\alpha}) \overset{\lambda \cdot}{\to} T_{M_{\alpha}}\iota (M_{\alpha}) \qquad
&(\alpha = j) \\
&\operatorname{id}_{T_{M_{\alpha}}\iota (M_{\alpha})} \qquad
&(\alpha \ne j).
\end{matrix}
\right.
$$

Conversely, we can recover the original actions $\{\mu_j\}$ by $\{\tau_j\}$
in the following way.
$$
\mu_j(\lambda) := \underset{M_j \subseteq M_\alpha}{\prod} \tau_\alpha(\lambda)
\qquad (j=1,2,\dots,l).
$$
Hence both multi-actions on $\widetilde{X}$ associated with $\{\mu_j\}$ and $\{\tau_j\}$ are equivalent.

\begin{es} Let us see some example of multi-actions.
\begin{enumerate}
\item (Majima) Let $X={\mathbb C}^2$
with coordinates $(z_1, z_2)$ and let $M_1 = \{z_1 = 0\}$ and $M_2 = \{z_2 = 0\}$. Then
\begin{eqnarray*}
\mu:(z_1,z_2,t_1,t_2) & \mapsto & \left(c_1z_1,c_2z_2,{t_1 \over c_1},{t_2 \over c_2}\right) \\
\tau_1:(z_1,z_2,t_1,t_2) & \mapsto & (\lambda z_1,z_2,\imin \lambda t_1,t_2) \\
\tau_2:(z_1,z_2,t_1,t_2) & \mapsto & (z_1,\lambda z_2,t_1,\imin \lambda t_2).
\end{eqnarray*}

\item (Takeuchi)
Let $X={\mathbb R}^3$ with coordinates $(x_1, x_2, x_3)$ and let
$M_1 = \{0\}$, $M_2 = \{x_2 = x_3 = 0\}$ and $M_3 = \{x_3 = 0\}$. Then
\begin{eqnarray*}
\mu:(x_1,x_2,x_3,t_1,t_2,t_3) & \mapsto & \left(c_1x_1,c_1c_2x_2,c_1c_2c_3x_3,{t_1 \over c_1},{t_2 \over c_2},{t_3 \over c_3}\right) \\
\tau_1:(x_1,x_2,x_3,t_1,t_2,t_3) & \mapsto & (\lambda x_1,x_2,x_3,\imin \lambda t_1,\lambda t_2,t_3) \\
\tau_2:(x_1,x_2,x_3,t_1,t_2,t_3) & \mapsto & (x_1,\lambda x_2,x_3,t_1,\imin \lambda t_2,
 \lambda t_3) \\
\tau_3:(x_1,x_2,x_3,t_1,t_2,t_3) & \mapsto & (x_1,x_2, \lambda x_3,t_1,t_2,
\imin \lambda t_3).
\end{eqnarray*}
\item (Mixed)
Let $X={\mathbb R}^3$ with coordinates $(x_1, x_2, x_3)$ and let
$M_1 = \{0\}$, $M_2 = \{x_2 = 0\}$ and $M_3 = \{x_3 = 0\}$. Then
\begin{eqnarray*}
\mu:(x_1,x_2,x_3,t_1,t_2,t_3) & \mapsto & \left(c_1x_1,c_1c_2x_2,c_1c_3x_3,{t_1 \over c_1},{t_2 \over c_2},{t_3 \over c_3}\right) \\
\tau_1:(x_1,x_2,x_3,t_1,t_2,t_3) & \mapsto & (\lambda x_1,x_2,x_3,\imin \lambda t_1,
 \lambda t_2,\lambda t_3) \\
\tau_2:(x_1,x_2,x_3,t_1,t_2,t_3) & \mapsto & (x_1, \lambda x_2,x_3,t_1,
\imin \lambda t_2,t_3) \\
\tau_3:(x_1,x_2,x_3,t_1,t_2,t_3) & \mapsto & (x_1,x_2, \lambda x_3,t_1,t_2,
\imin \lambda t_3).
\end{eqnarray*}
\end{enumerate}
\end{es}

\end{section}

\begin{section}{Multi-cones}\label{3}

Let $q \in \underset{1 \le j \le \ell}{\bigcap} M_j$ and
$p_j = (q;\, \xi_j)$ be a point in $T_{M_j}\iota(M_j)$ ($j=1,2,\dots,\ell$).
We set
$$
p = p_1 \underset{X}{\times} \dots \underset{X}{\times} p_\ell
\in \underset{X,1\leq j \leq \ell}{\times}T_{M_j}\iota(M_j),
$$
and $\tilde{p}_j = (q;\, \tilde{\xi}_j) \in T_{M_j} X$ designates the
image of the point $p_j$ by the canonical embedding $T_{M_j} \iota (M_j) \hookrightarrow T_{M_j} X$.
We denote by $\operatorname{Cone}_{\chi, j}(p)$ ($j=1,2,\dots,\ell$)
the set of open conic cones in
$(T_{M_j}X)_q \simeq {\R}^{n - {\rm dim} M_j}$ that contain the point
$\tilde{\xi}_j \in (T_{M_j} X)_q \simeq {\R}^{n - {\rm dim} M_j}$.

\begin{df}
We say that an open set $G\subset (TX)_q$ is a multi-cone along $\chi$
with direction to
$p \in \left(\underset{X,1\leq j\leq \ell}{\times}T_{M_j}\iota(M_j)\right)_q$ if
$G$ is written in the form
$$
G = \underset{1\leq j \leq\ell}{\bigcap} \pi_{j,\,q}^{-1}(G_j) \qquad
G_j \in \operatorname{Cone}_{\chi,j}(p)
$$
where $\pi_{j,\,q}: (TX)_q \to (T_{M_j}X)_q$ is the canonical projection.
We denote by $\operatorname{Cone}_\chi(p)$ the set of multi-cones along $\chi$
with direction to $p$.
\end{df}

\begin{es} We now give three typical examples of multi-cones.
\begin{enumerate}
\item (Majima) Let $X={\mathbb C}^2$
with coordinates $(z_1, z_2)$ and let $M_1 = \{z_1 = 0\}$ and $M_2 = \{z_2 = 0\}$.
Then $\operatorname{Cone}_{\chi}(p)$ for $p = (0,0;\,1,1)$ is nothing but
the set of multi sectors along $Z_1 \cup Z_2$ with their direction to $(1,1)$.
\item (Takeuchi)
Let $X={\mathbb R}^3$ with coordinates $(x_1, x_2, x_3)$ and let
$M_1 = \{0\}$, $M_2 = \{x_2 = x_3 = 0\}$ and $M_3 = \{x_3 = 0\}$.
For $p = (0,0,0;\, 1,1,1) \in T_{M_1}M_2 \underset{X}{\times} T_{M_2}M_3 \underset{X}{\times} T_{M_3}X$,
it is easy to see that
a cofinal set of $\operatorname{Cone}_{\chi}(p)$ is, for example, given by
the family of the sets
$$
\{(\xi_1, \xi_2, \xi_3);\,
\vert \xi_2 \vert + \vert \xi_3 \vert < \epsilon \xi_1,\,
\vert \xi_3 \vert < \epsilon \xi_2,\,
\xi_3 > 0
\}_{\epsilon > 0}
$$
\item (Mixed)
Let $X={\mathbb R}^3$ with coordinates $(x_1, x_2, x_3)$ and let
$M_1 = \{0\}$, $M_2 = \{x_2 = 0\}$ and $M_3 = \{x_3 = 0\}$.
For $p = (0,0,0;\, 1,1,1) \in T_{M_1}(M_2 \cap M_3) \underset{X}{\times} T_{M_2}X \underset{X}{\times} T_{M_3}X$,
a cofinal set of $\operatorname{Cone}_{\chi}(p)$ is, for example, given by
the family of the sets
$$
\{(\xi_1, \xi_2, \xi_3);\,
\vert \xi_2 \vert + \vert \xi_3 \vert < \epsilon \xi_1,\,
\xi_2 > 0,\,
\xi_3 > 0
\}_{\epsilon > 0}
$$
\end{enumerate}
\end{es}

\end{section}

\begin{section}{Multi-normal cone}\label{4}

\begin{df} Let $Z$ be a subset of $X$. The multi-normal cone to $Z$ along $M_1,\ldots,M_\ell$ is the set
$$
C_\chi(Z)=\overline{\imin{\widetilde{p}}(Z)} \cap S.
$$
\end{df}

For any $q \in X$,  there exists an isomorphism
$$
\psi: X \simeq (TX)_q
$$
near $q$ and $\psi(q) = (q;\,0)$ that satisfies $\psi(M_j) = (TM_j)_q$
for any $j=1,\dots,\ell$.
The existence of such a $\psi$ is guaranteed by Proposition \ref{prop:canonical-coordinate}.

\begin{lem}\label{lem:multi cone}
Let $Z$ be a subset of $X$.
We have the following equivalence:
$p \notin C_\chi(Z)$

if and only if
there exist an open subset $\psi(q) \in U \subset (TX)_q$
and a multi-cone $G \in \operatorname{Cone}_{\chi}(p)$ such that
$$
\psi(Z) \cap G \cap U = \emptyset
$$
holds.
\end{lem}
\dim\ \ The problem is local. Hence we identify $X \simeq (TX)_q \simeq {\mathbb R}^n$ by $\psi$, and
we consider, in what follows, $\operatorname{Cone}_{\chi}(p)$ as the set of cones defined in $X$.
We may assume $q=0$.
Recall that $\hat{I}_j \subset \{1,2,\dots,n\}$ is defined as
$$
\hat{I}_j := I_j \setminus \left(\underset{I_k \varsubsetneqq I_j}{\bigcup} I_k\right).
$$
The equivalent class $\hat{I}_j$ corresponds to $T_{M_j}\iota (M_j)$ for
$j \in \{1,\dots,\ell\}$,
and set
$B := \underset{1\leq j \leq \ell}{\bigcup} I_j =
\underset{1\leq j \leq \ell}{\bigcup} \hat{I}_j
\subset \{1,2,\dots,n\}.
$
It suffices to show the following claim
$$
p \in C_\chi(Z) \iff
\text{For any $p \in U$ and $G \in \operatorname{Cone}_{\chi}(p)$,  }\,
\psi(Z) \cap G \cap U \neq \emptyset.
$$

\

\par\noindent First we will show ($\Rightarrow$):
Assume that
$$
p = (q;\, \{\xi_i\}_{i \in B})
= \underset{X, 1\leq j \leq \ell}{\times} p_j \in C_\chi(Z)
$$
where $p_j$ is a point in $T_{M_j} \iota (M_j)$.
By the definition, we have a sequence
$$
p^{(m)} =
(x_1^{(m)}, \dots, x_n^{(m)};\, t_1^{(m)}, \dots, t_\ell^{(m)}) \in
\widetilde{p}^{-1}(Z) \subset \widetilde{X}
\qquad m=1,2, \dots
$$
satisfying that $x_i^{(m)} \to \xi_i$ for $i \in B$,  $x_i^{(m)} \to 0$ for $j \notin B$,
and $t_j^{(m)} \to 0$ for any $j$.
For $j \in \{1,2,\dots,\ell\}$, let us consider the commutative diagram
$$
\begin{matrix}

&&\underset{X,1\leq j \leq \ell}{\times}T_{M_j}\iota(M_j)
& \to  & T_{M_j}X \\
&&\downarrow & & \downarrow \\
&&\varphi:\widetilde{X} & \to &    \widetilde{X}_{M_{j}} \\
&&                         & \searrow & \downarrow \\
&&                         &           & X

\end{matrix}
$$
where the top horizontal arrow is given by the composition of morphisms
$$
\underset{X,1\leq j \leq \ell}{\times}T_{M_j}\iota(M_j)
\to T_{M_j}\iota(M_j) \underset{X}{\times} M
\hookrightarrow   T_{M_j}X
$$
and $\varphi: (x_1, \dots, x_n;\, t_1, \dots, t_\ell) \to
(\hat{x}_1, \dots, \hat{x}_n;\, \hat{t})$ is defined by
$$
\begin{array}{llll}
&\hat{t} &= \underset{M_\beta \subset M_j}{\prod} t_\beta, &\\
&\hat{x}_i &= t_{J_i} x_i &\qquad (i \notin I_j),\\
&\hat{x}_i &=
\left(\underset{\{\beta \in J_i; M_j \subsetneq M_\beta\}}{\prod} t_\beta\right) x_i
&\qquad(i \in I_j).
\end{array}
$$
Here the definitions of $J_i$, etc. were given in the proof
of Proposition \ref{prop:canonical-coordinate}.
Then $\varphi(p^{(m)})$ converges to $\overline{p}_j \in T_{M_j} X$ where
$\overline{p}_j$ is the image of $p_j$ by the canonical embedding $T_{M_j} \iota (M_j) \hookrightarrow T_{M_j} X$.
This implies $\overline{p}_j \in C_{M_j} (Z)$. Therefore it follows from the definition of a usual normal cone
that, for any cone $G_j \in \operatorname{Cone}_{\chi, j}(p)$,  we have
$\widetilde{p}_{M_j}(\varphi(p^{(m)})) \in \pi_{j,\,q}^{-1}(G_j)$ for any sufficiently large $m$.
This entails that, for any multi-cone $G \in \operatorname{Cone}_{\chi}(p)$,
we have $\widetilde{p}(p^{(m)}) \in G$ for any sufficiently large $m$, in particular,
we have $Z \cap G \cap U \neq \emptyset$ for any $U$.

\

\par\noindent Let us  show converse ($\Leftarrow$):  Set, for a subset $I \subset \{1,2,\dots, n\}$,
$$
\vert x \vert_I := \sum_{i \in I} \vert x_i \vert.
$$
Note that, if $I$ is empty,  we set $\vert x \vert_I = 1$.
Let
$$
p = (q;\, \xi) =
\underset{X,1\leq j \leq \ell}{\times} p_j
\in \underset{X,1\leq j \leq \ell}{\times}T_{M_j}\iota(M_j)
$$
where $p_j = (q;\, \xi_j) \in T_{M_j} \iota(M_j)$.
By the conic actions $\tau_j(\cdot)$, we may assume either
$\vert \xi_j \vert = \vert \xi \vert_{\hat{I}_j} = 1$
or
$\vert \xi_j \vert = \vert \xi \vert_{\hat{I}_j} = 0$ for $1\leq j \leq \ell$.

Let $\{G_j^{(m)}\}_{m=1,2,\dots}$ be a cofinal set of
$\operatorname{Cone}_{\chi,j}(p)$
and $\{U^{(m)}\}_{m=1,2,\dots}$ a set of fundamental neighborhoods of $q$. We set
$$
G^{(m)} := \underset{1\leq j \leq \ell}{\bigcap} \pi_{j,\,q}^{-1} G_j^{(m)} \qquad m = 1,2, \dots.
$$
Choose points  in $X$ as
$$
q^{(m)} \in Z \cap G^{(m)} \cap U^{(m)} \subset X \qquad m = 1,2, \dots,
$$
and define a sequence in $\widetilde{X}$ by
$
p^{(m)} := (q^{(m)};\, 1,\dots, 1).
$
Clearly we have $\widetilde{p}(p^{(m)}) \in Z \cap G^{(m)} \cap U^{(m)}$.
For each $1\leq j \leq \ell$, by taking a subsequence of $\{q^{(m)}\}$, we may assume
either $\vert q^{(m)} \vert_{\hat{I}_j} \ne 0$  for every $m$ or
$\vert q^{(m)} \vert_{\hat{I}_j} = 0$ for all $m$.
We divide the set $\{1,2,\dots,\ell\}$ into two sets $J'$ and $J''$ as follows:
$$
\begin{aligned}
J' &= \left\{j \in \{1,2,\dots,l\};\,
\vert \xi_j \vert = \vert \xi \vert_{\hat{I}_j} \ne 0\right\}, \\
J'' &= \{1,2,\dots, \ell\} \setminus J'.
\end{aligned}
$$
Note that $\vert q^{(m)} \vert_{\hat{I}_j} \ne 0$ holds for $j \in J'$.
Let us determine a sequence $\kappa^{(m)} = (\kappa_1^{(m)}, \dots, \kappa_\ell^{(m)})$
of positive real numbers that satisfies the following conditions.
\begin{enumerate}
\item $\kappa^{(m)} \to 0$,
	$\kappa^{(m)}_j = \vert q^{(m)} \vert_{\hat{I}_j}$ for $j \in J'$ and
$
\displaystyle\frac{\vert q^{(m)} \vert_{\hat{I}_j}}{\kappa^{(m)}_j} \to 0
$
for $j \in J''$.
\item For any pair $\alpha,\beta \in \{1,2,\dots,\ell\}$ with
$M_\alpha \subsetneq M_\beta$,
we have
$
\displaystyle\frac{\kappa^{(m)}_\beta} {\kappa^{(m)}_\alpha} \to 0.
$
\end{enumerate}
Set, for $j \in \{1,2,\dots,\ell\}$,
$$
d_j := \max \{k; M_j = M_{j_0} \subsetneq M_{j_1} \subsetneq \dots
\subsetneq M_{j_k},\, M_{j_1},\dots, M_{j_k} \in \chi\}
$$
and
$$
J'_k := J' \cup \{j \in J'';\, d_j \le k\}.
$$
Now we construct a sequence $\kappa^{(m)}$ by induction with respect to
$k$. For this purpose, we introduce
the following condition $\operatorname{Hypo}(j, k)$.
\begin{enumerate}

\item $\kappa_j^{(m)} \to 0$.
\item For $\beta' \in J'_k$ with
with $M_{\beta'} \subsetneq M_j$, we have
$\displaystyle\frac{\kappa^{(m)}_j}{\kappa^{(m)}_{\beta'}} \to 0$
\item For $\beta \in \{1,2,\dots,\ell\}$ with $M_j \subsetneq M_\beta$, we have
$\displaystyle\frac{\vert q^{(m)} \vert_{\hat{I}_\beta}}{\kappa^{(m)}_j} \to 0$.
Further we also have
$\displaystyle\frac{\vert q^{(m)} \vert_{\hat{I}_j}}{\kappa^{(m)}_j} \to 0$
if $j \in J''$.
\end{enumerate}

Assume $k = -1$, i.e., $J'_{-1} = J'$.
We set $\kappa^{(m)}_j = \vert q^{(m)} \vert_{\hat{I}_j}$
for $j \in J'$.
It is easy to see that the hypothesis $\operatorname{Hypo}(j, k)$ is satisfied
for any $j \in J'_{-1}$.
Indeed, as $\xi_j \ne 0$ and $q^{(m)} \in \pi_{j,\,q}^{-1}(G^{(m)}_j)$ hold,
there exists a sequence $\epsilon^{(m)} \to 0$
such that $\vert q^{(m)} \vert_{\hat{I}_\beta}
\le \epsilon^{(m)} \vert q^{(m)} \vert_{\hat{I}_j}$
for $\beta$ with $M_j \subsetneq M_\beta$. This implies 3.~of
$\operatorname{Hypo}(j, k)$. By the same reason, as $\beta' \in J'$,
2.~of $\operatorname{Hypo}(j, k)$ also holds.

Assume that we have constructed $\kappa^{(m)}_j$ for $j \in J'_k$ and
$\operatorname{Hypo}(j, k)$ holds for any $j \in J'_k$.
Let $j \in J''$ with $d_j = k+1$. Then we can determine
$\kappa^{(m)}_j$ so that the hypothesis $\operatorname{Hypo}(j, k)$ is satisfied
in the following way.
First note that
$\chi^*=\{M_{\beta'} \in \chi;\, \beta'  \in J'_k,\, M_{\beta'} \subsetneq M_j\}$
is a totally ordered set with respect to ``$\subset$'', in particular,
we have the maximal submanifold $M_{j^*} \in \chi^*$. If we can determine
$\kappa^{(m)}_j$ satisfying
$\displaystyle\frac{\kappa^{(m)}_j}{\kappa^{(m)}_{j^*}} \to 0$, then
$\displaystyle\frac{\kappa^{(m)}_j}{\kappa^{(m)}_{\beta'}} \to 0$ also holds
for any $M_{\beta'} \in \chi^*$ by induction hypothesis.
It follows from induction hypothesis again that
for any $\beta \in \{1,2,\dots,\ell\}$ with $M_{j^*} \subsetneq M_{\beta}$,
we have
$\displaystyle\frac{\vert q^{(m)} \vert_{\hat{I}_\beta}}{\kappa^{(m)}_{j^*}} \to 0$.
Therefore we can find $\kappa^{(m)}_j$ such that
$\displaystyle\frac{\kappa^{(m)}_j}{\kappa^{(m)}_{j^*}} \to 0$ and
$\displaystyle\frac{\vert q^{(m)} \vert_{\hat{I}_\beta}}{\kappa_j^{(m)}} \to 0$
for $\beta \in \{1,2,\dots,\ell\}$ with $M_j \subset M_{\beta}$ (note that
this contains the case $\beta = j)$.

By repeating the same procedure,
we obtain $\kappa^{(m)}_j$ for any $j \in J''$ with $d_j = k+1$.
As $I_j \cap I_{j'} = \emptyset$ for $j$ and $j'$ with $d_j = d_{j'}$,
we can conclude that
$\operatorname{Hypo}(j, k+1)$ holds for any $j \in J'_{k+1}$.
Hence we have obtained $\kappa_j^{(m)}$ for all $j \in \{1,2,\dots,\ell\}$.

Let us define points in $\widetilde{X}$ by
$$
\overline{p}^{(m)} := \left(\underset{j \in \{1,2,\dots,\ell\}}{\prod}
\tau_j \left( \frac{1}{\kappa_j^{(m)}}\right)\right) p^{(m)}.
$$
Note that $\widetilde{p}(\overline{p}^{(m)}) \in Z$ still holds.
Then the value of $j$-th coordinate $t_j$ of $\overline{p}^{(m)}$ is given by
that of
$$
\underset{ j \in \{1,2\dots,\ell\}}{\prod}
\mu_j\left( \frac{1}{\kappa^{(m)}_j}\right)
\underset{\beta \in J_{\supsetneq M_j}}{\prod} \mu_\beta
\left(\kappa^{(m)}_j\right) p^{(m)},
$$
which is equal to that of
$$
\mu_j\left( \frac{1}{\kappa^{(m)}_j}\right)
\underset{j \in J_{\supsetneq M_\beta}}{\prod} \mu_j
\left( \kappa^{(m)}_\beta\right)p^{(m)} =
\mu_j\left( \frac{1}{\kappa^{(m)}_j}\right)
\underset{\beta \in J_{\subsetneq M_j}}{\prod} \mu_j
\left(\kappa^{(m)}_\beta\right) p^{(m)}.
$$
Note that $J_{\subsetneq M_j}$ consists of at most 1 element.
If  $J_{\subsetneq M_j}$ is empty, then the
$j$-th component is
$
\kappa_j^{(m)}
$
which clearly tends to $0$ when $m \to \infty$.
If  $J_{\subsetneq M_j} = \{\beta\}$ for some $\beta$, then
it is given by
$
\displaystyle\frac{ \kappa^{(m)}_j }{\kappa^{(m)}_\beta}
$
which also tends to $0$ by the construction of $\kappa^{(m)}$.
As a result, we have $\overline{p}^{(m)} \to p$.
This completes the proof.
\qed

Given the family $\chi = \{M_1, \dots ,M_{\ell}\}$ and a sub-family $\chi_k := \{M_{j_1} ,\dots ,M_{j_k}\}$,
we have the natural maps
$$
\underset{X,1\leq j\leq\ell}{\times}T_{M_j}\iota_\chi(M_j) \hookleftarrow \left(\underset{X,1\leq i\leq k}{\times}T_{M_{j_i}}\iota_\chi(M_{j_i})\right) \underset{X}{\times} M \hookrightarrow \underset{X,1\leq i\leq k}{\times}T_{M_{j_i}}\iota_{\chi_k}(M_{j_i}),
$$
where $M=\bigcap_{j=1}^\ell M_j$. We set for short
$$
\underset{M,1\leq i\leq k}{\times}T_{M_{j_i}}\iota_\chi(M_{j_i}) := \left(\underset{X,1\leq i\leq k}{\times}T_{M_{j_i}}\iota_\chi(M_{j_i})\right) \underset{X}{\times} M.
$$

\begin{cor} \label{cor: restriction multi-cones}
Let $k \le \ell$ and $\{j_1,\dots,j_k\}$ be a subset of $\{1,2,\dots,\ell\}$. Set $\chi_k=\{M_{j_1},\ldots,M_{j_k}\}$.

Let $Z$ be a subset of $X$. Then we have
$$
C_\chi (Z) \cap \underset{M,1\leq i\leq k}{\times}T_{M_{j_i}}\iota_\chi(M_{j_i})
 = C_{\chi_k} (Z) \cap \underset{M,1\leq i\leq k}{\times}T_{M_{j_i}}\iota_\chi(M_{j_i}).
$$
\end{cor}
\dim\ \ Let us prove $\subseteq$. Suppose that $p \in \underset{M,1\leq i\leq k}{\times}T_{M_{j_i}}\iota_\chi(M_{j_i})$ does not belong to $C_{\chi_k}(Z)$. By Lemma \ref{lem:multi cone} there exists an open subset $\psi(q) \in U \subset (TM)_q$
and a multi-cone $G' \in \operatorname{Cone}_{\chi_k}(p)$ such that
$
\psi(Z) \cap G' \cap U = \emptyset
$
holds. For $\alpha \notin \{j_1,\ldots,j_k\}$, set
$G_\alpha = \{(T_{M_\alpha} X)_q\}$. Hence $G:=\left(\underset{\alpha \notin \{j_1,\dots,j_k\}}{\bigcap} \imin {\pi_\alpha} G_\alpha\right) \cap G' = G' \in \operatorname{Cone}_{\chi}(p)$ and $\psi(Z) \cap G \cap U = \emptyset$. By Lemma \ref{lem:multi cone} $p \notin C_\chi(Z)$.

