# Title

Wenhao Ou  Wenhao Ou, , Institute of Mathematics, Academy of Mathematics and
Systems Science, Chinese Academy of Sciences, Beijing, 100190, China
<EMAIL_ADDRESS>
# Orbifold modifications of complex analytic varieties

Wenhao Ou  Wenhao Ou, , Institute of Mathematics, Academy of Mathematics and
Systems Science, Chinese Academy of Sciences, Beijing, 100190, China
<EMAIL_ADDRESS>
###### Abstract.

We prove that if $X$ is a compact complex analytic variety, which has quotient
singularities in codimension 2, then there is a projective bimeromorphic
morphism $f\colon Y\to X$, such that $Y$ has quotient singularities, and that
the indeterminacy locus of $f^{-1}$ has codimension at least 3 in $X$. As an
application, we deduce the Bogomolov-Gieseker inequality on orbifold Chern
classes for stable reflexive coherent sheaves on a compact Kähler variety
which has quotient singularities in codimension 2.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Bimeromorphic transforms
  4. 4 Modification to double-point singularities
  5. 5 Further improvement of the defining equations
  6. 6 Local construction of covering spaces
  7. 7 Modification around an irreducible singular locus
  8. 8 Proof of the modification theorem
  9. 9 The Bogomolov-Gieseker inequality

##  1\. Introduction

The theory of holomorphic vector bundles is a central object in algebraic
geometry and analytic geometry. The notion of stable vector bundles on
complete curves was introduced by Mumford in [Mum63]. Such notion of stability
was then extended to torsion-free sheaves on any projective manifolds (see
[Tak72], [Gie77]), and is now known as slope stability. An important property
of stable vector bundles is the following Bogomolov-Gieseker inequality,
involving the Chern classes of the vector bundle.

######  Theorem 1.1.

Let $X$ be a projective manifold of dimension $n$, let $H$ be an ample
divisor, and let ${\mathcal{F}}$ be a $H$-stable vector bundle of rank $r$ on
$X$. Then

$\Big{(}c_{2}({\mathcal{F}})-\frac{r-1}{2r}c_{1}({\mathcal{F}})^{2}\Big{)}\cdot
H^{n-2}\geqslant 0.$

When $X$ is a surface, the inequality was proved in [Bog78]. In higher
dimension, one may apply Mehta-Ramanathan theorem in [MR82] to reduce to the
case of surfaces, by taking hyperplane sections. Later in [Kaw92], as a part
of the proof for the three-dimensional abundance theorem, Kawamata extended
the inequality to reflexive sheaves on projective surfaces with quotient
singularities. The same technique then allows us to deduce the Bogomolov-
Gieseker inequality for reflexive sheaves on projective varieties which have
quotient singularities in codimension 2.

On the analytic side, let $(X,\omega)$ be a compact Kähler manifold, and
$({\mathcal{F}},H)$ a Hermitian holomorphic vector bundle on $X$. Lübke proved
that if $h$ satisfies the Einstein condition, then the following inequality
holds (see [Lüb82]),

$\int_{X}\Big{(}c_{2}({\mathcal{F}},h)-\frac{r-1}{2r}c_{1}({\mathcal{F}},h)^{2}\Big{)}\wedge\omega^{n-2}\geqslant
0.$

It is now well understood that if ${\mathcal{F}}$ is slope stable, then it
admits a Hermitian-Einstein metric. The case when $X$ is a complete curve was
proved by Narasimhan-Seshadri in [NS65], the case of projective surfaces was
proved by Donaldson in [Don85], and the case of arbitrary compact Kähler
manifolds was proved by Uhlenbeck-Yau in [UY86]. Simpson extended the
existence of Hermitian-Einstein metric to stable Higgs bundles, on compact and
certain non compact Kähler manifolds, see [Sim88]. Furthermore, in [BS94],
Bando-Siu introduced the notion of admissible metrics and prove the existence
of admissible Hermitian-Einstein metrics on stable reflexive sheaves.

Comparing with the algebraic version, it is natural to expect a Bogomolov-
Gieseker type inequality, for stable coherent reflexive sheaves on a compact
Kähler variety, which has at most quotient singularities in codimension 2, see
for example [CHP23]. When the underlying space has quotient singularities
only, for example when it is a surface, an orbifold version of Donaldson-
Uhlenbeck-Yau theorem was proved by Faulk in [Fau22]. As a consequence, a
Bogomolov-Gieseker type inequality holds in this case. However, for a general
Kähler variety, we are not able to take hyperplane sections. So the algebraic
method does not apply. It was suggested in [CGN+23] that the existence of
orbifold modifications would imply the inequality. The main objective of this
paper is to study this problem. We prove the following theorem.

######  Theorem 1.2.

Let $X$ be a compact complex analytic variety. Assume that $X$ has quotient
singularities in codimension 2. That is, there is a closed analytic subset $V$
of codimension at least 3, such that $X\setminus V$ has quotient
singularities. Then there is a projective bimeromorphic morphism $f\colon Y\to
X$ such that $Y$ has quotient singularities, and that the indeterminacy locus
of $f^{-1}$ has codimension at least 3 in $X$.

By combining the theorem with the existence of orbifold Hermitian-Einstein
metrics in [Fau22], and by using a continuity argument, we deduce the
following Bogomolov-Gieseker inequality.

######  Theorem 1.3.

Let $(X,\omega)$ be a compact Kähler variety of dimension $n$, which has
quotient singularities in codimension 2. Let $\mathcal{F}$ be a
$\omega$-stable reflexive coherent sheaf of rank $r$ on $X$. Then we have the
following inequality on orbifold Chern classes of $\mathcal{F}$,

$\Big{(}2r\hat{c}_{2}(\mathcal{F})-(r-1)\hat{c}_{1}(\mathcal{F})^{2}\Big{)}\cdot[\omega]^{n-2}\geq
0.$

For the precise definition of the Chern classes $\hat{c}$, we refer to Section
9. We note that when $X$ is projective, a stronger version of Theorem 1.2 was
proved by Xu in [LT19]. The proof approximately proceeds as follows. We let
$X^{\circ}$ be the open subset on which $X$ has quotient singularities. Then
by considering the frame bundle on the local orbifold charts, we can obtain a
smooth quasi-projective variety $p\colon Y\to X$, such that $X=Y/G$, where
$G=\mathrm{GL}_{n}(\mathbb{C})$ and $n$ is the dimension of $X$ (see [Kre09]).
The action of $G$ on $Y$ only has finite stabilizers. There exists a
$G$-equivariant projective compactification $\overline{Y}$ of $Y$, such that
$G$ acts linearly on $\overline{Y}$. Then up to applying Kirwan’s partial
resolution in [Kir85], one may assume that the strictly semi-stable locus in
$\overline{Y}$ is empty. Then the GIT quotient $\overline{Y}//G$ is a
projective compactification of $X^{\circ}$ which has quotient singularities
only. Finally, by using equivariant resolution of singularities to the
birational map $\overline{Y}//G\dashrightarrow X$, there is a projective
compactification $Z$ of $X^{\circ}$ such that $Z$ has quotient singularities
and the natural birational map $Z\dashrightarrow X$ is a morphism. The main
difficulty to adapt this method in the setting of complex analytic varieties
is that, the theory of GIT relies essentially on the existence of an ample
line bundle, which is not the case if $X$ is not projective.

In this paper, we take a different approach and do not consider actions of
positive dimensional groups. It consists of several steps as we will sketch
them as follows. Let $X$ be a compact complex analytic variety which has
quotient singularities in codimension 2. Let $S$ be the codimension 2 part of
the singular locus and let $S_{1},...,S_{r}$ be the irreducible components.

In the first step, by blowing up intersections of components of $S$, and then
by taking an appropriate bimeromorphic model using MMP, we reduce to the case
when the $S_{e}$ are pairwise disjoint. Furthermore, $X$ has klt
singularities.

In the second step, around every point $x\in X$, we take the index-one cover
$V$ of the canonical sheaf $\omega_{X}$. The advantage we get is that $V$ has
canonical singularities. In particular, it has hypersurface singularities in
codimension 2. Since in general, index-one covers only exist locally, and $V$
is singular, we introduce the notion of orbi-varieties, which is a
straightforward extension of orbifolds.

Now we work locally around every point of $S$. In the third step, we assume
that $X$ has canonical singularities, and will blowup $X$ so that it has
hypersurface singularities. This is the main objective of Section 4.

Next in the fourth step, we blowup further to make the defining equation of
$X$ be in a very similar shape as the ones for Du Val singularities (that is,
surface canonical singularities). With this shape of defining equation, we may
think $X$ as a family of Du Val singularities. This will be done in Section 5.

In the fifth step, in Section 6, by using covering spaces which resolve Du Val
singularities, we construct covering spaces on $X$ so that we resolve the
singularities at general points of $S$. In general, such a covering space will
produce new and mysterious singularities, over the divisorial branched locus,
which can also be regarded as the degeneracy locus of the Du Val
singularities.

In the last step, we first show in Section 7 that the previous local
constructions can be carried out globally, with the language of orbi-
varieties. Then in Section 8, we take some dlt modification on $X$, so that we
envelop the divisorial branched locus into a dlt reduced divisor. An advantage
is that, dlt pairs are closed to snc pairs, and we have better understanding
for finite covers branched along a reduced dlt divisor. With such a
modification, we can deduce Theorem 1.2.

Acknowledgment. The author is grateful to Omprokash Das for conversations. The
author is supported by the National Key R&D Program of China (No.
2021YFA1002300).

##  2\. Preliminaries

We first fix some notation and prove some elementary results in this section.
Throughout this paper, the symbol $\mathbb{D}$ stands for a disc contained in
$\mathbb{C}$ centered at $0$. We will write $\mathbb{D}_{r}$ if we need to
stress the radius $r$ of $\mathbb{D}$. We denote by $\mathbf{0}_{n}$ for the
origin of $\mathbb{C}^{n}$. By a polydisc of dimension $n$, we refer to
$\mathbb{D}^{n}$, which is always assumed centered at the origin of
$\mathbb{C}^{n}$.

###  2.A. Complex analytic varieties and their singularities

A complex analytic variety $X$ is a reduced and irreducible complex space. We
will denote by $X_{\operatorname{\textsubscript{\rm sm}}}$ its smooth locus
and by $X_{\operatorname{\textsubscript{\rm sing}}}$ its singular locus. A
smooth complex analytic variety is also called a complex manifold.

We say that $X$ has quotient singularities in codimension $2$ if there is a
closed analytic subset $V\subseteq X$ of codimension at least $3$, such that
$X\setminus V$ has quotient singularities. We refer to [KM98, Section 2.3] for
the notion of terminal, canonical, klt and dlt singularities. As shown in
[GK20, Lemma 5.8] (see also [GKKP11, Proposition 9.3]), if $X$ has klt
singularities, then it has quotient singularities in codimension $2$.

