1 Introduction
Let 
$X$ be a complex manifold and 
$Y$ be a complex submanifold of codimension 
$r$. Our interest is in the analytic structure of a neighborhood of 
$Y$ when the normal bundle 
$N_{Y/X}$ is unitary flat. We say that a holomorphic vector bundle 
$E$ on 
$Y$ is unitary flat if
$$\begin{eqnarray}E\in \text{Image}\,(H^{1}(Y,U(r))\rightarrow H^{1}(Y,\text{GL}_{r}({\mathcal{O}}_{Y}))),\end{eqnarray}$$ or equivalently, the transition matrices of 
$E$ can be chosen to be 
$U(r)$-valued locally constant functions, where 
$U(r)$ is the set of 
$r\times r$ unitary matrices. A unitary flat vector bundle 
$E$ admits a flat connection whose monodromy 
$\unicode[STIX]{x1D70C}_{E}$ is a unitary representation of the fundamental group 
$\unicode[STIX]{x1D70B}_{1}(Y,\ast )$ of 
$Y$ (see Section 2.1 for the details). Our interest is in a sort of the linearizability problem of a neighborhood of 
$Y$. In other words, we are interested in comparing a neighborhood of 
$Y$ in 
$X$ and of the zero section in 
$N_{Y/X}$. One main goal of this paper is to investigate the existence of a holomorphic foliation 
${\mathcal{F}}$ of codimension 
$r$ on a neighborhood of 
$Y$ which includes 
$Y$ as a leaf with 
$\text{Hol}_{{\mathcal{F}},Y}=\unicode[STIX]{x1D70C}_{N_{Y/X}}$, where 
$\text{Hol}_{{\mathcal{F}},Y}$ is the holonomy of 
${\mathcal{F}}$ along 
$Y$.
In [Reference Arnol’dA], Arnol’d studied a neighborhood of an elliptic curve 
$Y$ by applying a linearizing technique as in [Reference SiegelSi]. In [Reference UedaU], Ueda studied the case where 
$Y$ is any compact complex curve and 
$r=1$. For such a pair 
$(Y,X)$, Ueda posed the Ueda class 
$u_{n}(Y,X)\in H^{1}(Y,N_{Y/X}^{-n}):=H^{1}(Y,{\mathcal{O}}_{Y}(N_{Y/X}^{\otimes -n}))$ as an obstruction of the linearization of a neighborhood of 
$Y$ in 
$n$-jet along 
$Y$ (
$n\geqslant 1$). The pair 
$(Y,X)$ is of infinite type if 
$u_{n}(Y,X)=0$ holds for each 
$n\geqslant 1$. When 
$(Y,X)$ is of infinite type, Ueda generalized the result of Arnol’d. He showed that an infinite type pair 
$(Y,X)$ admits the foliation 
${\mathcal{F}}$ as above if 
$N_{Y/X}$ is a torsion element of 
${\mathcal{P}}(Y):=\text{Image}\,(H^{1}(Y,U(1))\rightarrow H^{1}(Y,{\mathcal{O}}_{Y}^{\ast }))$, or satisfies the following Diophantine-type condition: there exists a constant 
$A>0$ such that 
$d(\mathbb{I}_{Y}^{(1)},N_{Y/X}^{m})\geqslant (2m)^{-A}$ holds for any 
$m\geqslant 1$. Here 
$\mathbb{I}_{Y}^{(1)}$ is the holomorphically trivial line bundle on 
$Y$ and 
$d$ is an invariant distance on 
${\mathcal{P}}(Y)$ ([Reference UedaU, Theorem 3], see [Reference UedaU, Section 4.1] for the details). Note that the proof of [Reference UedaU, Theorem 3] works not only when 
$Y$ is a curve, but also when 
$Y$ is a compact complex manifold of arbitrary dimension if the codimension of 
$Y$ is one.
We generalize Ueda’s theory to the case where the codimension 
$r$ of 
$Y$ is greater than 
$1$. We will define the obstruction class
$$\begin{eqnarray}u_{n}(Y,X)\in H^{1}(Y,N_{Y/X}\otimes S^{n+1}N_{Y/X}^{\ast })\end{eqnarray}$$ as a straightforward generalization of the Ueda class and generalize [Reference UedaU, Theorem 3] to the pair 
$(Y,X)$ of infinite type with 
$N_{Y/X}\in {\mathcal{E}}_{0}^{(r)}(Y)\cup {\mathcal{E}}_{1}^{(r)}(Y)$. Here we denote by 
$S^{n+1}N_{Y/X}^{\ast }$ the 
$n+1$th symmetric tensor bundle, 
${\mathcal{E}}_{0}^{(r)}(Y)$ the set 
$\{E_{\unicode[STIX]{x1D70C}}\mid \#(\text{Image}\,\unicode[STIX]{x1D70C})<\infty \}$, and by 
${\mathcal{E}}_{1}^{(r)}(Y)$ the set
$$\begin{eqnarray}\displaystyle & & \displaystyle \{\!E_{\unicode[STIX]{x1D70C}}|\unicode[STIX]{x1D70B}^{\ast }E_{\unicode[STIX]{x1D70C}}\in {\mathcal{S}}_{A}^{(r)}(\widetilde{Y})\nonumber\\ \displaystyle & & \displaystyle \text{for some finite normal covering}~\unicode[STIX]{x1D70B}:\widetilde{Y}\rightarrow Y~\text{and}~A>0\!\},\nonumber\end{eqnarray}$$ where 
$E_{\unicode[STIX]{x1D70C}}$ is the unitary flat vector bundle of rank 
$r$ which corresponds to a unitary representation 
$\unicode[STIX]{x1D70C}$ of 
$\unicode[STIX]{x1D70B}_{1}(Y,\ast )$ (see Section 2.1 for the correspondence) and 
${\mathcal{S}}_{A}^{(r)}(\widetilde{Y})$ is the set
$$\begin{eqnarray}\displaystyle & & \displaystyle \bigg\{\!\bigoplus _{\unicode[STIX]{x1D706}=1}^{r}L_{\unicode[STIX]{x1D706}}\bigg|L_{\unicode[STIX]{x1D706}}\in {\mathcal{P}}(\widetilde{Y}),\nonumber\\ \displaystyle & & \displaystyle d\biggl(\mathbb{I}_{\widetilde{Y}}^{(1)},~\bigotimes _{\unicode[STIX]{x1D706}=1}^{r}L_{\unicode[STIX]{x1D706}}^{a_{\unicode[STIX]{x1D706}}}\biggr)\geqslant \frac{1}{(2|a|)^{A}}~\text{for}~a=(a_{\unicode[STIX]{x1D706}})_{\unicode[STIX]{x1D706}}\in \mathbb{Z}^{r}~\text{with}~|a|\geqslant 1\bigg\}\nonumber\end{eqnarray}$$ (
$|a|:=a_{1}+a_{2}+\cdots +a_{r}$).
Theorem 1.1. Let 
$X$ be a complex manifold and 
$Y$ be a compact complex submanifold of codimension 
$r$ with 
$N_{Y/X}\in {\mathcal{E}}_{0}^{(r)}(Y)\cup {\mathcal{E}}_{1}^{(r)}(Y)$. Assume that the pair 
$(Y,X)$ is of infinite type (i.e. 
$u_{n}(Y,X)=0$ for each 
$n\geqslant 1$). Then the following holds:
(i) There uniquely exists a nonsingular holomorphic foliation
${\mathcal{F}}$ of codimension $r$ on some neighborhood $V$ of $Y$ which includes $Y$ as a leaf with $\text{Hol}_{{\mathcal{F}},Y}=\unicode[STIX]{x1D70C}_{N_{Y/X}}$.(ii) Suppose that there exists a hypersurface
$S$ such that $Y\subset S$ and $N_{Y/S}$ is unitary flat. Then there exists a nonsingular holomorphic foliation ${\mathcal{G}}_{S}$ of codimension $1$ on $V$ with the following properties by shrinking $V$ if necessary: ${\mathcal{G}}_{S}$ includes $S\cap V$ as a leaf with $U(1)$-linear holonomy, and each leaf of ${\mathcal{F}}$ is holomorphically immersed into a leaf of ${\mathcal{G}}_{S}$. Especially, $S\cap V$ is the union of leaves of ${\mathcal{F}}$.
Note that the assertion (i) in Theorem 1.1 implies the existence of a deformation family of 
$Y$ when 
$N_{Y/X}\in {\mathcal{E}}_{0}^{(r)}(Y)$. Note also that, for each unitary flat subbundle 
$E\subset N_{Y/X}$, the assertion (i) implies the existence of a submanifold 
$Z\subset V$ with 
$Y\subset Z$ and 
$N_{Y/Z}=E$ (see Remark 2.6). Especially, when 
$N_{Y/X}$ admits a direct decomposition to 
$r$ unitary flat line bundles, it is observed from Theorem 1.1(i) that 
$Y$ is realized as a transversal intersection of 
$r$ nonsingular hypersurfaces of a neighborhood of 
$Y$ in 
$X$ if 
$N_{Y/X}\in {\mathcal{E}}_{0}^{(r)}(Y)\cup {\mathcal{E}}_{1}^{(r)}(Y)$ and the pair 
$(Y,X)$ is of infinite type. As the pair 
$(Y,X)$ is always of infinite type when 
$H^{1}(Y,N_{Y/X}\otimes S^{n+1}N_{Y/X}^{\ast })=0$ for each 
$n\geqslant 1$, Theorem 1.1(i) can be applied to the case where, for example, 
$Y$ is an elliptic curve and 
$N_{Y/X}\in {\mathcal{E}}_{1}^{(r)}(Y)$. In this sense, Theorem 1.1(i) can be also regarded as a generalization of the result of Arnol’d for elliptic curves.
Theorem 1.1(ii) can be applied to the semipositivity problem (the existence problem of a 
$C^{\infty }$ Hermitian metric with semipositive curvature) on a holomorphic line bundle, since the assertion (ii) implies the unitary flatness of the line bundle 
$[S]$ on 
$V$.
Corollary 1.2. Let 
$X$ be a complex manifold of dimension 
$n$ and 
$L$ be a holomorphic line bundle on 
$X$. Take 
$D_{1},D_{2},\ldots ,D_{n-1}\in |L|$. Assume that 
$C:=\bigcap _{\unicode[STIX]{x1D706}=1}^{n-1}D_{\unicode[STIX]{x1D706}}$ is a smooth elliptic curve, 
$L|_{C}\in {\mathcal{E}}_{1}^{(1)}(C)$, and 
$\{D_{\unicode[STIX]{x1D706}}\}_{\unicode[STIX]{x1D706}=1}^{n-1}$ intersects transversally along 
$C$. Then 
$L$ is semipositive (i.e. 
$L$ admits a 
$C^{\infty }$ Hermitian metric with semipositive curvature).
Note that 
$L$ as in Corollary 1.2 has 
$C$ as a stable base locus: 
$C=\text{SB}(L):=\bigcap _{m\geqslant 1}\text{Bs}\,|L^{m}|$. Corollary 1.2 can be applied to the example of the blow-up of a del Pezzo manifold at a general point as follows:
Corollary 1.3. Let 
$(V,L)$ be a del Pezzo manifold of degree 
$1$ (i.e. 
$V$ is a projective manifold of dimension 
$n$ and 
$L$ is an ample line bundle on 
$V$ with 
$K_{V}^{-1}\cong L^{n-1}$ and the self-intersection number 
$(L^{n})$ is equal to 
$1$), and 
$C\subset V$ be an intersection of general 
$n-1$ elements of 
$|L|$. For each point 
$q\in C$ with 
$L|_{C}\otimes [-q]\in {\mathcal{E}}_{1}^{(0)}(C)\cup {\mathcal{E}}_{1}^{(1)}(C)$, the anticanonical bundle of the blow-up of 
$V$ at 
$q$ is semipositive.
We remark that Corollary 1.3 can be regarded as a generalization of the known phenomena for the blow-up of 
$\mathbb{P}^{2}$ at general nine points ([Reference Arnol’dA], [Reference BrunellaB], [Reference UedaU], see also [Reference DemaillyD2, Section 1]), or the blow-up of 
$\mathbb{P}^{3}$ at general eight points ([Reference KoikeK2, Corollary 1]. Note that this result can be re-proved by using [Reference Koike and OgawaKO, Theorem 1.4, Remark 3.12], which is a corrected form of [Reference KoikeK2, Theorem 1]).
The organization of the paper is as follows. In Section 2, we summarize some fundamental facts and notations on the unitary flat vector bundles on a compact complex manifold and local defining functions of compact submanifolds. In Section 3, we give the definitions of the obstruction class 
$u_{n}(Y,X)$ and the type of the pair 
$(Y,X)$. In Section 4, we prove Theorem 1.1. In Section 5, we show Corollary 1.2. In Section 6, we give some examples. Here we will prove Corollary 1.3. In Section 7, we list some remaining problems.
2 Preliminaries
2.1 Unitary flat vector bundles on compact complex manifolds
Let 
$Y$ be a compact complex manifold and 
$E$ be a holomorphic vector bundle on 
$Y$. We say that 
$E$ is unitary flat if
$$\begin{eqnarray}E\in \text{Image}\,(H^{1}(Y,U(r))\rightarrow H^{1}(Y,\text{GL}_{r}({\mathcal{O}}_{Y}))).\end{eqnarray}$$ It means that, for a suitable choice of an open covering 
$\{U_{j}\}$ of 
$Y$ and a local frame 
$e_{j}=(e_{j}^{1},e_{j}^{2},\ldots ,e_{j}^{r})$ of 
$E$ on each 
$U_{j}$, the transition matrix 
$T_{jk}$ of 
$\{(U_{j},e_{j})\}$ on each 
$U_{jk}:=U_{j}\cap U_{k}$ can be a locally constant function values in 
$U(r)$: i.e. for some 
$T_{jk}\in U(r)$, it holds that 
$e_{j}=T_{jk}e_{k}$, or equivalently, 
$e_{j}^{\unicode[STIX]{x1D706}}=\sum _{\unicode[STIX]{x1D707}=1}^{r}(T_{jk})_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D706}}\cdot e_{k}^{\unicode[STIX]{x1D707}}$. Here we denote by 
$(T_{jk})_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D706}}$ the 
$(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707})$th entry of 
$T_{jk}$. For a unitary flat vector bundle 
$E$, we can define a unitary flat metric 
$h$ on 
$E$ by regarding each 
$e_{j}$ as an orthonormal frame. By using this 
$h$, we obtain:
Lemma 2.1. Let 
$a_{j,\unicode[STIX]{x1D706}}:U_{j}\rightarrow \mathbb{C}$ be a holomorphic function. Assume that 
$\{(U_{j},\sum _{\unicode[STIX]{x1D706}=1}^{r}a_{j,\unicode[STIX]{x1D706}}\cdot e_{j}^{\unicode[STIX]{x1D706}})\}$ glue up to define a holomorphic global section 
$a$ of 
$E$. Then 
$a_{j,\unicode[STIX]{x1D706}}$ is a locally constant function on each 
$U_{j}$.
Proof. (See also the proof of [Reference SeshadriSe, Section 1, Proposition 1]) By applying the maximal principle to the psh (plurisubharmonic) function 
$|a|_{h}^{2}$, we obtain 
$|a|_{h}^{2}\equiv C$ for some constant 
$C$. As it holds that 
$|a_{j,\unicode[STIX]{x1D706}}|^{2}=C-\sum _{\unicode[STIX]{x1D706}\not =\unicode[STIX]{x1D707}}|a_{j,\unicode[STIX]{x1D707}}|^{2}$ on each 
$U_{j}$, we conclude that 
$|a_{j,\unicode[STIX]{x1D706}}|^{2}$ is pluriharmonic for each 
$\unicode[STIX]{x1D706}=1,2,\ldots ,r$, which proves the lemma.◻
By considering the monodromy of the Chern connection of 
$h$, we obtain a unitary representation 
$\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}_{E}:\unicode[STIX]{x1D70B}_{1}(Y,\ast )\rightarrow U(r)$. Conversely, for a given unitary representation 
$\unicode[STIX]{x1D70C}:\unicode[STIX]{x1D70B}_{1}(Y,\ast )\rightarrow U(r)$, we can construct a unitary flat vector bundle 
$E_{\unicode[STIX]{x1D70C}}$ by
$$\begin{eqnarray}E_{\unicode[STIX]{x1D70C}}:=\widetilde{Y}\times \mathbb{C}^{r}/{\sim}_{\unicode[STIX]{x1D70C}},\end{eqnarray}$$ where 
$\widetilde{Y}\rightarrow Y$ is the universal covering of 
$Y$ and 
${\sim}_{\unicode[STIX]{x1D70C}}$ is the relation defined by 
$(z,v){\sim}_{\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D6FE}z,\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6FE})v)$ for each 
$(z,v)\in \widetilde{Y}\times \mathbb{C}^{r}$ and 
$\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D70B}_{1}(Y,\ast )$.
Proposition 2.2. The above gives 
$1:1$-correspondence between the image of the natural map 
$H^{1}(Y,U(r))\rightarrow H^{1}(Y,\text{GL}_{r}({\mathcal{O}}_{Y}))$ and the set 
$\{\unicode[STIX]{x1D70C}:U(r)\text{-representation of}~\unicode[STIX]{x1D70B}_{1}(Y,\ast )\}/\sim$, where 
$\unicode[STIX]{x1D70C}\sim \unicode[STIX]{x1D70C}^{\prime }$ means that there exists 
$A\in \text{GL}_{r}(\mathbb{C})$ such that 
$A^{-1}\cdot \unicode[STIX]{x1D70C}\cdot A=\unicode[STIX]{x1D70C}^{\prime }$ holds.
For proving Proposition 2.2, we need the following:
Lemma 2.3. Let 
$E$ and 
$F$ be unitary flat vector bundles on 
$Y$. Assume that 
$E$ and 
$F$ are isomorphic to each other as holomorphic vector bundles. Then the image of 
$E$ and 
$F$ by the natural map
$$\begin{eqnarray}H^{1}(Y,U(r))\rightarrow H^{1}(Y,\text{GL}_{r}(\mathbb{C}))\end{eqnarray}$$coincide with each other.
Proof. The lemma is shown by applying Lemma 2.1 to a global section of the unitary flat vector bundle 
$\text{Hom}(E,F)\cong E^{\ast }\otimes F$. See the proof of [Reference SeshadriSe, Section 1, Proposition 1] for the details.◻
Proof of Proposition 2.2.