\

Let us prove $\supseteq$. Suppose that $p \in \underset{M,1\leq i\leq k}{\times}T_{M_{j_i}}\iota_\chi(M_{j_i})$ does not belong to $C_\chi(Z)$. By Lemma \ref{lem:multi cone} there exists an open subset $\psi(q) \in U \subset (TM)_q$
and a multi-cone $G \in \operatorname{Cone}_{\chi}(p)$ such that
$
\psi(Z) \cap G \cap U = \emptyset
$
holds. For $\alpha \notin \{1,\ldots,k\}$, $p_{j_\alpha}=(q;0)$ and we have
$\operatorname{Cone}_{\chi, j_\alpha}(p) = \{(T_{M_{j_\alpha}} X)_q\}$. Hence $G':=G \cap \left(\underset{X,1\leq j\leq k}{\times}T_{M_j}\iota_{\chi_k}(M_j)\right) \in \operatorname{Cone}_{\chi_k}(p)$ and $\psi(Z) \cap G' \cap U = \emptyset$. By Lemma \ref{lem:multi cone} $p \notin C_{\chi_k}(Z)$.
\qed

\begin{df} (i) Let $S$ be a subset of $X$. We set $\RP_i S=\mu_i(S,\RP).$
If $U \in \op(X)$, then $\RP_i U \in \op(X)$ since $\mu_i$ is open for each $i=1,\ldots,\ell$.

(ii)  Let $S$ be a subset of $X$. Let $J \subset \{1,\ldots,\ell\}$. We set $$\RP_JS=\RP_{i_1}\cdots\RP_{i_k}S=\mu_{i_1}(\cdots\mu_{i_k}(S,\RP),\dots,\RP), \ \ i_1,\ldots i_k \in J.$$
We set $(\RP)^\ell S=\RP_{\{1,\ldots,\ell\}}S=\mu(S,(\RP)^\ell).$
If $U \in \op(X)$, then $\RP_J U \in \op(X)$ since $\mu_i$ is open for each $i \in \{1,\ldots,\ell\}$.

(iii) Let $S$ be a subset of $X$. We say that $S$ is $(\RP)^\ell$-conic ($\ell$-conic for short) if
$S=(\RP)^\ell S$. In other words, $S$ is invariant by the action of
$\mu_i$, $i=1,\dots,\ell$.

\end{df}

\begin{df} (i)  We say that a subset $S$ of $X$ is $\RP_i$-connected if
$S \cap \RP_i x$ is connected for each $x \in S$.

(ii) We say that a subset $S$ of $X$ is $(\RP)^\ell$-connected if
there exists a permutation $\sigma:\{1,\ldots,\ell\}\to\{1,\ldots,\ell\}$ such that
\begin{equation*}
 \begin{cases}
 \text{$S$ is $\RP_{\sigma(1)}$-connected,}\\
 \text{$\RP_{\sigma(1)}S$ is $\RP_{\sigma(2)}$-connected,}\\
 \text{\ \ $\vdots$}\\
 \text{$\RP_{\sigma(1)}\cdots\RP_{\sigma(\ell-1)}S$ is $\RP_{\sigma(\ell)}$-connected}.
  \end{cases}
\end{equation*}
\end{df}

The proof of the following is almost the same of that of Proposition 4.1.3 of \cite{KS90}, and we shall not repeat it.

\begin{prop}\label{m4.1.3} Let $V$ be a $(\RP)^\ell$-conic open subset of $S$.
\begin{itemize}
\item[(i)] Let $W$ be an open neighborhood of $V$ in $\widetilde{X}$, and let $U=\widetilde{p}(W \cap \Omega)$. Then $V \cap C_\chi(X \setminus U)=\emptyset$.
\item[(ii)] Let $U$ be an open subset of $X$ such that $V \cap C_\chi(X \setminus U)=\emptyset$. Then $\imin {\widetilde{p}}(U) \cup V$ is an open neighborhood of $V$ in $\overline{\Omega}$.
\end{itemize}
\end{prop}

\begin{prop}\label{m4.1.4} Let $V$ be a $(\RP)^\ell$-conic subanalytic open subset of $S$. Then any subanalytic neighborhood $W$
of $V$ in $\widetilde{X}$ contains $\widetilde{W} \in
\op(\widetilde{X}_{sa})$ such that:
\begin{equation} \label{mpitilde}
\begin{cases}

\text{(i) $\widetilde{W} \cap \Omega$ is $(\RP)^\ell$-connected,} \\

\text{(ii) $\RP_1 \cdots \RP_{\ell}(\widetilde{W} \cap \Omega)=\imin {\widetilde{p}}(\widetilde{p}(W \cap \Omega))$ is subanalytic in $X$.}
\end{cases}
\end{equation}
\end{prop}
\dim\ \ Let $\chi_{i}=\{M_1,\dots,M_{i-1},M_{i+1},\dots,M_\ell\}$ let $\widetilde{X}_{i}$ be the normal deformation of $X$ with respect to $\chi_i$ and let $p_{i}:\widetilde{X} \to \widetilde{X}_{i}$. Let $V$ be a conic subanalytic open subset of $S$. \\

(i) We first prove that any subanalytic neighborhood $W$
of $V$ in $\widetilde{X}$ contains $\widetilde{W} \in
\op_{sa}(\widetilde{X})$ such that:
\begin{equation} \label{m1}
\begin{cases}
\text{(i) the fibers of  $\widetilde{p}_{i}:\widetilde{W} \cap
\{t_i>0\} \to
\widetilde{X}_{i}$ are connected,} \\
\text{(ii) $\widetilde{p}_{i}(\widetilde{W} \cap \{t_i>0\})$ is
subanalytic in $X$.}
\end{cases}
\end{equation}
Let $W \in
\op(\widetilde{X}_{sa})$ be a
neighborhood of $V$. Up to shrink $W$ we
may suppose $V=W \cap \{t_i=0\}$. Set
$X'=\widetilde{X} \setminus (M_i \times
\R)$, $S=X'/\RP$. Then $\alpha:X' \to S$
is an $\RP$-bundle and $\varphi: S \to \widetilde{X}_i$
is proper. Consider a continuous
subanalytic section of $\{t_i=0\} \to \{t_i=0\}/\RP$ (the $i$-th component of the action), extend it to a continuous subanalytic section $\sigma$ of $X' \to S$
and set $$ W'=\bigcup_{x \in
\alpha(W)}W'_x \cap \{t_1>0\}, $$ where $W'_x$
denotes the connected component of
$\imin\alpha(x)\cap W$ containing
$\sigma(x)$. By construction, the fibers
of $W' \to S$ are connected and $W'$ is an
open neighborhood of $V \setminus M_i$. Let
$$ W''=\bigcup_{x \in M_i \cap W}W''_x \cap \{t_i>0\}, $$
where $W''_x$ denotes the connected
component of $(\{x\} \times \R) \cap W$
intersecting $M$. Up to shrink $W''$,
$\widetilde{W}=W' \cup W'' \cup (W
\setminus \{t_i>0\})$ is an open neighborhood
of $V$ and satisfies \eqref{m1}(i).

 Let us see that $\widetilde{W}$ is
subanalytic and satisfies \eqref{m1} (ii). We may
reduce to the case $X=\R^n$, $M_i=\{0\}
\times \R^{n-m} \subset X$,
$\widetilde{X}=\R^{n+1}$, $\{t_i=0\}=\R^n
\times \{0\} \subset \widetilde{X}$,
$\{t_i>0\}=\R^n \times \RP \subset
\widetilde{X}$. So that $(x',x'',t)
\cdot c \mapsto (cx',x'',\imin c t)$ is
the action of $\RP$ on $\widetilde{X}$
and
 $\widetilde{p}:\{t_i>0\} \to X$ is the map
 $(x',x'',t) \mapsto
(tx',x'')$. In this situation $X'=\R^{n+1}
\setminus (M_i \times \R) \simeq
\mathbb{S}^{\ell-1} \times \RP \times
\R^{n-m} \times \R$. Moreover $X'/\RP
\simeq \mathbb{S}^{\ell-1} \times
\R^{n-\ell} \times \R$, indeed $$ X'=S
\times \RP =\{(ci(\vartheta),x'',s\imin
c),\;(\vartheta,x'',s) \in S,\; c \in \RP\},
$$ where $i:\mathbb{S}^{n-1}
\hookrightarrow \R^n$ denotes the
embedding. The
section $\sigma:S \iso S \times \{1\} \subset X'$ is a
globally subanalytic subset of $X'$. Let
us consider the globally subanalytic (even
semialgebraic) homeomorphism $\psi:\{t_i>0\}
\to \{t_i>0\}$ defined by
$\psi(x',x'',t)=(tx',x'',\imin t)$. Then
$\pi \circ \psi=\widetilde{p}_{i}$, where
$\pi:\R^n \times \RP \to \R^n$ is the
projection. The set $\psi(W \cap \{t_i>0\})$
is still subanalytic.  Let $p \in \R^n$.
Then $\imin \pi(p) \cap \psi(W \cap X'
\cap \{t_i>0\})$ is a disjoint union of
intervals. Let us consider the interval
$(m(p),M(p))$, $m(p)<M(p) \in \R \cup \{\pm\infty\}$
intersecting $\psi(\sigma\cap\Omega)$. Then
$\psi(W')=\{(p,r) \in \psi(W \cap X' \cap
\Omega) ;\; m(p)<r<M(p)\}$.

The set $\psi(W'\cap\{t_i>0\})$ is open subanalytic (it is a consequence of Proposition 1.2, Chapter 6 of \cite{VD98}).
Moreover, up to shrink $W$ we may suppose
that $\widetilde{p}_{i}(W' \cap
\{t_i>0\})=\widetilde{p}(\sigma \cap W \cap \{t_i>0\})$
which is subanalytic. Indeed, since we are working in a local chart, we may assume that $W \cap \sigma$ is globally subanalytic. Then $\RP (W' \cap \{t_i>0\}) = \imin {p_i}(\widetilde{p}_{i}(W' \cap
\{t_i>0\}))$ is subanalytic. Let $x'' \in
\R^{n-m}$. Then $(\{x''\} \times \R)
\cap W'' $ is a disjoint union of (bounded, up to shrink $W$)
intervals. Let us consider the interval
$(m(x''),M(x''))$, $m(x'')<M(x'') \in \R$
containing $0$. Then $W''=\{r \in W \cap
(M_i \times \R) ;\; m(x'')<r<M(x'')\}$.

The set $W''$ is subanalytic (it is a consequence of Proposition 1.2, Chapter 6 of \cite{VD98}).
Moreover (up to shrink $W''$) $\widetilde{p}(W'' \cap
\Omega)=W'' \cap M_i$, which is subanalytic.\\

(ii) Let us find $\widetilde{W}$ satisfying \eqref{mpitilde}. Let us argue by induction on $\ell$. The case $\ell=1$ follows from (i). Let us treat the general case. Set $S_\ell=\{t_\ell=0\}$, $\Omega_\ell=\{t_1,\dots,t_{\ell-1}>0\}$. By the induction hypothesis we can find a subanalytic neighborhood $W'$ of $V$ which is $(\RP)^{\ell-1}$-connected and such that $\RP_1 \cdots \RP_{\ell-1}(W' \cap \Omega_\ell)$ is subanalytic. Let us apply (i) with $\RP_1 \cdots \RP_{\ell-1}(W' \cap S_\ell \cap \Omega_\ell)$ instead of $V$ and $\RP_1 \cdots \RP_{\ell-1}(W' \cap \Omega_\ell)$ instead of $W$. Then we find $W''$ subanalytic $(\RP)^{\ell-1}$-conic and $\RP_\ell$-connected satisfying \eqref{m1}. The set
$\widetilde{W}=(W' \cap W'') \cup (W
\setminus \Omega)$ is an open neighborhood
of $V$ and satisfies \eqref{mpitilde}. \qed

\end{section}

\begin{section}{Morphisms between multi-normal deformations}\label{5}

Let $X$ and $Y$ be real analytic manifolds of dimension
$n$ and $m$ respectively
and let
$\chi^M=\{M_j\}_{j=1}^\ell$, $\chi^N=\{N_j\}_{j=1}^\ell$ be
families of smooth closed submanifolds of $X$ and $Y$
satisfying \eqref{eq:geometric-condition} respectively.
We define the $I^M_{j}$ as in 2. of
Proposition \ref{prop:canonical-coordinate}
and $J^M_{i}$ as in \eqref{eq:J_k}
(and similarly for $I^N_j$ and $J^N_i$).

Let $f:X \to Y$ be a morphism of real analytic manifolds such that
$f(M_j) \subseteq N_j$, $j=1,\ldots,\ell$.

We want to extend $f$ to a morphism $\widetilde{f}:\widetilde{X} \to \widetilde{Y}$. This is done by repeatedly employing the the usual construction
of a morphism between normal deformations,
i.e. we extend $f$ to  $\widetilde{f}_1:\widetilde{X}_{M_1} \to \widetilde{Y}_{N_1}$,
then we extend $\widetilde{f}_1$ to $\widetilde{f}_{1,2}:\widetilde{X}_{M_1,M_2} \to \widetilde{Y}_{N_1,N_2}$.
Then we can define recursively $\widetilde{f}:\widetilde{X} \to \widetilde{Y}$
by extending the morphism $\widetilde{f}_{1,\ldots,\ell-1}:\widetilde{X}_{M_1,\ldots,M_{\ell-1}} \to \widetilde{Y}_{N_1,\ldots,N_{\ell-1}}$
to the normal deformations with respect to $M_\ell$ and $N_\ell$ respectively.

In a local coordinate system set
$$
f(x_1,\ldots,x_n)=(f_1(x_1,\ldots,x_n),\ldots,f_m(x_1,\ldots,x_n)).
$$

Then, outside of $t_1t_2\ldots t_\ell = 0$,
we can define a morphism $\widetilde{f}:\widetilde{X}\to\widetilde{Y}$ $$
(x_1,\dots,x_n;\,t_1,\dots,t_\ell) \to
(y_1,\dots,y_m;\,\hat{t}_1,\dots,\hat{t}_\ell)
$$
by setting
$$
\left\{
\begin{aligned}
&\hat{t}_{J^N_k}y_k=f_k(t_{J^M_1}x_1,\ldots,t_{J^M_n}x_n)
\qquad (k = 1,2,\dots,m),\\
&\hat{t}_j = t_j \qquad (j = 1,2,\dots,\ell).
\end{aligned}
\right.
$$

In order to understand the restriction of $\widetilde{f}$ to $t_1t_2\cdots t_\ell=0$, we follow the notations of the proof of Proposition \ref{prop:zero section}. For any $k=1,2,\dots,m$, we set
$$
I^N(k) := \underset{k \in I^N_j}{\bigcap} I^N_j = \underset{j \in J^N_k}
{\bigcap} I^N_j.
$$
Let $k \in \underset{1 \le j \le \ell}{\bigcup} I^N_j$, and let
$j(k) \in \{1,2,\dots,\ell\}$ such that $I^N(k) = I^N_{j(k)}$.
Then, for any $j \in J^N_k$, we get $f(M_j) \subseteq N_{j(k)}$
because of $f(M_j) \subset N_j \subset N_{j(k)}$. We set
$$
I=\bigcup_{j \in J^N_k} I^M_j,\,\, M_I=\bigcap_{j \in J^N_k} M_j.
$$
By expanding $f_k(x_1,\dots,x_n)$ along the submanifold $M_I$,
we obtain
$$
f_k(x_1,\dots,x_n) =
\sum_{i \in I}\left.\frac{\partial f_k}{\partial x_i}\right|_{M_I} x_i
+ \frac{1}{2}\sum_{i_1,i_2 \in I}
\left.
\frac{\partial^2 f_k}{\partial x_{i_1}\partial x_{i_2}}\right|_{M_I}
x_{i_1}x_{i_2}
+ \dots,
$$
as $f_k\vert_{M_I} = 0$ holds due to $f(M_I) \subset N_{j(k)}$.
Then we get, on $t_1\dots t_\ell \ne 0$,

\begin{equation}{\label{eq:morphism-expansions}}
y_k =
\sum_{i \in I}\left.\frac{\partial f_k}{\partial x_i}\right|_{M_I}
\frac{t_{J^M_i}}{t_{J^N_k}}x_i
+ \frac{1}{2}\sum_{i_1,i_2 \in I}
\left.
\frac{\partial^2 f_k}{\partial x_{i_1}\partial x_{i_2}}\right|_{M_I}
\frac{t_{J^M_{i_1}}t_{J^M_{i_2}}}{t_{J^N_k}}x_{i_1}x_{i_2}
+ \dots.
\end{equation}
Let $i_1, \dots, i_p$ be a sequence $(i_1, \dots, i_p \in I)$.
\begin{itemize}
\item[(i)] If there exists $j \in J^N_k$ such that
$\{i_1,\ldots,i_p\} \cap I^M_j=\emptyset$, as
$f_k \vert_{M_j} = 0$ and derivatives
$\displaystyle\frac{\partial}{\partial x_{i_1}}$,
$\ldots$,
$\displaystyle\frac{\partial}{\partial x_{i_p}}$
are tangent to $M_j$, we have
$$
\left.{\partial^p f_k \over \partial x_{i_1} \ldots \partial x_{i_p}}\right|_{M_I}=0.
$$
\item[(ii)] Suppose that $\{i_1, \ldots, i_p\} \bigcap I^M_j \ne \emptyset$
holds for each $j \in J^N_k$. This implies that
$I^M_j$ contains some $i_q$ in $\{i_1,\ldots, i_p\}$,
which is equivalent to saying that
$j \in J^M_{i_q}$. Therefore any $j \in J^N_k$ belongs to
$J^M_{i_q}$ with some $q \in \{1,\dots,p\}$, from which we obtain
$$
J^N_k \subset J^M_{i_1} \cup \cdots \cup J^M_{i_p}.
$$
Now we have two cases.
    \begin{itemize}
    \item[(a)] If some pair of $J^M_{i_1}$, $\dots$, $J^M_{i_p}$
is not disjoint or if $J^N_k \subsetneqq J^M_{i_1} \sqcup \cdots \sqcup J^M_{i_p}$,
then $t_{J^M_{i_1}} \cdots t_{J^M_{i_p}} \imin {t_{J^N_k}} \to 0$
when $t \to 0$.
    \item[(b)] If $J^N_k = J^M_{i_1} \sqcup \cdots \sqcup J^M_{i_p}$ holds,
	    then the term with its indices $i_1,\dots,i_p$
	    in (\ref{eq:morphism-expansions})
	    becomes, by letting $t$ to $0$,
         $$
 \left.\frac{1}{p!}{\partial^p f_k \over \partial x_{i_1} \cdots \partial x_{i_p}}\right|_{M_I}x_{i_1} \cdots x_{i_p}.
         $$
     \end{itemize}
\end{itemize}
From these observations, the morphism $\tilde{f}$ is
described by, on $t_1= \dots = t_\ell = 0$,
         $$
	 y_k = \sum_{J^M_{i_1} \sqcup \cdots \sqcup J^M_{i_p} = J^N_k} \left.\frac{1}{p!}{\partial^p f_k \over \partial x_{i_1} \cdots \partial x_{i_p}}\right|_{M}x_{i_1} \cdots x_{i_p}
 \qquad (k\in \underset{1 \le j \le \ell}{\bigcup} I^N_j).
         $$
Here $M:=M_1 \cap \dots \cap M_\ell$.  Note that if there is no $\{i_1, \dots, i_p\}$ with
$J^N_k = J^M_{i_1} \sqcup \cdots \sqcup J^M_{i_p}$, then we set $y_k: = 0$.

Let us study the condition $J^N_k = J^M_{i_1} \sqcup \cdots \sqcup J^M_{i_p}$.
For this purpose, we introduce two definitions.
Let $k \in \underset{1 \le j \le \ell}{\bigcup} I^N_j$ and set
$$
\sup{}_\subset J^N_k := \left\{j \in J^N_k;\,
\text{$M_j$ is a maximal submanifold in
$\{M_\beta\}_{\beta \in J^N_k}$}\right\}
$$
and
$$
\inf{}_\subset J^N_k := \left\{j \in J^N_k;\,
\text{$M_j$ is a minimal submanifold
in $\{M_\beta\}_{\beta \in J^N_k}$}\right\}.
$$

Note that $\#\inf_\subset J^N_k \le \#\sup_\subset J^N_k$ holds.

For example, let us consider 3-closed submanifolds $\{M_1, M_2, M_3\}$
in $X$ and $\{N_1, N_2, N_3\}$ in $Y$. Let $f: X \to Y$ be
a morphism satisfying $f(M_j) \subset N_j$ ($j=1,2,3$). We assume that
$N_3 \subset N_2 \subset N_1$, $M_3 \subset M_1$, $M_3 \subset M_2$,
$M_1$ and $M_2$ intersect transversely. Then, for $k \in I^N_1$,
$J^N_{k} = \{1,2,3\}$ and
$\sup{}_\subset J^N_k := \{1,2\}$ and
$\inf{}_\subset J^N_k := \{3\}$.

\begin{lem}
Let $k \in \underset{1 \le j \le \ell}{\bigcup} I^N_j$ and $\{i_1, \dots,i_p\}$ a subset of $\underset{1 \le j \le \ell}{\bigcup}I^M_j$.
Then
$J^N_k = J^M_{i_1} \sqcup \cdots \sqcup J^M_{i_p}$ holds if and only if
we have
$$
p= \#\inf{}_\subset J^N_k = \#\sup{}_\subset J^N_k
$$
and there exists
a bijection $\sigma: \{1,2\dots,p\} \to \sup_\subset J^N_k$ satisfying
$$
i_\alpha \in \hat{I}^M_{\sigma(\alpha)} \qquad (\alpha \in \{1,2,\dots,p\}).
$$
\end{lem}
\dim \ \
We first show the claim that
if $\#\inf_\subset J^N_k < \#\sup_\subset J^N_k$,
then there exists no $\{i_1, \dots, i_p\}$ with
$J^N_k = J^M_{i_1} \sqcup \cdots \sqcup J^M_{i_p}$.
Assume that there exists $\{i_1, \dots, i_p\}$ with
$J^N_k = J^M_{i_1} \sqcup \cdots \sqcup J^M_{i_p}$.
Then it follows from  $\#\inf_\subset J^N_k < \#\sup_\subset J^N_k$ that we can
find indices $j$, $j'$, $j''$ in $J^N_k$ satisfying $I^M_{j'} \subset I^M_{j}$,
$I^M_{j''} \subset I^M_{j}$ and $I^M_{j'} \cap I^M_{j''} = \emptyset$.
By the assumption, there exist $\alpha'$ and $\alpha''$ in $\{1,2,\dots,p\}$
such that $j' \in J^M_{i_{\alpha'}}$ and $j'' \in J^M_{i_{\alpha''}}$.
As $I^M_{j'} \cap I^M_{j''} = \emptyset$, we have $\alpha' \ne \alpha''$.
On the other hand,  $I^M_{j'} \subset I^M_{j}$ and $I^M_{j''} \subset I^M_{j}$
implies $j \in J^M_{i_{\alpha'}}$ and $j \in J^M_{i_{\alpha''}}$, which contradicts
that $J^M_{i_{\alpha'}}$ and $J^M_{i_{\alpha''}}$ are disjoint. Hence we have obtained
the claim.

\medskip

In what follows, we assume
$$
\#\inf{}_\subset J^N_k = \#\sup{}_\subset J^N_k.
$$
Suppose that $\{i_1, \dots, i_p\}$ satisfies the condition
$J^N_k = J^M_{i_1} \sqcup \cdots \sqcup J^M_{i_p}$.
Let $j \in \sup{}_\subset J^N_k$. Then some $J^M_{i_q}$ ($q \in \{1,2,\dots,p\}$)
contains $j$, which implies $i_q \in I^M_j$.
Further we can show that
$i_q$ belongs to $\hat{I}^M_j$. If $i_q \in I^M_j \setminus \hat{I}^M_j$,
then there exists $j'$ with $i_q \in I^M_{j'}$ and
$M_j \subsetneq M_{j'}$ by the definition of $\hat{I}^M_j$,
from which we have $j' \in J^M_{i_q} \subset J^N_k$. This contradicts
$j \in \sup{}_{\subset} J^N_k$. Hence $i_q \in \hat{I}^M_j$ and
we have obtained that the set $\{i_1, \dots, i_p\}$
contains at least one index
that belongs to $\hat{I}^M_j$ for any $j \in \sup{}_{\subset} J^N_k$.
Further two or more indices in $\hat{I}^M_j$ cannot belongs to
$\{i_1, \dots, i_p\}$ at the same time because any pair of
$J^M_{i_1}$, $\cdots$, $J^M_{i_p}$ is disjoint.

Now we show $\{i_1, \dots, i_p\}$ consists of only these indices.
Let $i$ be an element in $\{i_1, \dots, i_p\}$.
Choosing $j' \in J^M_i$, we can find $j \in \sup{}_{\subset} J^N_k$
with $M_{j'} \subset M_j$. Then, by the above argument,
there exists $i_q \in \hat{I}^M_j$ which belongs to
$\{i_1, \dots, i_p\}$.
As $i_q \in I^M_j \subset I^M_{j'}$,  we have $j' \in J^M_{i_q}$,
which implies $J^M_{i_q} \cap J^M_{i} \ne \emptyset$.
Since each pair is disjoint, we have $i = i_q$.

Therefore we can find a bijection $\sigma: \{1,2,\dots,p\} \to \sup_{\subset} J^N_k$
such that $i_\alpha \in \hat{I}^M_{\sigma(\alpha)}$ ($\alpha \in \{1,2,\dots,p\}$).

Conversely if such a $\sigma$ exists, then
$J^N_k = J^M_{i_1} \sqcup \cdots \sqcup J^M_{i_p}$ easily follows
from $\#\inf{}_\subset J^N_k = \#\sup{}_\subset J^N_k$.
\qed

By this lemma, for $k \in \underset{1 \le j \le \ell}{\bigcup}I^N_j$ with
$\#\inf{}_\subset J^N_k = \#\sup{}_\subset J^N_k$, we
have
\begin{equation}{\label{eq:multi-morphism-zero}}
y_k = \sum_{i_1 \in \hat{I}^M_{j_1},\, \dots,\, i_p \in \hat{I}^M_{j_p}}
 \left.{\partial^p f_k \over \partial x_{i_1} \cdots \partial x_{i_p}}\right|_{M}x_{i_1} \cdots x_{i_p}
\end{equation}
where $p = \#\sup{}_\subset J^N_k$ and $\{j_1, \dots, j_p\} = \sup{}_\subset J^N_k$.
And we have $y_k = 0$
for $k \in \underset{1 \le j \le \ell}{\bigcup} I^N_j$
with $\#\inf{}_\subset J^N_k \ne \#\sup{}_\subset J^N_k$.