Let $X$ be a complex analytic varieties of dimension $n$. Let $S\subseteq
X_{\operatorname{\textsubscript{\rm sing}}}$ be an irreducible component.
Assume that $S$ has codimension 2 in $X$, and that $X$ has quotient
singularities at general points of $S$. Then as shown in [GK20, Lemma 5.8]
(see also [GKKP11, Proposition 9.3]), there is a proper closed analytic subset
$Z$ of $S$, such that the following property holds. For every point $x\in
S\setminus Z$, there is an open neighborhood $U$ of $x$ in $X$, such that
$(x\in U)\cong\mathbb{D}^{n-2}\times(o\in V)$, where $(o\in V)$ is a klt
surface singularity. In the next lemma, we show that the singularity type
$(o\in V)$ is the same as the one of any local surface intersecting $S$
transversally at $x$. Since $S\setminus Z$ is connected, these types are all
the same on $S\setminus Z$. Therefore, we will say that $X$ has the same type
of singularities at points of $S\setminus Z$. And if for example $(o\in V)$ is
a Du Val singularity of type $A_{r}$, then we say that $X$ has $A_{r}$-type
singularities at points of $S\setminus Z$.

######  Lemma 2.1.

With the notation above, if $T\subseteq U$ is a local surface intersecting $S$
transversally at $x$, then $(x\in T)$ is isomorphic to $(o\in V)$ as surface
singularities.

###### Proof.

We may assume hat $U=\mathbb{D}^{n-2}\times(o\in V)$. Then $S\subseteq U$ is
identified with $\mathbb{D}^{n-2}\times\\{o\\}$ Let $\mu\colon\widetilde{V}\to
V$ be the minimal resolution, and let $\rho\colon\widetilde{U}\to U$ be the
desingularization where $\widetilde{V}=\mathbb{D}^{n-2}\times\widetilde{V}$.
Let $\widetilde{T}$ be the strict transform of $T$ in $\widetilde{U}$, and let
$\tilde{x}\in\widetilde{T}$ be a point lying over $x$. With the product
structure $\widetilde{U}=\mathbb{D}^{n-2}\times\widetilde{V}$, we can write
$\tilde{x}=(\mathbf{0}_{n-2},\tilde{v})$, where $\tilde{v}$ is in the
exceptional divisor of $\mu$. Let $\sigma\colon S\to\widetilde{T}$ be a
section, which is the composition of the following sequence,

$S=\mathbb{D}^{n-2}\times\\{o\\}\to\mathbb{D}^{n-2}\times\\{\tilde{v}\\}\hookrightarrow\widetilde{T}.$

Since $T$ meets $S$ transversally at $x$, we deduce that, at $\tilde{x}$, the
Zariski tangent space of $\widetilde{T}$ and the Zariski tangent space of
$\sigma(S)$, both viewed as the subspace of the Zariski tangent space of
$\widetilde{U}$, intersect transversally. Since
$\operatorname{dim}\,\widetilde{T}+\operatorname{dim}\,\sigma(S)=\operatorname{dim}\,\widetilde{U}$,
and since $\sigma(S)$ is smooth, we deduce that $\widetilde{T}$ is smooth at
$\tilde{x}$. Therefore, we obtain that $\widetilde{T}$ is smooth. Furthermore,
by the adjunction formula, we see that $\widetilde{T}\to T$ is the minimal
resolution. This implies that $(x\in T)$ is isomorphic to $(o\in V)$ as
surface singularities. ∎

In the following lemma, we recall a characterization of different Du Val
singularities by their defining equations (see [KM98, 4.25]).

######  Lemma 2.2.

Let $(o\in X)$ be a germ of Du Val singularity, defined in a neighborhood of
the origin of $\mathbb{C}^{3}$ by an equation of the shape

$F(x,y,z)=x^{2}+F_{2}(y,z)+F_{3}(y,z)+R(y,z)=0,$

where $F_{2}$ and $F_{3}$ are homogeneous polynomials in $(y,z)$ of degree 2
and 3 respectively, and the $R=0\operatorname{mod}\,(y,z)^{4}$. Then the
following properties hold.

  1. (1)

$F_{2}\neq 0$ if and only if $(o\in X)$ is of type $A_{r}$ for some $r\geq 1$.

  2. (2)

$F_{2}=0$ and $F_{3}$ has three pairwise coprime factors if and only if $(o\in
X)$ is of type $D_{4}$.

  3. (3)

$F_{2}=0$ and $F_{3}$ has exactly two coprime factors if and only if $(o\in
X)$ is of type $D_{r}$ for some $r\geq 5$.

  4. (4)

$F_{2}=0$ and $F_{3}$ is a cube if and only if $(o\in X)$ is of type $E_{r}$
for $r=6,7,8$.

Furthermore, in the last case, if we can write

$F_{(}x,y,z)=x^{2}+u\cdot y^{3}+u_{a}\cdot yz^{a}+u_{b}\cdot z^{b},$

where $a\geq 3$ and $b\geq 4$ are integers, and $u$ is a unit, $u_{a}$ and
$u_{b}$ are units or zero, then the following assertions hold.

  1. (5)

$b=4$ and $u_{b}\neq 0$ if and only of $(o\in X)$ is of type $D_{6}$.

  2. (6)

$u_{b}=0$ or $b\geq 5$, and $u_{a}\neq 0$, and $a=3$ if and only if $(o\in X)$
is of type $D_{7}$.

  3. (7)

$u_{a}=0$ or $a\geq 4$, and $u_{b}\neq 0$, and $b=5$ if and only if $(o\in X)$
is of type $D_{8}$.

###### Proof.

The item (1) follows from the Step 3 of [KM98, 4.25]. For the item (2) and
(3), as in Step 4 of [KM98, 4.25], we see that $(o\in X)$ is of type $D_{r}$
for some $r\geq 4$ if and only if $F_{2}=0$ and $F_{3}$ is not a cube. In
particular, up to a linear change of coordinates, we can write

$F_{3}(y,z)=z(\alpha z+y)(\beta z+y)$

where $\alpha,\beta\neq 0$ are complex numbers. Applying [KM98, (4.24.3)], we
can write

$F_{3}(y,z)+R(y,z)=(z+p(y,z))(\alpha\beta
y^{2}+(\alpha+\beta)yz+z^{2}+q(y,z)),$

where $p,q$ are holomorphic functions whose terms have degree at least 2 and 3
respectively. Replacing $z$ by $z-p(y,z)$, we can assume that

$F_{3}(y,z)+R(y,z)=z(y^{2}+(\alpha+\beta)yz+\alpha\beta z^{2}+q(y,z)).$

If $\alpha=\beta$, then replacing $y$ by $y-\alpha z$, we can assume that

$\displaystyle F_{3}(y,z)+R(y,z)=z(y^{2}+q(y,z)).$

Applying the Weierstrass preparation theorem for $y^{2}+q(y,z)$ with respect
to $y$, and noting that $q(y,z)=0\operatorname{mod}\,(y,z)^{3}$, we have

$y^{2}+q(y,z)=(\mathrm{unit})\cdot(y^{2}+yu(z)+v(z))$

such that $u(z)=0\operatorname{mod}\,z^{2}$ and
$v(z)=0\operatorname{mod}\,z^{3}$. Replacing $y$ by $y-\frac{1}{2}u(z)$, we
can write

$F_{3}(y,z)+R(y,z)=z(y^{2}+w(z)),$

where $w(z)=0\operatorname{mod}\,z^{3}$. Therefore, we have

$F_{3}(y,z)+R(y,z)=(\mathrm{unit})\cdot zy^{2}+(\mathrm{unit})\cdot z^{s},$

for some integer $s\geq 4$. It follows that $(o\in X)$ is of type $D_{s+1}$.
This proves the item (3).

If $\alpha\neq\beta$, replacing $y$ by $y-\frac{1}{2}(\alpha+\beta)z$, we can
write

$F_{3}(y,z)+R(y,z)=z(y^{2}+v(z)\cdot z^{2}+q(y,z)),$

where $v(z)$ is a unit and $q(y,z)=0\operatorname{mod}\,(y,z)^{3}$. The same
argument as above shows that $(o\in X)$ is of type $D_{4}$. We hence deduce
the item (2).

Now we assume that $F_{3}$ is a cube. Then the item (4) is a consequence of
the items (1)-(3). The item (5) and (7) follows from the Step 7 of [KM98,
4.25], and the item (6) follows from the Step 8 of [KM98, 4.25]. This
completes the proof of the lemma. ∎

In the proof of the main theorem, we apply several local constructions around
a component of the singular locus. The following lemma enable us to extend
local morphisms to the whole variety.

######  Lemma 2.3.

Let $X$ be a complex analytic space and $U\subseteq X$ and open subspace.
Assume that there is a bimeromorpihc proper morphism $p\colon U^{\prime}\to U$
such that the indeterminacy locus $Z$ of $p^{-1}$ is compact. Then $p$ extends
to some bimeromorpihc proper morphism $p\colon X^{\prime}\to X$, which is an
isomorphism outside $U^{\prime}$. Assume further that there is some
$p$-exceptional divisor which is $p$-ample over $U$, then $p$ is projective
over $X$.

###### Proof.

Let $W=X\setminus Z$. Since $p$ is an isomorphism over $U\cap W$, we deduce
that $U^{\prime}$ and $W$ glue together along $U\cap W$ to a complex analytic
space $X^{\prime}$, and that $p$ extends to a proper bimeromorphic morphism
$p\colon X^{\prime}\to X$. If there is such a divisor $H$ ample over $U$, then
it is naturally a divisor in $X^{\prime}$ since $Z$ is proper. Note that it is
also $p$-ample over $X$. Thus $p\colon X^{\prime}\to X$ is projective. ∎

###  2.B. Finite morphisms between complex analytic varieties

In this subsection, we summarize some results concerning finite morphisms
between complex analytic varieties. Let $f\colon Y\to X$ be such a finite
morphism. Then branched locus of $f$ is the smallest closed subset $Z\subseteq
X$, such that $f$ is étale over $X\setminus Z$. By the divisorial branched
locus of $f$, we refer to the codimension 1 part of $Z$. The morphism $f$ is
said to be quasi-étale if its branched locus has codimension at least $2$. By
abuse of notation, we say that $f$ is Galois if its restriction over
$X\setminus Z$ is Galois. The following theorem is due to Grauert-Remmert.

######  Theorem 2.4.

Let $X$ be a complex analytic variety, and let $X^{\circ}\subseteq X$ be a
dense Zariski open subset. Assume that $X^{\circ}$ is normal and we have a
finite étale morphism $p\colon Y\to X^{\circ}$. Then $p$ extends to a finite
morphism $p\colon Z\to X$ with $Z$ normal, which is unique up to isomorphism.