Let 
$E$ be a unitary flat vector bundle. Take 
$\{(U_{j},e_{j})\}$ and 
$\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}_{E}$ as above. Let 
$\{(U_{j},f_{j}=(f_{j}^{1},f_{j}^{2},\ldots ,f_{j}^{r}))\}$ be another local frame of 
$E$ with 
$f_{j}=S_{jk}f_{k}$ on each 
$U_{jk}$ (
$S_{jk}\in U(r)$), and 
$\unicode[STIX]{x1D70C}^{\prime }:\unicode[STIX]{x1D70B}_{1}(Y,\ast )\rightarrow U(r)$ be the monodromy defined by using 
$f_{j}$ as an orthonormal frame on each 
$U_{j}$. By Lemma 2.3, we can take 
$A_{j}\in \text{GL}_{r}(\mathbb{C})$ for each 
$j$ with 
$A_{j}S_{jk}=T_{jk}A_{k}$. Then, for each loop 
$\unicode[STIX]{x1D6FE}$ of 
$Y$ with a base point 
$\ast \in U_{j}$, we can calculate that 
$\unicode[STIX]{x1D70C}([\unicode[STIX]{x1D6FE}])=A_{j}\cdot \unicode[STIX]{x1D70C}^{\prime }([\unicode[STIX]{x1D6FE}])\cdot A_{j}^{-1}$, which proves 
$\unicode[STIX]{x1D70C}\sim \unicode[STIX]{x1D70C}^{\prime }$.
Conversely, let 
$\unicode[STIX]{x1D70C}$ and 
$\unicode[STIX]{x1D70C}^{\prime }$ be two 
$U(r)$-representations of 
$\unicode[STIX]{x1D70B}_{1}(Y,\ast )$ with 
$\unicode[STIX]{x1D70C}\sim \unicode[STIX]{x1D70C}^{\prime }$. Take 
$A\in \text{GL}_{r}(\mathbb{C})$ such that 
$A^{-1}\cdot \unicode[STIX]{x1D70C}\cdot A=\unicode[STIX]{x1D70C}^{\prime }$. Define the map 
$F:\widetilde{Y}\times \mathbb{C}^{r}\rightarrow \widetilde{Y}\times \mathbb{C}^{r}$ by 
$F(z,v):=(z,A^{-1}\cdot v)$. Then it is easily observed that 
$F$ induces an isomorphism 
$E_{\unicode[STIX]{x1D70C}}\cong E_{\unicode[STIX]{x1D70C}^{\prime }}$, which proves the proposition.◻
Remark 2.4. The definition of the relation 
${\sim}$ in Proposition 2.2 can be replaced by the following one: we say 
$\unicode[STIX]{x1D70C}\sim \unicode[STIX]{x1D70C}^{\prime }$ if there exists 
$U\in U(r)$ such that 
$U^{-1}\cdot \unicode[STIX]{x1D70C}\cdot U=\unicode[STIX]{x1D70C}^{\prime }$ holds. It is because, for each 
$A\in \text{GL}_{r}(\mathbb{C})$ and 
$S\in U(r)$ with 
$A^{-1}\cdot S\cdot A\in U(r)$, it holds that 
$A^{-1}\cdot S\cdot A=U_{A}^{-1}\cdot S\cdot U_{A}$, where 
$U_{A}$ is the unitary part of the polar decomposition 
$A=U_{A}\cdot P_{A}$. Therefore, by the same argument as in the proof of Proposition 2.2, we obtain that the image of 
$H^{1}(Y,U(r))\rightarrow H^{1}(Y,\text{GL}_{r}({\mathcal{O}}_{Y}))$ is naturally isomorphic to 
$H^{1}(Y,U(r))$, or equivalently, the natural map 
$H^{1}(Y,U(r))\rightarrow H^{1}(Y,\text{GL}_{r}({\mathcal{O}}_{Y}))$ is injective.
Remark 2.5. Here we give another (more direct) proof of the injectivity of the natural map 
$i:H^{1}(Y,U(r))\rightarrow H^{1}(Y,\text{GL}_{r}({\mathcal{O}}_{Y}))$, which was taught by Professor Tetsuo Ueda. Let 
$E:=\{(U_{jk},T_{jk})\}$ and 
$F:=\{(U_{jk},S_{jk})\}$ be elements of 
$H^{1}(Y,U(r))$ with 
$i(E)=i(F)$. By Lemma 2.3, we can take 
$A_{j}\in \text{GL}_{r}(\mathbb{C})$ for each 
$j$ such that 
$S_{jk}A_{k}=A_{j}T_{jk}$ holds. Denote by 
$A_{j}=P_{j}U_{j}$ the polar decomposition of 
$A_{j}$, where 
$P_{j}$ is the positive definite Hermitian part and 
$U_{j}$ is the unitary part. Then we have 
$(S_{jk}P_{k}S_{jk}^{-1})\cdot (S_{jk}U_{k})=P_{j}\cdot (U_{j}T_{jk})$. By the uniqueness of the polar decomposition, we obtain 
$S_{jk}U_{k}=U_{j}T_{jk}$.
Remark 2.6. Let 
$F$ be a holomorphic subbundle of a unitary flat vector bundle 
$E$ on 
$Y$. Then it follows from the same argument as in the proof of Lemma 2.3 that 
$F$ is a unitary flat subbundle of 
$E$: i.e. 
$F$ is the unitary flat vector bundle which corresponds to a unitary subrepresentation of 
$\unicode[STIX]{x1D70C}_{E}$.
2.2 Local defining functions
Let 
$X$ be a complex manifold 
$X$ and 
$Y$ be a compact complex submanifold of codimension 
$r$ with unitary flat normal bundle. Take a sufficiently fine open covering 
$\{U_{j}\}$ of 
$Y$. In this paper, we always assume that 
$\#\{U_{j}\}<\infty$ and that 
$U_{j}$ and 
$U_{jk}$ are simply connected and Stein for each 
$j$ and 
$k$. Denote by 
$z_{j}$ a coordinate of 
$U_{j}$. Take a sufficiently small tubular neighborhood 
$V$ of 
$Y$ in 
$X$ and an open covering 
$\{V_{j}\}$ of 
$V$ with 
$V_{j}\cap Y=U_{j}$ for each 
$j$. By shrinking 
$V$ and 
$V_{j}$’s if necessary, we may assume that 
$U_{jk}\not =\emptyset$ if and only if 
$V_{jk}\not =\emptyset$.
Take a defining functions system 
$w_{j}=(w_{j}^{1},w_{j}^{2},\ldots ,w_{j}^{r})$ of 
$U_{j}$ in 
$V_{j}$. We regard 
$(z_{j},w_{j})$ as a coordinates system of 
$V_{j}$. Note that, here we denote by the same letter 
$z_{j}$ an extension of 
$z_{j}$ to 
$V_{j}$. In what follows, we always use the same 
$z_{j}$’s even though we often change 
$w_{j}$’s and shrink 
$V$ and 
$V_{j}$’s. More precisely, we fix a local projection 
$p_{j}:V_{j}\rightarrow U_{j}$ and, for any function 
$f$ defined on 
$U_{j}$, we always use the pull back 
$p_{j}^{\ast }$ for extending 
$f$ to 
$V_{j}$ and denote 
$p_{j}^{\ast }f$ by the same letter 
$f(z_{j})$.
As 
$N_{Y/X}$ is unitary flat, we can take a local frame 
$e_{j}=(e_{j}^{1},e_{j}^{2},\ldots e_{j}^{r})$ of the conormal bundle 
$N_{Y/X}^{\ast }$ on 
$U_{j}$ with 
$e_{j}=T_{jk}e_{k}$ for each 
$j$, 
$k$ (
$T_{jk}\in U(r)$). By changing 
$w_{j}$ if necessary, we may assume that 
$dw_{j}=e_{j}$ holds on each 
$U_{j}$ (consider a new defining functions system 
$M_{j}(z_{j})\cdot w_{j}$ if 
$e_{j}=M_{j}(z_{j})\cdot dw_{j}|_{U_{j}}$). In what follows, we always assume this condition for the system 
$\{w_{j}\}$. Then it follows that the expansion of the function 
$(\sum _{\unicode[STIX]{x1D707}=1}^{r}(T_{jk})_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D706}}\cdot w_{k}^{\unicode[STIX]{x1D707}})|_{V_{jk}}$ in the variables 
$w_{j}$ is in the form of 
$\sum _{\unicode[STIX]{x1D707}=1}^{r}(T_{jk})_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D706}}\cdot w_{k}^{\unicode[STIX]{x1D707}}=w_{j}^{\unicode[STIX]{x1D706}}+O(|w_{j}|^{2})$, where we denote by 
$O(|w_{j}|^{2})$ the higher order terms. Let us denote this expansion by
$$\begin{eqnarray}\mathop{\sum }_{\unicode[STIX]{x1D707}=1}^{r}(T_{jk})_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D706}}\cdot w_{k}^{\unicode[STIX]{x1D707}}=w_{j}^{\unicode[STIX]{x1D706}}+\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|\geqslant 2}f_{kj,\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D706}}(z_{j})\cdot w_{j}^{\unicode[STIX]{x1D6FC}},\end{eqnarray}$$ where 
$\unicode[STIX]{x1D6FC}=(\unicode[STIX]{x1D6FC}_{1},\unicode[STIX]{x1D6FC}_{2},\ldots ,\unicode[STIX]{x1D6FC}_{r})\in (\mathbb{Z}_{{\geqslant}0})^{r}$, 
$|\unicode[STIX]{x1D6FC}|:=\unicode[STIX]{x1D6FC}_{1}+\unicode[STIX]{x1D6FC}_{2}+\cdots +\unicode[STIX]{x1D6FC}_{r}$, and 
$w_{j}^{\unicode[STIX]{x1D6FC}}:=\prod _{\unicode[STIX]{x1D706}=1}^{r}(w_{j}^{\unicode[STIX]{x1D706}})^{\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}}$. We also denote this expansion by
$$\begin{eqnarray}T_{jk}w_{k}=w_{j}+\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|\geqslant 2}f_{kj,\unicode[STIX]{x1D6FC}}(z_{j})\cdot w_{j}^{\unicode[STIX]{x1D6FC}},\end{eqnarray}$$where
$$\begin{eqnarray}w_{j}=\left(\begin{array}{@{}c@{}}w_{j}^{1}\\ w_{j}^{2}\\ \vdots \\ w_{j}^{r}\end{array}\right),\quad f_{kj,\unicode[STIX]{x1D6FC}}=\left(\begin{array}{@{}c@{}}f_{kj,\unicode[STIX]{x1D6FC}}^{1}\\ f_{kj,\unicode[STIX]{x1D6FC}}^{2}\\ \vdots \\ f_{kj,\unicode[STIX]{x1D6FC}}^{r}\end{array}\right).\end{eqnarray}$$ We denote by 
$e_{j}^{\ast }=(e_{j,1}^{\ast },e_{j,2}^{\ast },\ldots ,e_{j,r}^{\ast })$ the dual of 
$e_{j}$ and regard it as a local frame of 
$N_{Y/X}$. For each 
$\unicode[STIX]{x1D6FC}$ with 
$|\unicode[STIX]{x1D6FC}|=n$, we denote by 
$e_{j}^{\unicode[STIX]{x1D6FC}}$ the local section 
$\prod _{\unicode[STIX]{x1D706}=1}^{r}(e_{j}^{\unicode[STIX]{x1D706}})^{\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}}$ of the symmetric tensor bundle 
$S^{n}N_{Y/X}^{\ast }$. Then 
$\{e_{j}^{\unicode[STIX]{x1D6FC}}\}\text{}_{|\unicode[STIX]{x1D6FC}|=n}$ forms a local frame of 
$S^{n}N_{Y/X}^{\ast }$ on 
$U_{j}$. On each 
$U_{jk}$, it holds that 
$e_{j}^{\unicode[STIX]{x1D6FC}}=\prod _{\unicode[STIX]{x1D706}=1}^{r}(\sum _{\unicode[STIX]{x1D707}=1}^{r}(T_{jk})_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D706}}\cdot e_{k}^{\unicode[STIX]{x1D707}})^{\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}}$. Let us denote by 
$\unicode[STIX]{x1D70F}_{jk,\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D6FC}}$ the coefficient of 
$e_{k}^{\unicode[STIX]{x1D6FD}}$ in the expansion of the right hand side: i.e.
$$\begin{eqnarray}e_{j}^{\unicode[STIX]{x1D6FC}}=\mathop{\sum }_{|\unicode[STIX]{x1D6FD}|=n}\unicode[STIX]{x1D70F}_{jk,\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D6FC}}\cdot e_{k}^{\unicode[STIX]{x1D6FD}}.\end{eqnarray}$$Remark 2.7. Note that the matrix 
$(\unicode[STIX]{x1D70F}_{jk,\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D6FC}})$ need not be unitary when 
$r>1$, however the vector bundle 
$S^{n}N_{Y/X}^{\ast }$ itself is unitary flat. Here we explain the unitary flat structure of 
$S^{n}N_{Y/X}^{\ast }$ induced from the orthonormal frames 
$\{(U_{j},e_{j})\}$ of 
$N_{Y/X}^{\ast }$. Let us consider the local sections
$$\begin{eqnarray}\{e_{j}^{\unicode[STIX]{x1D706}_{1}}\otimes e_{j}^{\unicode[STIX]{x1D706}_{2}}\otimes \cdots \otimes e_{j}^{\unicode[STIX]{x1D706}_{n}}|\unicode[STIX]{x1D706}_{p}\in \{1,2,\ldots ,r\}\}\end{eqnarray}$$ of 
$\bigotimes ^{n}N_{Y/X}^{\ast }:=N_{Y/X}^{\ast }\otimes N_{Y/X}^{\ast }\otimes \cdots \otimes N_{Y/X}^{\ast }$ and regard it as a local frame on each 
$U_{j}$. Then, as the transition matrix on 
$U_{jk}$ is equal to 
$S_{jk}\otimes S_{jk}\otimes \cdots \otimes S_{jk}\in U(r^{n})$, this local frame can be regarded as an orthonormal frame of the unitary flat metric induced from that of 
$N_{Y/X}^{\ast }$. By regarding each symmetric section of 
$\bigotimes ^{n}N_{Y/X}^{\ast }$ as a section of 
$S^{n}N_{Y/X}^{\ast }$ in the usual manner, we can regard 
$S^{n}N_{Y/X}^{\ast }$ as a unitary flat subbundle of 
$\bigotimes ^{n}N_{Y/X}^{\ast }$ with an orthonormal frame 
$\{\sqrt{n!/\unicode[STIX]{x1D6FC}!}\cdot e_{j}^{\unicode[STIX]{x1D6FC}}\}\text{}_{\unicode[STIX]{x1D6FC}}$ on each 
$U_{j}$, which induces the unitary flat structure of 
$S^{n}N_{Y/X}$ (
$\unicode[STIX]{x1D6FC}!:=\prod _{\unicode[STIX]{x1D706}=1}^{r}\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}!$).
3 The obstruction classes and the type of the pair $(Y,X)$
3.1 Definition of the obstruction classes
Take 
$\{(U_{j},z_{j})\},\{(V_{j},(z_{j},w_{j}))\},\{e_{j}\}$, and 
$\{(U_{jk},T_{jk})\}$ as in Section 2.2. In this section, we will define the obstruction class 
$u_{n}(Y,X)$ as a straightforward generalization of the Ueda class.
Definition 3.1. We say that the system 
$\{(V_{j},w_{j})\}$ is of type 
$n$ (
$n\geqslant 1$) if the coefficient function 
$f_{kj,\unicode[STIX]{x1D6FC}}$ in the expansion (1) is equal to 
$0$ for any 
$\unicode[STIX]{x1D6FC}$ with 
$|\unicode[STIX]{x1D6FC}|\leqslant n$ on each 
$U_{jk}$.
Let 
$\{(V_{j},w_{j})\}$ be a system of type 
$n$. Then, by definition, the expansion (1) can be written as follows: 
$T_{jk}w_{k}=w_{j}+\sum _{|\unicode[STIX]{x1D6FC}|\geqslant n+1}f_{kj,\unicode[STIX]{x1D6FC}}(z_{j})\cdot w_{j}^{\unicode[STIX]{x1D6FC}}$. For
$$\begin{eqnarray}f_{kj,n+1}=\left(\begin{array}{@{}c@{}}f_{kj,n+1}^{1}\\ f_{kj,n+1}^{2}\\ \vdots \\ f_{kj,n+1}^{r}\end{array}\right):=\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|=n+1}\left(\begin{array}{@{}c@{}}f_{kj,\unicode[STIX]{x1D6FC}}^{1}\\ f_{kj,\unicode[STIX]{x1D6FC}}^{2}\\ \vdots \\ f_{kj,\unicode[STIX]{x1D6FC}}^{r}\end{array}\right)\cdot e_{j}^{\unicode[STIX]{x1D6FC}},\end{eqnarray}$$we can show the following:
Lemma 3.2. The system 
$\{(U_{jk},\sum _{\unicode[STIX]{x1D706}=1}^{r}e_{j,\unicode[STIX]{x1D706}}^{\ast }\otimes f_{kj,n+1}^{\unicode[STIX]{x1D706}})\}$ satisfies the 
$1$-cocycle condition: i.e.
$$\begin{eqnarray}\bigg\{\biggl(U_{jk},\mathop{\sum }_{\unicode[STIX]{x1D706}=1}^{r}e_{j,\unicode[STIX]{x1D706}}^{\ast }\otimes f_{kj,n+1}^{\unicode[STIX]{x1D706}}\biggr)\bigg\}\in {\check{Z}}^{1}(Y,N_{Y/X}\otimes S^{n+1}N_{Y/X}^{\ast }).\end{eqnarray}$$Proof. The lemma can be shown by summing the expansions of 
$T_{jk}w_{k}-w_{j}$, 
$T_{jk}\cdot (T_{k\ell }w_{\ell }-w_{k})$, and 
$T_{j\ell }\cdot (T_{\ell j}w_{j}-w_{\ell })$ on 
$V_{jk\ell }$ and comparing the terms with 
$w_{j}^{\unicode[STIX]{x1D6FC}}$ of the both hand sides for each 
$\unicode[STIX]{x1D6FC}$ with 
$|\unicode[STIX]{x1D6FC}|=n+1$.◻
Definition 3.3. For a system 
$\{(V_{j},w_{j})\}$ of type 
$n$, we denote by 
$u_{n}(Y,X)=u_{n}(Y,X;\{w_{j}\})$ the class 
$[\{(U_{jk},\sum _{\unicode[STIX]{x1D706}=1}^{r}e_{j,\unicode[STIX]{x1D706}}^{\ast }\otimes f_{kj,n+1}^{\unicode[STIX]{x1D706}})\}]\in H^{1}(Y,N_{Y/X}\otimes S^{n+1}N_{Y/X}^{\ast })$ and call it the 
$n$th obstruction class.
Lemma 3.4. Let 
$\{(V_{j},w_{j})\}$ be a system of type 
$n$ with 
$dw_{j}|_{U_{j}}=e_{j}$ for each 
$j$. Assume 
$u_{n}(Y,X;\{w_{j}\})=0$. Then there exists a system 
$\{(U_{j},\widehat{w}_{j})\}$ of type 
$n+1$ with 
$d\widehat{w}_{j}|_{U_{j}}=e_{j}$ for each 
$j$.