Let $j \in \{1,2,\dots,l\}$.
Then, for any $k$ and $k'$ in $\hat{I}^N_j$, as
$J^N_k = J^N_{k'}$ holds, we have
$\inf{}_\subset J^N_k = \inf{}_\subset J^N_{k'}$ and
$\sup{}_\subset J^N_k = \sup{}_\subset J^N_{k'}$.
This implies that $\inf{}_\subset J^N_k$ and
$\sup{}_\subset J^N_k$ do not depend on a choice of
$k$ that belongs to $\hat{I}^N_j$.
We denote them
by $\inf{}_\subset \hat{I}^N_j$ and
$\sup{}_\subset \hat{I}^N_j$ respectively for short.

For $j \in \{1,2,\dots,l\}$
with $\#\inf{}_\subset \hat{I}^N_j= \#\sup{}_\subset \hat{I}^N_j$,
by considering
the equation (\ref{eq:multi-morphism-zero})
for indices $k \in \hat{I}^N_j$, we have
$$
\varphi_j:
\left(\underset{X,\, \beta \in \sup_\subset \hat{I}^N_j}{\times} T_{M_\beta}
\iota(M_\beta)\right)
\underset{X}{\times} M
\to
T_{N_{j}}\iota(N_{j})
\underset{Y}{\times} N.
$$
where $N=N_1 \cap \dots \cap N_\ell$.
Although $\varphi_j$ is not a map between vector bundles,
we still have
$$
\varphi_j\left(\tau^X_{j_1}(\lambda_{j_1})\ldots\tau^X_{j_p}(\lambda_{j_p}) p\right)
= \tau^Y_{j}(\lambda_{j_1}\ldots\lambda_{j_p})\varphi_j(p)
$$
where $\{j_1, \dots, j_p\} = \sup{}_\subset \hat{I}^N_j$ and
$\tau_\beta^X$ and $\tau_\beta^Y$
denote the action $\tau_\beta$ on each spaces $X$ and $Y$ respectively.
This implies that the image of a multi-conic set is conic and
the inverse image of a conic set is multi-conic.

We denote by the same symbol $\varphi_j$ the map
$$
\underset{X,\, 1 \le \beta \le \ell}{\times} T_{M_\beta}
\iota(M_\beta)
\to
\left(\underset{X,\, \beta \in \sup_\subset \hat{I}^N_j}{\times} T_{M_\beta}
\iota(M_\beta)\right)
\underset{X}{\times} M
\overset{\varphi_j}{\to}
T_{N_{j}}\iota(N_{j})
\underset{Y}{\times} N.
$$
Then we conclude that the restriction of $\tilde{f}$ to $\{t_1 = \dots = t_\ell = 0\}$ is a sum of $\varphi_j$
for $j=1,2.\dots,l$ with
$\#\inf_\subset \hat{I}^N_j = \#\sup_\subset \hat{I}^N_j$,
and each $\varphi_j$ is given by ($\ref{eq:multi-morphism-zero}$).
\

\begin{es} Set $Y=\CC^2$, let $X=\{(z_1, z_2, \xi_1, \xi_2) \in \CC^2 \times \mathbb{P}_\CC^1,\;
        \xi_2z_1 = \xi_1 z_2\}$ and let
\begin{eqnarray*}
\pi:X & \to & \CC^2 \\
\{(z_1,z_2,\xi_1,\xi_2)\} & \mapsto & (z_1,z_2)
\end{eqnarray*}
be the desingularization map.  Let $N_1=\{0\}$, $N_2=\{z_2=0\}$, $M_1=\imin \pi(0)$, $M_2=\{\xi_2=0\}$. Locally on $X$, for example on $U_1:=\{\xi_1 \neq 0\}$, set $\lambda=\displaystyle{\xi_2\over\xi_1}$, then $z_2=\displaystyle{\xi_2\over\xi_1}z_1$ and we have an homeomorphism
\begin{eqnarray*}
\psi:\CC^2 & \iso & U_1 \\
(\lambda,z_1) & \mapsto & (z_1,\lambda z_1,1,\lambda).
\end{eqnarray*}
We have $\imin \psi(M_1)=\{z_1=0\}$ and $\imin \psi(M_2)=\{\lambda=0\}$. We still denote them by $M_1,M_2$. The map $f:=\pi|_{U_1} \circ \psi$ is given by $(\lambda,z_1) \mapsto (\lambda z_1,z_1)$. Let us consider $\widetilde{f}$. On the zero section, let $(w_1,w_2)$ be the coordinates of $\CC^2 \simeq S'$, the zero section of $\widetilde{Y}$. We have $J^N_1=J^M_1=\{1\}$, $J^N_2=\{1,2\}=J^M_1 \sqcup J^M_2$, hence
\begin{eqnarray*}
w_1 & = & {\partial f \over z_1} z_1 = z_1 \\
w_2 & = & {\partial^2f \over \partial \lambda \partial z_1}\lambda z_1 = \lambda z_1
\end{eqnarray*}
which is a conic map with respect to the $(\RP)^2$-actions on $S$ and $S'$.
\end{es}

One can ask when the morphism thus obtained becomes a bundle map.
Suppose
\begin{enumerate}
 \item $f(M_j) \subset N_j$ for any $1 \le j \le \ell$.
 \item $M_{j'} \subset M_j$ if and only if $N_{j'} \subset N_j$ for $1 \le j,j' \le \ell$.
\end{enumerate}

\begin{prop}
Under the conditions 1.~and 2.~above, in the zero section $\{t_1 = \dots = t_\ell =0\}$ the morphism $\tilde{f}$ of
multiple-normal deformations is a bundle map.
\end{prop}
\dim\ \ For any $k \in \{1,2,\dots,m\}$,
we have $\#\inf{}_{\subset} J^N_k = \#\sup{}_{\subset} J^N_k = 1$.
Therefore, by (\ref{eq:multi-morphism-zero}), the map $\varphi_j$ becomes
a bundle map and the result follows. \qed

\begin{oss}\label{oss: chi'<chi} More generally, the morphism exists for the
case $\#\chi^N \leq \#\chi^M$.

\

Let $f: X \to Y$ and
$\chi^M=\{M_1,\dots,M_{\ell}\}$ in $X$
and $\chi^N=\{N_1,\dots, N_{\ell'}\}$ in $Y$
for $\ell' \leq \ell$.

\

Then the situation decomposes into the following:
$$
(X;\, M_1,\dots,M_\ell) \overset{\operatorname{Id}}{\longrightarrow}
(X;\, M_1,\dots,M_{\ell'}) \overset{f}{\longrightarrow}
(Y;\, N_1,\dots,N_{\ell'})
$$
For the second arrow, we have already constructed the morphism.
We will construct the morphism for the first arrow.
In what follows, we assume that
$Y=X$, $f=\operatorname{Id}$, $\ell>\ell'$ and $N_j$ = $M_j$.

\

Locally the morphism between multi-normal deformations
$$
(x_1,\dots,x_n;\, t_1,\dots,t_\ell) \to
(x'_1,\dots,x'_n;\, t'_1,\dots,t'_{\ell'})
$$
is given by:
$$
\begin{array}{lll}
t'_j &= t_{\kappa_{\chi^M,\chi^N}(j)}
&\qquad (1 \le j \le \ell'), \\
x'_i &= t_{J_{\chi^M,\chi^N,\, i}} x_i &\qquad (1 \le i \le n).
\end{array}
$$
Here $\kappa_{\chi^M,\chi^N}$ is the map from $\{1,2,\dots,\ell'\}$ to subsets of
$\{1,2,\dots,\ell\}$ defined by
$$
\kappa_{\chi^M,\chi^N}(j) :=
\left\{\beta \in \{1,2,\dots,\ell\};\,
\left(\underset{N_k \subsetneq N_j}{\bigcup} N_k\right)
\subsetneq f(M_\beta) \subset N_j\right\}
$$
and, for $1 \le i \le n$,
$$
J_{\chi^M,\chi^N,\, i} :=
J^M_i \setminus \left(\underset{j \in J^N_i}{\bigcup} \kappa_{\chi^M,\chi^N}(j)\right)
$$
with
$J^M_i: = \{j \in \{1,2,\dots,\ell\};\, i \in I^M_j\}$
and
$J^N_i: = \{j \in \{1,2,\dots,\ell'\};\, i \in I^N_j\}$.

For example,
if $\ell' = 2$ and $M_\ell \subsetneq M_{\ell-1} \subsetneq \dots \subsetneq M_1$ are
satisfied, as $\kappa_{\chi^M,\chi^N}(1) = \{1\}$,
$\kappa_{\chi^M,\chi^N}(2) = \{2,\dots,\ell\}$,
$J_{\chi^M,\chi^N,i} = J^M_i$  $(i \notin I^M_2)$ and
$J_{\chi^M,\chi^N,i} = \emptyset$  $(i \in I^M_2)$,
the local morphism is given by
$$
\begin{array}{llllll}
&t'_1 = t_1, & t'_2 = t_2\dots t_\ell, &\qquad \\
&x'_i = t_{J^M_i} x_i\, (i \notin I^M_2),\,
&x'_i = x_i \, (i \in I^M_2).
\end{array}
$$

We can certainly glue these locally defined morphisms (by the definition
of a multi-normal deformation)
and obtain the morphism between multi-normal deformed manifolds
(say $\widetilde{X}$ and $\widetilde{Y}$) globally. Moreover,
as $J^M_i = \underset{j \in J^N_i}{\sqcup}\kappa_{\chi^M,\chi^N}(j)$ is satisfied by
definition, the following diagram commutes.
$$
\begin{matrix}
\widetilde{X} & \to & \widetilde{Y} \\
            & \searrow & \downarrow \\
            &          &   X=Y
\end{matrix}
$$
Since $\iota(M_j) \subset \iota(N_j)$ holds ($j=1,\dots,\ell'$),
we have the canonical injection
$$
T_{M_j}\iota(M_j) \hookrightarrow T_{N_j}\iota(N_j) \qquad (j \le \ell').
$$
These injections and the zero map on
$
T_{M_j}\iota(M_j)
$
for $j > \ell'$
induce the bundle map over $M:=\underset{1 \le j \le \ell}{\bigcap} M_j$

$$
\varphi: \underset{X,1\leq j\leq \ell}{\times} T_{M_j}\iota(M_j)
\to
M \underset{Y}{\times}
\left(
\underset{Y,1\leq j\leq\ell'}{\times} T_{N_j}\iota({N_j})
\right)
$$

In this case, on the zero section $\{t=0\}$, the morphism from $\widetilde{X}$ to
$\widetilde{Y}$ coincides with $\varphi$.
\end{oss}

\end{section}

\begin{section}{Multi-specialization}\label{6}

Let $X$ be a real analytic manifold with $\operatorname{dim}{X} = n$, and let
$\chi = \{M_1,\dots,M_\ell\}$ be
a family of $\ell$-closed submanifolds satisfying H1-H2-H3. Let $\widetilde{X}$ be the normal deformation of $X$ with respect to the family $\chi$ and consider the diagram \eqref{multi normal deformation}. For an exposition on $(\RP)^\ell$-conic sheaves see the Appendix.

\begin{lem} Let $F \in D^b(k_{X_{sa}})$. There is a natural isomorphism
$$
\imin s R\Gamma_\Omega \imin p F \simeq s^!(\imin p F)_\Omega.
$$
\end{lem}

\dim\ \ We prove the assertion in several step. For $I,J \subseteq \{1,\dots,\ell\}$, $I \cap J = \emptyset$, set $S_I=\{t_i=0,\,i \in I\}$ and $\Omega_J=\{t_j>0,\,j\in J\}$.

(i) We show that if $\sharp J < \ell$, then $(R\Gamma_{\Omega_J}(\imin pF)_\Omega)|_S=0$. We may reduce to $F \in \mod_{\rc}(k_X)$. It follows because for any $(\RP)^\ell$-connected neighborhood $V$ (for example a ball of radius $\varepsilon>0$) of $p \in S$ and for any $W \subset V$ $(\RP)^\ell$-connected containing $V \setminus \Omega$
$$R\Gamma(V \cap \Omega_I;\imin pF) \simeq R\Gamma(W \cap \Omega_I;\imin pF)$$
since $\imin pF$ is conic. This implies
$$(R\Gamma_{\Omega_I}(\imin pF)_{\Omega_I\setminus\Omega})|_S \simeq (R\Gamma_{\Omega_I}\imin pF)|_S.$$

(ii) We show that if $\sharp (I \cup J) < \ell$, then $(R\Gamma_{S_I \cap \Omega_J}(\imin pF)_\Omega)|_S=0$. We argue by induction on $\sharp I$. If $\sharp I=0$ then it follows by (i) that $(R\Gamma_{\Omega_J}(\imin pF)_\Omega)|_S=0$. Suppose that it is true for $\sharp I \leq \ell-2$. Let $i_0 \in I$. We have the distinguished triangle
$$
\dt{(R\Gamma_{S_I \cap \Omega_J}(\imin pF)_\Omega)|_S}{(R\Gamma_{S_{I \setminus \{i_0\}} \cap \Omega_J}(\imin pF)_\Omega)|_S}{(R\Gamma_{S_{I \setminus \{i_0\}} \cap \Omega_{J \cup \{i_0\}}}(\imin pF)_\Omega)|_S}.
$$
The second and the third terms of the  triangle are zero since $\sharp (I \setminus \{i_0\}) \leq \ell-2$. Then the first one is zero.

(iii) We show that if $I \cup J=\ell$ then $(R\Gamma_{S_I \cap \Omega_J}(\imin p F)_\Omega)|_S \simeq (R\Gamma_S(\imin p F)_\Omega)|_S[\sharp J]$. We argue by induction on $\sharp J$. Let $j_0 \in J$. We have the distinguished triangle
$$
\dt{(R\Gamma_{S_{I \cup \{j_0\}} \cap \Omega_{J \setminus \{j_0\}}}(\imin pF)_\Omega)|_S}{(R\Gamma_{S_I \cap \Omega_{J \setminus \{j_0\}}}(\imin pF)_\Omega)|_S}{(R\Gamma_{S_I \cap \Omega_J}(\imin pF)_\Omega)|_S}
$$
where the second term is zero by (ii). If $\sharp J=1$ the result follows immediately.
Suppose that it is true for $\sharp J \leq \ell-1$. Then $(R\Gamma_{S_{I \cup \{j_0\}} \cap \Omega_{J \setminus \{j_0\}}}(\imin pF)_\Omega)|_S[1] \simeq (R\Gamma_{S_I \cap \Omega_J}(\imin pF)_\Omega)|_S$ by (ii) and $(R\Gamma_{S_{I \cup \{j_0\}} \cap \Omega_{J \setminus \{j_0\}}}(\imin p F)_\Omega)|_S \simeq (R\Gamma_S(\imin p F)_\Omega)|_S[\sharp J-1]$ by the induction hypothesis.

(iv) By (iii) we obtain $s^!(\imin p F)_\Omega[\ell] \simeq \imin sR\Gamma_\Omega\imin pF$. It remains to show that $(\imin p F)_\Omega[\ell] \simeq (p^!F)_\Omega$. We may reduce to the case $F \simeq \lind i \rho_*F_i$ with $F_i\in\mod_{\rc}(k_X)$. In this case we have
\begin{eqnarray*}
H^{k+\ell}(\imin p F)_\Omega & \simeq & \lind i\rho_*H^{k+\ell}(\imin p F_i)_\Omega \\
& \simeq & \lind i \rho_*H^k(p^!F_i)_\Omega \\
& \simeq & H^k(p^!F)_\Omega
\end{eqnarray*}
where the second isomorphism follows since $\imin {\widetilde{p}} [\ell] \simeq \widetilde{p}^!$ in $\mod(k_X)$. \qed

\begin{df} The (multi-)specialization along $\chi$ is the functor
\begin{eqnarray*}
\nu^{sa}_\chi:D^b(k_{X_{sa}}) & \to & D^b(k_{S_{sa}}) \\
F & \mapsto & \imin s\mathrm{R}\Gamma_\Omega\imin p F.
\end{eqnarray*}
\end{df}

\begin{teo} \label{m4.2.3} Let $F \in D^b(k_{X_{sa}})$.
\begin{itemize}
\item[(i)] $\nu^{sa}_\chi F \in D^b_{(\RP)^\ell}(k_{S_{sa}})$ (see Definition \ref{def:conic sheaf}).
\item[(ii)] Let $V$ be a $(\RP)^\ell$-conic subanalytic open subset of $S$. Then:
$$H^j(V;\nu^{sa}_\chi F) \simeq \lind U H^j(U;F),$$
where $U$ ranges through the family of $\op(X_{sa})$ such that
$C_\chi(X \setminus U) \cap V=\emptyset$.
\end{itemize}
\end{teo}
\dim\ \ (i) We may reduce to the case $F \in \mod(k_{X_{sa}})$.
Hence $F=\lind i \rho_* F_i$ with $F_i \in \mod_{\rc}(k_{X_{sa}})$
for each $i$.
We have $\imin p \lind i \rho_* F_i \simeq \lind i \rho_* \imin p
F_i$ and $\imin p F_i$ is $\R$-constructible and $(\RP)^\ell$-conic for each
$i$. Hence $\imin p F$ is $(\RP)^\ell$-conic. Since the functors
$\mathrm{R}\Gamma_\Omega$ and $\imin s$ send $(\RP)^\ell$-conic sheaves to $(\RP)^\ell$-conic sheaves
we obtain $\imin s \mathrm{R}\Gamma_\Omega \imin pF=\nu^{sa}_\chi F \in
D^b_{(\RP)^\ell}(k_{S_{sa}})$.

(ii) Let $U \in \op(X_{sa})$ such that $V \cap C_\chi(X \setminus
U)=\varnothing$. We have the chain of morphisms
\begin{eqnarray*}
\mathrm{R}\Gamma(U;F) & \to & \mathrm{R}\Gamma(\imin p(U);\imin p F) \\
& \to & \mathrm{R}\Gamma(\imin p(U) \cap \Omega;\imin p F) \\
& \to & \mathrm{R}\Gamma(\imin {\widetilde{p}}(U) \cup V;\mathrm{R}\Gamma_\Omega\imin p F) \\
& \to & \mathrm{R}\Gamma(V;\nu^{sa}_\chi F)
\end{eqnarray*}
where the third arrow exists since $\imin {\widetilde{p}}(U) \cup
V$ is a neighborhood of $V$ in $\overline{\Omega}$ by Lemma
\ref{m4.1.3} (ii). Let us show that it is an isomorphism.
Let $V$ be a conic open subanalytic subset of $S$. We have
\begin{eqnarray*}
H^k(V;\nu^{sa}_\chi F) & \simeq & \lind W H^k(W;\mathrm{R}\Gamma_\Omega\imin p F) \\
& \simeq & \lind W H^k(W \cap \Omega;\imin p F),
\end{eqnarray*}
where $W$ ranges through the family of subanalytic open
neighborhoods of $V$ in $X$. By Lemma \ref{m4.1.4} we may assume
that $W$ satisfies \eqref{mpitilde}

Since $\imin p F$ is $(\RP)^\ell$-conic, we have
\begin{eqnarray*}
H^k(W \cap \Omega;\imin p F) & \simeq & H^k(\imin p(p(W \cap
\Omega));\imin p F) \\
& \simeq & H^k(p(W \cap \Omega) \times \{(1)_\ell\};\imin p F) \\
& \simeq & H^k(p(W \cap \Omega);F),
\end{eqnarray*}
where $(1)_\ell=(1,\dots,1) \in (\RP)^\ell$. The second isomorphism follows since every subanalytic
neighborhood of $p(W \cap \Omega) \times \{(1)_\ell\}$ contains an
$(\RP)^\ell$-connected subanalytic neighborhood (the proof is similar to
that of Lemma \ref{m4.1.4}). By Lemma \ref{m4.1.3} (i) we have that
$p(W \cap \Omega)$ ranges through the family of subanalytic open
subsets $U$ of $X$ such that $V \cap C_\chi(X \setminus
U)=\emptyset$ and we obtain the result.
\qed

Remember that a sheaf $F \in \mod(k_{X_{sa}})$ is said to be quasi-injective if the restriction morphism $\Gamma(U;F) \to \Gamma(V;F)$ is surjective for each $U,V \in \op^c(X_{sa})$ with $U \supseteq V$.

\begin{cor} Let $F \in \mod(k_{X_{sa}})$ be quasi-injective. Then $F$ is $\nu^{sa}_\chi$-acyclic.
\end{cor}
\dim\ \ The result follows from Theorem \ref{m4.2.3} and the fact that quasi-injective sheaves ate $\Gamma(U;.\cdot)$-acyclic for each $U \in \op(X_{sa})$. \qed

\begin{cor} Let $F \in D^b(k_{X_{sa}})$ and let $p \in S$. Then
$$
(\imin \rho H^j\nu^{sa}_\chi F)_p \simeq \lind W H^j(W;F),
$$
where $W$ ranges through the family $\operatorname{Cone}\chi(p)$.
\end{cor}
\dim\ \ The result follows since for any subanalytic conic neighborhood $V$ of $p$, any $U \in \op(X_{sa})$ such that $C_\chi(X \setminus U) \cap V=\emptyset$ contains $W \in \operatorname{Cone}_\chi(p)$. \qed

Remember that
$$
\underset{M,1\leq i\leq k}{\times}T_{M_{j_i}}\iota_\chi(M_{j_i}) := \left(\underset{X,1\leq i\leq k}{\times}T_{M_{j_i}}\iota_\chi(M_{j_i})\right) \underset{X}{\times} M.
$$

\begin{cor} Let $F \in D^b(k_{X_{sa}})$. Let $k \le \ell$ and $\{j_1,\dots,j_k\}$ be a subset of $\{1,2,\dots,\ell\}$.

Set $\chi_k=\{M_{j_1},\dots,M_{j_k}\}$.
Then we have
$$
(\nu^{sa}_\chi F)|_{\underset{M,1\leq i\leq k}{\times}T_{M_{j_i}}\iota_\chi(M_{j_i})}
\simeq (\nu^{sa}_{\chi_k}F)|_{\underset{M,1\leq i\leq k}{\times}T_{M_{j_i}}\iota_\chi(M_{j_i})}.
$$
\end{cor}
\dim\ \

Set $Z=\underset{M,1\leq i\leq k}{\times}T_{M_{j_i}}\iota_\chi(M_{j_i})$. On $Z$ $\{\mu_1,\dots,\mu_\ell\}$ and $\{\mu_{j_1},\dots,\mu_{j_k}\}$ coincide.

Let $V$ be a $(\RP)^\ell$-conic globally subanalytic open subset of $Z$. Let $U \in \op(X_{sa})$ such that $V \cap C_\chi(X \setminus U)=\emptyset$. Let $W$ be the complement of $C_\chi(X \setminus U)$. Then $W$ is a $(\RP)^\ell$-conic open subanalytic neighborhood of $V$ such that $C_\chi(X \setminus U) \cap W=\emptyset$.
Then for $j \in \Z$
\begin{eqnarray*}
H^j(V;\nu^{sa}_\chi F) & \simeq & \lind W H^j(W;\nu^{sa}_\chi F) \\
& \simeq & \lind U H^k(U;F) \\
& \simeq & \lind {U'} H^k(U';F),
\end{eqnarray*}
where $W$ ranges through the family of $(\RP)^\ell$-conic open subanalytic neighborhoods of $V$, $U \in \op(X_{sa})$ is such that $C_\chi(X\setminus U) \cap W=\emptyset$ and $U' \in \op(X_{sa})$ is such that $C_{\chi}(X\setminus U') \cap V=\emptyset$. Remark that the first isomorphism is a consequence of the fact that $\nu^{sa}_\chi F$ is conic and the fact that by Lemma \ref{lem: Rpl neighborhood}
every open subanalytic neighborhood of $V$ contains a $(\RP)^\ell$-connected one. In the same way we see that
\begin{eqnarray*}
H^j(V;\nu^{sa}_{\chi_k} F)

& \simeq & \lind {U'} H^k(U';F),
\end{eqnarray*}
where $U' \in \op(X_{sa})$ is such that $C_{\chi_k}(X\setminus U') \cap V=\emptyset$. To show the result it is enough to see that
$$
C_\chi(X \setminus U) \cap V = \emptyset \ \ \Longleftrightarrow \ \ C_{\chi_k}(X \setminus U) \cap V = \emptyset.
$$
This follows from Corollary \ref{cor: restriction multi-cones}, which says that
$$
C_\chi(X \setminus U) \cap Z = C_{\chi_k}(X \setminus U) \cap Z.
$$
Hence we have the isomorphism
$$
H^j(V;\nu^{sa}_\chi F) \simeq
H^j(V;\nu^{sa}_{\chi_k} F)
$$
for each $(\RP)^\ell$-conic globally subanalytic open subset of $Z$. By Lemma \ref{lem: RPl decomposition} and the fact that both $\nu^{sa}_\chi F$ and $\nu^{sa}_{\chi_k} F$ are multi-conic  implies $(\nu^{sa}_\chi F)|_Z \simeq (\nu^{sa}_{\chi_k} F)|_Z$.
\qed

Let $f:X \to Y$ be a morphism of real analytic manifolds, $\chi^M=\{M_1,\ldots,M_\ell\}$, $\chi^N=\{N_1,\ldots,N_\ell\}$ two families of closed analytic submanifolds of $X$ and $Y$ respectively satisfying the hypothesis H1-H2-H3. Suppose that $f(M_i) \subseteq N_i$, $i=1,\ldots,\ell$.
 We call $T_\chi f$ the induced map on the zero section. Let $J \subseteq \{1,\ldots,\ell\}$ and set $\chi_J=\{M_{j_1},\ldots,M_{j_k}\}$, $j_k \in J$. The map $T_{\chi_J}f$ denotes the restriction of $\widetilde{f}$ to $\{t_{j_k}=0,\,j_k\in J\}$.

\begin{prop} Let $F \in D^b(k_{X_{sa}})$.

\begin{itemize}
\item[(i)] There exists a commutative diagram of canonical
morphisms
$$
\xymatrix{R(T_\chi f)_{!!}\nu^{sa}_{\chi^M} F  \ar[d] \ar[r] & \nu^{sa}_{\chi^N}Rf_{!!}F \ar[d] \\
R(T_\chi f)_*\nu^{sa}_{\chi^M} F & \nu^{sa}_{\chi^N}Rf_*F. \ar[l]}
$$
\item[(ii)] Moreover if $\supp F \to Y$ and $T_{\chi_J}f:C_{M_{j_1}\dots M_{j_k}}(\supp F) \to
\{t_{j_1}=\cdots=t_{j_k}=0\}$

for each $J = \{j_1,\dots,j_k\}$ are proper, and if $\supp F \cap \imin f(N_j) \subseteq M_j$, $j \in \{1,\ldots,\ell\}$, then
the above morphisms are isomorphisms.
\end{itemize}
\end{prop}
\dim\ \ (i) The existence of the arrows is done as in \cite{KS90} Proposition 4.2.4.