###### Proof.

Let $r\colon X^{\prime}\to X$ be the normalization. By assumption, $r$ is an
isomorphism over $X^{\circ}$. Hence up to replacing $X$ by $X^{\prime}$, we
may assume that it is normal. In this case, the theorem is proved in [GR71,
Theorem XII.5.4]. ∎

Let $X$ be a complex analytic variety and let $D_{1},...,D_{m}$ be pairwise
distinct irreducible divisors in $X$. Assume that there is an integer
$r_{i}>0$, such that $r_{i}D_{i}$ is defined by $s_{i}=0$, where $s_{i}$ is a
holomorphic function on $X$. Then for any integer $n>0$, we can construct the
cyclic cover $X[\sqrt[nr_{1}]{s_{1}},...,\sqrt[nr_{m}]{s_{m}}]$ as follows. We
consider the $\mathcal{O}_{X}$-algebra $\mathcal{A}$ defined by

$\mathcal{A}=\mathcal{O}_{X}[T_{1},...,T_{m}]/(T_{1}^{nr_{1}}-s_{1},...,T_{m}^{nr_{m}}-s_{m}).$

Then we define

$X[\sqrt[nr_{1}]{s_{1}},...,\sqrt[nr_{m}]{s_{m}}]=\mathrm{Spec}_{\mathcal{O_{X}}}\,\mathcal{A},$

and define

$X[\sqrt[nr_{1}]{s_{1}},...,\sqrt[nr_{m}]{s_{m}}]^{nor}$

as its normalization. Let $p\colon
X[\sqrt[nr_{1}]{s_{1}},...,\sqrt[nr_{m}]{s_{m}}]^{nor}\to X$ be the natural
morphism.

It is convenient that we allow $D_{i}=0$ in the previous construction. In this
case, $s_{i}$ is a unit and taking the $(nr_{i})$-root $T_{i}$ of $s_{i}$
induces an étale cover of $X$. We also observe that the cyclic cover locally
depends only on the divisors $D_{i}$ and on $n$, and is independent on the
choice of the integers $r_{i}$ and the functions $s_{i}$, in the sense of the
following lemma.

######  Lemma 2.5.

With the notation above, let $x\in X$ be a point and $U\subseteq X$ an open
neighborhood of $x$. Assume that for each $i$, $l_{i}D_{i}|_{U}$ is also
defined by $t_{i}=0$, where $l_{i}\geq 1$ is an integer and $t_{i}$ is a
holomorphic function on $U$. Then, up to shrinking $U$, every connected
component of $U[\sqrt[nl_{1}]{t_{1}},...,\sqrt[nl_{m}]{t_{m}}]^{nor}$ is
isomorphic to every connected component of $p^{-1}(U)$.

###### Proof.

We remark that both $l_{i}$ and $r_{i}$ are divisible by the Cartier index of
$D_{i}$ at $x$. Hence, up to shrinking $U$, it is enough to prove the case
when $l_{i}$ is the Cartier index of $D_{i}$ at $x$. Then there are integers
$d_{i}\geq 1$ and unit functions $\eta_{i}$ on $U$, such that
$r_{i}=d_{i}l_{i}$ and $s_{i}=\eta_{i}\cdot(t_{i})^{d_{i}}$. Up to shrinking
$U$, we may assume that $\eta_{i}$ admits a $(nr_{i})$-root $\gamma_{i}$. Then
the $\mathcal{O}_{U}$-algebra

$\mathcal{O}_{U}[T_{1},...,T_{m}]/(T_{1}^{nr_{1}}-s_{1},...,T_{m}^{nr_{m}}-s_{m}).$

is isomorphic to the product of copies of

$\mathcal{O}_{U}[T_{1},...,T_{m}]/(T_{1}^{nl_{1}}-\gamma_{1}^{nl_{1}}\cdot
t_{1},...,T_{m}^{nl_{m}}-\gamma_{m}^{nl_{m}}\cdot t_{m}).$

Since each $\gamma_{i}$ is a unit function, we deduce that

$\displaystyle\mathcal{O}_{U}[T_{1},...,T_{m}]/(T_{1}^{nl_{1}}-\gamma_{1}^{nl_{1}}\cdot
t_{1},...,T_{m}^{nl_{m}}-\gamma_{m}^{nl_{m}}\cdot t_{m})$ $\displaystyle\cong$
$\displaystyle\mathcal{O}_{U}[T_{1},...,T_{m}]/(T_{1}^{nl_{1}}-t_{1},...,T_{m}^{nl_{m}}-t_{m}).$

This implies the lemma. ∎

The following lemma summarizes two elementary properties on cyclic covers.

######  Lemma 2.6.

With the notation above, we assume that $X$ is normal. We set
$X^{\prime}=X[\sqrt[nr_{1}]{s_{1}},...,\sqrt[nr_{m}]{s_{m}}]^{nor}$. Then the
following properties hold.

  1. (1)

If $D_{i}$ is reduced and if $D^{\prime}_{i}=p^{-1}(D_{i})$, then
$D^{\prime}_{i}$ is Cartier. Furthermore, the ramification index of $p$ at
general points of every component of $D^{\prime}_{i}$ is equal to $n$.

  2. (2)

If $(X,\sum_{i=1}^{m}D_{i})$ is snc, then so is
$(X^{\prime},\sum_{i=1}^{m}D^{\prime}_{i})$.

  3. (3)

If $(X,\sum_{i=1}^{m}D_{i})$ is dlt, then so is
$(X^{\prime},\sum_{i=1}^{m}D^{\prime}_{i})$.

###### Proof.

For the item (1), we first prove the statement on the ramification indices of
$p$. Since $X$ is normal and $D_{i}$ is reduced, we deduce that $(X,D_{i})$ is
a snc pair at a general point $x\in D_{i}$. Then in a neighborhood $U$ of $x$
in $X$, every $s_{j}$ is a unit if $j\neq i$. Furthermore, $D_{i}$ is defined
by $t_{i}=0$ for some holomorphic function $t_{i}$ on $U$, and
$s_{i}=\eta\cdot t_{i}^{r_{i}}$ for some unit $\eta$ on $U$. By Lemma 2.5, up
to shrinking $U$, the preimage $p^{-1}(U)$ is a disjoint union of copies of
$Y$, where $Y=U[\sqrt[n]{t_{i}}]$. Since $Y\to U$ is totally branched along
$D_{i}\cap U$, we deduce that the ramification index of $p$ at general points
of every component of $D^{\prime}_{i}$ is equal to $n$.

We will now prove that $D_{i}^{\prime}$ is Cartier. We notice that $T_{i}$ is
a holomorphic function on $X^{\prime}$ and its zero locus is equal to
$D_{i}$’. It is enough to show that the vanishing order of $T_{i}$ along every
component of $D_{i}^{\prime}$ is 1. By definition, $T_{i}^{nr_{i}}=s_{i}$. The
vanishing order of $s_{i}$ along $D_{i}$ is $r_{1}$. From the result of the
previous paragraph, it follows that the vanishing order of $s_{i}$, viewed as
a holomorphic function on $X^{\prime}$, along every component of
$D_{i}^{\prime}$ is equal to $nr_{i}$. Thus the vanishing order of $T_{i}$
along every component of $D_{i}^{\prime}$ is 1.

For the item (2), let $D=\sum_{i=1}^{m}D_{i}$ and $D^{\prime}=p^{-1}(D)$. We
first observe the following fact. Let $x\in X$ be a point. Since $D_{i}$ is
smooth, it is irreducible at $x$. Then in an open neighborhood of $x$, $X$ is
isomorphic to a polydisc and $D$ is isomorphic to the sum of some coordinate
hyperplanes. Then we deduce that $(X^{\prime},D^{\prime})$ is log smooth at
every point $x^{\prime}\in X^{\prime}$ lying over $x$.

Next we will show that every irreducible component of $D^{\prime}$ is normal.
Without loss of the generality, we only need to show that $D^{\prime}_{1}$ is
normal. Assume by contradiction that there is a non-normal locus
$V^{\prime}\subseteq D^{\prime}_{1}$. Then $V^{\prime}$ is a non-klt center of
$(X^{\prime},D^{\prime}_{1})$ by [KM98, Proposition 5.51]. By [KM98,
Proposition 5.20], this implies that $V^{\prime}$ is an irreducible component
of the preimage of a non-klt center $V$ of the pair $(X,D_{1})$. Since $D_{1}$
is smooth. we can only have $V=\emptyset$. This is a contradiction.

In the last step, let $S$ be a stratum of $D^{\prime}$ and let $x^{\prime}\in
S$. We have seen that there is an open neighborhood $U^{\prime}$ of
$x^{\prime}$ on which $(X^{\prime},D^{\prime})$ is log smooth. Since the
irreducible components of $D^{\prime}$ are normal, up to shrinking
$U^{\prime}$, their restrictions on $U^{\prime}$ are either empty or
irreducible. Hence $S|_{U^{\prime}}$ is a stratum of
$D^{\prime}|_{U^{\prime}}$. It follows that $S$ is smooth at $x^{\prime}$.
Thus $(X^{\prime},D^{\prime})$ is snc.

For the item (3), by the definition of dlt pairs, there is a Zariski closed
subset $W\subseteq X$ such that $(X,D)$ is snc over $X^{\circ}:=X\setminus W$.
Furthermore, $W$ does not contain any non-klt center of $(X,D)$. Let
$W^{\prime}=p^{-1}(W)$ and let $X^{\prime\circ}=p^{-1}(X^{\circ})$. Then the
item (2) implies that $(X^{\prime},D^{\prime})$ is snc on $X^{\prime\circ}$.
Furthermore, by [KM98, Proposition 5.20] and its proof, if $Z^{\prime}$ is a
non-klt center of $(X^{\prime},D^{\prime})$, then $p(Z^{\prime})$ is a non-klt
center of $(X,D)$. Hence $W^{\prime}$ does not contain any non-klt center of
$(X^{\prime},D^{\prime})$. Therefore $(X^{\prime},D^{\prime})$ is dlt. This
completes the proof of the lemma. ∎

In the following lemma, we will use the basechange by a cyclic cover to kill
the divisorial branched locus of a given finite morphism.

######  Lemma 2.7.

Let $(X,\Delta)$ be a reduced dlt pair. Let $D_{1},...,D_{m}$ be the
irreducible components of $\Delta$. Assume that $r_{i}D_{i}$ is defined by
$s_{i}=0$ for some holomorphic function $s_{i}$ on $X$ and for some integer
$r_{i}\geq 1$. Let $\pi\colon Y\to X$ be a finite cover whose divisorial
branched locus is contained in $\Delta$. Let $N$ be the lcm of all the
coefficients in $\pi^{*}D_{i}$, $i=1,...,m$. Then for any integer $n$
divisible by $N$, the following properties hold. Let
$V=X[\sqrt[nr_{1}]{s_{1}},...,\sqrt[nr_{m}]{s_{m}}]^{nor}$. We denote by $W$
the normalization of $Y\times_{X}V$. Then the natural morphism $\mu\colon W\to
V$ is quasi-étale. Moreover, if $\pi^{-1}(X\setminus\Delta)$ is smooth, then
$W$ is smooth in codimension 2.