Proof. From the assumption 
$u_{n}(Y,X;\{w_{j}\})=0$, we can take
$$\begin{eqnarray}f_{j}=\left(\begin{array}{@{}c@{}}f_{j}^{1}\\ f_{j}^{2}\\ \vdots \\ f_{j}^{r}\end{array}\right)=\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|=n+1}\left(\begin{array}{@{}c@{}}f_{j,\unicode[STIX]{x1D6FC}}^{1}\\ f_{j,\unicode[STIX]{x1D6FC}}^{2}\\ \vdots \\ f_{j,\unicode[STIX]{x1D6FC}}^{r}\end{array}\right)\cdot e_{j}^{\unicode[STIX]{x1D6FC}}\end{eqnarray}$$such that
$$\begin{eqnarray}f_{j,\unicode[STIX]{x1D6FC}}-\mathop{\sum }_{|\unicode[STIX]{x1D6FD}|=n+1}T_{jk}f_{k,\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D70F}_{kj,\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D6FD}}=f_{kj,\unicode[STIX]{x1D6FC}}\end{eqnarray}$$ holds on 
$U_{jk}$ for each 
$\unicode[STIX]{x1D6FC}$ with 
$|\unicode[STIX]{x1D6FC}|=n+1$. Define a new system 
$\{\widehat{w}_{j}\}$ by
$$\begin{eqnarray}\widehat{w}_{j}^{\unicode[STIX]{x1D706}}:=w_{j}^{\unicode[STIX]{x1D706}}+\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|=n+1}f_{j,\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D706}}(z_{j})\cdot w_{j}^{\unicode[STIX]{x1D6FC}}.\end{eqnarray}$$ Then it follows from a simple computation that the system 
$\{\widehat{w}_{j}\}$ is of type 
$n+1$ with 
$d\widehat{w}_{j}|_{U_{j}}=dw_{j}|_{U_{j}}$, which proves the lemma.◻
Remark 3.5. Here we consider the case where 
$N_{Y/X}$ admits a direct decomposition 
$N_{Y/X}=N_{1}\oplus N_{2}\oplus \cdots \oplus N_{r}$ such that each 
$N_{\unicode[STIX]{x1D706}}$ is a unitary flat line bundle on 
$Y$. It follows from Schur’s lemma that such 
$N_{\unicode[STIX]{x1D706}}$’s are unique up to ordering and isomorphism (note that, as we mentioned in Remark 2.4, two unitary flat line bundles are isomorphic to each other if and only if the corresponding unitary representations coincide, see also [Reference UedaU, Proposition 1(2)]). In this case, the transition matrix 
$T_{jk}$ is written in the form 
$T_{jk}=\text{diag}\,(t_{jk}^{1},t_{jk}^{2},\ldots ,t_{jk}^{r})$ (
$t_{jk}^{\unicode[STIX]{x1D706}}\in U(1)$). Then it holds that
$$\begin{eqnarray}\unicode[STIX]{x1D70F}_{kj,\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D6FC}}=\left\{\begin{array}{@{}ll@{}}t_{jk}^{-\unicode[STIX]{x1D6FC}}:=\mathop{\prod }_{\unicode[STIX]{x1D706}=1}^{r}(t_{jk}^{\unicode[STIX]{x1D706}})^{-\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}}\quad & (\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FC})\\ 0\quad & (\text{otherwise}),\end{array}\right.\end{eqnarray}$$which induces a direct decomposition
$$\begin{eqnarray}N_{Y/X}\otimes S^{n+1}N_{Y/X}^{\ast }=\bigoplus _{\unicode[STIX]{x1D706}=1}^{r}\bigoplus _{|\unicode[STIX]{x1D6FC}|=n+1}N_{\unicode[STIX]{x1D706}}\otimes N_{\unicode[STIX]{x1D6FC}}^{-1}\end{eqnarray}$$ (
$N_{\unicode[STIX]{x1D6FC}}:=\bigotimes _{\unicode[STIX]{x1D706}=1}^{r}N_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}}$). Accordingly, we have a decomposition
$$\begin{eqnarray}u_{n}(Y,X;\{w_{j}\})=(u_{\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D706}}(Y,X;\{w_{j}\}))_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6FC}}\in \bigoplus _{\unicode[STIX]{x1D706}=1}^{r}\bigoplus _{|\unicode[STIX]{x1D6FC}|=n+1}H^{1}(Y,N_{\unicode[STIX]{x1D706}}\otimes N_{\unicode[STIX]{x1D6FC}}^{-1})\end{eqnarray}$$ of the 
$n$th obstruction class in this case. It is easily observed that 
$u_{\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D706}}(Y,X;\{w_{j}\})=[\{(U_{jk},f_{kj,\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D706}})\}]\in H^{1}(Y,N_{\unicode[STIX]{x1D706}}\otimes N_{\unicode[STIX]{x1D6FC}}^{-1})$.
3.2 Well-definedness of the obstruction classes and the type of the pair $(Y,X)$
Take 
$\{U_{j}\},\{V_{j}\}$, 
$\{e_{j}\},\{w_{j}\}$, and 
$\{T_{jk}\}$ as in Section 2.2. In this subsection, we study the dependence of the 
$n$th obstruction class 
$u_{n}(Y,X;\{w_{j}\})$ on a system 
$\{(V_{j},w_{j})\}$ of type 
$n$.
Lemma 3.6. Let 
$\{(V_{j},w_{j})\}$ and 
$\{(V_{j},\widehat{w}_{j})\}$ be systems of type 
$n$ such that 
$dw_{j}=d\widehat{w}_{j}=e_{j}$ holds on each 
$U_{j}$. Then, 
$u_{n}(Y,X;\{w_{j}\})=u_{n}(Y,X;\{\widehat{w}_{j}\})$.
Proof. Let
$$\begin{eqnarray}\displaystyle T_{jk}w_{k} & = & \displaystyle w_{j}+\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|\geqslant n+1}f_{kj,\unicode[STIX]{x1D6FC}}(z_{j})\cdot w_{j}^{\unicode[STIX]{x1D6FC}},\nonumber\\ \displaystyle T_{jk}\widehat{w}_{k} & = & \displaystyle \widehat{w}_{j}+\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|\geqslant n+1}\widehat{f}_{kj,\unicode[STIX]{x1D6FC}}(z_{j})\cdot \widehat{w}_{j}^{\unicode[STIX]{x1D6FC}}\nonumber\end{eqnarray}$$ be the expansions as in (1). It holds from the assumption 
$dw_{j}=d\widehat{w}_{j}$ that the expansion of 
$\widehat{w}_{j}$ in 
$w_{j}$ is in the form of 
$\widehat{w}_{j}^{\unicode[STIX]{x1D706}}=w_{j}^{\unicode[STIX]{x1D706}}+\sum _{|\unicode[STIX]{x1D6FC}|\geqslant 2}a_{j,\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D706}}(z_{j})\cdot w_{j}^{\unicode[STIX]{x1D6FC}}$, which in what follows we will denote by
$$\begin{eqnarray}\widehat{w}_{j}=w_{j}+\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|\geqslant 2}a_{j,\unicode[STIX]{x1D6FC}}(z_{j})\cdot w_{j}^{\unicode[STIX]{x1D6FC}}.\end{eqnarray}$$ Let 
$\unicode[STIX]{x1D708}_{0}$ be the maximum of the set of all 
$\unicode[STIX]{x1D708}\in \mathbb{Z}_{{\geqslant}2}$ such that 
$a_{j,\unicode[STIX]{x1D6FC}}\equiv 0$ holds for any 
$\unicode[STIX]{x1D6FC}$ with 
$|\unicode[STIX]{x1D6FC}|<\unicode[STIX]{x1D708}$ for each 
$j$. When 
$\unicode[STIX]{x1D708}_{0}>n+1$, it follows from 
$\widehat{w}_{j}=w_{j}+O(|w_{j}|^{n+2})$ that 
$f_{kj,\unicode[STIX]{x1D6FC}}=\widehat{f}_{kj,\unicode[STIX]{x1D6FC}}$, which proves the lemma. When 
$\unicode[STIX]{x1D708}_{0}=n+1$, we can calculate that
$$\begin{eqnarray}\displaystyle T_{jk}\widehat{w}_{k}-\widehat{w}_{j} & = & \displaystyle \mathop{\sum }_{|\unicode[STIX]{x1D6FD}|=n+1}\biggl(f_{kj,\unicode[STIX]{x1D6FD}}-a_{j,\unicode[STIX]{x1D6FD}}+T_{jk}\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|=n+1}a_{k,\unicode[STIX]{x1D6FC}}\cdot \unicode[STIX]{x1D70F}_{kj,\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D6FC}}\biggr)\cdot w_{j}^{\unicode[STIX]{x1D6FD}}\nonumber\\ \displaystyle & & \displaystyle +\,O(|w_{j}|^{n+2}).\end{eqnarray}$$By comparing the coefficients, we obtain the equation
$$\begin{eqnarray}\displaystyle & & \displaystyle \biggl[\unicode[STIX]{x1D6FF}\bigg\{\biggl(U_{j},\mathop{\sum }_{\unicode[STIX]{x1D706}=1}^{r}\mathop{\sum }_{|\unicode[STIX]{x1D6FD}|=n+1}a_{j,\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D706}}\cdot e_{j,\unicode[STIX]{x1D6FD}}^{\ast }\otimes e_{j}^{\unicode[STIX]{x1D6FD}}\biggr)\bigg\}\biggr]\nonumber\\ \displaystyle & & \displaystyle \quad =u_{n}(Y,X;\{w_{j}\})-u_{n}(Y,X;\{\widehat{w}_{j}\})\nonumber\end{eqnarray}$$ in 
${\check{Z}}^{1}(\{U_{j}\},N_{Y/X}\otimes S^{n+1}N_{Y/X}^{\ast })$, which proves the lemma. Finally, we will show the lemma for 
$\unicode[STIX]{x1D708}_{0}=\unicode[STIX]{x1D708}<n+1$ by assuming the lemma for 
$\unicode[STIX]{x1D708}_{0}\geqslant \unicode[STIX]{x1D708}+1$. As we may assume that 
$2\leqslant \unicode[STIX]{x1D708}\leqslant n$, it holds from the calculation as (2) that 
$\{(U_{j},\sum _{\unicode[STIX]{x1D706},|\unicode[STIX]{x1D6FD}|=\unicode[STIX]{x1D708}}a_{j,\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D706}}\cdot e_{j,\unicode[STIX]{x1D706}}^{\ast }\otimes e_{j}^{\unicode[STIX]{x1D6FD}})\}$ glue up to define a global section of 
$N_{Y/X}\otimes S^{\unicode[STIX]{x1D708}}N_{Y/X}^{\ast }$. Therefore, by Lemma 2.1 and Remark 2.7, it turns out that 
$a_{j,\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D706}}$ is a constant function on 
$U_{j}$ for each 
$\unicode[STIX]{x1D6FC}$ with 
$|\unicode[STIX]{x1D6FC}|=\unicode[STIX]{x1D708}$. Define a new system 
$\{v_{j}\}$ by 
$v_{j}^{\unicode[STIX]{x1D706}}:=\widehat{w}_{j}^{\unicode[STIX]{x1D706}}-\sum _{|\unicode[STIX]{x1D6FC}|=\unicode[STIX]{x1D708}}a_{j,\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D706}}w_{j}^{\unicode[STIX]{x1D6FC}}$. It is easy to see that 
$u_{n}(Y,X;\{\widehat{w}_{j}\})=u_{n}(Y,X;\{v_{j}\})$ holds (use 
$T_{jk}w_{k}=w_{j}+O(|w_{j}|^{n+1})$ and the fact that each 
$a_{j,\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D706}}$ is a constant). As 
$u_{n}(Y,X;\{w_{j}\})=u_{n}(Y,X;\{v_{j}\})$ holds from the lemma for 
$\unicode[STIX]{x1D708}_{0}\geqslant \unicode[STIX]{x1D708}+1$, we obtain the equation 
$u_{n}(Y,X;\{\widehat{w}_{j}\})=u_{n}(Y,X;\{w_{j}\})$.◻
Proposition 3.7. Let 
$\{(U_{j},e_{j})\}$ be a local frame of 
$N_{Y/X}^{\ast }$ as in Section 2.2. Then one and only one of the following holds:
(i) There exists
$n\geqslant 1$ and a system $\{w_{j}\}$ of type $n$ with $dw_{j}|_{U_{j}}=e_{j}$ and $u_{n}(Y,X;\{w_{j}\})\not =0$. In this case, there is no system $\{\widehat{w}_{j}\}$ of type $\unicode[STIX]{x1D708}$ with $d\widehat{w}_{j}|_{U_{j}}=e_{j}$ for any $\unicode[STIX]{x1D708}>n$.(ii) For each
$n\geqslant 1$, there exists a system $\{w_{j}\}$ of type $n$ with $dw_{j}|_{U_{j}}=e_{j}$ and $u_{n}(Y,X;\{w_{j}\})=0$.
Proof. Let 
$\{w_{j}\}$ be a system of type 
$n$ with 
$dw_{j}|_{U_{j}}=e_{j}$ for each 
$j$. Then, by Lemmas 3.4 and 3.6, we obtain that 
$u_{n}(Y,X;\{w_{j}\})=0$ if and only if there exists a system 
$\{\widehat{w}_{j}\}$ of type 
$n+1$ with 
$d\widehat{w}_{j}|_{U_{j}}=e_{j}$, which shows the proposition.◻
Definition 3.8. We define the type of the pair 
$(Y,X)$ as follows: 
$\text{type}\,(Y,X):=n$ for the case of Proposition 3.7(i), and 
$\text{type}\,(Y,X):=\infty$ for the case of Proposition 3.7(ii).
Lemma 3.9. 
$\text{type}\,(Y,X)$ does not depend on the choice of 
$\{e_{j}\}$.
Proof. Let 
$\{(U_{j},e_{j})\}$ and 
$\{(U_{j},\widehat{e}_{j})\}$ be local frames of 
$N_{Y/X}^{\ast }$ with 
$e_{j}=T_{jk}e_{k}$ and 
$\widehat{e}_{j}=\widehat{T}_{jk}\widehat{e}_{k}$ on each 
$U_{jk}$ (
$T_{jk},\widehat{T}_{jk}\in U(r)$). Assume that there exists a system 
$\{w_{j}\}$ of type 
$n$ with 
$dw_{j}|_{U_{j}}=e_{j}$. By Proposition 3.7, it is sufficient to show the existence of a system 
$\{\widehat{w}_{j}\}$ of type 
$n$ with 
$d\widehat{w}_{j}|_{U_{j}}=\widehat{e}_{j}$.
Let
$$\begin{eqnarray}T_{jk}w_{k}=w_{j}+\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|\geqslant n+1}f_{kj,\unicode[STIX]{x1D6FC}}(z_{j})\cdot w_{j}^{\unicode[STIX]{x1D6FC}}\end{eqnarray}$$ be the expansion (1) for the system 
$\{w_{j}\}$. From Lemma 2.3, we can take 
$M_{j}\in \text{GL}_{r}(\mathbb{C})$ with 
$\widehat{e}_{j}=M_{j}\cdot e_{j}$. Note that 
$M_{j}T_{jk}=\widehat{T}_{jk}M_{k}$ for each 
$j,k$. Define a new system 
$\{\widehat{w}_{j}\}$ by 
$\widehat{w}_{j}^{\unicode[STIX]{x1D706}}:=\sum _{\unicode[STIX]{x1D707}=1}^{r}(M_{j})_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D706}}\cdot w_{j}^{\unicode[STIX]{x1D707}}$. Then it clearly holds that 
$d\widehat{w}_{j}=\widehat{e}_{j}$. We can calculate that
$$\begin{eqnarray}\displaystyle \widehat{T}_{jk}\widehat{w}_{k} & = & \displaystyle \widehat{w}_{j}+\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|\geqslant n+1}M_{j}\cdot f_{kj,\unicode[STIX]{x1D6FC}}\cdot \mathop{\prod }_{\unicode[STIX]{x1D706}=1}^{r}\biggl(\mathop{\sum }_{\unicode[STIX]{x1D707}=1}^{r}(M_{j}^{-1})_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D706}}\cdot \widehat{w}_{j}^{\unicode[STIX]{x1D707}}\biggr)^{\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D708}}}\nonumber\\ \displaystyle & = & \displaystyle \widehat{w}_{j}+O(|\widehat{w}_{j}|^{n+1}),\nonumber\end{eqnarray}$$which proves the lemma. ◻
4 Proof of Theorem 1.1
4.1 Outline
Let 
$\{U_{j}\},\{V_{j}\}$, 
$\{e_{j}\}$, 
$\{T_{jk}\}$, and 
$\{w_{j}\}$ be as in Section 2.2. We will prove Theorem 1.1 based on the same idea as in the proof of [Reference UedaU, Theorem 3] and [Reference Koike and OgawaKO, Theorem 1.4]. We will construct a new system 
$\{u_{j}\}$ as the solution of a functional equation
$$\begin{eqnarray}w_{j}=u_{j}+\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|\geqslant 2}F_{j,\unicode[STIX]{x1D6FC}}(z_{j})\cdot u_{j}^{\unicode[STIX]{x1D6FC}},\end{eqnarray}$$where the coefficient functions
$$\begin{eqnarray}F_{j,\unicode[STIX]{x1D6FC}}(z_{j})=\left(\begin{array}{@{}c@{}}F_{j,\unicode[STIX]{x1D6FC}}^{1}(z_{j})\\ F_{j,\unicode[STIX]{x1D6FC}}^{2}(z_{j})\\ \vdots \\ F_{j,\unicode[STIX]{x1D6FC}}^{r}(z_{j})\end{array}\right)\end{eqnarray}$$ are holomorphic functions which we will construct in Section 4.4 so that 
$\{u_{j}\}$ exists and satisfies 
$T_{jk}u_{k}=u_{j}$ on a neighborhood of 
$U_{jk}$ for each 
$j,k$ (note that it follows from the inverse function theorem that there exists a unique solution 
$u_{j}$ if 
$\sum _{|\unicode[STIX]{x1D6FC}|\geqslant 2}F_{j,\unicode[STIX]{x1D6FC}}(z_{j})\cdot u_{j}^{\unicode[STIX]{x1D6FC}}$ has a positive radius of convergence). After taking such a solution 
$\{u_{j}\}$, we obtain Theorem 1.1(i) by considering a foliation 
${\mathcal{F}}$ whose leaves are locally defined by “
$u_{j}=$ (constant).”
Theorem 1.1(ii) is also shown by considering the same functional equation (3). Under the assumption of Theorem 1.1(ii), we will construct an initial system 
$\{w_{j}\}$ so that 
$\{w_{j}^{1}=0\}=V_{j}\cap S$ in Section 4.2. Starting from such an initial system, we will see in Section 4.4 that one can choose coefficient functions 
$\{F_{j,\unicode[STIX]{x1D6FC}}\}$ so that the following additional property holds for each 
$n\geqslant 2$:
- (Property)$_{n}$
- $F_{j,\unicode[STIX]{x1D6FC}}^{1}\equiv 0$ holds for any $\unicode[STIX]{x1D6FC}$ with $|\unicode[STIX]{x1D6FC}|=n$ and $\unicode[STIX]{x1D6FC}_{1}=0$.