(ii) If $\widetilde{p}^{-1}(\supp F)$ is proper over $\widetilde{X}$, then all the morphisms are isomorphisms. We have to prove that for a closed subset $Z$ of $X$, the restriction of $\widetilde{f}$ to $\overline{\imin {\widetilde{p}}(Z)}$ is proper, if $Z \to Y$ and $C_{M_{j_1}\dots M_{j_k}}(\supp F) \to
\{t_{j_1}=\cdots=t_{j_k}=0\}$

for $k\leq \ell$ are proper, and if $Z \cap \imin f(N_j) \subseteq M_j$, $j \in \{1,\ldots,\ell\}$. We argue by induction on $\sharp \chi$. If $\sharp \chi=1$ this is Proposition 4.2.4 of Sheaves on Manifolds. Suppose it is true for $\sharp\chi\leq\ell-1$. It follows from the hypothesis that the fibers of $\widetilde{f}$ restricted to $\overline{\widetilde{p}^{-1}(Z)}$ are compact (if $t_{j_1}=\cdots=t_{j_k}=0$ this is a consequence of the fact that $C_{M_{j_1}\dots M_{j_k}}(\supp F) \to
\{t_{j_1}=\cdots=t_{j_k}=0\}$

 is proper). Then it remains to prove that it is a closed map. Let $\{u_n\}_{n\in\N}$ be a sequence in $\overline{\imin{\widetilde{p}}(Z)}$ and suppose that $\{\widetilde{f}(u_n)\}_{n\in\N}$ converges. We shall find a convergent subsequence of $\{u_n\}_{n\in\N}$. We may also assume that $\{\widetilde{p}(u_n)\}_{n\in\N}$ converges. The map $\imin{\widetilde{p}}(Z) \setminus \{t_1=\cdots=t_\ell=0\} \to \widetilde{Y} \setminus \{t_1=\cdots=t_\ell=0\}$ is proper. Indeed, let $K$ be a compact subset of $\widetilde{Y} \setminus \{t_1=\cdots=t_\ell=0\}$, and

reduce to the case that $K$ is contained in $\{c \leq t_j \leq d\} \subset \widetilde{Y}$, $c,d>0$, $j \in \{1,\dots,\ell\}$. Suppose without loss of generality that $j=1$. Then $K_1:=\widetilde{p}_{N_1}(K)$ is a compact subset of $\widetilde{Y}_{N_2\ldots N_\ell}$. Let us identify $K_1$ with $\imin{\widetilde{p}_{N_1}}(K_1) \cap \{t_1=1\}$. Then $\imin{\widetilde{f}}(K_1)$ is compact by the induction hypothesis. Hence $\imin{\widetilde{f}}(K) \subseteq \mu_1([c,d],\imin{\widetilde{f}}(K_1))$ is compact since it is closed and contained in a compact subset.
We may assume that $\{\widetilde{f}(u_n)\}_{n\in\N}$ converges to a point of $\{t_1=\cdots=t_\ell=0\}$. Then $\{\widetilde{p}(u_n)\}_{n\in\N}$ converges to a point of $Z \cap \imin f(N_1 \cap \cdots \cap N_\ell) \subseteq M_1 \cap \cdots \cap M_\ell$.

\

Taking local coordinates systems of $X$ and $Y$, let $u_n=(x_{1n},\ldots,x_{mn},t_{1n},\ldots,t_{\ell n})$, $t_{jn}>0$, $j=1,\dots,\ell$. Then $t_{jn} \to 0$, $j=1,\dots,\ell$ and $t_{J^M_in}x_{in}\to 0$, $i=1,\ldots,m$. It is enough to show that $\{|x_{in}|\}_{n\in\N}$ is bounded for each $i=1,\dots,m$. We argue by contradiction. Suppose without loss of generality that $|x_{1n}|\to+\infty$ and that $\{x_{1n}/|x_n|\}_{n\in\N}$ is not infinitesimal.

Set $u_n^{(\beta)}=(x_n^{(\beta)},t_n^{(\beta)})$, where $x_{in}^{(\beta)}=t_{in\beta}x_{in}$ with $t_{in\beta}=t_{\beta n}$ if $i \in I_\beta$ and $t_{in\beta}=1$ if $i \notin I_\beta$, and where $t^{(\beta)}_{jn}=t_{jn}$ if $j \neq \beta$ and $t^{(\beta)}_{\beta n}=1$. Then $u_n$ belongs to $\imin{\widetilde{p}}(Z) \setminus \{t_1=\cdots=t_\ell=0\}$ and $\{\widetilde{f}(u_n^{(\beta)})\}_{n\in\N}$ converges. Since $\widetilde{f}$ is proper on $\imin{\widetilde{p}}(Z) \setminus \{t_1=\cdots=t_\ell=0\}$, then $\{|x_n^{(\beta)}|\}_{n\in\N}$ is bounded.

(a) Suppose that there exists $\beta \in \{1,\dots,\ell\}$ such that $1 \notin I_\beta$. The sequence $\{|x_n^{(\beta)}|\}_{n\in\N}$ is bounded. In particular $\{|x_{1n}|\}_{n\in\N}$ is bounded which is a contradiction.

(b) Suppose that $1 \in I_j$ for $j=1,\dots,\ell$. Suppose without loss of generality that $I_1$ is the biggest $I_j$ containing $1$. Set $\widetilde{u}_n=(\widetilde{x}_n,\widetilde{t}_n)$ where $\widetilde{x}_{in}=x_{in}/|x_n|$ if $i \in I_1$ and $\widetilde{x}_{in}=x_{in}$ otherwise and where $\widetilde{t}_{1n}=t_{1n}|x_n|$, $\widetilde{t}_{jn}=t_{jn}$ if $j \neq 1$. Remark that $|t_{1n}x_n|=|t_{1n}x_{1n}|||x_n|/x_{1n}|$ is bounded since the sequences $\{|x^{(1)}_{1n}|\}_{n\in\N}$ and $\{x_{1n}/|x_n|\}_{n\in\N}$ are bounded. By extracting a subsequence $\{\widetilde{u}_n\}_{n\in\N}$ converges to a non-zero vector $v$ and $\widetilde{u}_n$ belongs to $\imin{\widetilde{p}}(Z)$. On the other hand $\widetilde{f}(u_n)$ converges and hence $f_k(t_{J^M_1n}x_{1n},\dots,t_{J^M_mn}x_{mn})/t_{J^N_kn}|x_n|$ converges to $0$ for each $k$ such that $J^N_k \ni 1$. Moreover $\widetilde{f}(\mu_2(v,\lambda))=\widetilde{f}(v)$ for each $\lambda \in \RP$ which contradicts the fact that the fibers of $\overline{\imin{\widetilde{p}}(Z)} \to \widetilde{Y}$ are compact.

\qed

\begin{prop} Let $F \in D^b(k_Y)$.

\begin{itemize}
\item[(i)] There exists a commutative diagram of canonical
morphisms
$$
\xymatrix{\omega_{S_X/S_Y}\otimes\imin {(T_\chi f)}\nu^{sa}_{\chi^N} F  \ar[d] \ar[r] & \nu^{sa}_{\chi^M}(\omega_{S_X/S_Y}\otimes \imin f F) \ar[d] \\
T_{\chi} f^!\nu^{sa}_{\chi^N} F & \nu^{sa}_{\chi^M}f^!F. \ar[l]}
$$
\item[(ii)]

The above morphisms are ismomorphisms on the open sets where $T_{\chi_J} f$ is smooth for each $J \subseteq \{1,\ldots,\ell\}$.
\end{itemize}
\end{prop}
\dim\ \ (i) The existence of the arrows is done as in \cite{KS90} Proposition 4.2.5. When (ii) is satisfied the function $\widetilde{f}$ is smooth at any point of the boundary of $\Omega$ and all the above morphisms become isomorphisms.
\qed

\end{section}

\begin{section}{Multi-asymptotic expansions}\label{7}

Let $X$ be the $n$-dimensional complex vector space $\CC^n$ with coordinates
$(z_1, z_2, \dots, z_n)$ and $\chi = \{Z_1, \dots, Z_{\ell}\}$
a family of $\ell$-complex submanifolds
defined by
$$
Z_j := \{z \in X;\, z_{i_{j, 1}} = \dots = z_{i_{j, {m_j}}} = 0\}
$$
where $I_j = \{i_{j, 1},
\dots, i_{j, {m_j}}\}$ is a subset of $\{1,2,\dots,n\}$ and
$\{I_j\}_{j=1,2,\dots,\ell}$
satisfies conditions given in Proposition \ref{prop:canonical-coordinate}
(the conditions H1 and H2), that is,
$$
I_j \cap I_{j'} \ne \emptyset \Longrightarrow I_j \subset I_{j'} \text { or }
I_{j'} \subset I_{j} \qquad (j, j' \in \{1,2,\dots,\ell\}).
$$
We set, for $1 \le j \le \ell$,
$$
\hat{I}_j := I_j \setminus
\left(\underset{I_k \subsetneq I_j}{\bigcup} I_k \right)
$$
We also assume $\hat{I}_j \ne \emptyset$ for $j = 1,2,\dots, \ell$ (the condition H3).
Then we have $T_{Z_j}\iota(Z_j)\simeq Z_j \times \CC^{\#\hat{I}_j}$ and
the equality
$$
\underset{1 \le j \le \ell}{\bigcup} I_j = \hat{I}_1 \sqcup \hat{I}_2 \sqcup \dots
\sqcup \hat{I}_\ell,
$$
which follows from conditions H1,  H2 and H3.

For a subset $I = \{i_1, i_2, \dots, i_m\}$ of $\{1,2,\dots,n\}$,
we denote by $z_I$ the coordinates $(z_{i_1}, z_{i_2}, \dots, z_{i_m})$ and
by $\CC^I$ the complex space $\CC^{\#I}$ with coordinates $z_I$.
The map $\pi_{I}$ designates the projection from $X$ to $\CC^I$.
Let $\mu_j(z,\lambda): X \times \R \to X$ ($j=1,2,\dots,\ell$) be an action defined by
$$
\mu_j(z_1, \dots, z_n,\lambda) = (\lambda_{j1}z_1, \dots, \lambda_{jn}z_n)
$$ where $\lambda_{ji} = \lambda$
if $i \in I_j$ and $\lambda_{ji} = 1$ otherwise.
Let $U$ be an open subset in $X$.
We say that $U$ is a {\it{$\mu$-star shape}} if $U$ is non-empty and
$\mu_j(x_0,[0,1]) \subset U$ holds for any $x_0 \in U$ and $1 \le j \le \ell$.
An open ball with its center being the origin is, for example, a $\mu$-star shape.

Let $U$ be an open convex set in $X$ which is a $\mu$-star shape, and let
$G_j$ ($j=1,2, \dots,\ell$) be an open proper convex cone in $\CC^{\hat{I}_j}$.
In what follows, we only consider a proper cone, which means that
we work outside the zero section.

We define an open proper convex multi-cone
$S(U,\{G_j\}, \epsilon)$ ($\epsilon > 0$) in $X$
as follows. We first set
$$
V_j :=
\left\{
z \in U;\,
z_{\hat{I}_j} \in G_j, \,
\vert z_{I_j \setminus \hat{I}_j} \vert < \epsilon \vert z_{\hat{I}_j} \vert
\right\} \subset X \qquad (j=1,2,\dots,\ell).
$$
Here if $I_j \setminus \hat{I}_j = \emptyset$, then we set
$\vert z_{I_j \setminus \hat{I}_j} \vert = 0$, i.e., no condition.
Then we define
$$
S(U,\{G_j\}, \epsilon) := V_1 \cap V_2 \cap \dots \cap V_\ell \subset X.
$$

By convention, the case with no submanifold $Z_j$ is allowed.
In this case, we set $S(U, \{G_j\}, \epsilon) := U$.

Note that $S:=S(U, \{G_j\}, \epsilon)$ is non-empty and
$$
S \cap \left(\underset{1\le j \le \ell}{\bigcup} Z_j\right) = \emptyset
$$
follows from the condition H1, H2 and H3.
For example, $S$ gives just
a poly-sector when ${\underset{1 \le j \le \ell}{\bigcup} Z_j}$ forms a
normal-crossing divisor. The multi-cone $S$ plays the same
geometrical role as that of a sector in an asymptotic expansion.

Let $\Z_\ell$ denote the set of integer $\{1,2,\dots, \ell\}$ and
$\Powl$ be the set of all the subsets of
$\Z_\ell$ except for the empty set.
For a $J = \{j_1, j_2, \dots, j_m\} \in \Powl$ and
a proper convex multi-cone $S:=S(U, \{G_j\}, \epsilon)$,
we define
$$
\begin{aligned}
Z_{J} &:= Z_{j_1} \cap Z_{j_2} \cap \dots \cap Z_{j_m}, \\
I_{J} &:= I_{j_1} \cup I_{j_2} \cup \dots \cup I_{j_m}, \\
S_{J} &:= \operatorname{Int}_{Z_J}\left(\overline{S} \cap Z_{J}\right)
\end{aligned}
$$
where $\overline{A}$ is the closure of a set $A$ in $X$ and
$\operatorname{Int}_{Z_J}(B)$ denotes the interior of
a set $B$ in $Z_J$.  We often write $S_J$ by $S_j$ if $J = \{j\}$ from now on.
We give some fundamental properties of a multi-cone which are needed later.
\begin{lem}{\label{lemma:fundamental-geo}}
For $S := S(U,\{G_j\}, \epsilon)$ and $J \in \Powl$,
we have the followings.
\begin{enumerate}
\item $S_J$ is also a non-empty open proper convex multi-cone in $Z_J$
for submanifolds $\{Z_j \cap Z_J\}_{j \in J^*}$. To be more precise,
we have in $Z_J$
\begin{equation}{\label{eq:subsector-set-def}}
S_J = S(U\cap Z_J, \{G_j\}_{j \in J^*}, \epsilon)
\end{equation}
where $J^* = \{j \in \Z_\ell;\, Z_J \nsubseteq Z_j\}$ and
$I^*_j = I_j \setminus I_J$ $(j \in J^*)$ which defines submanifold $Z_j \cap Z_J$.
Note that $\{Z_j \cap Z_J\}_{j \in J^*}$ also satisfies the conditions
H1, H2 and H3 in $Z_J$.
\item $\overline{S_J} = \overline{S} \cap Z_J$ and $Z_{J'} \cap S_J = \emptyset$
for $J' \in \Powl$ with $Z_{J} \nsubseteq Z_{J'}$.
\item $\{S_J\}_{J \in \Powl}$
covers the edge $\overline{S} \cap \left(\underset{1\le j \le \ell}{\bigcup} Z_j\right) \cap U$ of $S$, i.e.,
\begin{equation}
\overline{S} \cap \left(\underset{1\le j \le \ell}{\bigcup} Z_j\right) \cap U=
\underset{J \in \Powl}{\bigcup} S_J.
\end{equation}
\end{enumerate}
\end{lem}
\dim\ \
Note that $Z_J \nsubseteq Z_j$ is equivalent to saying that
$\hat{I}_j \cap I_{J} = \emptyset$ under the conditions H1 and H2.
We also note an elementary fact that
$\overline{A \cap B} = \overline{A} \cap \overline{B}$ if
$A$ and $B$ are open convex sets with $A \cap B \ne \emptyset$.

We first show 1.~of the lemma.
As each $V_j$ is open convex and $V_1 \cap \dots \cap V_\ell$ is not empty,
we have $\overline{S} = \overline{V}_1 \cap \dots \cap \overline{V}_\ell$. Therefore
we have
$$
S_J = \operatorname{Int}_{Z_J}(\overline{V}_1 \cap \dots \cap \overline{V}_\ell \cap Z_{J})
=
\operatorname{Int}_{Z_J}(\overline{V}_1 \cap Z_{J}) \cap
\dots \cap
\operatorname{Int}_{Z_J}(\overline{V}_\ell \cap Z_{J})
$$
and
$$
\operatorname{Int}_{Z_J}(\overline{V}_j \cap Z_{J})
=\left\{
\begin{aligned}
V_j \cap Z_{J}   \qquad & (\hat{I}_j \cap I_{J} = \emptyset),\\
U \cap Z_{J}     \qquad & (\text{otherwise}),
\end{aligned}
\right.
$$
from which we have
$$
S_J = \underset{\hat{I}_j \cap I_J = \emptyset}{\bigcap} (V_j \cap Z_J).
$$
Hence $S_{J}$ is
the multi-cone $S(U \cap Z_{J}, \{G_j\}_{j \in J^*}, \epsilon)$
in $Z_J$.

Next we show 2.~of the lemma. As
$
\overline{V}_j \cap Z_{J} = \overline{U} \cap Z_{J}
$
if $\hat{I}_j \cap I_{J} \ne \emptyset$, we have
$$
\overline{S_J}
= \overline{\underset{\hat{I}_j \cap I_{J} = \emptyset}{\bigcap} V_j \cap Z_{J}}
= \underset{\hat{I}_j \cap I_{J} = \emptyset}{\bigcap} \overline{V_j} \cap Z_{J}
= \underset{1 \le j \le \ell}{\bigcap} \overline{V_j} \cap Z_{J}
= \overline{S} \cap Z_{J}.
$$
For $Z_J \nsubseteq Z_{J'}$,
we can find $j \in J'$ satisfying $Z_J \nsubseteq Z_j$. Then $j$ belongs to
$J^*$ by definition,
from which $(Z_j \cap Z_J) \cap S_J = \emptyset$
because $S_J$ is an open proper convex
multi-cone in $Z_J$ by 1.~of the lemma.

Noticing that $\overline{S_J} = \overline{S} \cap Z_J$ and $S_J$ is still
a multi-cone, 3.~of the lemma can be shown by induction.
\qed

The following properties of $S := S(U, \{G_j\}, \epsilon)$ are easily verified.
\begin{enumerate}
\item For any $j$ and $z \in S$, we have $\mu_j(z,(0,1]) \subset S$.
\item For any $\tilde{z} \in S_j$ ($1 \le j \le \ell$),
	there exists a point $z \in S$ satisfying
	$\mu_j(z,0) = \tilde{z}$.
\end{enumerate}

For a $J = \{j_1, j_2, \dots, j_m\} \in \Powl$,
we set
$$
\mu_J(z,\lambda_{j_1}, \lambda_{j_2}, \dots, \lambda_{j_m})
=\mu_{j_1}(\mu_{j_2}(\dots\mu_{j_m}(z,\lambda_{j_m}),\dots,\lambda_{j_2}),\lambda_{j_1}).
$$
Then the above properties (1) and (2) also hold for these multi-actions.
\begin{lem}{\label{lemma:mutli-cone-geometric}}
Let $J \in \Powl$ and $S := S(U, \{G_j\}, \epsilon)$.
\begin{enumerate}
	\item For any $z \in S$, we have
		$\mu_J(z, (0,1], \dots, (0,1]) \subset S$.
	\item For any $\tilde{z} \in S_J$, there exists $z \in S$ with
		$\mu_J(z,0,\dots,0) = \tilde{z}$.
\end{enumerate}
\end{lem}
\dim\ \
The claim 1.~is easy. We prove 2.~by induction with respect to the number of elements
of $J$. Assume $\#J > 1$ and fix an element $k \in J$.

The set $S_k$ is
the multi-cone $S(U \cap Z_k, \{G_j\}_{j \in J^*}, \epsilon)$
in $Z_k$.
Here $J^*$ was given in (\ref{eq:subsector-set-def}).
Noticing that $Z_k \subset Z_j$ if $\hat{I}_j \cap I_{k} \ne \emptyset$,

we have
$$
S_{J} = (S_k)_{J^* \cap J}.
$$
Then, by induction hypothesis, we can find $z_0 \in S_k$ satisfying
$$
\tilde{z} = \mu'_{J^* \cap J}(z_0,0,\dots,0)
$$
where $\mu'_{J^* \cap J}$
is the multi action in $Z_k$ defined by $\{I^*_j\}_{j \in J^*}$.
We can also find $z_1 \in S$ with $\mu_k(z_1,0) = z_0$.
As the restriction of
$\mu_{J^* \cap J}$ to $Z_k$ coincides with $\mu'_{J^* \cap J}$, we have
$$
\tilde{z} = \mu_{(J^* \cap J) \cup \{k\}}(z_1,0,\dots,0).
$$
Since $\mu_j(z,0)$ ($j \in J$) is the identity map on $Z_J$, we finally obtained
$\tilde{z} = \mu_{J}(z_1,0,\dots,0)$, which completes the proof.
\qed

Thanks to the lemma, we have
\begin{cor}{\label{cor:multi-cone-domain}}
Let $J \in \Powl$ and $S := S(U, \{G_j\}, \epsilon)$.
We have $\pi_J(S) = S_J$ where $\pi_J: X \to Z_J$ is the
canonical projection defined by $(z) \mapsto (z_{I_J^{\operatorname{C}}})$
$($$I_J^{\operatorname{C}}$ is the complement set of $I_J$$)$.
\end{cor}
\dim\ \
As $\pi_J(z) = \mu_J(z,0,\dots,0)$ holds, the inclusion
$S_J \subset \pi_J(S)$ comes from 2.~of Lemma \ref{lemma:mutli-cone-geometric}.
Let us show a converse inclusion. Let $\tilde{z} \in \pi_J(S)$ with
$\tilde{z} = \pi_J(z')$ ($z' \in S$). Then
$z'_\lambda := \mu_J(z', \lambda, \lambda, \dots, \lambda)$
$(0 < \lambda \le 1)$ is contained in $S$, and
$$
\lim_{\lambda \to 0^+} z'_\lambda = \mu_J(z',0,\dots,0) = \pi_J(z') = \tilde{z}.
$$
Moreover, as $\mu_J(z,0,\dots,0)$ is an open map,
an open neighborhood of $z'$ contained in $S$ is mapped to
an open neighborhood of $\tilde{z}$ in $Z_j$ by $\mu_J(z,0,\dots,0)$.
This implies $\tilde{z} \in S_{J}$.
\qed

Let $J = \{j_1, \dots, j_m\} \in \Powl$, and let
$\lambda = (\lambda_1, \dots, \lambda_\ell)$ be
variables of parameters for the multi-action
$\mu:=\mu_{\{1,2,\dots,\ell\}}$ in $X$.
We denote by $\Z_{\ge 0}^{J}$ the subset of $\Z_{\ge 0}^\ell$
consisting of $\beta = (\beta_1, \beta_2, \dots, \beta_\ell) \in \Z_{\ge 0}^\ell$ with
$\beta_j = 0$ if $j \notin J$. Note that $\Z_{\ge 0}^J$ is isomorphic
to $\Z_{\ge 0}^{\#J}$.

We set
$\partial_{\lambda}^\beta :=
\displaystyle\frac{\partial^\beta}{\partial \lambda^\beta}$
for $\beta \in \Z_{\ge 0}^\ell$. When $\beta$ belongs to $\Z_{\ge 0}^J$,
we sometimes denote it by $\partial_{\lambda, J}^\beta$ to emphasize the fact
that $\beta$ is an element of $Z_{\ge 0}^J$.

For $\beta \in \Z^J_{\ge 0}$, we introduce the polynomial of
the variables $z_{I_J}$ with constant coefficients by
\begin{equation}
T^\beta_{J}(z_{I_J}) :=
\left.\frac{1}{\beta!}\exp(-\mu(z,\lambda))
\partial^\beta_{\lambda, J}
\exp(\mu(z,\lambda))\right|_{\lambda = e_{\lambda,J}}.
\end{equation}
Here $e_{\lambda, J} \in \CC^\ell$ is the point $(\lambda_1, \dots, \lambda_\ell)$
with $\lambda_j = 0$ ($j \in J$) and $\lambda_j = 1$ ($j \notin J$) and
the exponential function $\exp$ on $\CC^n$ is defined by
$\exp(z_1,\dots,z_n):= e^{z_1} \dots e^{z_n}$ as usual.

Then, for $N = (n_1,n_2, \dots, n_\ell) \in \Z_{\ge 0}^\ell$, we define
a polynomial of $z_{I_J}$ with constant coefficients by
\begin{equation}
T^{<N}_{J}(z_{I_J}) := \sum_{\beta <_J N,\, \beta \in \Z_{\ge 0}^J} T^\beta_{J}(z_{I_J})
\end{equation}
where $\beta <_J N$ if and only if $\beta_k < n_k$ for any $k \in J$.
Note that if there exists no index $\beta <_J N$, then we set
$T^{<N}_{J}(z_{I_J}) := 0$ as usual convention.