###### Proof.

We fix a natural number $n$ divisible by $N$. Let $p\colon V\to X$ be the
natural morphism. Then the branched locus of $p$ is exactly $\Delta$. Let
$H_{i}=p^{-1}(D_{i})$ for $i=1,...,m$ and let $H=\sum_{i=1}^{m}H_{i}$. Then
each $H_{i}$ is Cartier and $p^{*}D_{i}=nH_{i}$ by Lemma 2.6. The latter
condition implies that $\mu$ is étale over general points of every component
of $H_{i}$. As a consequence, $\mu$ is quasi-étale.

Suppose that $\pi^{-1}(X\setminus\Delta)$ is smooth. We assume by
contradiction that there is a codimension 2 irreducible component $Z\subseteq
W_{\operatorname{\textsubscript{\rm sing}}}$. Since $p$ is étale over
$X\setminus\Delta$, we deduce that $W\to Y$ is étale over
$\pi^{-1}(X\setminus\Delta)$. Hence $Z$ is contained in
$(p\circ\mu)^{-1}(\Delta)$. Then $\mu(Z)\subseteq H$. We note that $(V,H)$ is
dlt by Lemma 2.6. Since $H$ is Cartier, by [Kol92, Proposition 16.6], we
deduce that $V$ is smooth around general points of $\mu(Z)$. By the Zariski’s
purity theorem, we conclude that $W$ is smooth around general points of $Z$.
This is a contradiction. ∎

We note that, in the previous lemma, if $\pi$ is Galois, then the number $N$
divides the degree of $\pi$. We will need the following elementary result.

######  Lemma 2.8.

Let $S$ be a polydisc with coordinates $(T_{1},...,T_{n},...)$.

$F(y,s)=y^{3}+a(s)y^{2}+b(s)y+c(s)$

be a cubic polynomial whose coefficients are holomorphic functiosn on $S$. Let
$X$ be the hypersurface defined by $F(y,s)=0$ and let $\pi\colon X\to S$ be
the natural projection. Assume that, either the divisorial branched locus of
$\pi$ is contained in $H=\\{T_{1}\cdots T_{n}=0\\}$, or the branched locus is
$S$.

Let $\overline{S}=S[\sqrt[6]{T_{1}},...,\sqrt[6]{T_{n}}]$ and let
$\pi\colon\overline{S}\to S$ be the natural morphism. We define

$G(y,\overline{s})=y^{3}+(a\circ\pi)(\overline{s})y^{2}+(b\circ\pi)(\overline{s})y+(c\circ\pi)(\overline{s})$

as the pullback of $F$ on $\overline{S}$. Then there is a factorization

$G(y,\overline{s})=(y-p(\overline{s}))(y-q(\overline{s}))(y-r(\overline{s}))$

for some holomorphic functions $p,q,r$ on $\overline{S}$.

###### Proof.

First we assume that the branched locus of $\pi$ is the entire space $S$. In
this case, $X$ has either one or two irreducible components. If there is only
one component, then with the reduced structure, it is bimeromorphic to $S$.
Since $S$ is smooth, it follows that $X_{red}$ is isomorphic to $S$. Hence
$F(y,s)=(y-p(s))^{3}$ for some holomorphic function $p$ on $S$. If there are
two components, then the same argument e shows that
$F(y,s)=(y-p(s))^{2}(y-q(s))$ some holomorphic functions $p$ and $q$ on $S$.
In conclusion, in this case, the function $F$ already admits a factorization.
Hence its pullback $G$ also admits a factorization.

Now we assume that the branched locus of $\pi$ is a proper subset of $S$.
Since the degree of $\pi$ is 3, we see that the multiplicity of $\pi$ along
any prime divisor in $X$ is contained in $\\{1,2,3\\}$. Let $\overline{X}$ be
the hypersurface defined by $G$ and let $Y$ be its normalization. Then the
natural finite morphism $Y\to\overline{S}$ is quasi-étale. The Zariski’s
purity theorem implies that it is indeed étale. Since $\overline{S}$ is simply
connected, we deduce that $Y$ has three connected components. Hence
$\overline{X}$ has three irreducible components. The same argument as in the
previous paragraph implies that each component is isomorphic to
$\overline{S}$. Therefore, Then there is a factorization

$G(y,\overline{s})=(y-p(\overline{s}))(y-q(\overline{s}))(y-r(\overline{s}))$

for some holomorphic functions $p,q,r$ on $\overline{S}$. ∎

###  2.C. Notion of orbi-varieties

We introduce the notion of orbi-varieties. It is a straightforward extension
of the notion of complex orbifold (see for example [CR02, Section 4]), just by
allowing orbifold charts to have singularities. Particularly, a complex
orbifold is an orbi-variety whose charts are all smooth.

######  Definition 2.9.

We take the following definitions.

  1. (1)

Let $U$ be a connected Hausdorff space. An orbifold chart of $U$ is a triple
$(\widetilde{U},G,\pi)$ such that $\widetilde{U}$ is a complex analytic
variety of dimension $n$, that $G$ is a finitie group acting faithfully and
holomorphically on $\widetilde{U}$, and that $\pi\colon\widetilde{U}\to U$ is
a continuous map inducing a homeomorphism from $\widetilde{U}/G$ to $U$.

  2. (2)

Let $i\colon U^{\prime}\hookrightarrow U$ be an open subset, and
$(\widetilde{U}^{\prime},G^{\prime},\pi^{\prime})$ an orbifold chart of
$U^{\prime}$. An injection from
$(\widetilde{U}^{\prime},G^{\prime},\pi^{\prime})$ to $(\widetilde{U},G,\pi)$
consists of an open embedding
$\varphi\colon\widetilde{U}^{\prime}\to\widetilde{U}$ and a group monomorphism
$\lambda\colon G^{\prime}\to G$ such that $\varphi$ is equivariant with
respect to $\lambda$, and that $i\circ\pi^{\prime}=\pi\circ\varphi$.

  3. (3)

Let $X$ be a connected second countable Hausdorff space. An orbifold atlas on
$X$ is a collection $\mathcal{U}=\\{(\widetilde{U},G,\pi)\\}$ of orbifold
charts of open subsets of $X$ which covers $X$ and is compatible in the
following sense. For any two orbifold charts $(\widetilde{U},G,\pi)$ and
$(\widetilde{U}^{\prime},G^{\prime},\pi^{\prime})$ in $\mathcal{U}$, of open
subsets $U$ and $U^{\prime}$ respectively, for any point $x\in U\cap
U^{\prime}$, there is an open neighborhood $V$ of $x$ with an orbifold chart
$(\widetilde{V},H,\rho)$ such that there are injections from
$(\widetilde{V},H,\rho)$ to $(\widetilde{U},G,\pi)$ and
$(\widetilde{U}^{\prime},G^{\prime},\pi^{\prime})$. In this case, we say that
$\mathfrak{X}=(\widetilde{U},G,\pi)$ is an orbi-variety with quotient space
$X$.

For simplicity, we may write $\mathfrak{X}=(X_{i},G_{i})_{i\in I}$ or just
$\mathfrak{X}=(X_{i},G_{i})$ for an orbi-variety. A main objective that we
introduce the notion of orbi-variety is to reduce the study of klt
singularities to the one of canonical singularities.

######  Example 2.10.

Assume that $X$ is a complex analytic variety with klt singularities. Then for
every point $x\in X$, there is an open neighborhood $U(x)$, and a smallest
positive integer $k(x)$, such that $(\omega_{X}^{\otimes
k(x)})^{**}\cong\mathcal{O}_{U(x)}$. Let $V(x)\to U(x)$ be the cyclic cover
induced by this isomorphism and let $G(x)$ be the Galois group. Then $V(x)$
has canonical singularities. Furthermore, the collection $(V(x),G(x))$ induces
an orbi-variety with quotient space $X$.

######  Lemma 2.11.

Let $\mathfrak{X}=(X_{i},G_{i},\pi_{i})_{i\in I}$ be an orbi-variety with
quotient space $X$. Assume that $f_{i}\colon Y_{i}\to X_{i}$ is a finite
Galois $G_{i}$-invariant morphism. Let $H_{i}$ be the Galois group of the
morphism from $Y_{i}$ to its image in $X$. Then $(Y_{i},H_{i})_{i\in I}$
induces an orbi-variety if and only if the following properties hold. For any
point $x\in\pi_{i}(X_{i})\cap\pi_{j}(X_{j})$, there is an open neighborhood
$U$ of $x$, there is an orbifold chart $(V,G)$ of $U$, with injections
$\iota_{i}\colon(V,G)\to(X_{i},G_{i})$ and
$\iota_{j}\colon(V,G)\to(X_{j},G_{j})$, such that every connected component of
$V\times_{X_{i}}Y_{i}$ is isomorphic to every connected component of
$V\times_{X_{j}}Y_{j}$ over $V$.

###### Proof.

We only need to prove the “if” direction. Let $W_{i}$ be a connected component
of $V\times_{X_{i}}Y_{i}$. Then there is a subgroup $H^{\prime}_{i}$ of
$H_{i}$ such that $(W_{i},H^{\prime}_{i})$ is an orbifold chart over $U$.
Similarly, we define $(W_{j},H^{\prime}_{j})$. The assumption implies that
$(W_{i},H^{\prime}_{i})$ is isomorphic to $(W_{j},H^{\prime}_{j})$. Hence we
obtain injections of orbifold charts $(W_{i},H^{\prime}_{i})\to(Y_{i},H_{i})$
and $(W_{i},H^{\prime}_{i})\to(Y_{j},H_{j})$. In another word, the condition
of the item (3) in Definition 2.9 is satisfied. Hence $(Y_{i},H_{i})_{i\in I}$
induces an orbi-variety. ∎

Let $\mathfrak{X}=(X_{i},G_{i})$ be a orbi-variety with quotient space $X$. An
orbi-divisor $\mathfrak{D}$ is a collection of divisors $(D_{i})$ such that
each $D_{i}$ is a $G_{i}$-invariant Weil divisor in $X_{i}$, and that they are
compatible along the overlaps. The images of $D_{i}$ in $X$ then defines a
divisor $E$ in $X$. The orbi-divisor $\mathfrak{D}$ is called a snc orbi-
divisor if the following property holds. For any stratum $V$ of $E$, the
preiamge $\pi_{i}^{-1}(V)$ is smooth in $X_{i}$. This is equivalent to that
$\pi_{i}^{-1}(E)$ is a snc divisor and that for any strata $V$ of $E$, the
preiamge $\pi_{i}^{-1}(V)$ is a disjoint union of strata of $\pi_{i}^{-1}(E)$.