Then it holds that the solution 
$\{u_{j}\}$ of the functional equation (3) also satisfies 
$\{u_{j}^{1}=0\}=V_{j}\cap S$ for each 
$j$. By considering a foliation 
${\mathcal{G}}_{S}$ whose leaves are locally defined by “
$u_{j}^{1}=$ (constant),” we obtain Theorem 1.1(ii).
Remark 4.1. It may seem that the foliations 
${\mathcal{F}}$ we will construct in the proof of Theorem 1.1(i) and (ii) are different from each other at first sight. Actually, the solutions 
$u_{j}$’s we will obtain are different from each other. However, the foliation 
${\mathcal{F}}$ itself does not depend on such differences. It can be shown by the following fact, which is obtained by the same arguments as in Lemma 3.6 and Proposition 3.9: Let 
$\{u_{j}\}$ and 
$\{\widehat{u}_{j}\}$ be systems with 
$u_{j}=T_{jk}u_{k}$ and 
$\widehat{u}_{j}=\widehat{T}_{jk}\widehat{u}_{k}$ on each 
$V_{jk}$ (
$T_{jk},\widehat{T}_{jk}\in U(r)$). Then there exist 
$M_{j}\in \text{GL}_{r}(\mathbb{C})$ and 
$a_{j,\unicode[STIX]{x1D6FC}}\in \mathbb{C}^{r}$ for each 
$j$ and 
$\unicode[STIX]{x1D6FC}$ such that 
$\widehat{u}_{j}=M_{j}\cdot (u_{j}+\sum _{|\unicode[STIX]{x1D6FC}|\geqslant 2}a_{j,\unicode[STIX]{x1D6FC}}\cdot u_{j}^{\unicode[STIX]{x1D6FC}})$ holds on each 
$V_{j}$.
4.2 Construction of the initial system $\{w_{j}\}$
We can use any system 
$\{w_{j}\}$ with 
$dw_{j}|_{U_{j}}=e_{j}$ as an initial system for the proof of Theorem 1.1(i). In what follows, we explain the construction of the initial system 
$\{w_{j}\}$ under the assumption of (ii).
By Remark 2.6 and the complete reducibility of the unitary representation, it follows that the short exact sequence 
$0\rightarrow N_{Y/S}\rightarrow N_{Y/X}\rightarrow N_{1}\rightarrow 0$ splits, where 
$N_{1}:=N_{S/X}|_{Y}$. Let 
$e_{j}^{1}$ be a local frame of 
$N_{1}^{-1}|_{U_{j}}$ with 
$e_{j}^{1}=t_{jk}^{1}e_{k}^{1}$ on 
$U_{jk}$ (
$t_{jk}^{1}\in U(1)$), and 
$e_{j}^{\prime }=(e_{j}^{2},e_{j}^{3},\ldots ,e_{j}^{r})$ be a local frame of 
$N_{Y/S}^{\ast }|_{U_{j}}$ with 
$e_{j}^{\prime }=S_{jk}e_{k}^{\prime }$ on 
$U_{jk}$ (
$S_{jk}\in U(r-1)$). Take a defining function 
$w_{j}^{1}$ of 
$V_{j}\cap S$ in 
$V_{j}$ for each 
$j$. By a simple argument, one can choose 
$\{w_{j}^{1}\}$ so that 
$dw_{j}^{1}=e_{j}^{1}$ holds on each 
$U_{j}$.
Lemma 4.2. Let 
$w_{j}^{1}$ be a defining function of 
$V_{j}\cap S$ in 
$V_{j}$ with 
$dw_{j}^{1}|_{U_{j}}=e_{j}^{1}$ for each 
$j$, and 
$\{(V_{j}\cap S,v_{j})\}=\{(V_{j}\cap S,(v_{j}^{2},v_{j}^{3},\ldots ,v_{j}^{r}))\}$ be a local defining functions system of 
$Y\subset V\cap S$ with 
$dv_{j}|_{U_{j}}=e_{j}^{\prime }$ for each 
$j$. Then there exists a holomorphic function 
$w_{j}^{\unicode[STIX]{x1D706}}:V_{j}\rightarrow \mathbb{C}$ with 
$w_{j}^{\unicode[STIX]{x1D706}}|_{V_{j}\cap S}=v_{j}^{\unicode[STIX]{x1D706}}$ for each 
$\unicode[STIX]{x1D706}=2,3,\ldots ,r$ such that 
$w_{j}:=(w_{j}^{1},w_{j}^{2},\ldots ,w_{j}^{r})$ satisfies 
$dw_{j}=T_{jk}dw_{k}$ on each 
$U_{jk}$, where
Proof. Take a holomorphic function 
$w_{j}^{\unicode[STIX]{x1D706}}:V_{j}\rightarrow \mathbb{C}$ with 
$w_{j}^{\unicode[STIX]{x1D706}}|_{V_{j}\cap S}=v_{j}^{\unicode[STIX]{x1D706}}$ for each 
$\unicode[STIX]{x1D706}=2,3,\ldots ,r$. Then the transition matrix 
$D_{jk}:=(\unicode[STIX]{x2202}w_{j}^{\unicode[STIX]{x1D706}}/\unicode[STIX]{x2202}w_{k}^{\unicode[STIX]{x1D707}}|_{U_{jk}})$ of 
$\{dw_{j}\}$ can be written in the form of
where 
$a_{jk}^{\unicode[STIX]{x1D706}}(z_{j})$ is a holomorphic function defined on 
$U_{j}$. As
$$\begin{eqnarray}\displaystyle e_{j,1}^{\ast }\otimes e_{j}^{1}-e_{k,1}^{\ast }\otimes e_{k}^{1} & = & \displaystyle e_{j,1}^{\ast }\otimes e_{j}^{1}-\mathop{\sum }_{\unicode[STIX]{x1D706}=1}^{r}\mathop{\sum }_{\unicode[STIX]{x1D707}=1}^{r}(D_{kj})_{\unicode[STIX]{x1D706}}^{1}\cdot (D_{kj}^{-1})_{1}^{\unicode[STIX]{x1D707}}\cdot e_{j,\unicode[STIX]{x1D707}}^{\ast }\otimes e_{j}^{\unicode[STIX]{x1D706}}\nonumber\\ \displaystyle & = & \displaystyle -\mathop{\sum }_{\unicode[STIX]{x1D707}=2}^{r}t_{kj}^{1}a_{jk}^{\unicode[STIX]{x1D707}}\cdot e_{j,\unicode[STIX]{x1D707}}^{\ast }\otimes e_{j}^{1},\nonumber\end{eqnarray}$$ it holds that the extension class of the short exact sequence 
$0\rightarrow N_{Y/S}\rightarrow N_{Y/X}\rightarrow N_{1}\rightarrow 0$ is equal to 
$[\{(U_{jk},-\sum _{\unicode[STIX]{x1D707}=2}^{r}t_{kj}^{1}a_{jk}^{\unicode[STIX]{x1D707}}\cdot e_{j,\unicode[STIX]{x1D707}}^{\ast }\otimes e_{j}^{1})\}]$ via the natural isomorphism 
$\text{Ext}^{1}(N_{1},N_{Y/S})\cong H^{1}(Y,N_{1}^{-1}\otimes N_{Y/S})$. Thus we can take 
$\{(U_{j},(m_{j}^{2}(z_{j}),m_{j}^{3}(z_{j}),\ldots ,m_{j}^{r}(z_{j})))\}$ such that
$$\begin{eqnarray}\left(\begin{array}{@{}c@{}}m_{j}^{2}\\ m_{j}^{3}\\ \vdots \\ m_{j}^{r}\end{array}\right)-(t_{jk}^{1})^{-1}S_{jk}\left(\begin{array}{@{}c@{}}m_{k}^{2}\\ m_{k}^{3}\\ \vdots \\ m_{k}^{r}\end{array}\right)=(t_{jk}^{1})^{-1}\cdot \left(\begin{array}{@{}c@{}}a_{jk}^{2}\\ a_{jk}^{3}\\ \vdots \\ a_{jk}^{r}\end{array}\right)\end{eqnarray}$$ holds on each 
$U_{jk}$, since the short exact sequence splits. Let us consider
Then it holds that 
$M_{j}^{-1}T_{jk}M_{k}=D_{jk}$. Thus the lemma is shown by considering a new system 
$M_{j}w_{j}$.◻
In what follows, we use the system 
$\{w_{j}\}$ as in Lemma 4.2 and use orthonormal frame 
$e_{j}:=dw_{j}|_{U_{j}}$ whenever we consider under the assumption of Theorem 1.1(ii).
4.3 Preliminary observation for constructing $\{F_{j,\unicode[STIX]{x1D6FC}}\}\text{}_{\unicode[STIX]{x1D6FC}}$
In this subsection, we give a heuristic explanation of how to construct 
$\{F_{j,\unicode[STIX]{x1D6FC}}\}\text{}_{\unicode[STIX]{x1D6FC}}$. For this purpose, we compare the expansions of the function 
$(T_{jk}w_{k})|_{V_{jk}}$ in two manners by assuming that the solution 
$u_{j}$ of the functional equation (3) exists and satisfies 
$T_{jk}u_{k}=u_{j}$ on 
$V_{jk}$.
The first expansion is obtained by using the functional equation (3) on 
$V_{k}$ as follows:
$$\begin{eqnarray}\displaystyle T_{jk}w_{k} & = & \displaystyle T_{jk}u_{k}+\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|\geqslant 2}T_{jk}F_{k,\unicode[STIX]{x1D6FC}}(z_{k})\cdot u_{k}^{\unicode[STIX]{x1D6FC}}\nonumber\\ \displaystyle & = & \displaystyle u_{j}+\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|\geqslant 2}T_{jk}F_{k,\unicode[STIX]{x1D6FC}}(z_{k})\cdot \mathop{\sum }_{|\unicode[STIX]{x1D6FD}|=|\unicode[STIX]{x1D6FC}|}\unicode[STIX]{x1D70F}_{kj,\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D6FC}}\cdot u_{j}^{\unicode[STIX]{x1D6FD}}\nonumber\\ \displaystyle & = & \displaystyle u_{j}+\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|\geqslant 2}T_{jk}\biggl(F_{k,\unicode[STIX]{x1D6FC}}(z_{k}(z_{j},0))+\mathop{\sum }_{|\unicode[STIX]{x1D6FE}|\geqslant 1}F_{kj,\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FE}}(z_{j})\cdot w_{j}^{\unicode[STIX]{x1D6FE}}\biggr)\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\sum }_{|\unicode[STIX]{x1D6FD}|=|\unicode[STIX]{x1D6FC}|}\unicode[STIX]{x1D70F}_{kj,\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D6FC}}\cdot u_{j}^{\unicode[STIX]{x1D6FD}}\nonumber\\ \displaystyle & = & \displaystyle u_{j}+\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|\geqslant 2}\mathop{\sum }_{|\unicode[STIX]{x1D6FD}|=|\unicode[STIX]{x1D6FC}|}T_{jk}F_{k,\unicode[STIX]{x1D6FC}}(z_{k}(z_{j},0))\cdot \unicode[STIX]{x1D70F}_{kj,\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D6FC}}\cdot u_{j}^{\unicode[STIX]{x1D6FD}}\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|\geqslant 2}\mathop{\sum }_{|\unicode[STIX]{x1D6FE}|\geqslant 1}\mathop{\sum }_{|\unicode[STIX]{x1D6FD}|=|\unicode[STIX]{x1D6FC}|}T_{jk}F_{kj,\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FE}}(z_{j})\cdot w_{j}^{\unicode[STIX]{x1D6FE}}\cdot \unicode[STIX]{x1D70F}_{kj,\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D6FC}}\cdot u_{j}^{\unicode[STIX]{x1D6FD}},\nonumber\end{eqnarray}$$ where 
$F_{kj,\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FE}}$’s are the coefficients of the expansion
$$\begin{eqnarray}F_{k,\unicode[STIX]{x1D6FC}}(z_{k}(z_{j},w_{j}))=F_{k,\unicode[STIX]{x1D6FC}}(z_{k}(z_{j},0))+\mathop{\sum }_{|\unicode[STIX]{x1D6FE}|\geqslant 1}F_{kj,\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FE}}(z_{j})\cdot w_{j}^{\unicode[STIX]{x1D6FE}}.\end{eqnarray}$$ on 
$V_{jk}$. Denoting by
$$\begin{eqnarray}h_{1,jk,\unicode[STIX]{x1D6FC}}(z_{j})=\left(\begin{array}{@{}c@{}}h_{1,jk,\unicode[STIX]{x1D6FC}}^{1}(z_{j})\\ h_{1,jk,\unicode[STIX]{x1D6FC}}^{2}(z_{j})\\ \vdots \\ h_{1,jk,\unicode[STIX]{x1D6FC}}^{r}(z_{j})\end{array}\right)\end{eqnarray}$$ the coefficient of 
$u_{j}^{\unicode[STIX]{x1D6FC}}$ in the expansion of the function
$$\begin{eqnarray}\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|\geqslant 2}\mathop{\sum }_{|\unicode[STIX]{x1D6FE}|\geqslant 1}\mathop{\sum }_{|\unicode[STIX]{x1D6FD}|=|\unicode[STIX]{x1D6FC}|}T_{jk}F_{kj,\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FE}}\cdot \unicode[STIX]{x1D70F}_{kj,\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D6FC}}\cdot u_{j}^{\unicode[STIX]{x1D6FD}}\cdot \mathop{\prod }_{\unicode[STIX]{x1D706}=1}^{r}\biggl(u_{j}^{\unicode[STIX]{x1D706}}+\mathop{\sum }_{|\unicode[STIX]{x1D6FF}|\geqslant 2}F_{j,\unicode[STIX]{x1D6FF}}^{\unicode[STIX]{x1D706}}\cdot u_{j}^{\unicode[STIX]{x1D6FF}}\biggr)^{\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D706}}},\end{eqnarray}$$we obtain
$$\begin{eqnarray}T_{jk}w_{k}=u_{j}+\mathop{\sum }_{|\unicode[STIX]{x1D6FD}|\geqslant 2}\biggl(\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|=|\unicode[STIX]{x1D6FD}|}T_{jk}F_{k,\unicode[STIX]{x1D6FC}}(z_{k}(z_{j},0))\cdot \unicode[STIX]{x1D70F}_{kj,\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D6FC}}+h_{1,jk,\unicode[STIX]{x1D6FD}}(z_{j})\biggr)\cdot u_{j}^{\unicode[STIX]{x1D6FD}}.\end{eqnarray}$$The second expansion is obtained by using the expansion (1) as follows:
$$\begin{eqnarray}\displaystyle T_{jk}w_{k} & = & \displaystyle w_{j}+\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|\geqslant 2}f_{kj,\unicode[STIX]{x1D6FC}}(z_{j})\cdot w_{j}^{\unicode[STIX]{x1D6FC}}\nonumber\\ \displaystyle & = & \displaystyle \biggl(u_{j}+\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|\geqslant 2}F_{j,\unicode[STIX]{x1D6FC}}(z_{j})\cdot u_{j}^{\unicode[STIX]{x1D6FC}}\biggr)\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|\geqslant 2}f_{kj,\unicode[STIX]{x1D6FC}}(z_{j})\cdot \mathop{\prod }_{\unicode[STIX]{x1D706}=1}^{r}\biggl(u_{j}^{\unicode[STIX]{x1D706}}+\mathop{\sum }_{|\unicode[STIX]{x1D6FD}|\geqslant 2}F_{j,\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D706}}(z_{j})\cdot u_{j}^{\unicode[STIX]{x1D6FD}}\biggr)^{\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}}\nonumber\\ \displaystyle & = & \displaystyle u_{j}+\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|\geqslant 2}(F_{j,\unicode[STIX]{x1D6FC}}(z_{j})+h_{2,jk,\unicode[STIX]{x1D6FC}}(z_{j}))\cdot u_{j}^{\unicode[STIX]{x1D6FC}},\nonumber\end{eqnarray}$$where we are denoting by
$$\begin{eqnarray}h_{2,jk,\unicode[STIX]{x1D6FC}}(z_{j})=\left(\begin{array}{@{}c@{}}h_{2,jk,\unicode[STIX]{x1D6FC}}^{1}(z_{j})\\ h_{2,jk,\unicode[STIX]{x1D6FC}}^{2}(z_{j})\\ \vdots \\ h_{2,jk,\unicode[STIX]{x1D6FC}}^{r}(z_{j})\end{array}\right)\end{eqnarray}$$ the coefficient of 
$u_{j}^{\unicode[STIX]{x1D6FC}}$ in the expansion of the function
$$\begin{eqnarray}\mathop{\sum }_{2\leqslant |\unicode[STIX]{x1D6FE}|<n}f_{kj,\unicode[STIX]{x1D6FE}}\mathop{\prod }_{\unicode[STIX]{x1D706}=1}^{r}\biggl(u_{j}^{\unicode[STIX]{x1D706}}+\mathop{\sum }_{2\leqslant |\unicode[STIX]{x1D6FD}|<n}F_{j,\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D706}}\cdot u_{j}^{\unicode[STIX]{x1D6FD}}\biggr)^{\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D706}}}.\end{eqnarray}$$ By comparing these two expansions, it is observed that the coefficient functions 
$\{F_{j,\unicode[STIX]{x1D6FD}}\}$ should be chosen so that the equation
$$\begin{eqnarray}F_{j,\unicode[STIX]{x1D6FD}}(z_{j})-\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|=|\unicode[STIX]{x1D6FD}|}T_{jk}F_{k,\unicode[STIX]{x1D6FC}}(z_{k})\cdot \unicode[STIX]{x1D70F}_{kj,\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D6FC}}=h_{1,jk,\unicode[STIX]{x1D6FD}}(z_{j})-h_{2,jk,\unicode[STIX]{x1D6FD}}(z_{j})\end{eqnarray}$$ holds on 
$U_{jk}$ for each 
$\unicode[STIX]{x1D6FD}$. Note that this equation means that
$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D6FF}\bigg\{\biggl(U_{j},\mathop{\sum }_{\unicode[STIX]{x1D706},|\unicode[STIX]{x1D6FD}|=n}F_{j,\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D706}}\cdot e_{j,\unicode[STIX]{x1D706}}^{\ast }\otimes e_{j}^{\unicode[STIX]{x1D6FD}}\biggr)\bigg\}\nonumber\\ \displaystyle & & \displaystyle \quad =\bigg\{\biggl(U_{jk},\mathop{\sum }_{\unicode[STIX]{x1D706},|\unicode[STIX]{x1D6FC}|=n}(h_{1,jk,\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D706}}-h_{2,jk,\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D706}})\cdot e_{j,\unicode[STIX]{x1D706}}^{\ast }\otimes e_{j}^{\unicode[STIX]{x1D6FC}}\biggr)\bigg\}\nonumber\end{eqnarray}$$ holds in 
${\check{Z}}^{1}(\{U_{j}\},N_{Y/X}\otimes S^{n}N_{Y/X}^{\ast })$ for each 
$n\geqslant 2$. Accordingly, we need 
$[\{(U_{jk},H_{jk,n})\}]=0\in H^{1}(Y,N_{Y/X}\otimes S^{n}N_{Y/X}^{\ast })$ for each 
$n\geqslant 2$, where
$$\begin{eqnarray}H_{jk,n}:=\mathop{\sum }_{\unicode[STIX]{x1D706}=1}^{r}\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|=n}(h_{1,jk,\unicode[STIX]{x1D6FC}}-h_{2,jk,\unicode[STIX]{x1D6FC}})\cdot e_{j,\unicode[STIX]{x1D706}}^{\ast }\otimes e_{j}^{\unicode[STIX]{x1D6FC}},\end{eqnarray}$$ which we will actually show in the next subsection by using the assumption 
$\text{type}\,(Y,X)=\infty$.