We now give a concrete form of the polynomial $T^{<N}_{J}(z_{I_J})$.
Let $I=\{i_1, \dots,i_m\}$ be a subset of $\{1,2,\dots,n\}$.
We denote by $\Z^I_{\ge 0}$ the subset of $\Z^n_{\ge 0}$ whose
element is $(\alpha_1, \dots, \alpha_n)$ with $\alpha_i = 0$ if $i \notin I$.
For $\alpha \in \Z_{\ge 0}^l$,
a monomial $z^\alpha$ designates $z_1^{\alpha_1}\dots z_n^{\alpha_n}$. If
$\alpha$ belongs to $\Z^I_{\ge 0}$, we also write it by $z_I^\alpha$.
We set
$$
\vert \alpha \vert_{I} = \alpha_{i_1} + \dots + \alpha_{i_m}.
$$
\begin{lem}{\label{lem:T_N-polynomial}}
We have
\begin{equation}{\label{eq:lemma-polynomial-TN}}
T^{<N}_{J}(z_{I_J}) = \sum_{\alpha \in A(N)} \frac{1}{\alpha!}z_{I_J}^\alpha
\end{equation}
where the indices set $A(N)$ is given by
$$
A(N) := \{\alpha \in \Z^{I_J}_{\ge 0};\, \vert \alpha \vert_{I_j} < n_j
\text { for any $j \in J$}\}.
$$
\end{lem}
\dim \ \
Let $i \in I_J$ and
$\beta = (\beta_1, \dots, \beta_n) \in Z^J_{\ge 0}$
($\beta_j = 0$ if $j \notin J$). We first assume
$\beta_j \ge 1$ for any $j \in J$. Then we have
$$
\begin{aligned}
\frac{\partial}{\partial {z_i}}T^{\beta}_J
& =
\frac{\partial}{\partial {z_i}}
\left( \left.\frac{1}{\beta!}\exp(-\mu(z,\lambda))
\partial^\beta_{\lambda, J}
\exp(\mu(z,\lambda))\right|_{\lambda = e_{\lambda,J}}\right) \\
&=
\left.\frac{1}{\beta!}\exp(-\mu(z,\lambda))
\partial^\beta_{\lambda, J}
\left(
\underset{j \in J_i}{\prod}\lambda_j
\exp(\mu(z,\lambda))\right)\right|_{\lambda = e_{\lambda,J}} \\
&=
\left.\frac{1}{(\beta - e_{J_i})!}\exp(-\mu(z,\lambda))
\partial^{\beta - e_{J_i}}_{\lambda, J}
\exp(\mu(z,\lambda))\right|_{\lambda = e_{\lambda,J}}
= T^{\beta-e_{J_i}}_J
\end{aligned}
$$
where $e_{J_i} = (e_{J_i,\, 1}, \dots, e_{J_i,\, \ell}) \in \Z^l_{\ge 0}$
is determined
by $e_{J_i,\, j} = 1$ for $j \in J_i$ and $e_{J_i,\, j} = 0$ otherwise.
This formula also holds for any $\beta \in \Z^l_{\ge 0}$
if we set $T^\beta_J(z_{I_J}) := 0$
for $\beta = (\beta_1, \dots, \beta_l) \in \Z^J$
with some $\beta_j < 0$ ($j \in J$).
Hence we obtained, for $N=(n_1, \dots, n_\ell) \in \Z_{\ge 0}^l$,
$$
\frac{\partial}{\partial {z_i}} T^{<N}_J = T^{<N - e_{J_i}}_J.
$$

We prove the lemma by induction with $n = n_1 + \dots + n_\ell$.
If $n = 0$, as both sides of the equation
(\ref{eq:lemma-polynomial-TN}) are zero by definition,
the lemma is true.

Now we prove the lemma for a general $n > 0$.
Let us consider the system of partial differential equations
of an unknown function
$u(z_{I_J})$ defined by
\begin{equation}{\label{eq:partial-diff}}
\frac{\partial}{\partial {z_i}} u = T^{<N - e_{J_i}}_J \quad (i \in I_J)
\end{equation}
Then, by induction hypothesis, the right hand side of the above equation
is given by
$
\displaystyle\sum_{\alpha \in A(N-e_{J_i})} \frac{1}{\alpha!}z_{I_J}^\alpha.
$
Clearly both $u=T^{<N}_{J}(z_{I_J})$ and
$
u=\displaystyle\sum_{\alpha \in A(N)} \frac{1}{\alpha!}z_{I_J}^\alpha
$
satisfy the same equation (\ref{eq:partial-diff}).
The solution of (\ref{eq:partial-diff}) is uniquely determined
if the initial values at $z_{I_J} = 0$ is given.
It is easy to see that $T^{<N}_{J}(0) = 1$ if $A(N) \ne \emptyset$
and $T^{<N}_{J}(0) = 0$ if $A(N) = \emptyset$.
Hence we have obtained
(\ref{eq:lemma-polynomial-TN}) for $N$.
This completes the proof.
\qed

\begin{df}
Let $S:=(U, \{G_i\},\epsilon)$ be a multi-cone in $X$.
We say that $F:=\{F_{J}\}_{J \in \Powl}$
is a total family of coefficients of multi-asymptotic expansion
along ${\underset{1 \le j \le \ell}{\bigcup} Z_j}$ on $S$ if each $F_J$ consists of
a family of holomorphic functions
$\{f_{J, \alpha}(z_{I^{\operatorname{C}}_J})\}_{\alpha \in \Z_{\ge 0}^{I_J}}$
defined on $S_J$. Here $I^{\operatorname{C}}_J$ is the complement set
of $I_{J}$ $($note that $z_{I^{\operatorname{C}}_J}$ are the coordinates of
the submanifold $Z_J$$)$.
\end{df}

Let $F$ be a total family of coefficients defined in the above definition.
We introduce a map $\tau_{F, J}$ from polynomials of the variables $z_{I_J}$
to those with coefficients in holomorphic functions on $S_J$ in the following
way. Let $p(z_{I_J}) = \displaystyle\sum_\alpha c_\alpha z_{I_J}^\alpha$
be a polynomial of the variable $z_{I_J}$ with constant coefficients. Then
we define $\tau_{F, J}(p)(z)$ by replacing
a monomial $z_{I_J}^\alpha$ in $p(z_{I_J})$ with
$f_{J, \alpha}(z_{I_J^{\operatorname{C}}})z^\alpha_{I_J}$ where $f_{J, \alpha}$
is given in $F_J = \{f_{J, \alpha}\}$, that is,
\begin{equation}
\tau_{F, J}(p)(z) := \sum_\alpha c_\alpha f_{J,\alpha}(z_{I_J^{\operatorname{C}}})
z_{I_J}^\alpha.
\end{equation}
Note that $\tau_{F, J}(p)(z)$ is a holomorphic function on
$\pi_J^{-1}(S_J)$, in particular,
it is defined on $S \subset \pi_J^{-1}(S_J)$
by Corollary \ref{cor:multi-cone-domain}.
We set
\begin{equation}
\begin{aligned}
T^{\beta}_{J}(F;\, z) &= \tau_{F, J}(T^\beta_J),\\
T^{<N}_{J}(F;\, z) &:= \sum_{\beta <_J N} T^\beta_{J}(F;\, z),
\end{aligned}
\end{equation}
and
\begin{equation}
\begin{aligned}
\operatorname{App}^{<N}(F; z)
:= \sum_{J \in \Powl}
(-1)^{(\#J + 1)} T^{<N}_{J}(F;\, z).
\end{aligned}
\end{equation}
It follows from Lemma \ref{lem:T_N-polynomial} that,
if ${\underset{1 \le j \le \ell}{\bigcup} Z_j}$
forms a normal crossing divisor,
$\operatorname{App}^{<N}(F; z)$ coincides with one defined by
Majima in \cite{Ma84}.

\medskip

Let us recall the definitions of the sets $J_{\supsetneq Z_j}$ and
$J_{\subsetneq Z_j}$ for $j \in \Z_\ell$.
$$
\begin{aligned}
J_{\supsetneq Z_j} &:= \{k \in \Z_\ell;\,  Z_k \supsetneq
Z_j,\,
\text{ there is no $m$ with $Z_k \supsetneq Z_m \supsetneq Z_j$ } \}\\
J_{\subsetneq Z_j} &:= \{k \in \Z_\ell;\,  Z_k \subsetneq Z_j,\,
\text{ there is no $m$ with $Z_k \subsetneq Z_m \subsetneq Z_j$ } \}.
\end{aligned}
$$
Then the function $w_j: \Z^\ell_{\ge 0} \to \Z$ ($j = 1,2, \dots, \ell$)
is defined by
\begin{equation}
w_j(N) = n_j - \sum_{k \in J_{\supsetneq Z_j}} n_k
\end{equation}
for $N=(n_1,n_2, \dots,n_\ell) \in \Z^\ell_{\ge 0}$. Note that $w_j$ takes
not only positive values but also negative ones.

Let $S'$ be another open proper convex multi-cone $S(U',\{G'_j\}, \epsilon')$.
We say that $S'$ is properly contained in $S$ if
$\epsilon' < \epsilon$, $U'$ is relatively compact in $U$ and
$G'_j$ ($j=1,2, \dots,\ell$) is properly contained in $G_j$ as a conic cone.

\begin{df}
Let $f$ be a holomorphic function on $S = S(U, \{G_j\}, \epsilon)$.
We say that
$f$ is multi-asymptotically developable along
${\underset{1 \le j \le \ell}{\bigcup} Z_j}$  on $S$ if
there exists a total family $F$ of coefficients of
multi-asymptotic expansion along ${\underset{1 \le j \le \ell}{\bigcup} Z_j}$ on $S$
that satisfies the following condition.

For any open proper convex multi-cone $S'=S(U', \{G'_j\}, \epsilon')$
properly contained in $S$ and for any
$N = (n_1, \dots, n_\ell) \in {\mathbb Z}_{\ge 0}^\ell$,
there exists a constant $C_{S',N} > 0$ for which we have an estimate
\begin{equation}{\label{eq:multi-asymptotic-formula}}
\left\vert f(z) - \operatorname{App}^{<N}(F; z) \right\vert \le C_{S', N}
\underset{1 \le j \le \ell}{\prod}
\operatorname{dist}(z,\, Z_j)^{w_j(N)}
\qquad (z \in S').
\end{equation}
\end{df}

Let us see some typical examples.

\begin{es}
Let $X=\mathbb C^n$ with the coordinates
$(z_1, z_2, \dots, z_n) = (z_1, z_2, z')$ and
$Z_j$ $(j=1,2)$ submanifolds defined by $\{z_j = 0\}$ $($i.e. $I_j = \{j\}$$)$.
Let $G_j$ $(j=1,2)$ be a proper open sector in $\CC$ and $U_R:=B^1_R \times B^1_R
\times B^{n-2}_R$ where $B^k_R$ designates an open ball in $\CC^k$
with its radius $R > 0$ and center $0$.
Note that $U$ is a $\mu$-star shape.
In this case, the multi-cone $S=S(U_R,\{G_1, G_2\}, \epsilon)$ is nothing but
a poly-sector $(G_1 \times G_2 \times \CC^{n-2}) \cap U_R$ and
we have
$$
S_1 = (G_2 \cap B^1_R) \times B^{n-2}_R,\,
S_2 = (G_1 \cap B^1_R) \times B^{n-2}_R,\,
S_{\{1,2\}} = B^{n-2}_R.
$$
Let $F$ be a total family of coefficients of multi-asymptotic expansion, that is,
$$
\begin{aligned}
F &= (F_{\{1\}}, F_{\{2\}}, F_{\{1,2,\}}) \\
&=\left(
\left\{f_{\{1\},k}(z_2, z')\right\}_{k \ge 0},\,
\left\{f_{\{2\},k}(z_1, z')\right\}_{k \ge 0},\,
\left\{f_{\{1,2\},\alpha}(z')\right\}_{\alpha \in \Z_{\ge 0}^2}\right)
\end{aligned}
$$
where $f_{\{1\},k}$ $($resp. $f_{\{2\},k}$ and $f_{\{1,2,\},\alpha}$$)$ is
a holomorphic function on $S_1$ $($resp. $S_2$ and $S_{\{1,2\}}$$)$ respectively.
For $N=(n_1, n_2) \in \Z^2_{\ge 0}$, an asymptotic expansion
$\operatorname{App}^{<N}(F; z)$ is given by
$$
\begin{aligned}
	T_{\{1\}}^{<N}(F;\, z) &= \sum_{k < n_1} f_{\{1\},k}(z_2, z')\frac{z_1^k}{k!}, \\
	T_{\{2\}}^{<N}(F;\, z) &= \sum_{k < n_2} f_{\{2\},k}(z_1, z')\frac{z_2^k}{k!}, \\
	T_{\{1,2\}}^{<N}(F;\, z) &= \sum_{\alpha_1 < n_1, \alpha_2 < n_2}
	f_{\{1,2\},\alpha}(z') \frac{z_1^{\alpha_1}z_2^{\alpha_2}}{\alpha_1! \alpha_2!} \\
	\operatorname{App}^{<N}(F;\, z) &=
T_{\{1\}}^{<N}(F;\, z) + T_{\{2\}}^{<N}(F;\, z) - T_{\{1,2\}}^{<N}(F;\, z).
\end{aligned}
$$
As $w_j(N) = n_j$ and $\operatorname{dist}(z, Z_j) = \vert z_j \vert$ $(j=1,2)$,
a holomorphic function $f$ is multi-asymptotically developable to $F$ if,
for any poly-sector $S'$ properly contained in $S$ and
for any $N=(n_1, n_2) \in \Z_{\ge 0}^2$,
there exists a positive constant $C_{S',N}$ such that
$$
\left|
f(z) - \operatorname{App}^{<N}(F;\, z)
\right|
\le C_{S', N} \vert z_1 \vert^{n_1} \vert z_2 \vert^{n_2}
\qquad (z \in S').
$$
Hence our definition coincides with that of strongly asymptotic
developability established by Majima in \cite{Ma84}.
\end{es}

\begin{es}
Let $X=\mathbb C^n$ with the coordinates
$(z_1, z_2, \dots, z_n) = (z_1, z_2, z')$, and let
$Z_1 = \{z_1 = 0\}$ and $Z_2 = \{z_1 = z_2 = 0\}$.
In this case,  $I_1$ and $I_2$ are given by
$\{1\}$ and $\{1,2\}$ respectively, and we have
$\hat{I}_1 = \{1\}$ and $\hat{I}_2 = \{2\}$.
Let $G_1$ and $G_2$ be proper open sectors in $\CC$.
We set $U_R:=B^2_R \times B^{n-2}_R$, which is a $\mu$-star shape.
The multi-cone $S=S(U_R,\{G_1, G_2\}, \epsilon)$ is defined by
$$
\{(z_1, z_2, z') \in (G_1 \times G_2 \times \CC^{n-2}_{z'}) \cap U;\,
\vert z_1 \vert < \epsilon \vert z_2 \vert \}.
$$
Then we have
$$
S_1 = (G_2 \cap B^1_R) \times B^{n-2}_R,\,
S_2 = S_{\{1,2\}} = B^{n-2}_R.
$$
Let $F$ be a total family of coefficients of multi-asymptotic expansion, which consists of
$$
\begin{aligned}
F &= (F_{\{1\}}, F_{\{2\}}, F_{\{1,2,\}}) \\
&=\left(
\left\{f_{\{1\},k}(z_2, z')\right\}_{k \ge 0},\,
\left\{f_{\{2\},\alpha}(z')\right\}_{\alpha \in \Z^2_{\ge 0}},\,
\left\{f_{\{1,2\},\alpha}(z')\right\}_{\alpha \in \Z^2_{\ge 0}}\right)
\end{aligned}
$$
where $f_{\{1\},k}$ $($resp. $f_{\{2\},\alpha}$ and $f_{\{1,2,\},\alpha}$$)$ is
a holomorphic function on $S_1$ $($resp. $S_2$ = $S_{\{1,2\}}$$)$ respectively.
Then we have, for $N=(n_1, n_2) \in \Z^2_{\ge 0}$,
$$
\begin{aligned}
	T_{\{1\}}^{<N}(F;\, z) &= \sum_{k < n_1} f_{\{1\},k}(z_2, z')\frac{z_1^k}{k!}, \\
	T_{\{2\}}^{<N}(F;\, z) &= \sum_{\alpha_1 + \alpha_2 < n_2}
	 f_{\{2\},\alpha}(z')\frac{z_1^{\alpha_1}z_2^{\alpha_2}}{\alpha_1!\alpha_2!}, \\
	T_{\{1,2\}}^{<N}(F;\, z) &= \sum_{\alpha_1 < n_1,\, \alpha_1 + \alpha_2 < n_2}
	f_{\{1,2\},\alpha}(z') \frac{z_1^{\alpha_1}z_2^{\alpha_2}}{\alpha_1! \alpha_2!} \\
	\operatorname{App}^{<N}(F;\, z) &=
T_{\{1\}}^{<N}(F;\, z) + T_{\{2\}}^{<N}(F;\, z) - T_{\{1,2\}}^{<N}(F;\, z).
\end{aligned}
$$
It follows from the definition that we have
$$
w_1(N) = n_1 \text{ and } w_2(N) = n_2 - n_1
$$
and
$$
\operatorname{dist}(z, Z_1) = \vert z_1 \vert
\text{ and }
\operatorname{dist}(z, Z_2) \simeq \vert z_1 \vert + \vert z_2 \vert.
$$
Hence a holomorphic function $f$ is multi-asymptotically developable to $F$ if,
for any proper convex multi-cone $S'$ properly contained in $S$ and
for any $N=(n_1, n_2) \in \Z_{\ge 0}^2$,
there exists a positive constant $C_{S',N}$ such that
$$
\left|
f(z) - \operatorname{App}^{<N}(F;\, z)
\right|
\le C_{S', N} \vert z_1 \vert^{n_1}
(\vert z_1 \vert + \vert z_2 \vert)^{n_2 - n_1}
\qquad (z \in S').
$$
\end{es}

One of important features of multi-asymptoticity is stability for differentiations.
\begin{prop}{\label{prop:multi-asymptotic-derivative}}
Let $S:=(U, \{G_i\},\epsilon)$ be a proper convex multi-cone in $X$ and $f$ a holomorphic function
on $S$. If $f(z)$ is multi-asymptotically developable along
${\underset{1 \le j \le \ell}{\bigcup} Z_j}$ on
$S$, then any derivative of $f$ is also multi-asymptotically developable
along ${\underset{1 \le j \le \ell}{\bigcup} Z_j}$ on $S$.
\end{prop}
\dim\ \
It suffices to show that, for an $i \in \{1,2,\dots, n\}$,
$\displaystyle\frac{\partial f}{\partial z_i}$
is also multi-asymptotically developable. We need the lemma below.
Let $F:=\{F_{J}\}_{J \in \Powl}$
be a total family of coefficients of multi-asymptotic expansion with
$F_J = \{f_{J,\alpha}\}_{\alpha \in \Z_{\ge 0}^{I_J}}$.
\begin{lem}{\label{lemma:multi-asymptotic-derivative}}
Let $N=(n_1, n_2, \dots, n_\ell) \in \Z_{\ge 0}^\ell$.  Then we have
$$
\frac{\partial} {\partial z_i}
\operatorname{App}^{<N_+}(F;\, z)
= \operatorname{App}^{<N}(F';\, z).
$$
Here $N_+=(n'_1,\dots,n'_\ell)$ is determined by $n'_j = n_j + 1$ for $j$ with $i \in I_j$
and $n'_j = n_j$ otherwise. The total family $F'=\{F'_J\}$
is given by
$$
F'_J := \{f'_{J,\alpha}\} =
\left\{\displaystyle\frac{\partial f_{J,\alpha}}{\partial z_i}\right\}
\qquad (\text{for $J \in \Powl$ with $i \notin I_J$})
$$
and
$$
F'_J := \{f'_{J,\alpha}\} := \{f_{J, \alpha + e_{i}}\}
\qquad(\text{for $J \in \Powl$ with $i \in I_J$})
$$
where $e_i \in \Z_{\ge 0}^n$ is the unit vector whose $i$-th element is equal to $1$.
\end{lem}
\dim\ \
Let $J = (j_1, j_2, \dots, j_m) \in \Powl$.
For $J$ with $i \notin I_J$, as the polynomial $T_J^{<N}$ does not contain
the variable $z_i$ and $n'_j = n_j$ for $j \in J$, we have
$$
\frac{\partial}{\partial {z_i}} T^{<N_+}_J(F;\, z)
=
\frac{\partial}{\partial {z_i}} \tau_{F, J}\left(T^{<N_+}_J\right)
=
\tau_{F', J}\left(T^{<N_+}_J\right)
=
T^{<N}_J(F';\, z).
$$

Assume that $J$ satisfies $i \in I_J$.
Then, by the proof of Lemma \ref{lem:T_N-polynomial}, we have
$$
\frac{\partial}{\partial {z_i}} T^{<N_+}_J = T^{<N}_J.
$$
As
$$
\displaystyle\frac{\partial}{\partial z_i}\tau_{F,J}(p) =
\tau_{F',J}\left(\displaystyle\frac{\partial p}{\partial z_i}\right)
$$ holds for a polynomial $p(z_{I_J})$ of the variables $z_{I_J}$,
we have obtained
$$
\begin{aligned}
\frac{\partial}{\partial {z_i}} T^{<N_+}_J(F;\, z) &=
\frac{\partial}{\partial {z_i}} \tau_{F, J}(T^{<N_+}_J) =
\tau_{F', J}\left(\frac{\partial}{\partial {z_i}} T^{<N_+}_J\right) \\
&= \tau_{F', J}\left(T^{<N}_J\right)
= T^{<N}_J(F';\, z).
\end{aligned}
$$
\qed

We continue the proof of the proposition.
It suffices to consider the case $i \in I_{\{1,2,\dots,\ell\}}$.
By the lemma, we have
$$
\left|
\frac{\partial f}{\partial z_i} - \operatorname{App}^{<N}(F';\, z) \right|
=
\left| \frac{\partial}{\partial z_i}
\left( f(z) - \operatorname{App}^{<N_+}(F;\, z) \right) \right|
$$
Let $j_0$ be a unique integer with $i \in \hat{I}_{j_0}$ and
$S':=S(U', \{G'_j\}, \epsilon')$
(resp. $S'':=S(U'', \{G''_j\}, \epsilon'')$)
a proper convex multi-cone properly
contained in $S$ (resp. $S'$). Then, as
$\operatorname{dist}(z, Z_{j_0}) \le \epsilon''
\operatorname{dist}(z, Z_j)$ ($z \in S''$) holds
for $Z_j \subset Z_{j_0}$,
there exists a positive constant $\kappa > 0$
such that for any point $z^* \in S''$
\begin{equation}{\label{diff-caucy-domain}}
z^* + \left\{z \in \CC^n;\, z_k = 0\, (k \ne i), \,
\vert z_i \vert \le \kappa \min\{1,\, \operatorname{dist}(z^*, Z_{j_0})\}
\right\}
\subset S'.
\end{equation}
Since $f(z)$ is multi-asymptotically developable, there exists a constant
$C_{S', N_+}$ satisfying
$$
\left| h(z):=f(z) - \operatorname{App}^{<N_+}(F;\, z) \right|
\le C_{S', N_+}
\underset{1 \le j \le \ell}{\prod}
\operatorname{dist}(z,\, Z_j)^{w_j(N_+)} \qquad (z \in S').
$$
Taking (\ref{diff-caucy-domain}) into account
and applying Cauchy-integral formula to $h(z)$, we get
$$
\left| \frac{\partial h(z)}{\partial z_i} \right| \le
C\frac{
\underset{1 \le j \le \ell}{\prod}
\operatorname{dist}(z,\, Z_j)^{w_j(N_+)}}
{\operatorname{dist}(z, Z_{j_0})} \qquad (z' \in S'')
$$
for a constant $C > 0$.
As $w_{j_0}(N) = w_{j_0}(N_+) - 1$ and $w_j(N) = w_j(N_+)$ for $j \ne j_0$,
we finally obtained
$$
\left|
\frac{\partial f}{\partial z_i} - \operatorname{App}^{<N}(F';\, z) \right|
\le C
\underset{1 \le j \le \ell}{\prod} \operatorname{dist}(z,\, Z_j)^{w_j(N)}
\qquad (z \in S'').
$$
This completes the proof.
\qed

By this proposition and integration by parts, we can obtain the following theorem.
\begin{teo}{\label{theorem:multi-asymptotic}}
Let $S:=S(U, \{G_i\}, \epsilon)$ be a proper convex multi-cone and $f$ a holomorphic
function on $S$. Then the following conditions are equivalent.
\begin{enumerate}
\item $f$ is multi-asymptotically developable along ${\underset{1 \le j \le \ell}{\bigcup} Z_j}$
on $S$.
\item  For any open proper convex multi-cone $S':=S(U', \{G'_i\}, \epsilon')$ properly
contained in $S$ and for any $\alpha \in \Z_{\ge 0}^n$,
$\left|\displaystyle\frac{\partial^\alpha f}{\partial z_\alpha}\right|$ is
bounded on $S'$.
\item For any open proper convex multi-cone $S':=S(U', \{G'_i\}, \epsilon')$ properly
contained in $S$,
the holomorphic function $f\vert_{S'}$ on $S'$ can be extended to a $\C^\infty$-function
on $X_{\mathbb R}$ ($X_{\mathbb R}$ denotes the underlying real analytic manifold of $X$).
\end{enumerate}
\end{teo}
\dim\ \
We first show 1.~implies 2.
By (\ref{eq:multi-asymptotic-formula}) with $N=(0,\dots,0)$, we
obtain that $f$ is bounded on $S'$. Since each higher derivative of $f$
is still multi-asymptotically developable thanks to
Proposition \ref{prop:multi-asymptotic-derivative},
$\displaystyle\frac{\partial^\alpha f}{\partial z^\alpha}$
is also bounded for any
$\alpha \in \Z^n_{\ge 0}$ on $S'$.

\medskip
As $S'$ is convex and as $f$ is holomorphic (i.e.
$\displaystyle\frac{\partial^\alpha f}{\partial \bar{z}^\alpha} = 0$),
the claim 3.~follows from 2.~by the result of Whitney in \cite{Wh34}.
Clearly 3.~implies 2. Hence the claim 2.~and 3.~are equivalent.