######  Lemma 2.12.

Let $\mathfrak{X}=(X_{i},G_{i})$ be a normal orbi-variety with quotient space
$X$. Assume that $\mathfrak{D}$ is an orbi-divisor in $\mathfrak{X}$. Then
there are $G_{i}$-invariant projective bimeromorphic morphisms $f_{i}\colon
Y_{i}\to X_{i}$, such that $(Y_{i},G_{i})$ induces an orbi-variety
$\mathfrak{Y}$ with quotient space $Y$, that the preimage of $\mathfrak{D}$ in
$\mathfrak{Y}$ is a snc orbi-divisor. Moreover, the induced morphism $f\colon
Y\to X$ is projective with a $f$-ample and $f$-exceptional divisor.

###### Proof.

Let $E$ be the image of $\mathfrak{D}$ in $X$ and let $E_{i}=f_{i}^{-1}(E)$.
Let $p_{i}\colon X^{\prime}_{i}\to X_{i}$ be the functorial principalization
of the ideal sheaf of $E_{i}$, see Theorem 3.2 in the next section. Then up to
replacing $X_{i}$ by $X^{\prime}_{i}$, we can assume that $\pi_{i}^{-1}(E)$ is
a snc divisor in $X_{i}$.

We note that, if $K$ is a stratum of $E_{i}$ of minimal dimension, then for
any $g\in G_{i}$, either $g(K)=K$ or $g(K)\cap K=\emptyset$. Let $r$ be the
minimum of the dimensions of the strata of $E_{i}$ for all $i$. Let
$p_{i}\colon X^{\prime}_{i}\to X_{i}$ be the blowup of all strata of $E_{i}$
of dimension $r$. Then there is a natural action of $G_{i}$ on
$X^{\prime}_{i}$ so that $p_{i}$ is $G_{i}$-invariant. Furthermore,
$(X^{\prime}_{i},G_{i})$ induces a complex orbifold
$\mathfrak{X}^{\prime}=(X^{\prime}_{i},G_{i},\mu_{i})$ with quotient space
$X^{\prime}$. There is a natural projecitve bimeromorphic morphism $p\colon
X^{\prime}\to X$, with a $p$-ample and $p$-exceptional divisor. If
$\Delta\subseteq X^{\prime}$ is the exceptional set of $p$, then $\Delta$ is a
divisor and $\mu_{i}^{-1}(\Delta)$ is the exceptional set of $p_{i}$. In
particular, $\mu_{i}^{-1}(\Delta)$ is smooth and is the disjoint union of its
irreducible components.

We continue this procedure by blowing up $(r+1)$-dimensional strata of the
strict transform $(p_{i}^{-1})_{*}E_{i}$ in $X^{\prime}_{i}$. By induction, we
obtain projective bimeromorphic morphisms $f_{i}\colon Y_{i}\to X_{i}$ such
that the following properties hold.

  1. (1)

There is a natural action of $G_{i}$ on $Y_{i}$ so that $f_{i}$ is
$G_{i}$-equivariant.

  2. (2)

We have a complex orbifold $\mathfrak{Y}=(Y_{i},G_{i},\nu_{i})$ with quotient
space $Y$.

  3. (3)

There is an induced bimeromorphic morphism $f\colon Y\to X$ with a $f$-ample
and $f$-exceptional divisor.

  4. (4)

For any prime $f$-exceptional $\Gamma$, the preimage $\nu^{-1}(\Gamma)$ is
smooth.

  5. (5)

The strict transform $(f_{i}^{-1})_{*}E_{i}$ in $Y_{i}$ is smooth.

  6. (6)

$f_{i}^{-1}(E_{i})$ is a snc divisor.

For the preimage of $\mathfrak{D}$ in $\mathfrak{Y}$, if $E^{\prime}\subseteq
Y$ is its image, then the items (4) and (5) above imply that, for every
irreducible component of $\Gamma$ of $E^{\prime}$, the preimage
$\nu_{i}^{-1}(\Gamma)$ is smooth, hence is the disjoint union of its
irreducible components. We then deduce that for any stratum $V$ of
$E^{\prime}$, the preimage $\nu_{i}^{-1}(V)$ is a disjoint union of strata of
$f_{i}^{-1}(E_{i})$. Therefore, we conclude that $f_{i}$ satisfies the
conditions of the lemma. ∎

###  2.D. Minimal Model Programs for complex analytic spaces

Let $f\colon X\to Y$ be a projective surjective morphism between complex
analytic varieties. We assume that either $Y$ is compact or $Y$ is a germ
around a point $y\in Y$. Assume that $(X,\Delta)$ is a dlt pair and $X$ is
$\mathbb{Q}$-factorial. We recall that $X$ is $\mathbb{Q}$-factorial if every
Weil divisor in $X$ is $\mathbb{Q}$-Cartier, and some reflexive power
$(\omega_{X}^{\otimes k})^{**}$ with $k>0$ of the canonical sheaf $\omega_{X}$
is locally free. In this case, we write $K_{X}$ for the class of
$\frac{1}{k}c_{1}((\omega_{X}^{\otimes k})^{**})$. Then the $f$-relative cone
theorem and contraction theorem hold, see [Nak87, Section 4], [DHP22, Section
2.6] and [Fuj22, Section 7]. Furthermore, if $H$ is a $f$-ample
$\mathbb{Q}$-divisor, we can run a $f$-relative MMP for $(X,\Delta)$, with the
scaling of $H$, see [DHP22, Theorem 1.4] or [Fuj22, Theorem 1.7]. The
following lemma is essentially proved in [Bir12, Lemma 4.2].

######  Lemma 2.13.

With the notation above, assume that

$(X,\Delta)=(X_{0},\Delta_{0})\dashrightarrow(X_{1},\Delta_{1})\dashrightarrow\cdots$

is the sequence of $f$-relative MMP, with the scaling of $H$. Assume that
$\lambda_{0}\geq\lambda_{1}\geq\cdots$ is the sequence of the nef threshold
for $H_{i}$ with respect to $K_{X_{i}}+\Delta_{i}$. That it,

$\lambda_{i}=\inf\\{s\geq 0\ |\ K_{X_{i}}+\Delta_{i}+H_{i}\mbox{ is nef}\\}.$

Then either $\lambda:=\lim_{i}\lambda_{i}=0$, or the previous sequence of MMP
terminates.

###### Proof.

We assume that $\lambda>0$. Let $\mu=\frac{\lambda}{2}$. We first consider the
case when $(y\in Y)$ is a germ of complex analytic variety. In particular, we
may assume that $Y$ is Stein.

By [DHP22, Theorem 2.12], there is some rational number $\varepsilon>0$ such
that $\mu H+\varepsilon\Delta$ is $f$-ample. Then, by using [DHP22, Lemma
2.21], we can obtain an effective $\mathbb{Q}$-dvisor $D$ such that
$D\sim_{\mathbb{Q}}\mu H=\mu H+\varepsilon\Delta$ and that
$(X,(1-\varepsilon)\Delta+D)$ is klt.

Let $\Delta^{\prime}=(1-\varepsilon)\Delta+D$. We observe that

$K_{X}+\Delta^{\prime}\sim_{\mathbb{Q}}K_{X}+\Delta+\mu H.$

Therefore, the sequence $(X_{i})$ is also a sequence of $f$-relative MMP for
$K_{X}+\Delta^{\prime}$, with the scaling of $(\lambda_{0}-\mu)H$. Such a MMP
must terminates by [DHP22, Theorem 1.4].

In the case when $Y$ is a compact complex analytic variety, the previous
argument shows that the sequence $(X_{i})$ must be stationary in a
neighborhood of any point of $Y$. Since $Y$ is compact, it follows that the
MMP must terminates. ∎

We use the MMP to construct certian bimeromorphic models in the following two
lemmas.

######  Lemma 2.14.

Let $X$ be a compact complex analytic variety. Let $\\{S_{1},...,S_{r}\\}$ be
a collection of codimension 2 irreducible components of
$X_{\operatorname{\textsubscript{\rm sing}}}$, such that $X$ has quotient
singularities around general points of $S_{i}$ for all $i=1,...,r$. Then there
is a projective bimeormorphic morphism $f\colon Y\to X$ such that the
following properties hold.

  1. (1)

$f$ is an isomorphism over $X_{\operatorname{\textsubscript{\rm sm}}}$.

  2. (2)

$f$ is an isomorphism over general points of $S_{i}$ for all $i=1,...,r$.

  3. (3)

The codimension 2 part of $Y_{\operatorname{\textsubscript{\rm sing}}}$ is
$\bigcup f^{-1}_{*}S_{i}$.

  4. (4)

$Y$ has $\mathbb{Q}$-factorial klt singularities.

###### Proof.

Let $r\colon\widetilde{X}\to X$ be a projective desingularization which is an
isomorphism over $X_{\operatorname{\textsubscript{\rm sm}}}$. Let $E$ be the
reduced sum of all $r$-exceptional prime divisors whose centers in $X$ are one
of the $S_{i}$’s. Let $\varepsilon>0$ be a small enough rational number such
that $-(1-\epsilon)$ is smaller than the discrepancy of $K_{X}$ around general
points of $S_{i}$. Then, by [DHP22, Theorem 1.4], we can run a $r$-relative
MMP for the klt pair $(\widetilde{X},(1-\epsilon)E)$, with the scaling of
certain $r$-ample divisor, and obtain a projective bimeormorphic morphism
$f\colon Y\to X$. The negativity lemma shows that the prime divisors with
discrepancy $\leq 0$ for $K_{Z}$ which are contracted by
$\widetilde{X}\dashrightarrow Y$ are exactly the irreducible components of
$E$. This implies that $g$ is an isomorphism over general points of every
$S_{i}$. We note that, if $\Gamma$ is a prime divisor over $Y$ which has
discrepancy at most 0 for $K_{Y}$, then the discrepancy of $\Gamma$ for
$K_{\widetilde{X}}+(1-\varepsilon)E$ is at most 0. This implies that the
center of $\Gamma$ in $\widetilde{X}$ is contained in $E$. Therefore, $Y$ has
terminal singularities outside $\bigcup f^{-1}_{*}S_{i}$. Hence the
codimension dimension 2 part of $Y_{\operatorname{\textsubscript{\rm sing}}}$
is exactly $\bigcup f^{-1}_{*}S_{i}$. ∎

######  Lemma 2.15.

Let $(X,D)$ be a pair, where $X$ is a normal compact complex analytic variety.
Let $\\{S_{1},...,S_{r}\\}$ be the collection of all codimension 2 irreducible
components of $X_{\operatorname{\textsubscript{\rm sing}}}$, which are not
contained in $D$. Then there is a projective bimeormorphic morphism $f\colon
Y\to X$ such that the following properties hold.

  1. (1)

$f$ is an isomorphism over the snc locus of $(X,D)$ and over general points of
every $S_{i}$.

  2. (2)

The exceptional locus of $f$ is a divisor.

  3. (3)

$(Y,\Delta+f^{-1}_{*}D)$ is a $\mathbb{Q}$-factorial dlt pair, where $\Delta$
is the whole reduced $f$-exceptional divisor.