4.4 Inductive construction of $F_{j,\unicode[STIX]{x1D6FC}}$
Based on the observation in the previous subsection, we construct the coefficient functions 
$F_{j,\unicode[STIX]{x1D6FC}}$. In the following inductive construction, the following properties of 
$H_{jk,n}$ are essential: 
$H_{jk,2}=-\sum _{\unicode[STIX]{x1D706}=1}^{r}e_{j,\unicode[STIX]{x1D706}}^{\ast }\otimes f_{kj,2}^{\unicode[STIX]{x1D706}}$ holds and 
$H_{jk,n}$ depends only on 
$\{F_{j,\unicode[STIX]{x1D6FC}}\}\text{}_{|\unicode[STIX]{x1D6FC}|<n}$ for each 
$n\geqslant 3$. These properties are easily shown by the definition of 
$H_{jk,n}$.
Step $1$ (The construction of $\{F_{j,\unicode[STIX]{x1D6FC}}\}\text{}_{|\unicode[STIX]{x1D6FC}|=2}$)
For each 
$\unicode[STIX]{x1D6FD}$ with 
$|\unicode[STIX]{x1D6FD}|=2$, we take 
$\{(U_{j},F_{j,\unicode[STIX]{x1D6FD}})\}$ as a solution of the equation
$$\begin{eqnarray}F_{j,\unicode[STIX]{x1D6FD}}(z_{j})-\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|=2}T_{jk}F_{k,\unicode[STIX]{x1D6FC}}(z_{k})\cdot \unicode[STIX]{x1D70F}_{kj,\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D6FC}}=-f_{kj,\unicode[STIX]{x1D6FD}}.\end{eqnarray}$$ Note that there actually exists a solution of this equation, since 
$u_{1}(Y,X)=0$ holds. Strictly speaking, we choose appropriate solution 
$\{(U_{j},F_{j,\unicode[STIX]{x1D6FC}})\}$ of the above equation by using [Reference UedaU, Lemma 3] (=[Reference Kodaira and SpencerKS, Lemma 2]) or [Reference UedaU, Lemma 4] as we will explain the details in Section 4.5.
Claim 4.3. Fix 
$\{F_{j,\unicode[STIX]{x1D6FD}}\}\text{}_{|\unicode[STIX]{x1D6FD}|=2}$ as above. Then the following holds:
(a) For any choice of the remaining coefficient functions
$\{F_{j,\unicode[STIX]{x1D6FC}}\}\text{}_{|\unicode[STIX]{x1D6FC}|>2}$, the solution $\{u_{j}\}$ of the functional equation (3) is a system of type $2$ if exists.(b) Under the assumption of Theorem 1.1(ii), we can take
$\{F_{j,\unicode[STIX]{x1D6FD}}\}\text{}_{|\unicode[STIX]{x1D6FD}|=2}$ with $\text{(Property)}_{2}$.
Proof. (a) is shown by comparing the expansions of 
$(T_{jk}w_{k})|_{V_{jk}}$ in two manners as in the previous section. We skip the details here since the computation is almost the same as (and much easier than) that in the proof of Lemma 4.4 below.
Under the assumption of Theorem 1.1(ii), 
$f_{kj,\unicode[STIX]{x1D6FC}}^{1}\equiv 0$ holds for each 
$\unicode[STIX]{x1D6FC}$ with 
$\unicode[STIX]{x1D6FC}_{1}=0$, since 
$t_{jk}^{1}w_{k}^{1}$ is divisible by 
$w_{j}^{1}$ (recall that we are using an initial system 
$\{w_{j}\}$ as in Section 4.2 in this setting). Thus, by considering the decomposition 
$S^{m}N_{Y/X}^{\ast }=\bigoplus _{\ell =0}^{m}(N_{1}^{-\ell }\otimes S^{m-\ell }N_{Y/S}^{\ast })$, the defining equation of 
$\{F_{j,\unicode[STIX]{x1D6FC}}^{1}\}\text{}_{|\unicode[STIX]{x1D6FC}|=2,\unicode[STIX]{x1D6FC}_{1}=0}$ can be rewritten by the equation
$$\begin{eqnarray}\unicode[STIX]{x1D6FF}\biggl(\bigg\{\biggl(U_{j},\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|=2,\unicode[STIX]{x1D6FC}_{1}=0}F_{j,\unicode[STIX]{x1D6FC}}^{1}\cdot e_{j,1}^{\ast }\otimes e_{j}^{\unicode[STIX]{x1D6FC}}\biggr)\bigg\}\biggr)=0\end{eqnarray}$$ in 
${\check{Z}}^{1}(\{U_{j}\},N_{1}\otimes S^{2}N_{Y/S}^{\ast })$, which proves the assertion (b).◻
Step $(n-1)$ (The construction of $\{F_{j,\unicode[STIX]{x1D6FC}}\}\text{}_{|\unicode[STIX]{x1D6FC}|=n}$)
After choosing 
$F_{j,\unicode[STIX]{x1D6FC}}$ for each 
$\unicode[STIX]{x1D6FC}$ with 
$|\unicode[STIX]{x1D6FC}|<n$, we take 
$\{(U_{j},F_{j,\unicode[STIX]{x1D6FD}})\}\text{}_{|\unicode[STIX]{x1D6FD}|=n}$ as a solution of the equation
$$\begin{eqnarray}F_{j,\unicode[STIX]{x1D6FD}}(z_{j})-\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|=n}T_{jk}F_{k,\unicode[STIX]{x1D6FC}}(z_{k})\cdot \unicode[STIX]{x1D70F}_{kj,\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D6FC}}=h_{1,jk,\unicode[STIX]{x1D6FD}}-h_{2,jk,\unicode[STIX]{x1D6FD}}.\end{eqnarray}$$ Here we use the fact that 
$h_{1,jk,\unicode[STIX]{x1D6FD}}$ and 
$h_{2,jk,\unicode[STIX]{x1D6FD}}$ depend only on 
$\{F_{j,\unicode[STIX]{x1D6FC}}\}\text{}_{|\unicode[STIX]{x1D6FC}|<n}$. The existence of a solution 
$\{(U_{j},F_{j,\unicode[STIX]{x1D6FC}})\}$ is assured by Lemma 4.4(a) below. Strictly speaking, we choose appropriate 
$\{(U_{j},F_{j,\unicode[STIX]{x1D6FC}})\}$ from the solutions by using [Reference UedaU, Lemma 3] (=[Reference Kodaira and SpencerKS, Lemma 2]) or [Reference UedaU, Lemma 4] as we will explain the details in Section 4.5.
Lemma 4.4. Let 
$\{F_{j,\unicode[STIX]{x1D6FC}}\}\text{}_{|\unicode[STIX]{x1D6FC}|\leqslant n-1}$ be as in Step 
$(n-2)$. Then the following holds:
(a)
$[\{(U_{jk},H_{jk,n})\}]=0\in H^{1}(Y,N_{Y/X}\otimes S^{n}N_{Y/X}^{\ast })$.(b) Let
$\{F_{j,\unicode[STIX]{x1D6FC}}\}\text{}_{|\unicode[STIX]{x1D6FC}|=n}$ be as above. Then, for any choice of the remaining coefficient functions $\{F_{j,\unicode[STIX]{x1D6FC}}\}\text{}_{|\unicode[STIX]{x1D6FC}|>n}$, the solution $\{u_{j}\}$ of the functional equation (3) is of type $n$ if exists.(c) Under the assumption of Theorem 1.1(ii), we can take
$\{F_{j,\unicode[STIX]{x1D6FD}}\}\text{}_{|\unicode[STIX]{x1D6FD}|=n}$ with $\text{(Property)}_{n}$.
Proof. Fix 
$\{F_{j,\unicode[STIX]{x1D6FC}}\}\text{}_{|\unicode[STIX]{x1D6FC}|\geqslant n}$ and consider the solution 
$\{u_{j}\}$ of the functional equation (3). By Lemma 4.4(b) for Step 
$(n-2)$, it turns out that 
$\{u_{j}\}$ is of type 
$n-1$ and thus
$$\begin{eqnarray}u_{k}^{\unicode[STIX]{x1D6FC}}=\mathop{\prod }_{\unicode[STIX]{x1D706}=1}^{r}\biggl(\mathop{\sum }_{\unicode[STIX]{x1D707}=1}^{r}(T_{kj})_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D706}}u_{j}^{\unicode[STIX]{x1D707}}+O(|u_{j}|^{n})\biggr)^{\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D706}}}=\mathop{\sum }_{|\unicode[STIX]{x1D6FD}|=|\unicode[STIX]{x1D6FC}|}\unicode[STIX]{x1D70F}_{kj,\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D6FC}}\cdot u_{j}^{\unicode[STIX]{x1D6FD}}+O(|u_{j}|^{n+1})\end{eqnarray}$$ holds for each 
$\unicode[STIX]{x1D6FC}$ with 
$|\unicode[STIX]{x1D6FC}|\geqslant 2$. Consider the expansions
$$\begin{eqnarray}\displaystyle T_{jk}w_{k} & = & \displaystyle T_{jk}u_{k}+\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|\geqslant 2}T_{jk}F_{k,\unicode[STIX]{x1D6FC}}(z_{k})\cdot u_{k}^{\unicode[STIX]{x1D6FC}}\nonumber\\ \displaystyle & = & \displaystyle T_{jk}u_{k}+\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|\geqslant 2}T_{jk}F_{k,\unicode[STIX]{x1D6FC}}(z_{k})\cdot \mathop{\sum }_{|\unicode[STIX]{x1D6FD}|=|\unicode[STIX]{x1D6FC}|}\unicode[STIX]{x1D70F}_{kj,\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D6FC}}\cdot u_{j}^{\unicode[STIX]{x1D6FD}}+O(|u_{j}|^{n+1})\nonumber\\ \displaystyle & = & \displaystyle T_{jk}u_{k}+\mathop{\sum }_{2\leqslant |\unicode[STIX]{x1D6FD}|\leqslant n}\biggl(\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|=|\unicode[STIX]{x1D6FD}|}T_{jk}F_{k,\unicode[STIX]{x1D6FC}}(z_{k}(z_{j},0))\cdot \unicode[STIX]{x1D70F}_{kj,\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D6FC}}+h_{1,jk,\unicode[STIX]{x1D6FD}}(z_{j})\biggr)u_{j}^{\unicode[STIX]{x1D6FD}}\nonumber\\ \displaystyle & & \displaystyle +\,O(|u_{j}|^{n+1})\nonumber\end{eqnarray}$$and
$$\begin{eqnarray}\displaystyle T_{jk}w_{k}=u_{j}+\mathop{\sum }_{2\leqslant |\unicode[STIX]{x1D6FC}|\leqslant n}(F_{j,\unicode[STIX]{x1D6FC}}(z_{j})+h_{2,jk,\unicode[STIX]{x1D6FC}}(z_{j}))\cdot u_{j}^{\unicode[STIX]{x1D6FC}}+O(|u_{j}|^{n+1}). & & \displaystyle \nonumber\end{eqnarray}$$ Note that 
$h_{1,jk,\unicode[STIX]{x1D6FD}}$’s and 
$h_{2,jk,\unicode[STIX]{x1D6FC}}$’s appear in these expansions coincides with those defined in the previous subsection by equation (4). By comparing these, we obtain
$$\begin{eqnarray}\displaystyle & & \displaystyle T_{jk}u_{k}-u_{j}=\mathop{\sum }_{|\unicode[STIX]{x1D6FD}|=n}\biggl(F_{j,\unicode[STIX]{x1D6FD}}(z_{j})\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|=n}T_{jk}F_{k,\unicode[STIX]{x1D6FC}}(z_{k}(z_{j},0))\cdot \unicode[STIX]{x1D70F}_{kj,\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D6FC}}-h_{1,jk,\unicode[STIX]{x1D6FD}}(z_{j})+h_{2,jk,\unicode[STIX]{x1D6FD}}(z_{j})\biggr)\cdot u_{j}^{\unicode[STIX]{x1D6FD}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,O(|u_{j}|^{n+1}).\end{eqnarray}$$ By considering this equation (5) in the case where 
$F_{j,\unicode[STIX]{x1D6FC}}\equiv 0$ for each 
$\unicode[STIX]{x1D6FC}$ with 
$|\unicode[STIX]{x1D6FC}|\geqslant n$, we obtain 
$u_{n-1}(Y,X;\{u_{j}\})=[\{(U_{jk},-H_{jk,n})\}]$. Thus the assertion (a) follows from the assumption 
$u_{n-1}(Y,X)=0$. The assertion (b) also follows directly from equation (5).
Under the assumption of Theorem 1.1(ii), it follows from the argument as in the proof of Claim 4.3 that the defining equation of 
$\{F_{j,\unicode[STIX]{x1D6FC}}^{1}\}\text{}_{|\unicode[STIX]{x1D6FC}|=n,\unicode[STIX]{x1D6FC}_{1}=0}$ can be rewritten by the equation
$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D6FF}\biggl(\bigg\{\biggl(U_{j},\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|=n,\unicode[STIX]{x1D6FC}_{1}=0}F_{j,\unicode[STIX]{x1D6FC}}^{1}\cdot e_{j,1}^{\ast }\otimes e_{j}^{\unicode[STIX]{x1D6FC}}\biggr)\bigg\}\biggr)\nonumber\\ \displaystyle & & \displaystyle \quad =\bigg\{\biggl(U_{jk},\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|=n,\unicode[STIX]{x1D6FC}_{1}=0}(h_{1,jk,\unicode[STIX]{x1D6FC}}^{1}-h_{2,jk,\unicode[STIX]{x1D6FC}}^{1})\cdot e_{j,1}^{\ast }\otimes e_{j}^{\unicode[STIX]{x1D6FC}}\biggr)\bigg\}\nonumber\end{eqnarray}$$ in 
${\check{Z}}^{1}(\{U_{j}\},N_{1}\otimes S^{n}N_{Y/S}^{\ast })$. Thus it is sufficient for proving the assertion (c) to show 
$h_{1,jk,\unicode[STIX]{x1D6FC}}^{1}\equiv 0$ and 
$h_{2,jk,\unicode[STIX]{x1D6FC}}^{1}\equiv 0$ for each 
$\unicode[STIX]{x1D6FC}$ with 
$|\unicode[STIX]{x1D6FC}|=n$ and 
$\unicode[STIX]{x1D6FC}_{1}=0$, which can be easily checked from Lemma 4.4(c) in Step 
$(\unicode[STIX]{x1D708})$ for each 
$\unicode[STIX]{x1D708}\leqslant n-2$.◻
4.5 Norm estimate for $F_{j,\unicode[STIX]{x1D6FC}}$ in a special setting
In this subsection, we estimate the norm of 
$F_{j,\unicode[STIX]{x1D6FC}}$ in order to show the convergence of the functional equation (3). Here we treat a special case where 
$(Y,X)$ is as in Remark 3.5: i.e. 
$N_{Y/X}$ admits a direct decomposition 
$N_{Y/X}=N_{1}\oplus N_{2}\oplus \cdots \oplus N_{r}$. Note that, in this case, the defining equation of each 
$\{F_{j,\unicode[STIX]{x1D6FC}}\}$ is rewritten by the equation
$$\begin{eqnarray}\unicode[STIX]{x1D6FF}(\{(U_{j},F_{j,\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D706}})\})=\{(U_{jk},h_{1,jk,\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D706}}-h_{2,jk,\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D706}})\}\end{eqnarray}$$ in 
${\check{Z}}^{1}(Y,N_{\unicode[STIX]{x1D706}}\otimes N_{\unicode[STIX]{x1D6FC}}^{-\unicode[STIX]{x1D6FC}})$. We additionally assume that 
$N_{Y/X}$ is the holomorphically trivial vector bundle 
$\mathbb{I}_{Y}^{(r)}$ of rank 
$r$ or 
$N_{Y/X}\in {\mathcal{S}}^{(r)}(Y):=\bigcup _{A>0}{\mathcal{S}}_{A}^{(r)}(Y)$.
4.5.1 The case where $N_{Y/X}\cong \mathbb{I}_{Y}^{(r)}$
Here we consider the case where 
$N_{Y/X}\cong \mathbb{I}_{Y}^{(r)}$. Note that, in this case, 
$N_{\unicode[STIX]{x1D706}}\otimes N_{\unicode[STIX]{x1D6FC}}^{-\unicode[STIX]{x1D6FC}}\cong \mathbb{I}_{Y}^{(1)}$ holds for any 
$\unicode[STIX]{x1D706}$ and 
$\unicode[STIX]{x1D6FC}$. Fix 
$U_{j}^{\ast }\Subset U_{j}$ with 
$\bigcup _{j}U_{j}^{\ast }=Y$. Take a constant 
$K:=K(\mathbb{I}_{Y}^{(1)})$ as in [Reference UedaU, Lemma 3] (=[Reference Kodaira and SpencerKS, Lemma 2]): i.e. for any 
$1$-cocycle 
$a=\{(U_{jk},a_{jk})\}\in {\check{Z}}^{1}(\{U_{j}\},{\mathcal{O}}_{Y})$ with 
$\Vert a\Vert :=\max _{j,k}\sup _{U_{jk}}|a_{jk}|<\infty$ which is cohomologous to zero, there exists a 
$0$-cochain 
$b=\{(U_{j},b_{j})\}\in {\check{C}}^{0}(\{U_{j}\},{\mathcal{O}}_{Y})$ such that 
$a$ is the coboundary of 
$b$ and that 
$\Vert b\Vert :=\max _{j}\sup _{U_{j}}|b_{j}|\leqslant K\Vert a\Vert$. Take also a positive number 
$M$ larger than 
$\max _{j}\max _{\unicode[STIX]{x1D706}}\sup _{V_{j}}|w_{j}^{\unicode[STIX]{x1D706}}|$ and 
$\max _{jk}\max _{\unicode[STIX]{x1D706}}\sup _{V_{jk}}|w_{k}^{\unicode[STIX]{x1D706}}|$, and a sufficiently large positive number 
$R$ so that 
$\{(z_{j},w_{j})\mid z_{j}\in U_{j}\cap U_{k}^{\ast },|w_{j}|<R^{-1}\}\subset V_{k}$ holds for each 
$j,k$. By using these constants, let us consider the formal series 
$A(X)=A(X^{1},X^{2},\ldots ,X^{r})=\sum _{|\unicode[STIX]{x1D6FC}|\geqslant 2}A_{\unicode[STIX]{x1D6FC}}X^{\unicode[STIX]{x1D6FC}}$ defined by
$$\begin{eqnarray}\displaystyle A(X) & = & \displaystyle 2KA(X)\cdot \biggl(-1+\mathop{\prod }_{\unicode[STIX]{x1D706}=1}^{r}\frac{1}{1-R(X^{\unicode[STIX]{x1D706}}+A(X))}\biggr)\nonumber\\ \displaystyle & & \displaystyle +\,2KM\cdot \biggl(-1-R(X^{1}+X^{2}+\cdots +X^{r}+rA(X))\nonumber\\ \displaystyle & & \displaystyle +\mathop{\prod }_{\unicode[STIX]{x1D706}=1}^{r}\frac{1}{1-R(X^{\unicode[STIX]{x1D706}}+A(X))}\biggr).\end{eqnarray}$$ Note that, by the inductive argument on 
$|\unicode[STIX]{x1D6FC}|$, it is easy to see that each coefficient 
$A_{\unicode[STIX]{x1D6FC}}$ is determined uniquely and is a positive real number.