\medskip

Now we will show 3.~implies 1.
Assume that $f$ satisfies 3. In particular, any derivative of $f$
extends to $\overline{S'}$ and is bounded on $\overline{S'}$.
It follows from Lemma {\ref{lemma:mutli-cone-geometric}} that
for $z \in S'$ and $0 \le \lambda_j \le 1$ ($j=1,2,\dots,\ell$),
we get $\mu(z,\lambda) \in \overline{S'}$. Therefore,
for $N = (n_1, \dots, n_\ell) \in \Z^\ell_{\ge 0}$,
$$
\varphi_N(f;z) :=
\int_{0}^{\infty}\dots\int_{0}^{\infty}
\frac{\partial^N f(\mu(z,\lambda))}{\partial \lambda^N}
\underset{1 \le j \le \ell}{\prod}
K_{n_j - 1}(1 - \lambda_j) d\lambda_1 \dots d\lambda_\ell
$$
where
$$
K_n(t) :=
\left\{
\begin{aligned}
\delta(t)\qquad &(n = -1), \\
\frac{t_+^n}{n!}      \qquad &(n \ge 0)
\end{aligned}
\right.
$$
is well-defined on $S'$. We define
$e_J = (e_{J,1}, \dots, e_{J, \ell}) \in \Z^\ell$
by $e_{J, j} = 0$ if $j \in J$
and $e_{J, j} = 1$ otherwise. Then, by integration by parts, we have
$$
\varphi_N(f;z) = f(z) - \sum_{J \in \Powl}
(-1)^{\#J + 1}\sum_{\beta \in \Z^{J}, \beta<_J N}
\left.
\frac{1}{\beta!}\partial^\beta_{\lambda, J}f(\mu(z,\lambda))
\right\vert_{\lambda = e_J}.
$$
Hence it suffices to show  that $\varphi_N(f; z)$ has an estimation
which appears in a multi-asymptotic expansion.
Let us consider coordinates transformation of $\lambda$ as
$\lambda_j = \nu_j(z) \tilde{\lambda}_j$ ($j=1,2,\dots,\ell$) where
$\nu_j(z)$ is given by
$$
\nu_j (z):= \left\{
\begin{aligned}
	\frac{1}{\vert z_{I_j} \vert} \qquad&
	(J_{\subsetneq Z_j} = \emptyset), \\
	\frac{\vert z_{I_k} \vert }{\vert z_{I_j} \vert}
	\qquad&  (k \in J_{\subsetneq Z_j}).
\end{aligned}
\right.
$$
Note that $J_{\subsetneq Z_j}$ consists of at most one element.
Then we have
$$
\frac{\partial^N f(\mu(z,\lambda))}{\partial \lambda^N} =
\frac{1}{\underset{1\le j \le \ell}{\prod} \nu_j^{n_j}(z)}
\frac{\partial^N }{\partial \tilde{\lambda}^N}
f\left(\frac{z_1}{\vert z_{I_{j(1)}} \vert} \tilde{\lambda}_{J_1},
\dots,
\frac{z_n}{\vert z_{I_{j(n)}} \vert}  \tilde{\lambda}_{J_n}\right).
$$
Here, for $k \in \{1,2,\dots,n\}$, we denote by $j(k)$ the integer $j$
that satisfies $k \in \hat{I}_{j}$ and we set
$J_k = \{j \in \Z_\ell;\, k \in I_j\}$.
Since $\displaystyle{\frac{1}{\nu_j(z)}}$ is bounded on $S'$,
$\tilde{\lambda}_j$ is also bounded when
$0 \le \lambda_j \le 1$. Hence we get
$$
\sup_{\lambda \in [0,1]^\ell }
\left| \frac{\partial^N f(\mu(z,\lambda))}{\partial \lambda^N} \right| \le
\frac{C}{\underset{1\le j \le \ell}{\prod} \nu^{n_j}_j(z)} \qquad (z \in S')
$$
for some constant $C > 0$. By noticing
$$
\frac{1}{\underset{1\le j \le \ell}{\prod} \nu_j^{n_j}(z)} =
\underset{1\le j \le \ell}{\prod} \operatorname{dist}(z, Z_j)^{w_j(N)}
$$
we have obtained a multi-asymptotic expansion of $f$ with desired estimation.
The proof has been completed.
\qed

Let $S:=S(U, \{G_i\}, \epsilon)$ be a proper convex multi-cone
and $f$ a holomorphic function that is multi-asymptotically developable
along ${\underset{1 \le j \le \ell}{\bigcup} Z_j}$ on $S$.
Let $S^{(k)} := S(U^{(k)}, \{G^{(k)}_{i}\}, \epsilon^{(k)})$ ($k=1,2,3,\dots$)
be a family of
proper convex multi-cone properly contained in $S$, which satisfies
$$
S^{(1)} \subset S^{(2)} \subset S^{(3)} \subset \dots \text{ and }
\underset{k \ge 1}{\bigcup} S^{(k)} = S.
$$

Let $J \in \Powl$.
Then
$\left.
\displaystyle\frac{\partial^\alpha f}{\partial z^\alpha}
\right|_{S^{(k)}}$
has a unique Whitney extension $\varphi^{(k)}$ on $\overline{S^{(k)}}$
by the above theorem. As $\underset{k \ge 1}{\bigcup} S^{(k)}_J = S_J$,
a family $\left\{\varphi^{(k)}\vert_{S^{(k)}_J}\right\}$ determines
a function on $S_J$,  which is holomorphic.
We denote it by
$\left.
\displaystyle\frac{\partial^\alpha f}{\partial z^\alpha}
\right|_{S_J}.
$
Note that, as $S_J$ is a multi-cone in $Z_J$ for a family of submanifolds
$\chi_J := \{Z_j \cap Z_J\}_{j \in J^*}$
where $J^*$ is given by
$\{j \in \Z_\ell; Z_J \nsubseteq Z_j\}$,
it follows again from the theorem that the holomorphic function
$
\left.
\displaystyle\frac{\partial^\alpha f}{\partial z^\alpha}
\right|_{S_J}
$
is multi-asymptotically developable along
$\underset{j\in J^*}{\bigcup} (Z_j \cap Z_J)$ on $S_J$.

\begin{prop}{\label{prop:consistent-coeffs}}
Let $S:=S(U, \{G_i\}, \epsilon)$ be a proper convex multi-cone and
$f$ a holomorphic function on $S$. Assume that $f$ is multi-asymptotically
developable to a family
$F:=\{F_J\}_{J \in \Powl}$
of coefficients of multi-asymptotic expansion with
$F_J =\{f_{J,\alpha}\}_{\alpha \in \Z^{I_J}_{\ge 0}}$. Then we have
\begin{equation}{\label{eq:prop-boundary}}
\left.\frac{\partial^\alpha f}{\partial z^\alpha} \right|_{S_J} = f_{J, \alpha}
\qquad (J \in \Powl,\, \alpha \in \Z^{I_J}_{\ge 0}).
\end{equation}
In particular, coefficients of multi-asymptotic expansion are unique and they are
multi-asymptotically developable functions themselves.
\end{prop}
\dim\ \
It suffices to show the proposition for a proper convex multi-cone
$S':=S(U', \{G'_i\}, \epsilon')$ that is properly contained in $S$.
Moreover, by Proposition \ref{prop:multi-asymptotic-derivative} and
Lemma \ref{lemma:multi-asymptotic-derivative},
it enough to show (\ref{eq:prop-boundary}) with $\alpha = 0$.
We prove the lemma by induction with respect to the number of elements in $J$.
We use the same symbol $f$ for a unique continuous extension of $f$
on $\overline{S'}$ in what follows.

Let $J=\{j\}$.
By taking $N=(n_1,n_2,\dots,n_\ell)$ with $n_j = 1$ and
$n_k = 0$ ($k \ne j$), we have
\begin{equation}{\label{eq:prop-xxx-xxx-xxx}}
\vert f(z) - f_{J,0}(z_{I_J}) \vert \le C
\operatorname{dist}(z, Z_j)
\underset{k \in J_{\subsetneq Z_j}}{\prod}
\frac{1}{\operatorname{dist}(z, Z_k)}.
\end{equation}
For $\tilde{z}^* \in S'_J$, by Lemma \ref{lemma:mutli-cone-geometric},
we can find a point $z^* \in S'$ with $\mu_j(z^*,0) = \tilde{z}^*$.
Then there exists $\delta > 0$
such that $\operatorname{dist}(\mu_j(z^*,[0,1]), Z_k) > \delta$
for $k$ with $Z_k \subsetneq Z_j$ because of $\tilde{z}^* \notin Z_k$
(2.~of Lemma \ref{lemma:fundamental-geo}).
Hence, by putting $z = \mu(z^*,\lambda)$ into (\ref{eq:prop-xxx-xxx-xxx})
and letting $\lambda \to 0^+$, we get $f(\tilde{z}^*) = f_{J,0}(\tilde{z}^*)$.
This shows (\ref{eq:prop-boundary}) with $\alpha = 0$.

Let $J \in \Powl$ with $\#J > 1$,
and let
$J^*$ be a subset of $J$ consisting of $j \in J$
such that $Z_j$ is a minimal submanifold
in $\{Z_k\}_{k \in J}$ with respect to the order $\subset$.
Set $N=(n_1,n_2,\dots,n_\ell)$ with $n_j = 1$ if $j \in J$ and $n_j = 0$
otherwise. Then, by noticing
$\operatorname{dist}(z, Z_k) \le \epsilon' \operatorname{dist}(z, Z_j)$
if $Z_j \subsetneq Z_k$,
we have
\begin{equation}{\label{eq:tmp-tmp-xxx-yyy}}
\begin{aligned}
\Biggl\vert f(z) + \sum_{J' \subsetneq J, J' \ne \emptyset}
&(-1)^{\#J'}f_{J',0}(z_{I_{J'}})
+ (-1)^{\#J}f_{J,0}(z_{I_J}) \Biggr\vert  \\
&\le C
\underset{j \in J^*}{\prod}
\frac{\operatorname{dist}(z, Z_j)}
{\underset{k \in J_{\subsetneq Z_j}}{\prod}
\operatorname{dist}(z, Z_k)}.
\end{aligned}
\end{equation}
By induction hypothesis, for $J' \subsetneq J$, the coefficient
$f_{J',0}$ also has a unique continuous extension on $\overline{S'_{J'}}$ satisfying
$f_{J',0} = f\vert_{\overline{S'_{J'}}}$.
Hence, noticing $S'_J \subset \overline{S'_{J'}} \cap Z_J$
due to 2.~of Lemma \ref{lemma:fundamental-geo},
we have
$$
f(z) + \sum_{J' \subsetneq J, J' \ne \emptyset}(-1)^{\#J'}f_{J',0}(z_{I_{J'}})
= (-1)^{\#J+1}f(z)\qquad z \in S'_J.
$$
Let $\tilde{z} \in S'_J$ and $z^*$ a point in $S$ with
$\mu_J(0,\dots,0; z^*) = \tilde{z}$. Then,
by putting $z = \mu_J(z^*,\lambda,\dots,\lambda)$ into
(\ref{eq:tmp-tmp-xxx-yyy}) and letting $\lambda \to 0^+$, as
the right hand side of (\ref{eq:tmp-tmp-xxx-yyy}) tends to 0
by the same reason as that for $\#J = 1$,
we have $f_{J,0}(\tilde{z}) = f(\tilde{z})$.
This completes the proof.
\qed

Let us extend the notion of {\it{a consistent family of
coefficients of strongly asymptotic expansion}} defined by Majima in \cite{Ma84}
to our case.
Let $S:=S(U, \{G_i\}, \epsilon)$ be a proper convex multi-cone and
$F:=\{F_J\}_{J \in \Powl}$
a family of  coefficients of multi-asymptotic expansion with
$F_J =\{f_{J,\alpha}\}_{\alpha \in \Z^{I_J}_{\ge 0}}$.
For a proper convex cone $S':=S(U', \{G'_i\}, \epsilon')$ properly
contained in $S$, we can consider the natural restriction $F\vert_{S'}$ of $F$
to $S'$. Moreover we can also define the restriction of $F$ to a submanifold.
Let $J \in \Powl$ and $\alpha \in \Z_{\ge 0}^{I_J}$.
Then the restriction $F\vert_{J,\alpha}$ of $F$ to $Z_J$ is defined as follows.

Let $J^* := \{j \in \Z_\ell; Z_J \nsubseteq Z_j\}$ and
$I^*_j = I_j \setminus I_J$ ($j \in J^*$).
We assume $J^* \ne \emptyset$ and
denote by $S^*$ the proper convex multi-cone $S_J$ in $Z_J$.

Then for $K \in \P(J^*) \setminus \emptyset$,
a family
$F^*_{K, \alpha} := \{f^*_{K, \alpha, \gamma} \}_{\gamma \in \Z_{\ge 0}^{I^*_K}}$
of holomorphic functions on $S^*_K \subset Z_J$
is defined by
$$
f^*_{K, \alpha, \gamma} = f_{J \cup K, (\alpha, \gamma)}\vert_{Z_J}
$$
where $(\alpha, \gamma) \in \Z^{I_J}_{\ge 0} \times \Z^{I^*_K}_{\ge 0}$
can be considered as an element in $\Z^{I_{J \cup K}}_{\ge 0}$ by
the identification $\Z^{I_J} \times \Z^{I^*_K} = \Z^{I_{J \cup K}}$.
Then the restriction $F\vert_{J,\alpha}$ of $F$ is defined by a family
$\{F^*_{K,\alpha}\}_{K \in \P(J^*) \setminus \emptyset}$.

The restriction $F\vert_{J,\alpha}$ gives a family of coefficients of multi-asymptotic expansion along
submanifolds $\{Z_j \cap Z_J\}_{j \in J^*}$ in $Z_J$ on the proper convex
multi-cone $S^* = S(U \cap Z_J, \{G_j\}_{j \in J^*}, \epsilon) \subset Z_J$.
\begin{df}
Let $S:=S(U, \{G_i\}, \epsilon)$ be a proper convex multi-cone and
$F := \{F_J\}_{J \in \Powl}$
a family of  coefficients of multi-asymptotic expansion with
$F_J =\{f_{J,\alpha}\}_{\alpha \in \Z^{I_J}_{\ge 0}}$.

We say that $F$ is a consistent family of coefficients of multi-asymptotic expansion
along ${\underset{1 \le j \le \ell}{\bigcup} Z_j}$ on $S$ if the following conditions are satisfied.
\begin{enumerate}
\item For any $J \in \Powl$ with
$J^* := \{j \in \Z_\ell; Z_J \nsubseteq Z_j\} \ne \emptyset$
and for any $\alpha \in \Z^{I_J}_{\ge 0}$,
a holomorphic function $f_{J,\alpha}$ on $S_J$
is multi-asymptotically developable to the family $F\vert_{J,\alpha}$ along
$\{Z_j \cap Z_J\}_{j \in J^*}$ on $S_J$.
\item For any $J$ and $J' \in \Powl$ with $S_J = S_{J'}$, we have $F_J = F_{J'}$.
\end{enumerate}

\end{df}

The following corollary immediately follows from
Theorem \ref{theorem:multi-asymptotic} and
Proposition \ref{prop:consistent-coeffs}.
\begin{cor}
Let $S:=S(U, \{G_i\}, \epsilon)$ be a proper convex multi-cone and
$F$ a family of  coefficients of multi-asymptotic expansion along ${\underset{1 \le j \le \ell}{\bigcup} Z_j}$
on $S$. If some holomorphic function on $S$ is multi-asymptotically developable to $F$,
then $F$ is a consistent family of multi-asymptotic expansion.
\end{cor}

Let $S':=S(U', \{G'_i\}, \epsilon')$ be a proper convex multi-cone
properly contained in $S$, and let $F:= \{F_J\}$ be a consistent family
on $S$ with $F_J = \{f_{J,\alpha}\}_{\alpha \in \Z^{I_J}_{\ge 0}}$.

By Theorem \ref{theorem:multi-asymptotic}, each $F_J$ can be regarded
as a $  \C^\infty$-Whitney jet on $\overline{S'_J}$.
(see \cite{M66} for Malgrange's definition of a $\C^\infty$-Whitney jet).
As $\overline{S'_J} \cap \overline{S'_{J'}} = \overline{S'_{J \cup J'}}$ holds
($J$, $J'$ $\in \hat{\P}(\Z_\ell)$),
Whitney jets defined by $F_J$ and $F_{J'}$ coincide
on the set $\overline{S'_{J \cup J'}}$
by Proposition \ref{prop:consistent-coeffs}.

Therefore, as $\underset{J \in \hat{\P}(\Z_\ell)}{\bigcup} \overline{S'_J} =
\overline{S'} \cap \left(\underset{1\le j \le \ell}{\bigcup} Z_j\right)$ holds by Lemma \ref{lemma:fundamental-geo},
it follows from Theorem 5.5 of \cite{M66} that we obtain the Whitney
jet defined on $\overline{S'} \cap \left(\underset{1\le j \le \ell}{\bigcup} Z_j\right)$
whose restriction to $\overline{S'_J}$ is equal to one
defined by $F_J$.
Hence we have obtained the following proposition.

\begin{prop}{\label{prop:consisten-equiv}}
Let $S:=S(U, \{G_i\}, \epsilon)$ be a proper convex multi-cone and
$F:= \{F_J\}$ a family of  coefficients of multi-asymptotic expansion along
${\underset{1 \le j \le \ell}{\bigcup} Z_j}$ on $S$ with
$F_J = \{f_{J,\alpha}\}_{\alpha \in \Z^{I_J}_{\ge 0}}$.
Then the following conditions are equivalent.
\begin{enumerate}
\item $F$ is consistent.
\item For any proper convex cone $S':=S(U',\{G_i\}, \epsilon')$ properly contained in $S$,
the restriction $F\vert_{S'}$ of $F$ to $S'$ can be
extended to a $\C^\infty$-function on $X_{\R}$, that is,
there exists a $\C^\infty$-function $\varphi(z)$ on $X_{\R}$ such that for any
$J \in \Powl$
and $\alpha \in \Z^{I_J}_{\ge 0}$ we have
$
f_{J, \alpha} =
\left. \displaystyle\frac{\partial^\alpha \varphi}
{\partial z^\alpha}\right|_{S'_J}
$
for $z \in S'_J$.
\end{enumerate}
\end{prop}

\end{section}

\begin{section}{Multi-specialization and asymptotic expansions}\label{8}

Let $X$ be a real analytic manifold. We consider a slight generalization of the sheaf of Whitney $\C^\infty$-functions of \cite{KS01}. As usual, given $F \in D^b(\CC_{X})$ we set $D'F=\rh(F,\CC_X)$. Remember that an open subset $U$ of $X$ is locally cohomologically trivial (l.c.t. for short) if $D'\CC_U \simeq \CC_{\overline{U}}$.

\begin{df} Let $F \in \mod_{\rc}(\CC_X)$ and let $U \in \op(X_{sa})$. We define the presheaf $\CW_{X|F}$ as follows:
$$U \mapsto \Gamma(X;H^0D'\CC_U \otimes F \wtens \C^\infty_X).$$
\end{df}
Let $U,V \in \op(X_{sa})$, and consider the exact sequence
$$\exs{\CC_{U\cap V}}{\CC_U \oplus \CC_V}{\CC_{U\cup V}},$$
applying the functor $\ho(\cdot,\CC_X)=H^0D'(\cdot)$ we obtain
$$\lexs{H^0D'\CC_{U\cap V}}{H^0D'\CC_U \oplus H^0D'\CC_V}{H^0D'\CC_{U\cup V}},$$
applying the exact functors $\cdot \otimes F$, $\cdot\wtens \C^\infty_M$ and taking global sections
we obtain
$$\lexs{\CW_{X|F}(U\cup V)}{\CW_{X|F}(U) \oplus \CW_{X|F}(V)}{\CW_{X|F}(U\cap V)}.$$
This implies that $\CW_{X|F}$ is a sheaf on $X_{sa}$. Moreover if $U \in \op(X_{sa})$ is l.c.t., the morphism $\Gamma(X;\CW_X) \to \Gamma(U;\CW_{X|F})$ is surjective and $R\Gamma(U;\CW_{X|F})$ is concentrated in degree zero. Let $\exs{F}{G}{H}$ be an exact sequence in $\mod_{\rc}(\CC_X)$, we obtain an exact sequence in $\mod(\CC_{X_{sa}})$
\begin{equation}\label{exsFGH}
\exs{\CW_{X|F}}{\CW_{X|G}}{\CW_{X|H}}.
\end{equation}

We can easily extend the sheaf $\CW_{X|F}$ to the case of $F \in D^b_{\rc}(\CC_X)$, taking a finite resolution of $F$ consisting of locally finite sums $\oplus \CC_V$, with $V$ l.c.t. in $\op^c(X_{sa})$. In fact, the sheaves $\CW_{X|\oplus \CC_V}$ form a complex quasi-isomorphic to $\CW_{X|F}$ consisting of acyclic objects with respect to $\Gamma(U;\cdot)$, where $U$ is l.c.t. in $\op^c(X_{sa})$.

As in the case of Whitney $\C^\infty$-functions one can prove that, if $G \in D^b_{\rc}(\CC_X)$ one has
$$\imin \rho \rh(G,\CW_{F|X}) \simeq D'G \otimes F \wtens \C^\infty_X.$$

\begin{es} Setting $F=\CC_X$ we obtain the sheaf of Whitney $\C^\infty$-functions. Let $Z$ be a closed subanalytic subset of $X$. Then $\CW_{X|\CC_{X\setminus Z}}$ is the sheaf of Whitney $\C^\infty$-functions vanishing on $Z$ with all their derivatives.
\end{es}

\begin{nt} Let $S$ be a locally closed subanalytic subset of $X$. We set for short $\CW_{X|S}$ instead of $\CW_{X|\CC_S}$.
\end{nt}

Let $\chi=\{M_1,\ldots,M_\ell\}$ be a family of closed analytic submanifolds of $X$ satisfying H1-H2-H3, set $M= \bigcup_{i=1}^\ell M_i$ and consider $\widetilde{X}$. Consider the diagram \eqref{multi normal deformation}.

Set $F=\CC_{X\setminus M}$, $G=\CC_X$, $H=\CC_M$ in \eqref{exsFGH}. The exact sequence
$$\exs{\CW_{X|X\setminus M}}{\CW_X}{\CW_{X|M}}$$
induces an exact sequence
\begin{equation}\label{asym}
\exs{\nu^{sa}_\chi\CW_{X|X\setminus M}}{\nu^{sa}_\chi\CW_X}{\nu^{sa}_\chi \CW_{X|M}},
\end{equation}
in fact let $V$ be a l.c.t. conic subanalytic subset of $\widetilde{X}$ and $U \in \op(X_{sa})$ such that
$C_\chi(X \setminus U) \cap V=\varnothing$, then we can find a l.c.t. $U'\subset U$ satisfying the same property.

Applying the functor $\imin \rho$ to the exact sequence \eqref{asym} we obtain the exact sequence
$$
\exs{\imin \rho\nu^{sa}_\chi\CW_{X|X\setminus M}}{\imin \rho\nu^{sa}_\chi\CW_X}{\imin \rho \nu^{sa}_\chi \CW_{X|M}},
$$
where the surjective arrow is the map which associates to a function its asymptotic expansion.\\

Let $X$ be a complex manifold and let $X_\R$ denote the underlying real analytic manifold
of $X$. Let $\chi=\{Z_1,\ldots,Z_\ell\}$ be a family of complex submanifolds of $X$ satisfying H1-H2-H3 and set $Z= \bigcup_{i=1}^\ell Z_i$. Let $F \in D^b_{\rc}(\CC_X)$. We denote by $\OW_{X|F}$ the sheaf defined as follows:
$$
\OW_{X|F} := \rh_{\rho_!\D_{\overline{X}}}(\rho_!\OO_{\overline{X}},\CW_{X_\R|F}).
 $$
Let $\exs{F}{G}{H}$ be an exact sequence in $\mod_{\rc}(\CC_X)$. Then the exact sequence \eqref{exsFGH} gives rise to the distiguished triangle
\begin{equation}\label{dtFGH}
\dt{\OW_{X|F}}{\OW_{X|G}}{\OW_{X|H}}.
\end{equation}

Setting $F=\CC_{X\setminus Z}$, $G=\CC_X$, $H=\CC_Z$ in \eqref{dtFGH} and applying the functor of specialization, we have the distinguished triangle
\begin{equation}\label{dtColin}
\dt{\imin \rho \nu^{sa}_\chi\OW_{X|X \setminus Z}}{\imin \rho \nu^{sa}_\chi\OWX}{\imin \rho \nu^{sa}_\chi \OW_{X|Z}}.
\end{equation}
The sheaf $\imin \rho \nu^{sa}_\chi\OWX$ is concentrated in degree zero.
This follows from the following result of \cite{DS96}: if $U \in \op(X_{sa})$ is convex, then $R\Gamma(X;\CC_{\overline{U}} \wtens \OO_X)$ is concentrated in degree zero.

\begin{prop}\label{vanishing} Outside the zero section, the flat specialization along
$Z := Z_1 \cup \cdots \cup Z_\ell$
$$
\imin\rho\nu_\chi^{sa}\OW_{X|X \setminus Z}
$$
is concentrated in degree zero.
\end{prop}
\dim\ \
Set
$$I_j = \{i_{j,1},\dots,i_{j,m_j}\} \ \ (j = 1, 2, \dots , \ell).$$
Moreover, by a rotation of the coordinates, we may assume that
$$
\hat{I}_j = \{i_{j,1},\dots,i_{j,m'_j}\} \ \ (m'_j \leq m_j).
$$
We can identify $\CC^{{\hat{I}_j}}$ with $(T_{Z_j} \iota(Z_j))_0$. Let $p_j = (1, 0, \dots , 0) \in (T_{Z_j} \iota(Z_j))_0$.
Under the above identification, $p_j$ is the point in $\CC^{\hat{I}_j}$ with $z_{i_{j,1}} = 1$ and
$z_{i_{j,k}} = 0$  ($1 < k \leq m'_j$).
Let $G_j$ be closed proper convex cone in $\CC^{\hat{I}_j}$ with direction $p_j$, which is defined by
$$
\{(z_{j,1},\dots,z_{j,m'_j}) \in \CC^{\hat{I}_j};\; |z_{j,1}|\leq\epsilon,\; |\arg(z_{j,1})|\leq\epsilon,\; |z_{j,k}|\leq\epsilon|z_{j,1}|,\;
k\geq 2\}.
$$
Let us define $G$ as the product
$$
G:=G_1 \times G_2 \times \cdots \times G_\ell \times B_\epsilon,
$$
where $G_j \in \CC^{\hat{I}_j}$ and $B_\epsilon$ is a closed ball in $\CC^{n-\sum_{j=1}^\ell m'_j}$. Set
\begin{eqnarray*}
T &:=& \{z \in \CC^n;\;z_{i_{1,1}}=0\} \cup \cdots \cup \{z \in \CC^n;\; z_{i_{\ell,1}}=0\} \\
&=& \{z \in \CC^n;\; z_{i_{1,1}} \cdots z_{i_{\ell,1}}=0\}.
\end{eqnarray*}
As
$$
G_j \cap \{(z_{i_{j,1}},\dots,z_{i_{j,m'_j}}) \in \CC^{\hat{I}_j};\;z_{i_{j,1}}=0\} = \{0\}
$$
holds, we have
$$
G \setminus T = (G_1 \setminus \{0\}) \times \cdots \times (G_\ell \setminus \{0\}) \times B_\epsilon.
$$
Therefore, by Proposition 5.4 of \cite{KS96}
$R\Gamma(X,\CC_{G\setminus T} \wtens \OO_X)$ is quasi-isomorphic to
$$
R\Gamma(\CC^{\hat{I}_1};\CC_{G_1\setminus\{0\}}\wtens\OO_X) \widehat{\boxtimes} \cdots \widehat{\boxtimes} R\Gamma(\CC^{\hat{I}_\ell};\CC_{G_\ell\setminus\{0\}}\wtens\OO_X) \widehat{\boxtimes} R\Gamma(B_\epsilon;\CC_{B\epsilon} \wtens \OO_X),
$$
where $\widehat{\boxtimes}$ denotes the topological tensor product of \cite{Gr55}. It is well known that each $R\Gamma(\CC^{\hat{I}_j};\CC_{G_j\setminus\{0\}} \wtens\OO_X)$, $j=1,\dots,\ell$, is concentrated in degree zero. It is a consequence of the Borel-Ritt theorem.\\

Let us consider the the holomorphic map $f : X \to X$
$$
f(z_1,\dots,z_n) = (f_1(z), \dots, f_n(z)) :=
\left(\left( \prod_{j \in \hat{J}_1}z_{i_{j,1}}\right)z_1,\dots,\left(\prod_{j \in \hat{J}_n}z_{i_{j,1}}\right)z_n\right),
$$
where, for $k \in \{1, 2, \dots , n\}$,
$$
\hat{J}_k := \{j \in \{1, 2, \dots , \ell\};\; k \in I_j,\; k \notin \hat{I}_j\}.
$$
Clearly $f|_G$ is a proper map ($G$ is compact). We can also prove that $f$ induces
an isomorphism on $X \setminus T$.