###### Proof.

Let $\rho\colon Z\to X$ be a projective log resolution of $(X,D)$, which is an
isomorphism over the snc locus of $(X,D)$. Let $E$ be the whole reduced
$\rho$-exceptional divisor and let $\Gamma=E+\rho^{-1}_{*}D$. Then we can run
a $\rho$-relative MMP for the reduced dlt pair $(Z,\Gamma)$, with the scaling
of some $\rho$-ample divisor $H$ as follows,

$(Z,\Gamma)=(Z_{0},\Gamma_{0})\dashrightarrow(Z_{1},\Gamma_{1})\dashrightarrow\cdots.$

If the MMP terminates, then we let $Y$ be the output of the MMP and let
$f\colon Y\to X$ be the natural morphism. From the negativity lemma, we deduce
that $f$ is an isomorphism over general points of each $S_{i}$.

Otherwise, by Lemma 2.13, the nef threshold sequence $(\lambda_{j})$ of the
MMP tends to 0. Let $V_{i}$ be a local surface intersectiong $S_{i}$
transversally at a very general point $x_{i}$. Then the MMP induces a surface
MMP over $V_{i}$. Since $X$ has quotient singularities around general points
of $S_{i}$, we see that $V_{i}$ is a klt surface singularity. Since $D$ does
not contain $S_{i}$, we can pick $x_{i}\not\in D$. Then we deduce that there
is some integer $N(i)\geq 0$ such that $Z_{j}$ is isomorphic to $X$ at $x_{i}$
if $j\geq N(i)$. Let $k$ be the maximum of $\\{N(1),...,N(r)\\}$. Then
$Z_{k}\to X$ is an isomorphism over general points of every $S_{i}$. In this
case, we set $Y=Z_{k}$ and denote by $f\colon Y\to X$ the natural morphism.

In both cases, the $f$-exceptional locus is a divisor, and
$(Y,\Delta+f^{-1}_{*}D)$ is a $\mathbb{Q}$-factorial dlt pair, where $\Delta$
is the whole reduced $g$-exceptional divisor. Furthermore, $f$ is an
isomorphism over the snc locus of $(X,D)$ and over general points of every
$S_{i}$. This completes the proof of the lemma. ∎

##  3\. Bimeromorphic transforms

In this section, we collect some tools of bimeromorphic transforms that we
need for the proof of Theorem 1.2

###  3.A. Functorial resolution of singularities

Throughout this paper, a construction is called functorial if commutes with
local analytic isomorphisms. The functorial resolution of singularities
implies equivariant resolution of singularities. It also allow us to glue
local resolutions along the overlaps. We recall their statements from [Wło09]
as follows.

######  Theorem 3.1.

Let $X$ be a complex analytic variety. Then $X$ admits a functorial (or
canonical) desingularization $r\colon\widetilde{X}\to X$. More precisely,
$\widetilde{X}$ is a smooth complex analytic variety, $r$ is a projective
bimeromorphic morphism, which is a composition of blowups at smooth centers
contained in $X_{\operatorname{\textsubscript{\rm sing}}}$, the
$r$-exceptional locus is a snc divisor, and the construction of $r$ commutes
with local analytic isomorphisms.

######  Theorem 3.2.

Let $X$ be a complex analytic variety, and let $\mathcal{I}$ be a coherent
sheaf on $X$. Then $\mathcal{I}$ admits a functorial (or canonical)
principalization $r\colon\widetilde{X}\to X$. More precisely, $\widetilde{X}$
is a smooth complex analytic variety, $r$ is a projective bimeromorphic
morphism, which is a composition of blowups at smooth centers contained in the
cosupport of $\mathcal{I}$, $r^{*}\mathcal{I}$ is an invertible sheaf, and the
construction $r$ of commutes with local analytic isomorphisms.

We will need the following partial desingularization result.

######  Proposition 3.3.

Let $X$ be a complex analytic variety and let $S$ be the union of some
irreducible components of $X_{\operatorname{\textsubscript{\rm sing}}}$. Let
$\Delta$ be a reduced divisor in $X$, which does not contain any component of
$S$. Then there is a projective bimeromorphic morphism $f\colon Y\to X$
satisfying the following properties. Let $U=X\setminus S$ and $V=f^{-1}(U)$.

  1. (1)

Let $E$ be the $f$-exceptional locus. Then $E$ is a divisor and
$(V,(E+f^{-1}_{*}\Delta)|_{V})$ is a snc pair. In particular,
$Y_{\operatorname{\textsubscript{\rm sing}}}$ is contained in $f^{-1}(S)$.

  2. (2)

$f$ is an isomorphism over general points of every irreducible component of
$S$.

  3. (3)

$f$ is an isomorphism over the snc locus of $(X,\Delta)$.

###### Proof.

Let $r\colon Z\to X$ be a log resolution of $(X,\Delta)$ such that it can be
decomposed into a sequence

$Z=Z_{m}\to\cdots\to Z_{0}=X,$

where each $Z_{j+1}\to Z_{j}$ is the blowup of a smooth irreducible center
$C_{j}$ contained in the singular locus of $(X_{j},\Delta_{j})$, and
$\Delta_{j}$ is the sum of the strict transform of $\Delta$ in $X_{j}$ and the
whole reduced exceptional divisor of $X_{j}\to X$. We will construct by
induction a sequence of projective bimeromorphic morphisms

$Y=Y_{m}\to\cdots\to Y_{0}=X$

such that the following properties hold. The composition $f\colon Y\to X$
satisfies the properties of the proposition. In particular, we can define
$S_{j}$ the strict transform of $S$ in $Y_{j}$, and $Y_{j+1}\to Y_{j}$ is an
isomrophism over general points of every irreducible component of $S_{j}$. Let
$V_{j}$ be the preimage of $U$ in $Y_{j}$ and let $W_{j}$ be the preimage of
$U$ in $Z_{j}$. Then $V_{j}\cong W_{j}$ under the natural bimeromorphic map
$\psi_{j}\colon Z_{j}\dashrightarrow Y_{j}$ over $X$.

Assume that we have constructed $Y_{j}$. If $C_{j}\cap W_{j}=\emptyset$, then
we let $Y_{j+1}\to Y_{j}$ be the identity map. We note that $W_{j+1}\cong
W_{j}$. Hence $V_{j+1}\cong W_{j+1}$ under $\psi_{j+1}$.

If $C_{j}\cap W_{j}\neq\emptyset$, then we can define $D_{j}$ the strict
transform of $C_{j}$ via $\psi_{j}$. Let $Y_{j+1}\to Y_{j}$ the blowup of
$D_{j}$. It follows that $V_{j+1}\cong W_{j+1}$ under $\psi_{j+1}$. It remains
to prove that $Y_{j+1}\to Y_{j}$ is an isomorphism over general points of
every irreducible component of $S_{j}$. It is equivalent to show that $D_{j}$
does not contain any of these irreducible components. Assume by contradiction
that $D_{j}$ contains a component $F_{j}$ of $S_{j}$. Then the image $D$ of
$D_{j}$ in $X$ contains the one $F$ of $F_{j}$. We note that $D\cap
U\neq\emptyset$. On the one hand, by assumption, $F$ is an irreducible
component of $S$. On the other hand, $D$ is also the image of $C_{j}$ in $X$.
By assumption on $Z$, $D$ is contained in singular locus of $(X,\Delta)$. It
follows that there is an irreducible component $P$ of the singular locus of
$(X,\Delta)$ such that

$F\subseteq D\subseteq P.$

Since $\Delta$ does not contain $F$, we deduce that $P$ is an irreducible
component of $X_{\operatorname{\textsubscript{\rm sing}}}$. Then we must have
$F=P$. This is a contradiction as we have

$\emptyset\neq D\cap U\subseteq P\cap U=F\cap U=\emptyset.$

In conclusion, $Y_{j+1}\to Y_{j}$ is an isomorphism over general points of
every irreducible component of $S_{j}$. This completes the proof of the
proposition. ∎

###  3.B. Blowup at an ideal sheaf

Let $X$ be a complex analytic variety and $S\subseteq X$ an irreducible
subvariety. Then if $g\colon S^{\prime}\to S$ is a blowup at a subvariety or
more generally a blowup at an coherent ideal sheaf, then there is a natural
blowup $f\colon X^{\prime}\to X$ which extends $g$. Indeed, it is the blowup
of $X$ at the same center. As a consequence, if $g\colon S^{\prime}\to S$ is
now a composition of a sequence of functorial blowups, then there is a natural
morpihism $f\colon X^{\prime}\to X$ which extends $g$ and is functorial.

For an effective Cartier divisor $H\subseteq S$, the blowup of $S$ at $H$ is
just an isomorphism. However, we can blowup $X$ at $H$ and get a non trivial
morphism $f\colon X^{\prime}\to X$. In this case, $f$ induces an isomorphism
from the preimage $S^{\prime}$ of $S$ to $S$. We will focus on this operation
in the next subsection.

In the remainder of this subsection, we assume that $S$ is smooth of dimension
$n$. Let $o\in S\subseteq X$ be a point. We will shrink $S$ and $X$ around $o$
freely. Assume that there is a closed immersion from $X$ to an open subset $W$
of $\mathbb{C}^{n+N}$, which sends $o$ to the origin. Since $S$ is smooth, up
to shrinking $X$, we may assume that $S$ is a polydisc in $\mathbb{C}^{n}$ and
$W$ is an open neighborhood of $\mathbf{0}_{N}\times S$ in
$\mathbb{C}^{N}\times S$. We identify $S$ as a subset of $W$ via
$\mathbf{0}_{N}\times S\subseteq W$. We also assume that the inclusion
$S\subseteq X$ coincides with $\mathbf{0}_{N}\times S$. We denote by
$\mathcal{J}_{X}$ the ideal sheaf of $X$. Let $F(x_{1},...,x_{N},s)$ be a
holomorphic function on $W$, where $(x_{1},...,x_{N})$ are the coordinates of
$\mathbb{C}^{N}$ and $s\in S$ is a point. We assume that
$F\in\mathcal{J}_{X}$. In particular, we have $F(0,...,0,s)=0$ for all $s\in
S$. Then we can decompose

$F=F_{1}+F_{2}+\cdots,$

where $F_{d}(x_{1},...,x_{N},s)$ is a homogeneous polynomial of degree $d$ in
$(x_{1},...,x_{N})$, with coefficients as holomorphic functions on $S$. We
will blowup $W$ at closed subspaces contained in $S=\mathbf{0}_{N}\times S$
and then investigate the defining functions of the strict transform
$X^{\prime}$ of $X$, in a neighborhood of the strict transform $S^{\prime}$ of
$S$.