Lemma 4.5. The formal series 
$A(X)$ is convergent (i.e. 
$A(X)$ can be regarded as a holomorphic function defined on a neighborhood of the origin of 
$\mathbb{C}^{r}$).
Proof. Let us consider
$$\begin{eqnarray}\displaystyle & & \displaystyle P(X,Y):=-Q(X,Y)\cdot Y+2KY\cdot (-Q(X,Y)+1)\nonumber\\ \displaystyle & & \displaystyle \qquad +\,2KM\cdot (Q(X,Y)\cdot (-1-R(X^{1}+X^{2}+\cdots +X^{r}+rY))+1),\nonumber\end{eqnarray}$$ where 
$Q(X,Y):=\prod _{\unicode[STIX]{x1D706}=1}^{r}(1-R(X^{\unicode[STIX]{x1D706}}+Y))\in \mathbb{C}[X,Y]$. As 
$P(0,Y)=-Y+O(Y^{2})$, we can apply the implicit function theorem to obtain a holomorphic function 
$a(X)$ defined on a neighborhood of the origin of 
$\mathbb{C}^{r}$ with 
$a(0)=0$ and 
$P(X,a(X))\equiv 0$. This means that 
$a(X)$ satisfies equation (7) and thus we obtain 
$a(X)=A(X)$ on a neighborhood of the origin, which proves the lemma.◻
In what follows, we will show that one can choose the coefficient functions 
$\{F_{j,\unicode[STIX]{x1D6FC}}\}$ as in the previous section so that 
$A(X)$ is a dominating series of the function equation (3).
First, we will show the existence of the solution 
$\{(U_{j},F_{j,\unicode[STIX]{x1D6FC}})\}\text{}_{|\unicode[STIX]{x1D6FC}|=2}$ of the equation (6) with 
$\max _{\unicode[STIX]{x1D706}}\Vert \{(U_{j},F_{j,\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D706}})\}\Vert \leqslant A_{\unicode[STIX]{x1D6FC}}$ for each 
$\unicode[STIX]{x1D706}=1,2,\ldots ,r$ and 
$\unicode[STIX]{x1D6FC}$ with 
$|\unicode[STIX]{x1D6FC}|=2$ (
$\Vert \{(U_{j},F_{j,\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D706}})\}\Vert :=\max _{j}\sup _{U_{j}}|F_{j,\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D706}}|$). Note that 
$A_{\unicode[STIX]{x1D6FC}}=2KMR^{2}$ holds for each 
$\unicode[STIX]{x1D6FC}$ with 
$|\unicode[STIX]{x1D6FC}|=2$. By Cauchy estimate, we obtain the inequality 
$\max _{j}\sup _{U_{j}\cap U_{k}^{\ast }}|f_{kj,\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D706}}|\leqslant MR^{2}$. Combining this inequality and the argument as in [Reference UedaU, p. 599], we obtain 
$\Vert \{(U_{jk},f_{kj,\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D706}})\}\Vert \leqslant 2MR^{2}$ (
$\Vert \{(U_{j},f_{kj,\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D706}})\}\Vert :=\max _{j,k}\sup _{U_{jk}}|f_{kj,\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D706}}|$). Thus we obtain 
$\Vert \{(U_{j},F_{j,\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D706}})\}\Vert \leqslant A_{\unicode[STIX]{x1D6FC}}$ from the definition of the constant 
$K$.
Next, we will show the existence of the solution 
$\{(U_{j},F_{j,\unicode[STIX]{x1D6FC}})\}\text{}_{|\unicode[STIX]{x1D6FC}|=n}$ of equation (6) with 
$\max _{\unicode[STIX]{x1D706}}\Vert \{(U_{j},F_{j,\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D706}})\}\Vert \leqslant A_{\unicode[STIX]{x1D6FC}}$ for each 
$\unicode[STIX]{x1D706}=1,2,\ldots ,r$ and 
$\unicode[STIX]{x1D6FC}$ with 
$|\unicode[STIX]{x1D6FC}|=n$ by assuming the assertion for 
$|\unicode[STIX]{x1D6FC}|<n$. Take 
$\unicode[STIX]{x1D6FC}$ with 
$|\unicode[STIX]{x1D6FC}|=n$. Then, it follows from the inductive assumption that 
$\max _{j,k}\sup _{U_{j}\cap U_{k}^{\ast }}|h_{1,jk,\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D706}}|$ is bounded by the coefficient of 
$X^{\unicode[STIX]{x1D6FC}}$ in the expansion of
$$\begin{eqnarray}\mathop{\sum }_{2\leqslant |\unicode[STIX]{x1D6FE}|<n}\mathop{\sum }_{|\unicode[STIX]{x1D6FD}|\geqslant 1}\max _{j,k}\sup _{U_{j}\cap U_{k}^{\ast }}|F_{kj,\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D706}}|\cdot X^{\unicode[STIX]{x1D6FE}}\cdot \mathop{\prod }_{\unicode[STIX]{x1D706}=1}^{r}(X^{\unicode[STIX]{x1D706}}+A(X))^{\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D706}}}.\end{eqnarray}$$ By Cauchy estimate, it is bounded by the coefficient of 
$X^{\unicode[STIX]{x1D6FC}}$ in the expansion of
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{2\leqslant |\unicode[STIX]{x1D6FE}|<n}\mathop{\sum }_{|\unicode[STIX]{x1D6FD}|\geqslant 1}A_{\unicode[STIX]{x1D6FE}}R^{|\unicode[STIX]{x1D6FD}|}\cdot X^{\unicode[STIX]{x1D6FE}}\cdot \mathop{\prod }_{\unicode[STIX]{x1D706}=1}^{r}(X^{\unicode[STIX]{x1D706}}+A(X))^{\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D706}}}\nonumber\\ \displaystyle & & \displaystyle \quad =A(X)\cdot \biggl(-1+\mathop{\prod }_{\unicode[STIX]{x1D706}=1}^{r}\frac{1}{1-R(X^{\unicode[STIX]{x1D706}}+A(X))}\biggr).\nonumber\end{eqnarray}$$ From a similar argument, it can be seen that 
$\max _{j,k}\sup _{U_{j}\cap U_{k}^{\ast }}|h_{2,jk,\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D706}}|$ is bounded by the coefficient of 
$X^{\unicode[STIX]{x1D6FC}}$ in the expansion of
$$\begin{eqnarray}M\cdot \biggl(-1-R(X^{1}+X^{2}+\cdots +X^{r}+rA(X))+\mathop{\prod }_{\unicode[STIX]{x1D706}=1}^{r}\frac{1}{1-R(X^{\unicode[STIX]{x1D706}}+A(X))}\biggr).\end{eqnarray}$$ Thus, by the argument as in [Reference UedaU, p. 599] and the defining function (7) of 
$A(X)$, we obtain the inequality 
$\Vert \{(U_{jk},h_{1,jk,\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D706}}-h_{2,jk,\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D706}})\}\Vert \leqslant K^{-1}A_{\unicode[STIX]{x1D6FC}}$ for each 
$\unicode[STIX]{x1D706}$. Therefore the assertion follows from the definition of the constant 
$K$.
4.5.2 The case where $N_{Y/X}\in {\mathcal{S}}^{(r)}(Y)$
When 
$N_{Y/X}\in {\mathcal{S}}^{(r)}(Y)$, by using [Reference UedaU, Lemma 4] instead of [Reference UedaU, Lemma 3], the same arguments as in Section 4.5.1 can be carried out after replacing the defining function (7) of 
$A(X)$ with
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{|\unicode[STIX]{x1D6FC}|\geqslant 2}\unicode[STIX]{x1D700}_{|\unicode[STIX]{x1D6FC}|-1}^{-1}A_{\unicode[STIX]{x1D6FC}}X^{\unicode[STIX]{x1D6FC}}\nonumber\\ \displaystyle & & \displaystyle \quad =2A(X)\cdot \biggl(-1+\mathop{\prod }_{\unicode[STIX]{x1D706}=1}^{r}\frac{1}{1-R(X^{\unicode[STIX]{x1D706}}+A(X))}\biggr)\nonumber\\ \displaystyle & & \displaystyle \qquad +\,2M\cdot \biggl(-1-R(X^{1}+X^{2}+\cdots +X^{r}+rA(X))\nonumber\\ \displaystyle & & \displaystyle \qquad +\mathop{\prod }_{\unicode[STIX]{x1D706}=1}^{r}\frac{1}{1-R(X^{\unicode[STIX]{x1D706}}+A(X))}\biggr),\nonumber\end{eqnarray}$$where
$$\begin{eqnarray}\unicode[STIX]{x1D700}_{n}^{-1}:=\frac{1}{K}\min _{\unicode[STIX]{x1D6FC}\in \mathbb{Z}^{r},\,|\unicode[STIX]{x1D6FC}|=n}d(\mathbb{I}_{Y}^{(1)},N_{\unicode[STIX]{x1D6FC}})\end{eqnarray}$$ (
$K$ is the constant as in [Reference UedaU, Lemma 4], see also [Reference UedaU, Section 4.6] for the details). Thus, for proving the convergence of the functional equation (3), it is sufficient to see the convergence of the formal series 
$A(X)$ with the above new defining equation.
Consider 
$B(Y):=Y+A(Y,Y,\ldots ,Y)=Y+\sum _{n=2}^{\infty }B_{n}Y^{n}$, where 
$B_{n}=\sum _{|\unicode[STIX]{x1D6FC}|=n}A_{\unicode[STIX]{x1D6FC}}$ for each 
$n\geqslant 2$. As it can be easily seen that 
$A_{\unicode[STIX]{x1D6FC}}\geqslant 0$ holds for each 
$\unicode[STIX]{x1D6FC}$, we have 
$A_{\unicode[STIX]{x1D6FC}}\leqslant B_{|\unicode[STIX]{x1D6FC}|}$. Therefore, for showing the convergence of 
$A(X)$, it is sufficient to show that 
$B(Y)$ has a positive radius of convergence. By considering 
$X^{1}=X^{2}=\cdots =X^{r}=Y$, we obtain the defining function of 
$B(Y)$ as follows:
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{n=2}^{\infty }\unicode[STIX]{x1D700}_{n-1}^{-1}B_{n}Y^{n} & = & \displaystyle 2(B(Y)-Y)\cdot \biggl(-1+\frac{1}{(1-RB(Y))^{r}}\biggr)\nonumber\\ \displaystyle & & \displaystyle +\,2M\cdot \biggl(-1-rRB(Y)+\frac{1}{(1-RB(Y))^{r}}\biggr).\nonumber\end{eqnarray}$$ Also consider another formal series 
$\widehat{B}(Y)=Y+\sum _{n=2}^{\infty }\widehat{B}_{n}Y^{n}$ defined by
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{n=2}^{\infty }\unicode[STIX]{x1D700}_{n-1}^{-1}\widehat{B}_{n}Y^{n} & = & \displaystyle 2\widehat{B}(Y)\cdot \biggl(-1+\frac{1}{(1-R\widehat{B}(Y))^{r}}\biggr)\nonumber\\ \displaystyle & & \displaystyle +\,2M\cdot \biggl(-1-rR\widehat{B}(Y)+\frac{1}{(1-R\widehat{B}(Y))^{r}}\biggr).\nonumber\end{eqnarray}$$ As it clearly holds that 
$\widehat{B}_{n}\geqslant B_{n}$ for each 
$n\geqslant 2$, it is sufficient to show that 
$\widehat{B}(Y)$ has a positive radius of convergence. According to Siegel’s argument ([Reference SiegelSi], see also [Reference UedaU, Lemma 5]), it is sufficient to see the following two properties of 
$\{\unicode[STIX]{x1D700}_{n}\}$: (a) There exists a positive number 
$A$ such that 
$\unicode[STIX]{x1D700}_{n}<(2n)^{A}$ for any 
$n\geqslant 1$, and (b) 
$\unicode[STIX]{x1D700}_{n-m}^{-1}\leqslant \unicode[STIX]{x1D700}_{n}^{-1}+\unicode[STIX]{x1D700}_{m}^{-1}$ for any 
$n>m$. The property (a) directly follows from the assumption that 
$N_{Y/X}\in {\mathcal{S}}^{(r)}(Y)$. The property (b) can be shown by
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D700}_{n}^{-1}+\unicode[STIX]{x1D700}_{m}^{-1} & = & \displaystyle \frac{1}{K}(d(\mathbb{I}_{Y}^{(1)},N_{\unicode[STIX]{x1D6FC}^{(n)}})+d(\mathbb{I}_{Y}^{(1)},N_{\unicode[STIX]{x1D6FC}^{(m)}}))\nonumber\\ \displaystyle & {\geqslant} & \displaystyle \frac{1}{K}d(N_{\unicode[STIX]{x1D6FC}^{(n)}},N_{\unicode[STIX]{x1D6FC}^{(m)}})=\frac{1}{K}d(\mathbb{I}_{Y}^{(1)},N_{\unicode[STIX]{x1D6FC}^{(n)}-\unicode[STIX]{x1D6FC}^{(m)}}),\nonumber\end{eqnarray}$$ where 
$\unicode[STIX]{x1D6FC}^{(n)}\in \mathbb{Z}^{r}$ is an element which attains the minimum in the definition of 
$\unicode[STIX]{x1D700}_{n}^{-1}$.
4.6 End of the proof for the special setting
Assume that 
$N_{Y/X}$ is holomorphically trivial or 
$N_{Y/X}\in {\mathcal{S}}^{(r)}(Y)$ holds. Then, by choosing the coefficient functions 
$\{F_{j,\unicode[STIX]{x1D6FC}}\}$ as above, we can deduce from the inverse function theorem that there exists a solution 
$\{u_{j}\}$ of the functional equation (3). By shrinking 
$V$ if necessary, we may assume that 
$u_{j}$ is defined on 
$V_{j}$ for each 
$j$. From Lemma 4.4(b), it holds that the solutions satisfy 
$u_{j}=T_{jk}u_{k}$ on each 
$V_{jk}$. Thus the theorem for this special case follows from the arguments as we already explained in Section 4.1.
4.7 Proof for the general setting
When 
$N_{Y/X}\in {\mathcal{E}}_{0}^{(r)}(Y)$, consider 
$G:=\text{ker}\,\unicode[STIX]{x1D70C}$, where 
$\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}_{N_{Y/X}}$ is the unitary representation corresponding to 
$N_{Y/X}$. Fix a tubular neighborhood 
$V$ of 
$Y$ in 
$X$ and regard 
$G$ as a normal subgroup of 
$\unicode[STIX]{x1D70B}_{1}(V,\ast )$ by the natural isomorphism 
$\unicode[STIX]{x1D70B}_{1}(Y,\ast )\cong \unicode[STIX]{x1D70B}_{1}(V,\ast )$. From the assumption, there exists a finite normal covering 
$\unicode[STIX]{x1D70B}:\widetilde{V}\rightarrow V$ corresponding to 
$G\subset \unicode[STIX]{x1D70B}_{1}(V,\ast )$. Denote by 
$\widetilde{Y}$ the preimage 
$\unicode[STIX]{x1D70B}^{-1}(Y)$. Then it is clear from the construction that 
$N_{\widetilde{Y}/\widetilde{V}}=(\unicode[STIX]{x1D70B}|_{\widetilde{Y}})^{\ast }N_{Y/X}$ is holomorphically trivial. Let us denote 
$\unicode[STIX]{x1D70B}^{-1}(V_{j})$ by 
$\widetilde{V}_{j}$ and 
$\widetilde{V}_{j}\cap \widetilde{Y}$ by 
$\widetilde{U}_{j}$. We may assume that 
$\widetilde{U}_{j}$ is the union of 
$d$ copies of 
$U_{j}$, where 
$d$ is the degree of the map 
$\unicode[STIX]{x1D70B}$. Consider the local defining functions system 
$\{\widetilde{w}_{j}\}$ defined by 
$\widetilde{w}_{j}:=(\unicode[STIX]{x1D70B}|_{\widetilde{V}_{j}})^{\ast }w_{j}$.
By Lemma 4.6 below, 
$(\widetilde{Y},\widetilde{V})$ is of infinite type. Thus, from the result we showed in Section 4.6, we can solve the functional equation (3) with initial system 
$\{\widetilde{w}_{j}\}$ on each 
$\widetilde{V}_{j}$ to obtain a local defining functions system 
$\{\widetilde{u}_{j}\}$ of 
$\widetilde{Y}$ in 
$\widetilde{V}$ with 
$\widetilde{u}_{j}=T_{jk}\widetilde{u}_{k}$ on each 
$\widetilde{V}_{jk}$. Note that, as 
$\widetilde{w}_{j}=\widetilde{u}_{j}+O(|\widetilde{u}_{j}|^{2})$, it holds that 
$d\widetilde{u}_{j}|_{\widetilde{U}_{j}}=(\unicode[STIX]{x1D70B}|_{\widetilde{U}_{j}})^{\ast }e_{j}$.