Moreover we have
$$
p \in G \cap T \Longleftrightarrow f(p) \in f(G) \cap (Z_1 \cup \cdots \cup  Z_\ell).
$$
Therefore we get
$$
Rf_!\CC_{G\setminus T} = \CC_{f(G)\setminus (Z_1 \cup \cdots \cup Z_\ell)}.
$$
It follows from Theorem 5.7 of \cite{KS96} that we have
$$
R\Gamma(X;\rh_{\D_X}(\D_{X \stackrel{f}{\to} X},\CC_{G \setminus T} \wtens \OO_X) \simeq R\Gamma(X;\CC_{f(G)\setminus (Z_1 \cup \cdots \cup Z_\ell)} \wtens \OO_X).
$$
Set $\varphi := z_{i_{1,1}}z_{i_{2,1}}\cdots z_{i_{\ell,1}}$.
Let $\OO_{X,\varphi}$ denote the sheaf of meromorphic functions whose poles are contained in $\{\varphi = 0\}$
(i.e. $\OO_{X, \varphi} =
\OO_X[z_{i_{1,1}}^{-1},\, \dots,\, z_{i_{\ell,1}}^{-1}]$) and
we set $\D_{X,\varphi} := \OO_{X, \varphi} \underset{\OO_X}{\otimes} \D_{X}$.
As $\CC_{G\setminus T} \wtens \OO_X$ is a $\D_{X,\varphi}$-module and
$\D_{X \stackrel{f}{\to}X}$
is left
pseudo-coherent over $\D_X$, we have
$$
\rh_{\D_X}(\D_{X \stackrel{f}{\to}X}, \CC_{G\setminus T}
\wtens \OO_X) \simeq  \rh_{\D_{X,\varphi}}(\D_{X,\varphi} \underset{\D_X}{\otimes}
 \D_{X \stackrel{f}{\to}X},\CC_{G\setminus T} \wtens \OO_X).
$$
We also have
$$
\D_{X,\varphi} \underset{\D_X}{\otimes}\D_{X \stackrel{f}{\to}X}\simeq \D_{X,\varphi}.
$$
As a matter of fact, $\D_{X \stackrel{f}{\to}X}$ is given by
$$
\D_{X \stackrel{f}{\to}X} = \frac{\D_{X\times X}}
{\D_{X \times X}(w_1 - f_1(z),\,\dots,\, w_n - f_n(z),\,
\theta_1,\, \dots,\, \theta_n)}
$$
where $(w_1, \dots, w_n)$ is a system of coordinates of the second $X$ and
the vector field $\theta_k$ ($k=1,2,\dots,n$) is defined by
$$
\theta_k := \frac{\partial}{\partial z_k} +
\sum_{i = 1}^n \frac{\partial f_i}{\partial z_k}\frac{\partial}{\partial w_i}.
$$
We denote by $J_f$ the Jacobian of $f$. Since $f$ gives an isomorphism
outside $T$,
we have $J_f \ne 0$ outside $T$, which implies
$J_f = z_{i_{1,1}}^{\beta_1}\dots z_{i_{\ell,1}}^{\beta_l} h$ for some
$\beta_j \ge 0$
$j=1,\dots,\ell$ and
for some $h$ with $h(0) \ne 0$.
Hence, as $J_f$ has an inverse in $\OO_{X, \varphi}$, we conclude that
$\D_{X,\varphi} \underset{\D_X}{\otimes}\D_{X \stackrel{f}{\to}X}$
is a free $\D_{X,\varphi}$ module of rank $1$.

Hence we have obtained
$$
R\Gamma(X;\CC_{G\setminus T} \wtens \OO_X) \simeq R\Gamma(X;\CC_{f(G)\setminus(Z_1\cup\cdots\cup Z_\ell)} \wtens \OO_X).
$$
This implies that $R\Gamma(X;\CC_{f(G)\setminus(Z_1\cup\cdots\cup Z_\ell)} \wtens \OO_X)$ is also concentrated in degree zero. Since
$f(G)$ contains $\overline{S}$ for some multi-cone $S$, and since , for a given multi-cone $S$,
there exists a $G$ such that $\overline{S}$ contains $f(G)$, we have the required result. \qed

\begin{prop}\label{prop: exact sequence} The distinguished triangle \eqref{dtColin} induces an exact sequence outside the zero section
\begin{equation}\label{exspT}
\exs{\imin \rho H^0 \nu^{sa}_\chi\OW_{X|X \setminus Z}}{\imin \rho H^0 \nu^{sa}_\chi\OWX}{\imin \rho H^0 \nu^{sa}_\chi \OW_{X|Z}}.
\end{equation}
All the terms are concentrated in degree zero.

\end{prop}
\dim\ \ The exactness of the sequence \eqref{exspT} follows from Proposition \ref{vanishing} and the fact that outside the zero section $\imin \rho \nu^{sa}_\chi\OW_X$ is concentrated in degree zero. This also proves the vanishing of the cohomology of degree $\geq 1$ of the three terms.
\qed

We give a functorial construction of multi-asymptotically  developable
functions using Whitney holomorphic functions. Let $U$ be an open l.c.t. subanalytic subset in $X$.
Then $\Gamma(U;\CW_{X_\R}) \simeq \Gamma(X; \CC_{\overline{U}} \wtens C^\infty_{X_\R})$ is nothing
but the set of $\C^\infty$-Whitney jets on $\overline{U}$. Further
$H^0(U;\OW_X) \simeq H^0(X; \CC_{\overline{U}} \wtens \OO_X)$ consists of
$\C^\infty$-Whitney jets on $\overline{U}$ that satisfy the Cauchy-Riemann system.
Therefore, if a proper convex multi-cone $S:=S(U,\{G_j\}, \epsilon)$
is subanalytic, the set of holomorphic functions on $S$
that are multi-asymptotically developable along
$Z$ is equal to
$$
\varprojlim_{S' \subset\subset S}
\Gamma(X;\CC_{\overline{S'}} \wtens \OO_X)
$$
where $S'$ runs through the family of open subanalytic proper convex multi-cones
$S(U',\, \{G'_j\},\, \epsilon')$ properly contained in $S$.
Moreover, by Proposition \ref{prop:consisten-equiv}, the set of consistent families
of coefficients of multi-asymptotic expansion is given by
$$
\varprojlim_{S'\subset\subset S}
\Gamma(X; \CC_{\overline{S'} \cap Z} \wtens \OO_X).
$$

Let $W$ be an open convex subset in $Z_1 \cap \dots \cap Z_\ell$ and
$G_j$ an open proper convex cone in
$\CC^{\hat{I}_j}$

($j=1,2,\dots,\ell$).
Using these $W$ and $\{G_j\}$, we define
an $(\RP)^\ell$-conic open subset $V(W, \{G_j\})$ in
$\underset{X,1\le j \le \ell}{\times} \overset{\circ}{T}_{Z_j}\iota_{\chi} (Z_j)$
by
$$
W \times G_1 \times \dots \times G_\ell
\subset \left(\bigcap Z_j\right) \times \CC^{\hat{I}_1} \times \dots \CC^{\hat{I}_\ell}
\simeq \underset{X,1\le j \le \ell}{\times} T_{Z_j}\iota_{\chi} (Z_j)
$$
Note that a family of open sets of type $V(W, \{G_j\})$ forms a basis of
topology for $(\RP)^\ell$-conic sheaves on
$\underset{X,1\le j \le \ell}{\times} \overset{\circ}{T}_{Z_j}\iota(Z_j)$.
We define the presheaves
$\widetilde{\A}^{<0}_{\chi}$, $\widetilde{\A}_{\chi}$ and $\widetilde{\A}^{CF}_{\chi}$
on $\underset{X,1\le j \le \ell}{\times} \overset{\circ}
{T}_{Z_j}\iota(Z_j)$. For an $(\RP)^\ell$-conic open let $V:=V(W, \{G_j\})$, we set

$$
\begin{aligned}
\Gamma(V;\widetilde{\A}^{<0}_{\chi}) &=  \lind {U,\, \epsilon > 0}
\{f \in \OO(S(U,\, \{G_j\}, \epsilon));\, \text{$f$ is multi-asymptotically}\\
&\qquad \text{developable to zero along $Z$ on $S(U,\, \{G_j\}, \epsilon)$}\},
\end{aligned}
$$
$$
\begin{aligned}
\Gamma(V;\widetilde{\A}_{\chi})
&= \varinjlim_{U,\, \epsilon > 0}
\{f \in \OO(S(U,\, \{G_j\}, \epsilon));\, \text{$f$ is multi-asymptotically}\\
&\qquad \text{developable along $Z$ on $S(U,\, \{G_j\}, \epsilon)$}\},
\end{aligned}
$$
$$
\begin{aligned}
\Gamma(V;\widetilde{\A}^{CF}_{\chi})
&= \varinjlim_{U,\, \epsilon > 0}
\{F;\, \text{$F$ is a consistent family of coefficients of}\\
&\qquad \text{multi-asymptotic expansion along $Z$
on $S(U,\, \{G_j\}, \epsilon)$}\},
\end{aligned}
$$
where

$U$ ranges through the family open convex neighborhoods of $W$ which are a $\mu$-star shape. Let us consider the multi-specialization of Whitney holomorphic functions. We have
$$
\begin{aligned}
\Gamma(V;\imin \rho \nu^{sa}_\chi \OW_{X|X\setminus Z}) &= \lpro {V'}\,\lind {U'} \Gamma(U';\OW_{X|X\setminus Z}),
\end{aligned}
$$
$$
\begin{aligned}
\Gamma(V;\imin \rho \nu^{sa}_\chi \OW_X) &= \lpro {V'}\,\lind {U'} \Gamma(U';\OW_X),
\end{aligned}
$$
$$
\begin{aligned}
\Gamma(V;\imin \rho \nu^{sa}_\chi \OW_{X|Z}) &= \lpro {V'}\,\lind {U'} \Gamma(U';\OW_{X|Z}),
\end{aligned}
$$
where $V'$ ranges through the family of subanalytic open cones such that $\overline{V'} \setminus Z \subset V$, $U'$ ranges through the family of $\op(X_{sa})$ such that
$C_\chi(X \setminus U') \cap V'=\emptyset$.

The identity morphism induces morphisms of presheaves
\begin{eqnarray*}
\widetilde{\A}^{<0}_\chi & \to & \imin \rho \nu^{sa}_\chi \OW_{X|X\setminus Z}, \\
\widetilde{\A}_\chi & \to & \imin \rho \nu^{sa}_\chi \OW_X, \\
\widetilde{\A}^{CF}_\chi & \to & \imin \rho \nu^{sa}_\chi \OW_{X|Z}.
\end{eqnarray*}
The above morphisms are isomorphisms in the stalks (i.e. in the limit of multi-cones containing a given direction $p \in \underset{X,1\le j \le \ell}{\times} \overset{\circ}{T}_{Z_j}\iota_{\chi} (Z_j)$). Let $\A^{<0}_\chi$ (resp. $\A_\chi$, resp. $\A^{CF}_\chi$) be the sheaves associated to $\widetilde{\A}^{<0}_\chi$ (resp. $\widetilde{\A}_\chi$, resp. $\widetilde{\A}^{CF}_\chi$). We get \begin{eqnarray*}
\A^{<0}_\chi & \iso & \imin \rho \nu^{sa}_\chi \OW_{X|X\setminus Z}, \\
\A_\chi & \iso & \imin \rho \nu^{sa}_\chi \OW_X, \\
\A^{CF}_\chi & \iso & \imin \rho \nu^{sa}_\chi \OW_{X|Z}.
\end{eqnarray*}
By Proposition \ref{prop: exact sequence} outside the zero section we have the exact sequence of sheaves
$$
\exs{\A^{<0}_{\chi}}{\A_{\chi}}{\A^{CF}_{\chi}}.
$$
When $Z$ forms a normal crossing divisor,
these sheaves are nothing but sheaves of strongly asymptotically developable
functions defined by Majima in \cite{Ma84}.

\end{section}

\appendix
\begin{section}*{Appendix}
\end{section}
\begin{section}{Conic sheaves}

Let $k$ be a field. Let $X$ be a real analytic manifold endowed
with a subanalytic action $\mu$ of $\RP$. In other words we have a
subanalytic map
$$\mu: X \times \RP \to X,$$
which satisfies, for each $t_1,t_2 \in \RP$:
$$
  \begin{cases}
    \mu(x,t_1t_2)=\mu(\mu(x,t_1),t_2), \\
    \mu(x,1)=x.
  \end{cases}
$$
Note that $\mu$ is open, in fact let $U \in \op(X)$ and $(t_1,t_2)
\in \op(\RP)$. Then $\mu(U,(t_1,t_2))=\bigcup_{t \in
(t_1,t_2)}\mu(U,t)$, and $\mu(\cdot,t):X \to X$ is a homeomorphism
(with inverse $\mu(\cdot,\imin t)$). We have a diagram
$$\xymatrix{X \ar[r]^{\hspace{-5mm}j} & X \times \RP \ar@ <2pt>
[r]^{\hspace{0.3cm}\mu} \ar@ <-2pt> [r]_{\hspace{0.3cm}p}& X,}$$
where $j(x)=(x,1)$ and $p$ denotes the projection. We have $\mu
\circ j=p \circ j=\id$.

\begin{df} (i) Let $S$ be a subset of $X$. We set $\RP S=\mu(S,\RP).$
If $U \in \op(X)$, then $\RP U \in \op(X)$ since $\mu$ is open.

(ii) Let $S$ be a subset of $X$. We say that $S$ is conic if
$S=\RP S$. In other words, $S$ is invariant by the action of
$\mu$.

(iii) An  orbit of $\mu$ is the set $\RP x$ with $x \in X$.
\end{df}

We assume that the orbits of $\mu$ are contractible. For each $x \in X$ there are two possibilities: either $\RP x=x$ or $\RP x \simeq \R$.

\begin{df} We say that a subset $S$ of $X$ is $\RP$-connected if
$S \cap \RP x$ is connected for each $x \in S$.
\end{df}

\begin{lem} \label{lem: RP-connectedness} Let $S_1,S_2 \subset X$ and suppose that $S_2$ is conic. Then $\RP(S_1 \cap S_2)=\RP S_1 \cap S_2$. (ii) If $S_1$ and $S_2$ are $\RP$-connected, then $S_1 \cap S_2$ is $\RP$-connected.
\end{lem}
\dim\ \ (i) The inclusion $\subseteq$ is true since $\RP(S_1 \cap S_2) \subset \RP S_i$, $i=1,2$. Let us prove $\supseteq$. Let $x \in \RP S_1 \cap S_2$. Then there exists $a \in \RP$ such that $\mu(x,a) \in S_1$. Since $S_2$ is conic $\mu(x,a) \in S_2$ and the result follows.

(ii) Let $x_1,x_2 \in S_1 \cap S_2$. Then $x_i=\mu(x,a_i)$ for some $x \in X$ and some $a_i \in \RP$, $i=1,2$. Suppose $a_1 \leq a_2$. Since $S_1,S_2$ are $\RP$-connected, $i=1,2$ and the orbits of $\mu$ are either points or isomorphic to $\R$, we have $\mu(x,[a_1,a_2]) \in S_i$ $i=1,2$ and the result follows.
\qed

Let $X,Y$ topological spaces endowed with an action ($\mu_{X}$ and $\mu_{Y}$ respectively) of $\RP$.

\begin{df} A continuous function $f:X \to Y$ is said to be conic if for each $x \in X$, $a \in \RP$ we have
$f(\mu_{X}(x,a))=\mu_{Y}(f(x),a)$.
\end{df}

\begin{lem} \label{lem: conic maps} Let $f:X \to Y$ be a conic map. (i) Suppose that $S \subset Y$ is $\RP$-connected (resp. conic). Then $\imin f(S)$ is $\RP$-connected (resp. conic). (ii) Suppose that $Z \subset X$ is conic. Then $f(Z)$ is conic.
\end{lem}
\dim\ \ (i) Let $x_1,x_2 \in \imin f(S)$ and suppose that there exists $x \in X$ such that $x_1,x_2 \in \mu_X(x,\RP)$, i.e. $x_i=\mu_X(x,a_i)$, $a_i \in \RP$, $i=1,2$. Since $f$ is conic we have $f(x_i)=\mu_Y(f(x),a_i) \in \mu_Y(f(x),\RP) \cap S=f(\mu(x,\RP) \cap \imin f(S))$. Suppose $a_1 \leq a_2$. Since $S$ is $\RP$-connected $\mu(f(x),[a_1,a_2]) \subseteq S$, hence $\mu(x,[a_1,a_2]) \subseteq \imin f (f(\mu(x,[a_1,a_2])))=\imin f(\mu(f(x),[a_1,a_2])) \subseteq \imin f(S)$.

If $S$ is conic then $f(\mu_X(\imin f(S),\RP))=\mu_Y(f(\imin f(S)),\RP)=\mu_Y(S,\RP)=S$, hence $\mu(\imin f(S),\RP) =\imin f(S)$.

(ii) We have $\mu_Y(f(Z),\RP)=f(\mu(Z,\RP))=f(Z)$.
\qed

\begin{df} A sheaf of $k$-modules $F$ on $X_{sa}$ is conic if the restriction morphism
$\Gamma(\RP U;F) \to \Gamma(U;F)$ is an isomorphism for each
$\RP$-connected $U \in \op^c(X_{sa})$ with $\RP U \in
\op(X_{sa})$.
\begin{itemize}
\item[(i)]We denote by $\mod_{\RP}(k_{X_{sa}})$ the subcategory of
$\mod(k_{X_{sa}})$ consisting of conic sheaves.
\item[(ii)] We denote by $D^b_{\RP}(k_{X_{sa}})$,

the subcategory of $D^b(k_{X_{sa}})$ consisting of objects $F$
such that $H^j(F)$ belongs to $\mod_{\RP}(k_{X_{sa}})$ for all $j
\in \Z$.
\end{itemize}
\end{df}

Assume the hypothesis below:
\begin{equation}\label{hypsa}
  \begin{cases}

 \text{(i) for any $U \in \op^c_{sa}(X)$ we have $\RP U \in \op(X_{sa})$,}\\
 \text{(ii) for any $x \in X$\ \ the set $\RP x$ is contractible,}\\
 \text{(iii) there exists a covering $\{V_n\}_{n \in \N}$ of $X_{sa}$ such that}\\
 \text{\ \ $V_n$ is $\RP$-connected and $V_n \subset\subset V_{n+1}$ for each $n$}.
  \end{cases}
\end{equation}

The following result was proven in \cite{Pr07}.

\begin{prop}\label{RPU} Assume \eqref{hypsa}. Let $U \in \op(X_{sa})$ be $\RP$-connected and such that $\RP U \in \op(X_{sa})$. Let $F \in D^b_{\RP}(k_{X_{sa}})$. Then
$$
R\Gamma(\RP U;F) \iso R\Gamma(U;F).
$$
\end{prop}

\end{section}

\begin{section}{Multi-conic sheaves}

Let $X$ be a topological space
with $\ell$ actions $\{\mu_i\}_{i=1}^\ell$ of $\RP$ such that $\mu_i(\mu_j(x,t_j),t_i)=\mu_j(\mu_i(x,t_i),t_j)$. We have a map
\begin{eqnarray*}
\mu: X \times (\RP)^\ell & \to & X \\
(x,t_1,\dots,t_\ell) & \mapsto & \mu_1(\cdots\mu_\ell(x,t_\ell),\dots,t_1).
\end{eqnarray*}

\begin{df} (i) Let $S$ be a subset of $X$. We set $\RP_i S=\mu_i(S,\RP).$
If $U \in \op(X)$, then $\RP_i U \in \op(X)$ since $\mu_i$ is open for each $i=1,\ldots,\ell$.

(ii)  Let $S$ be a subset of $X$. Let $J \subset \{1,\ldots,\ell\}$. We set $$\RP_JS=\RP_{i_1}\cdots\RP_{i_k}S=\mu_{i_1}(\cdots\mu_{i_k}(S,\RP),\dots,\RP), \ \ i_1,\ldots i_k \in J.$$
We set $(\RP)^\ell S=\RP_{\{1,\ldots,\ell\}}S=\mu(S,(\RP)^\ell).$
If $U \in \op(X)$, then $\RP_J U \in \op(X)$ since $\mu_i$ is open for each $i \in \{1,\ldots,\ell\}$.

(iii) Let $S$ be a subset of $X$. We say that $S$ is $\ell$-conic if
$S=(\RP)^\ell S$. In other words, $S$ is invariant by the action of
$\mu_i$, $i=1,\dots,\ell$.

\end{df}

\begin{df} (i)  We say that a subset $S$ of $X$ is $\RP_i$-connected if
$S \cap \RP_i x$ is connected for each $x \in S$.