More precisely, let $\mathcal{I}$ be a coherent ideal sheaf on $S$ whose
cosupport is proper closed subset. Up to shrinking $X$, we assume that
$\mathcal{I}$ is globally generated. Let $h\colon W^{\prime}\to W$ be the
blowup of the ideal generated by $(x_{1},...,x_{N},\mathcal{I})$. Since the
cosupport of $\mathcal{I}$ is strictly contained in $S$, we can denote by
$X^{\prime}$ and $S^{\prime}$ the strict transforms of $X$ and $S$
respectively in $W^{\prime}$. Let $\mathcal{J}_{X^{\prime}}$ be the ideal
sheaf of $X^{\prime}$.

Then for every point $o\in S^{\prime}$, there is a Zariski neighborhood
$U^{\prime}\subseteq W^{\prime}$ of $o$, such that $U^{\prime}$ is isomorphic
to an neighborhood of $\mathbf{0}_{N}\times(S^{\prime}\cap U^{\prime})$ in
$\mathbb{C}^{N}\times(S^{\prime}\cap U^{\prime})$, and that
$h|_{U^{\prime}}\colon U^{\prime}\to W$ can be written in coordinates as

$(x^{\prime}_{1},...,x^{\prime}_{N},s^{\prime})\to(tx^{\prime}_{1},...,tx^{\prime}_{N},h(s^{\prime}))=(x_{1},...,x_{N},s),$

where $t$ is a holomorphic function on $S^{\prime}\cap U$, which is a
generator of the invertible ideal sheaf $(h|_{S^{\prime}})^{*}\mathcal{I}$ on
$S^{\prime}\cap U^{\prime}$. Furthermore, regarded as a function on
$U^{\prime}$, that is
$t(x^{\prime}_{1},...,x^{\prime}_{N},s^{\prime})=t(s^{\prime})$, $t$ is the
generator of the invertible ideal sheaf $h^{*}(x_{1},...,x_{N},\mathcal{I})$.
Indeed, we can let $U^{\prime}\subseteq W^{\prime}\setminus h^{-1}_{*}H$ and
$t=(h|_{S^{\prime}})^{*}\xi$, where $\xi$ is one of the generators of
$\mathcal{I}$, and $H$ is the zero locus of $\xi$, regarded as a function on
$W$.

If we pullback the function $F_{i}$ by $h|_{U^{\prime}}$, then it is of the
shape $t^{i}\cdot G_{i}(x^{\prime}_{1},...,x^{\prime}_{N},s^{\prime})$. Thus
if $m$ is the smallest integer such that $F_{m}\neq 0$, then the pullback of
$F$ is of the shape

$(h^{*}F)(x^{\prime}_{1},...,x^{\prime}_{N},s^{\prime})=t^{m}(G_{m}+tG_{m+1}+\cdots
t^{k-m}G_{k}+\cdots).$

When the coefficients of $F_{m}$ are not all contained in $\mathcal{I}$, then
$t$ does not divides $G_{m}$, and the function

$F^{\prime}(x^{\prime}_{1},...,x^{\prime}_{N},s^{\prime})=G_{m}+tG_{m+1}+\cdots
t^{k-m}G_{k}+\cdots$

is in $\mathcal{J}_{X^{\prime}}$.

We assume that every coefficient of $F_{m}$ belongs to $\mathcal{I}$. More
precisely, we can write

$F_{m}(x_{1},...,x_{N},s)=\sum_{I}a_{I}(s)x_{I},$

where the sum runs over all multi-index $I$ with $|I|=m$, such that
$a_{I}\in\mathcal{I}$ for all $I$. If we set
$a_{I}^{\prime}(s^{\prime})=a_{I}(h(s^{\prime}))$, then $t$ divides
$a_{I}^{\prime}$. In particular, we can write
$a_{I}^{\prime}(s^{\prime})=t\cdot b_{I}^{\prime}(s)$. Then the pullback of
$F_{m}$ can be written as

$(h^{*}F_{m})(x^{\prime}_{1},...,x^{\prime}_{N},s^{\prime})=\sum
a_{I}^{\prime}(s^{\prime})\cdot t^{m}\cdot x^{\prime}_{I}=t^{m+1}(\sum
b_{I}^{\prime}(s^{\prime})x^{\prime}_{I}).$

This implies that $G_{m}$ is divisible by $t$.

In addition, if $\mathcal{I}$ is exactly the ideal generated by the
coefficients $a_{I}$ of $F_{m}$, then, up to shrinking $U^{\prime}$, there is
some $b_{I}^{\prime}$ which is nowhere vanishing on $U^{\prime}\cap
S^{\prime}$. As a consequence, we can write
$(f^{*}F_{m})(x^{\prime}_{1},...,x^{\prime}_{N},s^{\prime})=t^{m+1}H_{m}(x^{\prime}_{1},...,x^{\prime}_{N},s^{\prime})$
such that $H_{m}$ is a nonzero homogeneous polynomial in
$(x^{\prime}_{1},...,x^{\prime}_{N})$ for any $s^{\prime}\in U^{\prime}\cap
S^{\prime}$. The function

$F^{\prime}(x^{\prime}_{1},...,x^{\prime}_{N},s^{\prime})=H_{m}+H_{m+1}+tH_{m+2}+\cdots+t^{k-m-1}H_{k}+\cdots$

is in $\mathcal{J}_{X^{\prime}}$.

###  3.C. Blowup at a Cartier divisor

Let $X$ be a complex analytic variety and $S\subseteq X$ an irreducible
subvariety. In the previous subsection, we have seen the blowup of $X$ at a
center strictly contained in $S$. In this subsection, we will specify the case
when the center if a Cartier divisor in $S$. We will perform this kind of
bimeromorphic transforms in Section 5 and Section 6.

In the remainder of this subsection, let $S$ be a smooth complex analytic
variety and let $H=H_{1}+\cdots H_{n}$ be a reduced snc divisor on $S$. We
assume that $S$ is isomorphic to a neighborhood of the origin in
$\mathbb{C}^{N}$ with $N\geq n$, and that $H_{i}$ is isomorphic to the
coordinate hyperplane $\\{T_{i}=0\\}$. We consider $W=\mathbb{D}^{3}\times S$.
Let $F(x,y,z,s)$ be a holomorphic function on $W$, where $(x,y,z)$ are the
coordinates of $\mathbb{D}^{3}$ and $s\in S$ is a point. We assume that

$F(x,y,z,s)=x^{2}+F_{2}(y,z,s)+F_{3}(y,z,s)\cdots F_{k}(y,z,s)+\cdots,$

where $F_{k}(y,z,s)$ is a homogeneous polynomial of degree $d$ in $(y,z)$,
with coefficients as holomorphic functions on $S$. Let $X\subseteq W$ be
subvariety defined by $F=0$.

Let $f\colon W^{\prime}\to W$ be the blowup at the ideal
$\mathcal{I}=(x,y,z,T_{i})$. Let $X^{\prime}$ and $S^{\prime}$ be the strict
transforms of $X$ and $S$ respectively in $W^{\prime}$. Then $f$ induces an
isomorphism from $S^{\prime}$ to $S$. On the open subset
$U^{\prime}=W^{\prime}\setminus f^{-1}_{*}(D_{i})$, where $D_{i}$ is the
divisor in $W$ defined by $T_{i}=0$, there is an isomorphism
$U^{\prime}\cong\mathbb{D}^{3}\times S$, such that $f|_{U^{\prime}}$ can be
written with coordinates as

$(x^{\prime},y^{\prime},z^{\prime},s)\to(T_{i}x,T_{i}y,T_{i}z,s).$

Moreover, the $f$-exceptional divisor $E$ is defined by $\\{T_{i}=0\\}$ on
$U^{\prime}$. We note that the pullback $f^{*}F$ is exactly divisible by
$T_{i}^{2}$. Thus, the defining function of $X^{\prime}$ on $U^{\prime}$
becomes

$F^{\prime}(x^{\prime},y^{\prime},z^{\prime},s)=x^{\prime
2}+F_{2}(y^{\prime},z^{\prime},s)+T_{i}F_{3}(y^{\prime},z^{\prime},s)\cdots
T_{i}^{k-2}F_{k}(y^{\prime},z^{\prime},s)+\cdots$

We will need to perform this operation several times in order to raise the
exponent of $T_{i}$ in the degree $k$ part of the defining function $F$ for
$k$ large.

More precisely, let $f\colon W^{\prime}\to W$ be the composition of the
following sequence

$W^{\prime}=W^{e}\to W^{e-1}\cdots W^{0}=W,$

such that each $\varphi^{k}\colon W^{k+1}\to W^{k}$ is the blowup of $W^{k}$
at the center $H_{i}^{k}$ for some $i\in\\{1,...,n\\}$, where $H_{i}^{k}$ is
the preimage of $H_{i}$ in $S^{k}$, and $S^{k}$ is the strict transform of $S$
in $W^{k}$. We say that there are $m$ blowups whose centers are $H_{i}$ if
there are $\varphi^{k_{1}},...,\varphi^{k_{m}}$ in the previous sequence, such
that $\varphi^{k_{l}}$ is the blowup of $W^{k_{l}}$ at the center
$H_{i}^{k_{l}}$ for all $l=1,...,m$. Let $X^{\prime}$ and $S^{\prime}$ be the
strict transforms of $X$ and $S$ respectively in $W^{\prime}$. Then $f$
induces an isomorphism from $S^{\prime}$ to $S$. Let
$T^{\prime}=\prod_{i=1}^{n}T_{i}^{m_{i}}$, where $m_{i}$ is the number of
blowups whose centers are $H_{i}$. Then there is an open neighborhood
$U^{\prime}$ of $S^{\prime}$, such that $f|_{U^{\prime}}$ can be written with
coordinates as

$(x^{\prime},y^{\prime},z^{\prime},s)\to(T^{\prime}x,T^{\prime}y,T^{\prime}z,s).$

Moreover, the $f$-exceptional divisor $E$ is defined by $\\{T^{\prime}=0\\}$
on $U^{\prime}$ as a closed subset. A defining function of $X^{\prime}$ on
$U^{\prime}$ is

$F^{\prime}(x^{\prime},y^{\prime},z^{\prime},s)=x^{\prime
2}+G_{2}(y^{\prime},z^{\prime},s)+T^{\prime}G_{3}(y^{\prime},z^{\prime},s)+\cdots(T^{\prime})^{k-2}G_{k}(y^{\prime},z^{\prime},s)+\cdots.$

By abuse of notation, we say that, after the blowups, the defining equation of
$X$ becomes

$F(x,y,z,s)=x^{2}+F_{2}(y,z,s)+T^{\prime}F_{3}(y,z,s)\cdots(T^{\prime})^{k-2}F_{k}(y,z,s)+\cdots.$

######  Remark 3.4.

We will call the previous operation a blowup of $X$ at the $H_{i}$’s. As
mentioned earlier, the objective of this operation is to raise the exponent of
$T_{i}$ in $F_{k}$. We observe that the number we raise for the exponent of
$T_{i}$ depends only on the number of blowups whose center is $H_{i}$. In
particular, the order of the blowups at the $H_{i}$’s does not matter for our
purpose.