Define a function 
$u_{j}$ on 
$V_{j}$ by
$$\begin{eqnarray}(\unicode[STIX]{x1D70B}|_{\widetilde{V}_{j}})^{\ast }u_{j}=\frac{1}{d}\mathop{\sum }_{\unicode[STIX]{x1D708}=1}^{d}i_{\unicode[STIX]{x1D708}}^{\ast }\widetilde{u}_{j},\end{eqnarray}$$ where 
$\{i_{1},i_{2},\ldots ,i_{d}\}$ is the set of deck transformations of 
$\unicode[STIX]{x1D70B}$. Clearly it holds that 
$du_{j}|_{U_{j}}=e_{j}$ and 
$\{u_{j}=0\}=U_{j}$ hold, which means that 
$\{u_{j}\}$ is a local defining functions system of 
$Y$. It is also easy to see that 
$u_{j}=T_{jk}u_{k}$ holds on each 
$V_{jk}$, which shows the assertion (i) for the case where 
$N_{Y/X}\in {\mathcal{E}}_{0}^{(r)}(Y)$. Under the assumption in Theorem 1.1(ii), it follows from 
$\boldsymbol{(Property)}_{n}$’s that we may assume that 
$\widetilde{u}_{j}^{1}$ is a defining function of 
$\widetilde{V}_{j}\cap \unicode[STIX]{x1D70B}^{-1}(S)$ in 
$\widetilde{V}_{j}$ with 
$d\widetilde{u}_{j}^{1}=\unicode[STIX]{x1D70B}^{\ast }((1+O(|v_{j}|))\cdot dw_{j}^{1})$ on each 
$\widetilde{V}_{j}\cap \unicode[STIX]{x1D70B}^{-1}(S)$. Therefore 
$u_{j}^{1}$ is a defining function of 
$V_{j}\cap S$ by shrinking 
$V$ if necessary, which proves the assertion (ii) for the case where 
$N_{Y/X}\in {\mathcal{E}}_{0}^{(r)}(Y)$.
When 
$N_{Y/X}\in {\mathcal{E}}_{1}^{(r)}(Y)$, there exists a finite normal covering 
$\unicode[STIX]{x1D70B}:\widetilde{Y}\rightarrow Y$ such that 
$(\unicode[STIX]{x1D70B}|_{\widetilde{Y}})^{\ast }N_{Y/X}\in {\mathcal{S}}^{(r)}(\widetilde{Y})$. The theorem for this case is shown by the same argument as above by using this map 
$\unicode[STIX]{x1D70B}$.
Lemma 4.6. Let 
$\unicode[STIX]{x1D70B},\widetilde{Y},\widetilde{V}$ be as above. Then 
$\text{type}\,(Y,X)=\text{type}\,(\widetilde{Y},\widetilde{V})$ holds.
Proof. Let
$$\begin{eqnarray}T_{jk}w_{k}=w_{j}+\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|\geqslant n+1}f_{kj,\unicode[STIX]{x1D6FC}}\cdot w_{j}^{\unicode[STIX]{x1D6FC}}.\end{eqnarray}$$ be the expansion (1) for the system 
$\{w_{j}\}$. Then, by pulling it back by 
$\unicode[STIX]{x1D70B}$, we obtain
$$\begin{eqnarray}T_{jk}\widetilde{w}_{k}=\widetilde{w}_{j}+\mathop{\sum }_{|\unicode[STIX]{x1D6FC}|\geqslant n+1}(\unicode[STIX]{x1D70B}|_{\widetilde{U}_{j}})^{\ast }f_{kj,\unicode[STIX]{x1D6FC}}\cdot \widetilde{w}_{j}^{\unicode[STIX]{x1D6FC}}\end{eqnarray}$$ on each 
$\widetilde{V}_{jk}$. Thus, 
$\{\widetilde{w}_{j}\}$ is a system of type 
$n$ and
$$\begin{eqnarray}u_{n}(\widetilde{Y},\widetilde{V};\{\widetilde{w}_{j}\})=(\unicode[STIX]{x1D70B}|_{\widetilde{Y}})^{\ast }u_{n}(Y,X;\{w_{j}\})\end{eqnarray}$$ holds. Therefore we obtain the lemma, since the map 
$(\unicode[STIX]{x1D70B}|_{\widetilde{Y}})^{\ast }:H^{1}(Y,N_{Y/X}\otimes S^{n}N_{Y/X}^{\ast })\rightarrow H^{1}(Y,N_{\widetilde{Y}/\widetilde{V}}\otimes S^{n}N_{\widetilde{Y}/\widetilde{V}}^{\ast })$ is injective.◻
5 Proof of Corollary 1.2
As 
$D_{\unicode[STIX]{x1D706}}$’s intersect to each other transversally along 
$C$, it follows that 
$N_{C/X}=\bigoplus _{\unicode[STIX]{x1D706}=1}^{n-1}N_{\unicode[STIX]{x1D706}}$ and 
$N_{C/D_{1}}=\bigoplus _{\unicode[STIX]{x1D706}=2}^{n-1}N_{\unicode[STIX]{x1D706}}$, where 
$N_{\unicode[STIX]{x1D706}}:=N_{D_{\unicode[STIX]{x1D706}}/X}|_{Y}$ for each 
$\unicode[STIX]{x1D706}=1,2,\ldots ,n-1$. Note that each 
$N_{\unicode[STIX]{x1D706}}$ is isomorphic to 
$L|_{C}$ and thus it is an element of 
${\mathcal{E}}_{1}^{(1)}(C)$. Thus, as 
$N_{C/X}\cong \oplus ^{r}L|_{C}\in {\mathcal{E}}_{1}^{(r)}(C)$ and 
$H^{1}(C,N_{\unicode[STIX]{x1D706}}\otimes N_{\unicode[STIX]{x1D6FC}}^{-1})\cong H^{1}(C,L|_{C}^{-|\unicode[STIX]{x1D6FC}|+1})=0$ hold, it follows that 
$(X,S:=D_{1},Y:=C)$ satisfies the assumption of Theorem 1.1(ii). Therefore we obtain that the line bundle 
$L=[D_{1}]$ admits a unitary flat metric 
$h_{V}$ on a neighborhood 
$V$ of 
$C$ (i.e. 
$h_{V}$ is a 
$C^{\infty }$ Hermitian metric on 
$L|_{V}$ whose Chern curvature is 
$0$).
On the other hand, 
$L$ admits a singular Hermitian metric 
$h_{\text{sing}}$ such that 
$h_{\text{sing}}|_{X\setminus C}$ is a 
$C^{\infty }$ Hermitian metric on 
$L|_{X\setminus C}$, 
$h_{\text{sing}}\rightarrow \infty$ holds when a point approaches to 
$C$, and that Chern curvature current of 
$h_{\text{sing}}$ is semipositive. Indeed, the Bergman type singular Hermitian metric 
$h_{\text{sing}}:=(\sum _{\unicode[STIX]{x1D706}=1}^{n-1}h_{\unicode[STIX]{x1D706}}^{-1})^{-1}$ satisfies these properties, where 
$h_{\unicode[STIX]{x1D706}}$ is a singular Hermitian metric on 
$L$ with 
$|f_{\unicode[STIX]{x1D706}}|_{h_{\unicode[STIX]{x1D706}}}^{2}\equiv 1$ for a section 
$f_{\unicode[STIX]{x1D706}}\in H^{0}(X,L)$ with 
$\text{div}(f_{\unicode[STIX]{x1D706}})=D_{\unicode[STIX]{x1D706}}$.
A 
$C^{\infty }$ Hermitian metric 
$h$ on 
$L$ with semipositive curvature can be constructed by using the regularized minimum construction for these two metrics 
$h_{V}$ and 
$h_{\text{sing}}$, which is the same construction as we used for proving [Reference KoikeK1, Corollary 3.4].
In what follows, we explain the detail of the regularized minimum construction. Take a positive constant 
$\unicode[STIX]{x1D700}$ and a 
$C^{\infty }$ nonnegative function 
$\unicode[STIX]{x1D703}:\mathbb{R}\rightarrow \mathbb{R}_{{\geqslant}0}$ with support in the interval 
$[-1,1]$ such that 
$\int _{\mathbb{R}}\unicode[STIX]{x1D703}(t)\,dt=1$ and 
$\int _{\mathbb{R}}t\unicode[STIX]{x1D703}(t)\,dt=0$ hold. Define a function 
$M:\mathbb{R}^{2}\rightarrow \mathbb{R}$ (“a regularized max function,” see also [Reference DemaillyD1, Section 5.E]) by
$$\begin{eqnarray}M(s_{1},s_{2}):=\unicode[STIX]{x1D700}^{-2}\int _{\mathbb{R}^{2}}\max \{s_{1}+t_{1},s_{2}+t_{2}\}\cdot \unicode[STIX]{x1D703}(t_{1}/\unicode[STIX]{x1D700})\cdot \unicode[STIX]{x1D703}(t_{2}/\unicode[STIX]{x1D700})\,dt_{1}\,dt_{2}.\end{eqnarray}$$ Then it is easily checked that this function is nondecreasing in all variables, 
$C^{\infty }$ convex on 
$\mathbb{R}^{2}$, and that 
$M(s_{1},s_{2})=\max \{s_{1},s_{2}\}$ holds if 
$|s_{1}-s_{2}|>2\unicode[STIX]{x1D700}$. Fix a relatively compact open neighborhood 
$V_{0}$ of 
$C$ in 
$V$, and an open covering 
$\{W_{j}\}$ of 
$X$. By fixing a local trivialization of 
$L$ on each 
$W_{j}$, we can regard the metrics 
$h_{V}$ and 
$h_{\text{sing}}$ restricted to 
$W_{j}\cap V$ and 
$W_{j}$ as nonnegative functions, say 
$e^{-\unicode[STIX]{x1D711}_{V,j}}$ and 
$e^{-\unicode[STIX]{x1D711}_{\text{sing},j}}$, respectively. Note that 
$\unicode[STIX]{x1D711}_{V,j}$ is pluriharmonic (and thus it is 
$C^{\infty }$ psh) function on 
$W_{j}\cap V$, and that 
$\unicode[STIX]{x1D711}_{\text{sing},j}$ is a psh function which is 
$C^{\infty }$ on 
$W_{j}\setminus C$ and 
$\unicode[STIX]{x1D711}_{\text{sing},j}(p)\rightarrow -\infty$ holds as 
$p\rightarrow C$. By the compactness of 
$\overline{V}\setminus V_{0}$, it holds for sufficiently large number 
$N$ that 
$\unicode[STIX]{x1D711}_{V,j}(p)+2\unicode[STIX]{x1D700}-N<\unicode[STIX]{x1D711}_{\text{sing},j}(p)$ for all 
$j$ and for all 
$p\in W_{j}\cap (\overline{V}\setminus V_{0})$. For such a number 
$N$, one can easily see that the function 
$\unicode[STIX]{x1D711}_{j}$ defined by
$$\begin{eqnarray}\unicode[STIX]{x1D711}_{j}(p):=\left\{\begin{array}{@{}ll@{}}M(\unicode[STIX]{x1D711}_{V,j}(p)-N,\unicode[STIX]{x1D711}_{\text{sing},j}(p))\quad & (p\in V)\\ \unicode[STIX]{x1D711}_{\text{sing},j}(p)\quad & (p\in X\setminus \overline{V})\end{array}\right.\end{eqnarray}$$ is a well-defined 
$C^{\infty }$ psh function on 
$W_{j}$ (here we used the properties of the function 
$M$ we mentioned above). A 
$C^{\infty }$ Hermitian metric 
$h$ on 
$L$ with semipositive curvature can be defined by 
$e^{-\unicode[STIX]{x1D711}_{j}}$ on each 
$W_{j}$.
6 Examples
6.1 Deformation spaces of projective manifolds
Let 
$B$ be a domain of 
$\mathbb{C}^{r}$ which includes the origin and 
$\unicode[STIX]{x1D70B}:X\rightarrow B$ be a deformation of projective manifolds: i.e. 
$X$ is a holomorphic manifold of dimension 
$n+r$ and 
$\unicode[STIX]{x1D70B}$ is a proper holomorphic surjective submersion whose fiber 
$\unicode[STIX]{x1D70B}^{-1}(x)$ is a projective manifold of dimension 
$n$ for each 
$x\in X$. Denote by 
$Y$ the central fiber 
$\unicode[STIX]{x1D70B}^{-1}(0)$. Let us assume that 
$Y$ is a smooth fiber for simplicity. In this case, 
$N_{Y/X}$ is holomorphically trivial.
Take a coordinate 
$x=(x^{1},x^{2},\ldots ,x^{r})$ of 
$\mathbb{C}^{r}$. Then, by considering a global defining functions system 
$w:=\{w^{\unicode[STIX]{x1D706}}:=\unicode[STIX]{x1D70B}^{\ast }x^{\unicode[STIX]{x1D706}}\}$ of 
$Y$, it is easily seen that the pair 
$(Y,X)$ is of infinite type. In this case, Theorem 1.1(i) is easily checked. Indeed, the foliation 
${\mathcal{F}}$ in this case is the one which is induced by the fibration 
$\unicode[STIX]{x1D70B}$. In what follows, we give a simple proof of Theorem 1.1(ii) for this fundamental example.
Let 
$S\subset X$ be a nonsingular hypersurface such that 
$Y\subset S$ and 
$N_{Y/S}$ is unitary flat. Let us consider the line bundle 
$[S]$. It holds that 
$[S]|_{Y}=[S]|_{S}|_{Y}=N_{S/X}|_{Y}$. As we have already mentioned in Section 4.2, it follows from Remark 2.6 and the complete reducibility of the unitary representations that 
$N_{S/X}|_{Y}$ is unitary flat line bundle. Therefore, 
$[S]|_{Y}$ is unitary flat and thus it is topologically trivial. As the first Chern class 
$c_{1}([S]|_{\unicode[STIX]{x1D70B}^{-1}(x)})$ depends continuously on 
$x$, it holds that 
$[S]|_{L}$ is also topologically trivial for each leaf 
$L$ of 
${\mathcal{F}}$. Assume that 
$L\not \subset S$. Then, as the divisor 
$S|_{L}$ is an effective divisor on a projective manifold such that the corresponding line bundle is topologically trivial, it follows that 
$S\cap L=\emptyset$. Therefore we obtain that 
$S=\unicode[STIX]{x1D70B}^{-1}(\overline{S})$ holds, where 
$\overline{S}:=\unicode[STIX]{x1D70B}(S)$. By shrinking 
$B$ and choosing appropriate 
$x$, we may assume that 
$\overline{S}=\{x=(x^{1},x^{2},\ldots ,x^{r})\in B\mid x^{1}=0\}$. Then we can construct the foliation 
${\mathcal{G}}_{S}$ as in Theorem 1.1(ii) by considering “
$w^{1}=$(constant).”
6.2 Projective bundles
Let 
$M$ be a compact complex manifold and 
$E$ be a holomorphic vector bundle on 
$M$ of rank 
$r+1$. Assume that there exists a subbundle 
$F\subset E$ of rank 
$r$ such that 
$F$ is a unitary flat vector bundle and the quotient bundle 
$L:=E/F$ is the holomorphically trivial line bundle. In this subsection, we consider the projective bundle 
$X:=\mathbb{P}(E)$ and the section 
$Y\subset X$ of 
$\unicode[STIX]{x1D70B}:X\rightarrow M$ defined by the natural map 
$p:E\rightarrow L$, where we are regarding 
$\mathbb{P}(E)$ as the relative hyperplane bundle.
Fix an open covering 
$\{U_{j}\}$ of 
$M$ and take a local frame 
$(e_{j}^{1},e_{j}^{2},\ldots ,e_{j}^{r})$ of 
$F$ with 
$e_{j}^{\unicode[STIX]{x1D706}}=\sum _{\unicode[STIX]{x1D707}=1}^{r}(S_{jk}^{-1})_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D706}}\cdot e_{k}^{\unicode[STIX]{x1D707}}$, or equivalently, 
$e_{j,\unicode[STIX]{x1D706}}^{\ast }=\sum _{\unicode[STIX]{x1D707}=1}^{r}(S_{jk})_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D707}}\cdot e_{k,\unicode[STIX]{x1D707}}^{\ast }$ on each 
$U_{jk}$ (
$S_{jk}\in U(r)$). By extending these appropriately, we obtain a local frame 
$e_{j}=(e_{j}^{0},e_{j}^{1},\ldots ,e_{j}^{r})$ of 
$E$ with 
$e_{j}^{\ast }=\widetilde{S}_{jk}e_{k}^{\ast }$ on each 
$U_{jk}$, where
Here the function 
$a_{jk,\unicode[STIX]{x1D706}}$ is a holomorphic function defined on 
$U_{jk}$ for each 
$\unicode[STIX]{x1D706}$. Fix a neighborhood 
$V_{j}$ of 
$\unicode[STIX]{x1D70B}^{-1}(U_{j})\cap Y$ in 
$\unicode[STIX]{x1D70B}^{-1}(U_{j})$ and a coordinate 
$z_{j}$ of 
$U_{j}$. For each 
$w_{j}=(w_{j}^{1},w_{j}^{2},\ldots ,w_{j}^{r})\in \mathbb{C}^{r}$, consider the map
$$\begin{eqnarray}(z_{j},w_{j})\mapsto \biggl[e_{j,0}^{\ast }(z_{j})+\mathop{\sum }_{\unicode[STIX]{x1D706}=1}^{r}w_{j}^{\unicode[STIX]{x1D706}}\cdot e_{j,\unicode[STIX]{x1D706}}^{\ast }(z_{j})\biggr]\end{eqnarray}$$ and regard 
$(z_{j},w_{j})$ as a coordinates system of 
$V_{j}$ by this map. Then we obtain
$$\begin{eqnarray}\displaystyle w_{k}^{\unicode[STIX]{x1D707}} & = & \displaystyle \frac{\mathop{\sum }_{\unicode[STIX]{x1D706}=1}^{r}(S_{jk})_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D707}}\cdot w_{j}^{\unicode[STIX]{x1D706}}}{1+\mathop{\sum }_{\unicode[STIX]{x1D706}=1}^{r}a_{jk,\unicode[STIX]{x1D706}}w_{j}^{\unicode[STIX]{x1D706}}}\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{\unicode[STIX]{x1D706}=1}^{r}(S_{jk})_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D707}}\cdot w_{j}^{\unicode[STIX]{x1D706}}-\mathop{\sum }_{\unicode[STIX]{x1D706}=1}^{r}\mathop{\sum }_{\unicode[STIX]{x1D708}=1}^{r}(S_{jk})_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D707}}\cdot a_{jk,\unicode[STIX]{x1D708}}w_{j}^{\unicode[STIX]{x1D706}}w_{j}^{\unicode[STIX]{x1D708}}+O(|w_{j}|^{3}),\nonumber\end{eqnarray}$$and thus
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{\unicode[STIX]{x1D707}=1}^{r}(S_{jk}^{-1})_{\unicode[STIX]{x1D707}}^{p}\cdot w_{k}^{\unicode[STIX]{x1D707}}=w_{j}^{p}-\mathop{\sum }_{\unicode[STIX]{x1D708}=1}^{r}a_{jk,\unicode[STIX]{x1D708}}w_{j}^{p}w_{j}^{\unicode[STIX]{x1D708}}+O(|w_{j}|^{3}) & & \displaystyle \nonumber\end{eqnarray}$$ on each 
$V_{jk}$, which can be regarded as the expansion (1) for the local defining functions system 
$\{w_{j}\}$ with the transition matrix 
$T_{jk}:=S_{jk}^{-1}$ of 
$N_{Y/X}$ (note that 
$N_{Y/X}^{\ast }\cong F$).