(ii) We say that a subset $S$ of $X$ is $(\RP)^\ell$-connected if
there exists a permutation $\sigma:\{1,\ldots,\ell\}\to\{1,\ldots,\ell\}$ such that
\begin{equation}\label{RPl connected}
 \begin{cases}
 \text{$S$ is $\RP_{\sigma(1)}$-connected,}\\
 \text{$\RP_{\sigma(1)}S$ is $\RP_{\sigma(2)}$-connected,}\\
 \text{\ \ $\vdots$}\\
 \text{$\RP_{\sigma(1)}\cdots\RP_{\sigma(\ell-1)}S$ is $\RP_{\sigma(\ell)}$-connected}.
  \end{cases}
\end{equation}
\end{df}

The following results follow from the case $\ell=1$.

\begin{lem} \label{lem: l-connectedness} (i) $S_1,S_2 \subset X$ and suppose that $S_2$ is $(\RP)^\ell$-conic. Then $(\RP)^\ell(S_1 \cap S_2)=(\RP)^\ell S_1 \cap S_2$. (ii) If moreover $S_1$ is $(\RP)^\ell$-connected then $S_1 \cap S_2$ is $(\RP)^\ell$-connected.
\end{lem}

\begin{oss} In (ii) of Lemma \ref{lem: l-connectedness} we have to assume that $S_2$ is $(\RP)^\ell$-conic. Indeed it is not true that the intersection of two $(\RP)^\ell$-connected is $(\RP)^\ell$-connected in general.
\end{oss}

Let $X,Y$ topological spaces endowed with $\ell$ actions $\{\mu_{Xi}\}_{i=1}^\ell$, $\{\mu_{Yi}\}_{i=1}^\ell$ of $\RP$.

\begin{df} A continuous function $f:X \to Y$ is said to be $\ell$-conic if for each $x \in X$, $a \in \RP$ we have
$f(\mu_{Xi}(x,a))=\mu_{Yi}(f(x),a)$, $i=1,\ldots,\ell$.
\end{df}

\begin{lem} \label{lem: l-conic maps} Let $f:X \to Y$ be a $\ell$-conic map. (i) Suppose that $S \subset Y$ is $(\RP)^\ell$-connected (resp. $(\RP)^\ell$-conic). Then $\imin f(S)$ is $(\RP)^\ell$-connected (resp. $(\RP)^\ell$-conic. (ii) Suppose that $Z \subset X$ is $(\RP)^\ell$-conic. Then $f(Z)$ is $(\RP)^\ell$-conic.
\end{lem}

\begin{df}\label{def:conic sheaf} A sheaf of $k$-modules $F$ on $X_{sa}$ is $\ell$-conic if it is conic with respect to each $\mu_i$.

\begin{itemize}
\item[(i)]We denote by $\mod_{(\RP)^\ell}(k_{X_{sa}})$ the subcategory of
$\mod(k_{X_{sa}})$ consisting of $\ell$-conic sheaves.
\item[(ii)] We denote by $D^b_{(\RP)^\ell}(k_{X_{sa}})$,

the subcategory of $D^b(k_{X_{sa}})$ consisting of objects $F$
such that $H^j(F)$ belongs to $\mod_{(\RP)^\ell}(k_{X_{sa}})$ for all $j
\in \Z$.
\end{itemize}
\end{df}

Let us assume the following hypothesis
\begin{equation}\label{hypsam}
  \begin{cases}
 \text{(i) every $U \in \op^c_{sa}(X)$ has a finite covering consisting }\\
 \text{\ \ of $(\RP)^\ell$-connected subanalytic open subsets,}\\
 \text{(ii) we have $\RP_J U \in \op(X_{sa})$ for any $U \in \op^c_{sa}(X)$}\\
 \text{\ \ and any $J \subset \{1,\ldots,\ell\}$,}\\
 \text{(iii) $\RP_i x$ is contractible for any $x \in X$ and any $i=1,\ldots,\ell$,}\\
 \text{(iv) there exists a covering $\{V_n\}_{n \in \N}$ of $X_{sa}$ such that}\\
 \text{\ \ $V_n$ is $(\RP)^\ell$-connected and $V_n \subset\subset V_{n+1}$ for each $n$}.
  \end{cases}
\end{equation}
In this situation the orbits of $\mu_i$, $i=1,\dots,\ell$ are either $\RP x \simeq \R$ or $\RP x = x$.\\

\begin{prop}\label{prop: RPUl} Let $U \in \op(X_{sa})$ be $(\RP)^\ell$-connected. Let $F \in D^b_{(\RP)^\ell}(k_{X_{sa}})$. Then
$$R\Gamma((\RP)^\ell U;F) \iso R\Gamma(U;F).$$
\end{prop}
\dim\ \ Suppose that $U$ satisfies \eqref{RPl connected}. By Proposition \ref{RPU} we have
$$R\Gamma(U;F) \simeq R\Gamma(\RP_{\sigma(1)}U;F) \simeq \cdots \simeq R\Gamma(\RP_{\sigma(1)}\cdots\RP_{\sigma(\ell)}U;F)$$
and $\RP_{\sigma(1)}\cdots\RP_{\sigma(\ell)}U=(\RP)^\ell U$. \qed

If $X$ satisfies \eqref{hypsam} (i)-(iv), then it follows from Proposition \ref{prop: RPUl} that for $\ell$-conic subanalytic sheaves it is enough to study the cohomology of the sections on $\ell$-conic open subsets.\\

Let $\{\mu_i\}_{i=1}^\ell$, $\ell \leq n$ be actions of $\RP$ on $\R^n$ defined by
$$
\mu_j:(x_1,\ldots,x_n,c_j) \mapsto (c_{1j}x_1,\ldots,c_{nj}x_n),
$$
where $c_{ij}=1$ if $j \notin I_i$, $c_{ij}=c_j \in \RP$ if $j \in I_i$. We assume the following hypothesis
\begin{equation}\label{H}
\text{Either $I_i \subset I_j$, $I_j \subset I_i$ or $I_i \cap I_j = \emptyset$ holds
($i,j = 1,2,\dots,\ell$).}
\end{equation}
\begin{equation} \label{H3}
\text{ $I_i \varsupsetneqq \bigcup_{I_j \subset I_i}I_j$\; for $i=1,2,\ldots,\ell$.}
\end{equation}

\begin{df} Let $i,j = 1,\ldots,\ell$. When $I_i \cap I_j = \emptyset$ the actions $\mu_i$ and $\mu_j$ ar said to be orthogonal and we write $\mu_i \bot \mu_j$.
\end{df}

\begin{lem}\label{lem: RPl decomposition}  Each globally subanalytic $U \in \op(\R^n_{sa})$ has a finite covering consisting of globally subanalytic $(\RP)^\ell$-connected open subsets.
\end{lem}
\dim\ \ We may assume that $U$ is connected. We will prove the assertion in several steps. In the rest of the proof we write for short subanalytic instead of globally subanalytic.\\

(i) First of all, as in \cite{Pr07}, let us prove the case $\ell=1$, i.e. there exists a finite open covering $\{U_\alpha\}$ of $U$ with $U_\alpha$ subanalytic and $\RP$-connected. Let us consider an action
\begin{eqnarray*}
\mu: \R^m \times \R^{n-m} \times \RP & \to & \R^n \\
(x_1,\ldots,x_m,x_{m+1},\ldots,x_n,\lambda) & \mapsto & (\lambda x_1,\ldots,\lambda x_m,x_{m+1},\ldots,x_n).
\end{eqnarray*}
Let us consider the morphism of manifolds
\begin{eqnarray*}
\varphi : \mathbb{S}^{m-1} \times \R \times \R^{n-m} & \to & \R^n\\
(\vartheta,r,z) & \mapsto & (r i(\vartheta),z),
\end{eqnarray*}
where $i:\mathbb{S}^{m-1} \hookrightarrow \R^m$ denotes the
embedding. The map $\varphi$ is proper and subanalytic. The subset
$\imin \varphi(U)$ is subanalytic in $\R^m
\times  \mathbb{S}^{n-1} \times \R$.
\begin{itemize}
\item[(${\rm i}_{\rm a}$)] Let us consider a cylindrical cell decomposition of $\imin\varphi(U \setminus Z)$ (for the definition see \cite{Co00}) with respect to the fibers of the projection $\pi:\mathbb{S}^{m-1} \times \R \times \R^{m-n} \to \mathbb{S}^{m-1} \times \R^{m-n}$. Each cell $D$ is defined by $(f,g):=\{(x,y);\;f(x)<y<g(y)\}$, where $f,g:D' \to \R$ are subanalytic functions and $D'$ is a cell of $\pi(U)$. Consider $h:D'\to \R$, $f<h<g$, for example $h={f+g \over 2}$. Extend the graph of $h$ to $\widetilde{h}$

on an open subanalytic neighborhood $U'$ of $D'$
such that the graph of $\widetilde{h}$ is contained in $U$. For $p \in U'$ set let $(m(p),M(p))$ be the connected component of $\imin {\pi}(p) \cap U$ containing $\widetilde{h}(p)$. The functions $m,M:U'\to\R$ are subanalytic and the open set $U_D=\bigcup_{p \in U'}(m(p),M(p))$ has the required properties.
In this way we obtain a finite covering $\{W_j\}_{j \in J}$ of $\imin \varphi(U \setminus Z)$ consisting of $\RP$-connected subanalytic open subsets.

\item[(${\rm i}_{\rm b}$)] Let $p \in \pi(\imin\varphi(U \cap Z))$. Then $\imin \pi(p) \cap U$ is a disjoint union of intervals. Let us consider the interval $(m(p),M(p))$, $m(p)<M(p) \in \R$ containing $0$. Set $W_Z=\{(p,r) \in U ;\; m(p)<r<M(p)\}$. The set $W_Z$ is open subanalytic, contains $\imin\varphi(U \cap Z)$ and its intersections with the fibers of $\pi$ are connected. Then $\varphi(W_Z)$ is an open $\RP$-connected subanalytic neighborhood of $U \cap Z$ and it is contained in $U$.
\end{itemize}
By (${\rm i}_{\rm a}$) there exists a finite covering $\{\varphi(W_j)\}_{j \in J}$ of $U \setminus Z$ consisting of $\RP$-connected subanalytic open subsets, and $\varphi(W_Z) \cup \bigcup_{j \in J}\varphi(W_j)=U$.\\

(ii) Suppose that $I_i \cap I_j = \emptyset$ for each $i,j \in \{1,\ldots,\ell\}$, i.e. all the actions are orthogonal. Let $U$ be $\RP_1$-connected and such that $\RP_1 \cdots \RP_{j-1} U$ is $\RP_j$-connected for each $j=1,\ldots,\ell-1$. We are going to find a finite cover $\{U_\alpha\}$ with $U_\alpha$ $\RP_1$-connected and such that $\RP_1 \cdots \RP_{j-1} U$ is $\RP_j$-connected for each $j=1,\ldots,\ell$.

For any subset $J=\{i_1,\ldots,i_k\}$ of $\{1,\ldots,\ell-1\}$ let us consider
$$
S_J := \left\{x \in \R^n;\; \left\|\sum_{i \in I_k}x_i^2=1\right\|,\; k \in J,\;x_j=0,\;j \in I_k,\; k \notin J\right\}.
$$
By construction $S_J$ is invariant under $\mu_\ell$. Let us consider $\RP_1 \cdots \RP_{\ell-1}U \cap S_J$. By (i) it admits a finite covering $\{V_{\beta_J}\}$ consisting of $\RP_\ell$-connected subanalytic open subsets of $S_J$. For each $\beta_J$ consider $U_{\beta_J}:=\RP_J(V_\beta \times \R^{n_J})$, where

$n_J=n-\sharp\bigcup_{i \in J}I_i$ (hence $\R^{n_J}$ is represented by the coordinates $x_k$, $k \notin \bigcup_{i \in J}I_i$). We want to show that $U \cap U_{\beta_J}$ has the desired properties. First remark that by Lemma \ref{lem: l-connectedness} (i) $\RP_1\cdots\RP_j(U \cap U_{\beta_J}) = \RP_1\cdots\RP_jU \cap U_{\beta_J}$ for each $j=1,\ldots,\ell-1$ since $U_{\beta_J}$ is conic with respect to $\mu_1,\ldots,\mu_{\ell-1}$ by construction.
In particular by Lemma \ref{lem: l-connectedness} it is $\RP_j$-connected for each $j=1,\ldots,\ell-1$.

Moreover $\RP_1\cdots\RP_{\ell-1}U \cap U_{\beta_J} = U_{\beta_J}$ which is $\RP_\ell$-connected by construction. Then $U \cap U_{\beta_J}$ has the desired properties and the result follows. Remark that this proof does not depend on the choice of the permutation $\sigma$ in \eqref{RPl connected}. \\

(iii) Let us consider the general case. We argue by decreasing induction on $\sharp J$, where
$$
J=\left\{i \in \{1,\ldots,\ell\},\; I_i \cap \bigcup_{i\neq j}I_j=\emptyset\right\},
$$
i.e. $J$ denotes the number of $\mu_i$ such that $\mu_i \bot \mu_j$ for each $j\neq i$. If $\sharp J=\ell$ the result follows by (ii). Suppose that the result is true for $\sharp J=1,\ldots,\ell$, we shall prove it for $\sharp J=0$, i.e. there are no actions which are orthogonal to all the other ones.

Thanks to \eqref{H} and \eqref{H3}, there exists $i \in \{1,\ldots,\ell\}$ such that $I_i \setminus \bigcup_{j\neq i}I_j \neq \emptyset$. Up to take a permutation of $\{1,\ldots,\ell\}$ we may assume $i=\ell$.
Let $U$ be $\RP_1$-connected and such that $\RP_1 \cdots \RP_{j-1} U$ is $\RP_j$-connected for each $j=1,\ldots,\ell-1$. We are going to find a finite cover $\{U_\alpha\}$ with $U_\alpha$ $\RP_1$-connected and such that $\RP_1 \cdots \RP_{j-1} U_\alpha$ is $\RP_j$-connected for each $j=1,\ldots,\ell$.
\begin{itemize}

\item[(${\rm iii}_{\rm a}$)] Given $i_0 \in I_\ell \setminus \bigcup_{j\neq \ell}I_j$ let us consider $U':=U \cap \{x_{i_0} \neq 0\}$ and consider the subanalytic homeomorphism $\psi$ of $\{x_{i_0} \neq 0\}$ given by
    $$
    \begin{cases}
    \text{$\psi(x_k)=\displaystyle{x_k\over x_{i_0}}$ \ \ if $k \in I_j \subset I_\ell$, $j \neq \ell$,} \\
    \text{$\psi(x_k)=x_k$ \ \ otherwise.}
    \end{cases}
    $$
    Set $\mu'_k=\mu_k \circ (\psi \times \id_{(\RP)^\ell})$. By construction we have $\mu'_\ell \bot \mu'_j$ for each $j \neq \ell$, hence by the induction hypothesis we may find a finite cover $\{V_\alpha\}$ of $\psi(U')$ with the required properties. Then $\{\imin \psi(V_\alpha)\}$ is a finite cover of $U'$ consisting of $(\RP)^\ell$-connected open subanalytic subsets.

\item[(${\rm iii}_{\rm b}$)]  By \eqref{H} $\bigcup_{I_j \subset I_\ell}I_j$ is a disjoint union $I_{i_1} \sqcup \cdots \sqcup I_{i_k}$, $\{i_1,\dots,i_k\} \subseteq \{1,\dots,\ell-1\}$.
    Set $x=(x_i)$, $i \in I_\ell \setminus \bigcup_{j \neq \ell}I_j$, $y=(x_j)$, $j \in I_{i_1} \sqcup \cdots \sqcup I_{i_k}$ $z=(x_k)$, $k \notin \bigcup_{i=1}^\ell I_i$. Assume that $I_\ell \setminus \bigcup_{j \neq \ell}I_j=\{1,\ldots,m\}$.

    Set $S_0:=\{x=0\}$, and consider $V_0:=U \cap S_0$. It is an open subanalytic subset of $S_0$, it is

    $\RP_1$-connected and such that $\RP_1 \cdots \RP_{j-1} V_0$ is $\RP_j$-connected for each $j=1,\ldots,\ell$. Indeed on $S_0$ $\mu_\ell$ is generated by $\mu_1,\ldots,\mu_{\ell-1}$ and hence $\RP_1\cdots\RP_{\ell-1} V_0$ is conic with respect to $\mu_\ell$.

Let us consider the morphism of manifolds
\begin{eqnarray*}
\varphi : \mathbb{S}^{m-1} \times \R \times \R^{n-m} & \to & \R^n\\
(\vartheta,r,y,z) & \mapsto & (r i(\vartheta),y,z),
\end{eqnarray*}
where $i:\mathbb{S}^{m-1} \hookrightarrow \R^m$ denotes the
embedding. Endow $\mathbb{S}^{m-1} \times \R \times \R^{n-m}$ with the actions $\mu'_j=\mu_j$, $j=1,\dots,\ell-1$, $\mu'_\ell(\vartheta,r,y,z,\lambda)=(\vartheta,\lambda r,\lambda y,z)$. Then $\varphi$ is a conic map.

Up to shrink $U$, we may suppose that $\imin \varphi(U)=\imin \varphi(U) \cap (\imin \varphi(V_0) \times \R)$. Denote by $\pi:\mathbb{S}^{m-1} \times \R \times \R^{n-m} \to \mathbb{S}^{m-1} \times \R^{n-m}$ the projection. Let $p \in \imin \varphi(\RP_1 \cdots \RP_{\ell-1}V_0)$. Then $\imin \pi(p) \cap \imin \varphi(\RP_1 \cdots \RP_{\ell-1}U)$ is a disjoint union of intervals. Let us consider the interval $(m(p),M(p))$, $m(p)<M(p) \in \R$ containing $0$. Set
$$
U_0=\varphi\left(\bigcup_p(m(p),M(p))\right),
$$
where $p \in \imin \varphi(\RP_1 \cdots \RP_{\ell-1}V_0)$.

The set $U_0$ is open subanalytic, contains $U \cap S_0$ and if $(x,y,z) \in U_0$, then $(0,y,z) \in U_0$. Moreover by Lemma \ref{lem: l-conic maps} (ii) $U_0$ is conic with respect to $\mu_1,\ldots,\mu_{\ell-1}$. We want to show that $U \cap U_0$ has the desired properties. First remark that by Lemma \ref{lem: l-connectedness} (i) $\RP_1\cdots\RP_j(U \cap U_0) = \RP_1\cdots\RP_jU \cap U_0$ for each $j=1,\ldots,\ell-1$ since $U_0$ is conic with respect to $\mu_1,\ldots,\mu_{\ell-1}$. In particular by Lemma \ref{lem: l-connectedness} (ii) it is $\RP_j$-connected for each $j=1,\ldots,\ell-1$. We prove that $\RP_1\cdots\RP_{\ell-1}U \cap U_{0} = U_{0}$ is $\RP_\ell$-connected. Suppose that $(x,y,z),(\lambda x,\lambda y,z) \in U_0$ for some $\lambda \in \RP$. By construction $(0,y,z),(0,\lambda y,z) \in U_0$. We may assume without loss of generality $\lambda >1$. We argue by contradiction. Suppose that there exists $1<\mu<\lambda$ such that $(\mu x,\mu y,z) \notin U_0$. Since $U_0$ is conic with respect to $\mu_{i_1},\dots,\mu_{i_k}$, for each $t \in \RP$ we have $(\mu x, t\mu y,z) \notin U_0$. Set $t=\lambda/\mu$. Then $(\mu x,\lambda y,z) \notin U_0$ which leads to a contradiction since by construction $(\mu x,\lambda y,z) \in U_0$ for each $\mu \in (0,\lambda)$.

\end{itemize}
Hence we have found coverings of $U \cap \{x_{i_k} \neq 0\}$ for each $i_k \in I_i \setminus \bigcup_{j \neq i}I_j$ and a neighborhood of $U \cap S_0$ with the required properties. Then the result follows.
\qed

\begin{lem} \label{lem: Rpl neighborhood} Let $Z$ be a globally subanalytic $(\RP)^\ell$-conic subset, and let $V$ be a globally subanalytic $(\RP)^\ell$-conic open subset of $Z$. Then each subanalytic neighborhood $W$ of $V$ in $\R^n$ contains $\widetilde{W} \in
\op(\R^n_{sa})$ such that:
\begin{equation} \label{mV}
\begin{cases}

\text{(i) $\widetilde{W}$ is $(\RP)^\ell$-connected,} \\

\text{(ii) $\RP_1 \cdots \RP_{\ell}\widetilde{W}$ is subanalytic.}
\end{cases}
\end{equation}
\end{lem}
\dim\ \ We argue by induction on the number $\ell$ of actions. Up to shrink $W$ we may suppose that $W \cap Z=V$.\\

(i) Let us consider the case $\ell=1$. Let us consider an action
\begin{eqnarray*}
\mu: \R^m \times \R^{n-m} \times \RP & \to & \R^n \\
(x_1,\ldots,x_m,x_{m+1},\ldots,x_n,\lambda) & \mapsto & (\lambda x_1,\ldots,\lambda x_m,x_{m+1},\ldots,x_n).
\end{eqnarray*}
There are two possibilities:
$$
\begin{cases}

\text{(${\rm i}_{\rm a}$) $V \not\subseteq \{0\} \times \R^{n-m}$,} \\

\text{(${\rm i}_{\rm b}$) $V \subseteq \{0\} \times \R^{n-m}$.}
\end{cases}
$$
Let us consider the morphism of manifolds
\begin{eqnarray*}
\varphi : \mathbb{S}^{m-1} \times \R \times \R^{n-m} & \to & \R^n\\
(\vartheta,r,z) & \mapsto & (r i(\vartheta),z),
\end{eqnarray*}
where $i:\mathbb{S}^{m-1} \hookrightarrow \R^m$ denotes the
embedding. Denote by $\pi:\mathbb{S}^{m-1} \times \R \times \R^{n-m} \to \mathbb{S}^{m-1} \times \R^{n-m}$ the projection.
\begin{itemize}
\item[(${\rm i}_{\rm a}$)] Let $p \in \imin\varphi(W) \cap \{r=1\}$. Up to shrink $W$, we may suppose that $\imin\varphi(W) \cap \{r=1\}$ is globally subanalytic. Then $\imin \pi(p) \cap W$ is a disjoint union of intervals. Let us consider the interval $(m(p),M(p))$, $m(p)<M(p) \in \R$ containing $1$. Set $W'=\{(p,r) \in \imin \varphi (W) ;\; m(p)<r<M(p)\}$. Remark that in the case $p \in \imin\varphi(V)$, if $0 \in (m(p),M(p))$, then $(m(p),M(p))=\R$ since $V$ is conic. The set $W'$ is open subanalytic, contains $\imin\varphi(V)$ and its intersections with the fibers of $\pi$ are connected. Then $\varphi(W')$ is an open $\RP$-connected subanalytic neighborhood of $V$ and it is contained in $W$. Moreover, since $\imin\varphi(W) \cap \{r=1\}$ is globally subanalytic, then $\RP \varphi(W')=\varphi(\RP W')=\varphi(\RP(W' \cap \{r=1\}))$ is globally subanalytic.
\item[(${\rm i}_{\rm b}$)] Let $p \in \imin\varphi(V) \subseteq \{r=0\}$. Then $\imin \pi(p) \cap W$ is a disjoint union of intervals. Let us consider the interval $(m(p),M(p))$, $m(p)<M(p) \in \R$ containing $0$. Set $W'=\{(p,r) \in U ;\; m(p)<r<M(p)\}$. The set $W'$ is open subanalytic, contains $\imin\varphi(V)$ and its intersections with the fibers of $\pi$ are connected. Then $\varphi(W')$ is an open $\RP$-connected subanalytic neighborhood of $V$ and it is contained in $W$. Moreover, $\RP \varphi(W')=\varphi(\RP W')=\varphi((W' \cap \{r=0\}) \times \R)=\varphi(\imin \pi(\imin\varphi(V)))$ is globally subanalytic.
\end{itemize}

(ii) Suppose that $I_i \cap I_j = \emptyset$ (i.e. $\mu_i \bot \mu_j$) for each $i,j \in \{1,\ldots,\ell\}$. Let $W$ be a subanalytic neighborhood of $V$ which is $\RP_1$-connected and such that $\RP_1\cdots\RP_{j-1}W$ is $\RP_{j}$-connected for $j=1,\dots,\ell-1$. Suppose that
\begin{eqnarray*}
\mu_\ell: \R^m \times \R^{n-m} \times \RP & \to & \R^n \\
(x_1,\ldots,x_m,x_{m+1},\ldots,x_n,\lambda) & \mapsto & (\lambda x_1,\ldots,\lambda x_m,x_{m+1},\ldots,x_n),
\end{eqnarray*}
and consider the morphism of manifolds
\begin{eqnarray*}
\varphi : \mathbb{S}^{m-1} \times \R \times \R^{n-m} & \to & \R^n\\
(\vartheta,r,z) & \mapsto & (r i(\vartheta),z),
\end{eqnarray*}
where $i:\mathbb{S}^{m-1} \hookrightarrow \R^m$ denotes the
embedding.
Following the construction of (i), $\RP_1\cdots\RP_{\ell-1}W$ contains a neighborhood $W'$ of $V$ which is $\RP_\ell$-connected and conic with respect to $\mu_1,\dots,\mu_{\ell-1}$. By Lemma \ref{lem: l-connectedness} (i) we have $\RP_1\cdots\RP_j(W \cap W')=\RP_1\cdots\RP_jW \cap W'$, $j=1,\dots,\ell-1$, hence by Lemma \ref{lem: l-connectedness} (ii) the set $W \cap W'$ has the desired properties.\\

(iii) Let us consider the general case. We argue by decreasing induction on $\sharp J$, where
$$
J=\left\{i \in \{1,\ldots,\ell\},\; I_i \cap \bigcup_{i\neq j}I_j=\emptyset\right\},
$$
i.e. $J$ denotes the number of $\mu_i$ such that $\mu_i \bot \mu_j$ for each $j\neq i$. If $\sharp J=\ell$ the result follows by (ii). Suppose that the result is true for $\sharp J=1,\ldots,\ell$, we shall prove it for $\sharp J=0$, i.e. there are no actions which are orthogonal to all the other ones.

Thanks to \eqref{H} and \eqref{H3}, there exists $i \in \{1,\ldots,\ell\}$ such that $I_i \setminus \bigcup_{j\neq i}I_j \neq \emptyset$. Up to take a permutation of $\{1,\ldots,\ell\}$ we may assume $i=\ell$.
By \eqref{H} $\bigcup_{I_j \subset I_\ell}I_j$ is a disjoint union $I_{i_1} \sqcup \cdots \sqcup I_{i_k}$, $\{i_1,\dots,i_k\} \subseteq \{1,\dots,\ell-1\}$.
Set $x=(x_i)$, $i \in I_\ell \setminus \bigcup_{j \neq \ell}I_j$, $y=(x_j)$, $j \in I_{i_1} \sqcup \cdots \sqcup I_{i_k}$ $z=(x_k)$, $k \notin \bigcup_{i=1}^\ell I_i$. Assume that $I_\ell \setminus \bigcup_{j \neq \ell}I_j=\{1,\ldots,m\}$.
Let us consider the morphism of manifolds
\begin{eqnarray*}
\varphi : \mathbb{S}^{m-1} \times \R \times \R^{n-m} & \to & \R^n\\
(\vartheta,r,y,z) & \mapsto & (r i(\vartheta),y,z),
\end{eqnarray*}
where $i:\mathbb{S}^{m-1} \hookrightarrow \R^m$ denotes the
embedding.

Let $W$ be a subanalytic neighborhood of $V$ which is $\RP_1$-connected and such that $\RP_1\cdots\RP_{j-1}W$ is $\RP_{j}$-connected for $j=1,\dots,\ell-1$. Following the construction of (i), $\RP_1\cdots\RP_{\ell-1}W$ contains a neighborhood $W'$ of $V$ which is $\RP_\ell$-connected and conic with respect to $\mu_1,\dots,\mu_{\ell-1}$. Then, arguing as in Lemma \ref{lem: RPl decomposition} ${\rm(iii)}_{\rm b}$ $W \cap W'$ has the desired properties.
\qed

\begin{teo} Let $\{\mu_i\}_{i=1}^\ell$, $\ell \leq n$ be actions of $\RP$ on $\R^n$ defined by
$$
\mu_j:(x_1,\ldots,x_n,c_j) \mapsto (c_{1j}x_1,\ldots,c_{nj}x_n),
$$
where $c_{ij}=1$ or $c_{ij}=c_j \in \RP$ satisfying \eqref{H} and \eqref{H3}. Then $\R^n$ satisfies \eqref{hypsam} (i)-(iv).
\end{teo}
\dim\ \ \eqref{hypsam} (i) follows from Lemma \ref{lem: RPl decomposition}, \eqref{hypsam} (ii) follows from the fact that $\mu_i$ is a globally subanalytic map for $i=1,\ldots,\ell$, hence if $U$ is globally subanalytic  $\mu_i(U,\RP)$ is still globally subanalytic,

(iii) and (iv) are trivial. \qed

\end{section}

\addcontentsline{toc}{section}{\textbf{References}}

config: arxiv