Set 
$\unicode[STIX]{x1D700}_{j}:=dw_{j}$. Then it follows from the above expansion that the first obstruction class 
$u_{1}(Y,X;\{w_{j}\})$ is defined by
$$\begin{eqnarray}-\mathop{\sum }_{p=1}^{r}\mathop{\sum }_{\unicode[STIX]{x1D708}=1}^{r}a_{jk,\unicode[STIX]{x1D708}}\cdot \unicode[STIX]{x1D700}_{j,p}^{\ast }\otimes \unicode[STIX]{x1D700}_{j}^{p}\cdot \unicode[STIX]{x1D700}_{j}^{\unicode[STIX]{x1D708}}.\end{eqnarray}$$ On the other hand, it follows from the arguments as in Lemma 4.2 that 
$-\sum _{\unicode[STIX]{x1D708}=1}^{r}a_{jk,\unicode[STIX]{x1D708}}\cdot \unicode[STIX]{x1D700}_{j}^{\unicode[STIX]{x1D708}}$ can be regarded as the extension class 
$\unicode[STIX]{x1D6FF}(1)\in H^{1}(Y,N_{Y/X}^{\ast })(\cong H^{1}(M,L^{-1}\otimes F))$ of the short exact sequence 
$0\rightarrow F\rightarrow E\rightarrow L\rightarrow 0$. Thus we can conclude that 
$u_{1}(Y,X)$ is the image of 
$\text{id}_{N_{Y/X}}\otimes \unicode[STIX]{x1D6FF}(1)\in H^{1}(Y,\text{End}(N_{Y/X})\otimes N_{Y/X}^{\ast })\cong H^{1}(Y,N_{Y/X}\otimes N_{Y/X}^{\ast }\otimes N_{Y/X}^{\ast })$ by the map
$$\begin{eqnarray}H^{1}(Y,N_{Y/X}\otimes N_{Y/X}^{\ast }\otimes N_{Y/X}^{\ast })\rightarrow H^{1}(Y,N_{Y/X}\otimes S^{2}N_{Y/X}^{\ast })\end{eqnarray}$$ induced from the natural map 
$N_{Y/X}^{\ast }\otimes N_{Y/X}^{\ast }\rightarrow S^{2}N_{Y/X}^{\ast }$. In what follows, we regard each section of 
$S^{2}N_{Y/X}^{\ast }$ as a symmetric section of 
$N_{Y/X}^{\ast }\otimes N_{Y/X}^{\ast }$ and 
$S^{2}N_{Y/X}^{\ast }$ as a unitary flat subbundle of 
$N_{Y/X}^{\ast }\otimes N_{Y/X}^{\ast }$. Then the natural map 
$N_{Y/X}^{\ast }\otimes N_{Y/X}^{\ast }\rightarrow S^{2}N_{Y/X}^{\ast }$ can be regarded as the map 
$\unicode[STIX]{x1D700}_{j}^{\unicode[STIX]{x1D707}}\otimes \unicode[STIX]{x1D700}_{j}^{\unicode[STIX]{x1D708}}\mapsto \text{Sym}(\unicode[STIX]{x1D700}_{j}^{\unicode[STIX]{x1D707}}\otimes \unicode[STIX]{x1D700}_{j}^{\unicode[STIX]{x1D708}}):=\frac{1}{2}(\unicode[STIX]{x1D700}_{j}^{\unicode[STIX]{x1D707}}\otimes \unicode[STIX]{x1D700}_{j}^{\unicode[STIX]{x1D708}}+\unicode[STIX]{x1D700}_{j}^{\unicode[STIX]{x1D708}}\otimes \unicode[STIX]{x1D700}_{j}^{\unicode[STIX]{x1D707}})$. By using this, it clearly holds that 
$u_{1}(Y,X)=0$ if and only if 
$s_{\ast }(\unicode[STIX]{x1D6FF}(1))=0\in H^{1}(Y,N_{Y/X}\otimes N_{Y/X}^{\ast }\otimes N_{Y/X}^{\ast })$ holds, where 
$s_{\ast }$ is the map induced from
$$\begin{eqnarray}s:N_{Y/X}^{\ast }\rightarrow N_{Y/X}\otimes N_{Y/X}^{\ast }\otimes N_{Y/X}^{\ast }:~\unicode[STIX]{x1D700}_{j}^{\unicode[STIX]{x1D708}}\mapsto \frac{1}{2}\mathop{\sum }_{\unicode[STIX]{x1D707}=1}^{r}\unicode[STIX]{x1D700}_{j,\unicode[STIX]{x1D707}}^{\ast }\otimes (\unicode[STIX]{x1D700}_{j}^{\unicode[STIX]{x1D707}}\otimes \unicode[STIX]{x1D700}_{j}^{\unicode[STIX]{x1D708}}+\unicode[STIX]{x1D700}_{j}^{\unicode[STIX]{x1D708}}\otimes \unicode[STIX]{x1D700}_{j}^{\unicode[STIX]{x1D707}}).\end{eqnarray}$$ As we can regard 
$N_{Y/X}^{\ast }$ as a unitary flat subbundle (and thus a direct component by the complete reducibility of the unitary representation) of 
$N_{Y/X}\otimes N_{Y/X}^{\ast }\otimes N_{Y/X}^{\ast }$ via 
$s$, we can conclude that 
$u_{1}(Y,X)=0$ holds if and only if 
$\unicode[STIX]{x1D6FF}(1)=0$ holds, or equivalently, 
$0\rightarrow F\rightarrow E\rightarrow L\rightarrow 0$ splits.
Therefore, it holds that the pair 
$(Y,X)$ is of type 
$1$ if 
$0\rightarrow F\rightarrow E\rightarrow L\rightarrow 0$ does not split. When 
$0\rightarrow F\rightarrow E\rightarrow L\rightarrow 0$ splits, it can be easily seen that 
$T_{jk}w_{k}=w_{j}$ holds on each 
$V_{jk}$, which shows that the pair 
$(Y,X)$ is of infinite type in this case.
6.3 The blow-up of a del Pezzo manifold at a general point (Proof of Corollary 1.3)
Let 
$(V,L)$ be a del Pezzo manifold of degree 
$1$: i.e. 
$V$ is a projective manifold of dimension 
$n$ and 
$L$ is an ample line bundle on 
$V$ with 
$K_{V}^{-1}\cong L^{n-1}$ and the self-intersection number 
$(L^{n})$ is equal to 
$1$. From [Reference FujitaF, 6.4], it holds that 
$\text{dim}\,H^{0}(V,L)=n$. Take general elements 
$D_{1},D_{2},\ldots ,D_{n}\in |L|$. By [Reference FujitaF, 4.2] and 
$(D_{1},D_{2},\ldots ,D_{n})=(L^{n})=1$, it holds that the intersection 
$\bigcap _{\unicode[STIX]{x1D706}=1}^{n}D_{\unicode[STIX]{x1D706}}$ is a point, which we denote by 
$p$. It is clear that 
$D_{\unicode[STIX]{x1D706}}$’s intersect each other transversally at 
$p$. From this fact and Bertini’s theorem, we may assume that each 
$D_{\unicode[STIX]{x1D706}}$ is nonsingular.
Consider a sequence of the subvarieties 
$V_{n}:=V,V_{n-1}:=D_{1},V_{n-2}:=D_{1}\cap D_{2},\ldots ,V_{1}:=D_{1}\cap D_{2}\ldots ,\cap D_{n-1}$. Denote by 
$L_{\unicode[STIX]{x1D706}}$ the restriction 
$L|_{V_{\unicode[STIX]{x1D706}}}$ for each 
$\unicode[STIX]{x1D706}=1,2,\ldots ,n-1$. Note that it follows from a simple inductive argument that 
$(V_{\unicode[STIX]{x1D706}},L_{\unicode[STIX]{x1D706}})$ is also a del Pezzo manifold of degree 
$1$ for each 
$\unicode[STIX]{x1D706}$. Especially, for 
$\unicode[STIX]{x1D706}=1$, it holds that 
$V_{1}$ is an elliptic curve and 
$\text{deg}\,L_{1}=1$. Take 
$q\in V_{1}$ and denote by 
$\unicode[STIX]{x1D70B}:X\rightarrow V$ the blow-up at 
$q$. Let us denote by 
$E$ the exceptional divisor, by 
$\widetilde{D}_{\unicode[STIX]{x1D706}}$ the strict transform 
$(\unicode[STIX]{x1D70B}^{-1})_{\ast }D_{\unicode[STIX]{x1D706}}$, and by 
$Y$ the strict transform 
$(\unicode[STIX]{x1D70B}^{-1})_{\ast }V_{1}$. Then it is clear that 
$\widetilde{D}_{1},\widetilde{D}_{2},\ldots ,\widetilde{D}_{n-1}\in |\widetilde{L}|$, where 
$\widetilde{L}:=\unicode[STIX]{x1D70B}^{\ast }L\otimes {\mathcal{O}}_{X}(-E)$, and that 
$\widetilde{D}_{\unicode[STIX]{x1D706}}$’s intersect each other transversally along 
$Y$. Thus we can apply Corollary 1.2 to this example to obtain Corollary 1.3.
6.4 An example of an infinite type pair which does not admit ${\mathcal{F}}$ as in Theorem 1.1
In [Reference UedaU, Section 5.4], Ueda constructed a pair 
$(C,S)$ of a surface 
$S$ and a compact curve 
$C$ of genus 
$g\geqslant 1$ embedded in 
$S$ with unitary flat normal bundle such that 
$(C,S)$ is infinite type, however there does not exist a foliation 
${\mathcal{F}}$ as in Theorem 1.1. Here we will construct such a pair for the case where the codimension 
$r$ is greater than 
$1$.
Let 
$(C,S)$ be as above with 
$g=1$. By shrinking 
$S$ to a tubular neighborhood of 
$C$ if necessary, we may assume 
$\unicode[STIX]{x1D70B}_{1}(C,\ast )\cong \unicode[STIX]{x1D70B}_{1}(S,\ast )$. Denote by 
$\unicode[STIX]{x1D70C}:=\unicode[STIX]{x1D70C}_{N_{C/S}}$ the unitary representation of 
$\unicode[STIX]{x1D70B}_{1}(C,\ast )$ corresponding to the unitary flat line bundle 
$N_{C/S}$ and by 
$\widetilde{S}\rightarrow S$ the universal covering of 
$S$. Set 
$X:=\widetilde{S}\times \mathbb{C}^{r-1}/\sim$, where 
${\sim}$ is the relation defined by
$$\begin{eqnarray}(z,(v_{1},v_{2},\ldots ,v_{r-1}))\sim (\unicode[STIX]{x1D6FE}z,(\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6FE})v_{1},\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6FE})v_{2},\ldots ,\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6FE})v_{r-1}))\end{eqnarray}$$ for each 
$(z,(v_{1},v_{2},\ldots ,v_{r-1}))\in \widetilde{S}\times \mathbb{C}^{r-1}$ and 
$\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D70B}_{1}(S,\ast )$. We denote by 
$Z$ the submanifold 
$\widetilde{S}\times \{(0,0,\ldots ,0)\}/\sim$ of 
$X$ and by 
$Y$ the submanifold 
$\widetilde{C}\times \{(0,0,\ldots ,0)\}/\sim$ of 
$Z$, where 
$\widetilde{C}\subset \widetilde{S}$ is the universal covering of 
$C$. Note that 
$(Y,Z)$ is naturally isomorphic to 
$(C,S)$. As 
$N_{Y/X}\cong N_{C/S}^{\oplus r}$ holds and 
$N_{C/S}$ is nontorsion, 
$H^{1}(Y,N_{Y/X}\otimes S^{n}N_{Y/X}^{\ast })=0$ holds for each 
$n\geqslant 2$. Therefore we obtain that 
$(Y,X)$ is of infinite type.
Assume that there exists a local defining functions system 
$\{(V_{j},w_{j})\}$ of 
$Y$ with 
$w_{j}=T_{jk}w_{k}$ on each 
$V_{jk}$ (
$T_{jk}\in U(r)$), where 
$\{V_{j}\}$ is as in Section 2.2. Take a local frame 
$\{e_{j}\}$ of 
$N_{Y/X}^{\ast }$ such that 
$e_{j}=t_{jk}e_{k}$ on each 
$U_{jk}(=Y\cap V_{jk})$, where 
$t_{jk}\in U(1)$ is a transition function for some local frames of 
$N_{C/S}^{\ast }$. Let 
$A_{j}:U_{j}\rightarrow \text{GL}_{r}(\mathbb{C})$ be a holomorphic function defined by 
$e_{j}=A_{j}\cdot dw_{j}|_{U_{j}}$. By considering 
$\{(U_{j},A_{j})\}$ as a global section of the vector bundle 
$\text{End}\,(N_{Y/X}^{\ast })$, it follows from Lemma 2.1 that each 
$A_{j}$ is a constant function (see also [Reference SeshadriSe, Section 1, Proposition 1]). Thus, by replacing 
$w_{j}$ with 
$A_{j}\cdot w_{j}$, we may assume that 
$T_{jk}=\text{diag}\,(t_{jk},t_{jk},\ldots ,t_{jk})$. For a fixed index 
$j_{0}$, it is clear that 
$w_{j_{0}}^{\unicode[STIX]{x1D706}}|_{V_{j_{0}}\cap Z}\not \equiv 0$ for some 
$\unicode[STIX]{x1D706}$. Without loss of generality, we may assume 
$\unicode[STIX]{x1D706}=1$. Set 
$f_{j}:=w_{j}^{1}|_{V_{j}\cap Z}$ for each 
$j$. Then, as 
$f_{j}=t_{jk}f_{k}$ on each 
$V_{jk}\cap Z$ and 
$f_{j_{0}}\not \equiv 0$, we obtain that 
$f_{j}\not \equiv 0$ for any 
$j$ and that the divisors 
$\text{div}(f_{j})$ glue up to define a divisor 
$D$ of 
$V\cap Z$ (
$V=\cup _{j}V_{j}$). Let 
$D=aY+\sum _{\unicode[STIX]{x1D708}=1}^{\ell }b_{\unicode[STIX]{x1D708}}W_{\unicode[STIX]{x1D708}}$ be the irreducible decomposition of 
$D$ (
$a,b_{\unicode[STIX]{x1D708}}>0$, note that we may assume 
$\ell <\infty$ by shrinking 
$V$ if necessary). As the line bundle 
$[D]$ is unitary flat, the intersection number 
$(D,Y)$ can be computed as follows: 
$(D,Y)=\text{deg}\,[D]|_{Y}=0$. The self-intersection number 
$(Y,Y)$ is also equal to 
$0$, since 
$(Y,Y)=\text{deg}\,N_{Y/Z}=\text{deg}\,N_{C/S}$. Therefore it holds that 
$(W_{\unicode[STIX]{x1D708}},Y)=0$ for each 
$\unicode[STIX]{x1D708}$, which means that we may assume that 
$D=aY$ by shrinking 
$V$ if necessary. Thus it holds that the system 
$\{(V_{j}\cap Z,f_{j})\}$ induces a foliation 
${\mathcal{F}}$ on a neighborhood of 
$C$ in 
$S$ as in Theorem 1.1, which contradicts to the property of the pair 
$(C,S)$.
7 Discussion
In this section, we list some remaining problems.
In [Reference UedaU, Section 3], the neighborhood structure of 
$Y$ is investigated also for the pair 
$(Y,X)$ of finite type for the case where 
$r=1$. According to [Reference UedaU, Theorem 1], 
$Y$ admits a fundamental system of strongly pseudoconcave neighborhoods. As an analogy, it seems to be natural to ask the following question for example.
Question 7.1. Let 
$X$ be a compact complex manifold and 
$Y\subset X$ be a compact complex submanifold with unitary flat normal bundle such that the pair 
$(Y,X)$ is of type 
$n<\infty$. What kind of psh functions do there exist on 
$X\setminus Y$?
One of the most interesting application of [Reference UedaU, Theorem 1] is the classification of the pairs 
$(Y,X)$ of finite type such that 
$X$ is a projective surface and 
$Y$ is an elliptic curve [Reference NeemanN, Section 6]. We are also interested in a higher dimensional analogy of this result:
Question 7.2. Classify the pairs 
$(Y,X)$ of finite type such that 
$X$ is a projective manifold and 
$Y$ is an elliptic curve with unitary flat normal bundle.
We are also interested in some concrete examples. In our context, the example of the blow-up of 
$\mathbb{P}^{2}$ at nine points is one of the most interesting examples, see [Reference Arnol’dA], [Reference BrunellaB], and [Reference DemaillyD2, Section 1]. The example we treated in Section 6.3 is a natural generalization of this example. From this point of view, it seems to be natural to ask the following:
Question 7.3. (Higher dimensional analogue of [Reference KoikeK3, Question 1.2])
Let 
$(V,L),C=V_{1},q,Y,X$ be as in Corollary 1.3 and Section 6.3. Is there a point 
$q\in C$ such that 
$K_{X}^{-1}$ admits no 
$C^{\infty }$ Hermitian metric with semipositive curvature, or that 
$Y$ does not admit a pseudoflat neighborhood system?
In [Reference KoikeK2], we studied the neighborhood structure of a submanifold 
$Y$ of 
$X$ with codimension 
$r=2$. Under the assumption of the existence of a hypersurface 
$S$ of 
$X$ with unitary flat normal bundle which includes 
$Y$ as a submanifold, we posed the obstruction classes 
$u_{n,m}(Y,S,X)\in H^{1}(N_{S/X}|_{Y}^{-n}\otimes N_{Y/S}^{-m})$. Thereafter, we found a mistake in the proof of [Reference KoikeK2, Theorem 1], which is corrected as [Reference Koike and OgawaKO, Theorem 1.4] by using a new obstruction classes 
$v_{n,m}(Y,S,X)\in H^{1}(N_{S/X}|_{Y}^{-n}\otimes N_{Y/S}^{-m+1})$.
Question 7.4. What is the relation between 
$(u_{n,m}(Y,S,X),v_{n,m}(Y,S,X))$ and 
$u_{n}(Y,X)$ we defined in Section 3.1?
Acknowledgments
The author would like to give heartful thanks to Prof. Tetsuo Ueda whose comments and suggestions were of inestimable value for my study. He thanks Prof. Kento Fujita who taught him an example of the blow-up of a del Pezzo manifold at a general point. He also thanks Prof. Noboru Ogawa for helpful comments and warm encouragements. He is supported by the Grant-in-Aid for Scientific Research (KAKENHI No. 28-4196) and the Grant-in-Aid for JSPS fellows.