######  Remark 3.5.

Assume that $F_{j}(y,z,s)=a(s)\cdot E_{j}(y,z,s)$, where the zero locus of
$a(s)$ is contained in $H$. Let $b(s)$ be a holomorphic function whose zero
locus is contained in $H$. By blowing up $X$ at the $H_{i}$’s several times,
the defining equation of $X$ becomes

$\displaystyle F(x,y,z,s)$ $\displaystyle=$ $\displaystyle
x^{2}+F_{2}(y,z,s)+\cdots+(T^{\prime})^{j-3}F_{j-3}$
$\displaystyle+(T^{\prime})^{j-2}\Big{(}a(s)E_{j}(y,z,s)+T^{\prime}\cdot\big{(}F_{j+1}(y,z,s)+T^{\prime}F_{j+2}(y,z,s)+\cdots\big{)}\Big{)}.$

By choosing $m_{i}$ greater than the vanishing order of $a(s)b(s)$ along
$H_{i}$ for all $i=1,...,n$, the function $a(s)b(s)$ divides $T^{\prime}$. As
a result, the defining equation of $X$ can be written as

$\displaystyle F(x,y,z,s)$ $\displaystyle=$ $\displaystyle
x^{2}+F_{2}(y,z,s)+\cdots+(T^{\prime})^{j-3}F_{j-3}$
$\displaystyle+(T^{\prime})^{j-2}a(s)\big{(}E_{j}(y,z,s)+b(s)E_{j+1}(y,z,s)+\cdots\big{)},$

where $E_{j+1},...$ are holomorphic functions. In other words, after the
blowups, we can write $F_{j}(y,z,s)=a^{\prime}(s)\cdot E_{j}(y,z,s)$, where
$a^{\prime}(s)=(T^{\prime})^{j-2}a(s)$. Furthermore, the zero locus of
$a^{\prime}(s)$ is contained in $H$, and $a^{\prime}(s)b(s)$ divides
$F_{j+1}+\cdots$. In conclusion, by blowing up $X$ and by replacing $a(s)$ by
$a^{\prime}(s)$, we can make $a(s)b(s)$ divides $F_{j+1}+\cdots$.

######  Remark 3.6.

Assume that $F(x,y,z,s)=x^{2}+G(y,z,s)$, where the terms of $G$ have degrees
at least 3 in $(y,z)$. Assume that

$G(y,z,s)=\sum_{i=0}^{3}R_{i}(y,z,s)\cdot y^{i}\cdot(q(s)z)^{3-i}+P(y,z,s),$

where the zero locus of $q(s)$ is contained in $H$. We blow up $X$ at each
$H_{i}$ for $m_{i}$ times, and we choose $m_{i}$ greater than the vanishing
order of $q(s)$ along $H_{i}$. Particularly, there is a holomorphic function
$b(s)$ such that $T^{\prime}=q(s)b(s)$. The defining equation of $X$ becomes

$\displaystyle F(x,y,z,s)$ $\displaystyle=$ $\displaystyle
x^{2}+(T^{\prime})^{-2}\cdot G(T^{\prime}y,T^{\prime}z,s)$ $\displaystyle=$
$\displaystyle
x^{2}+(T^{\prime})^{-2}\cdot\Big{(}\sum_{i=0}^{3}R_{i}(T^{\prime}y,T^{\prime}z,s)\cdot(T^{\prime})^{3}y^{i}\cdot(q(s)z)^{3-i}+P(T^{\prime}y,s)\Big{)}$
$\displaystyle=$ $\displaystyle
x^{2}+T^{\prime}\cdot\sum_{i=0}^{3}R_{i}(T^{\prime}y,T^{\prime}z,s)\cdot
y^{i}\cdot(q(s)z)^{3-i}+(T^{\prime})^{-2}\cdot P(T^{\prime}y,s).$

We note that $P^{\prime}(y,s)=(T^{\prime})^{-2}\cdot P(T^{\prime}y,s)$ is a
holomorphic function in $(y,s)$. Let

$R_{i}^{\prime}(y,q(s)z,s)=T^{\prime}R_{i}(T^{\prime}y,b(s)q(s)z,s)=T^{\prime}R_{i}(T^{\prime}y,T^{\prime}z,s),$

for $i=0,2,3$. Then we have

$F(x,y,z,s)=x^{2}+\sum_{i=0}^{3}R^{\prime}_{i}(y,q(s)z,s)\cdot
y^{i}\cdot(q(s)z)^{3-i}+P^{\prime}(y,s),$

which now can be viewed as a function in $(x,y,q(s)z,s)$. In conclusion, by
blowing up $X$, we can make $F(x,y,z,s)$ a holomorphic function in
$(x,y,q(s)z,s)$.

##  4\. Modification to double-point singularities

Let $(S,H)$ be a reduced snc pair, where $S$ is a complex analytic variety.
Let $o\in S$ be a point. We consider a holomorphic function $F$ defined in a
neighborhood of $(\mathbf{0}_{3},o)\in\mathbb{C}^{3}\times S$ of the shape

$F(x,y,z,s)=x^{2}+F_{2}(y,z,s)+F_{3}(y,z,s)+R(y,z,s),$

where $(x,y,z)$ are coordinates of $\mathbb{C}^{3}$ and $s\in S$, the
functions $F_{2}$ and $F_{3}$ are homogeneous polynomials in $(y,z)$, of
degrees $2$ and $3$ respectively, with coefficients as holomorphic functions
on $S$, and $R(y,z,s)=0\operatorname{mod}\,(y,z)^{4}$. We say that $F$ is of
standard form with respect to $(S,H)$ if the following property holds. Either
$F_{2}(y,z,s)=a(s)\cdot G_{2}(y,z,s)$, where $G_{2}(y,z,s)$ is a nonzero
polynomials in $(y,z)$ for any point $s\in S$ , and the zero locus of $a(s)$
is contained in $H$; or $F_{2}(y,z,s)=0$ and $F_{3}(y,z,s)=b(s)\cdot
G_{3}(y,z,s)$, where $G_{3}(y,z,s)$ is a nonzero polynomials in $(y,z)$ for
any point $s\in S$, and the zero locus of $b(s)$ is contained in $H$. The
objective of this section is to prove the following proposition.

######  Proposition 4.1.

Let $X$ be a normal complex analytic variety. Let $S$ be a codimension 2
irreducible component of $X_{\operatorname{\textsubscript{\rm sing}}}$. Assume
that there is a proper closed subset $C\subseteq S$ such that $X$ has same
type of canonical singularities at points of $S\setminus C$. Then there is a
projective bimeromorphic morphism $f\colon Y\to X$ such that the following
properties hold.

  1. (1)

$f$ is the composition of morphisms induced by blowing up $S$ at ideal
sheaves.

  2. (2)

If $S_{Y}$ is the strict transform of $S$ in $Y$, then $(S_{Y},H_{Y})$ is a
reduced snc pair for a divisor $H_{Y}$.

  3. (3)

For every point $o\in S_{Y}$, there is a neighborhood $U\subseteq Y$ of $o$,
such that $U_{Y}$ is isomorphic to a hypersurface in $\mathbb{D^{3}}\times
S_{Y}$, defined by an equation of standard form with respect to
$(S_{Y},H_{Y})$.

  4. (4)

$Y$ has the same type of canonical singularities at every point of
$S_{Y}\setminus H_{Y}$.

  5. (5)

The construction of $f$ is functorial.

We remark that the items (1)-(4) still hold if we blow up $Y$ at a stratum of
$H_{Y}$. The proof of the proposition is divided into three steps. In the
first one, we reduce to the case when $X$ has hypersurface singularities. In
the second step, we reduce to the case when $X$ has double-point
singularities. And we arrange the defining equation into standard form in the
last step.

###  4.A. Hypersurface singularities

In this subsection, we consider the following situation. Let $X$ be a complex
analytic variety and let $S\subseteq X_{\operatorname{\textsubscript{\rm
sing}}}$ be an irreducible component. Assume that $S$ is smooth of dimension
$n$. Let $W=\mathbb{D}^{N}\times S$. Assume that $X$ is isomorphic to the
closed subspace in $W$ defined by holomorphic functions
$f_{j}(x_{1},...,x_{N},s)$, where $j=1,...,k$, $(x_{1},...,x_{N})$ are the
coordinates of $\mathbb{C}^{N}$ and $s\in S$. Assume that the subset
$S\subseteq X$ coincides with $\\{\mathbf{0}_{N}\\}\times S$. We let

$\mathrm{d}_{x}f_{j}=\sum_{i=1}^{N}\frac{\partial f_{j}}{\partial
x_{i}}\mathrm{d}x_{i},$

and let $r(s)$ be the rank of
$(\mathrm{d}_{x}f_{j}(\mathbf{0}_{N},s))_{j=1,...,k}$, in the vector space
generated by $\mathrm{d}x_{1},...,\mathrm{d}x_{N}$.

######  Lemma 4.2.

With the notation above, the embedded dimension of $X$ at a point $s\in
S\subseteq X$ is equal to $N+n-r(s)$.

###### Proof.

Since $S$ is smooth, by definition, the embedded dimension of $X$ is equal to
$N+n-R(s)$, where $R(s)$ is the rank of
$(\mathrm{d}f_{j}(\mathbf{0}_{N},s))_{j=1,...,k}$, in the vector space
$\Omega^{1}_{W,(\mathbf{0}_{N},s)}$. We note that $f_{j}((\mathbf{0}_{N},s)=0$
for all $s\in S$ and for all $j=1,...,k$. Hence the partial derivatives of
$f_{j}$, with respect to the variables in $S$, are all zero along
$\\{\mathbf{0}_{N}\\}\times S$, for all $j=1,...,k$. It follows that $R(s)$ is
equal to $r(s)$. This completes the proof of the lemma. ∎

######  Lemma 4.3.

With the notation above, let $r=\operatorname{min}\\{r(s)\ |\ s\in S\\}$. In
another word, $N+n-r$ is the maximum of the embedded dimensions of $X$ at the
points of $S$. Assume that $X$ has embedded dimension smaller than $N+n-r$ at
general points of $S$. Then there is a projective bimeromorphic morphism
$f\colon Y\to X$ obtained by blowing up $X$ at ideal sheaves whose cosupport
are strictly contained in $S$, such that the maximum of the embedded
dimensions of $Y$ at the points of $S_{Y}$ is at most $N+n-r-1$, where $S_{Y}$
is the strict transform of $Y$. Moreover, $S_{Y}$ is smooth and $f$ is
functorial.

###### Proof.

Let $o\in S$ be a point at which the embedded dimension of $X$ is $N+n-r$. We
will first work locally around $o$. In particular, we are allowed to shrink
$S$ around $o$. Then, by showing that the operation in the lemma is
functorial, we can glue the local construction in the global setting.

Up to a linear change of coordinates, we may assume that

$\mathrm{d}_{x}f_{j}(\mathbf{0}_{N},o)=\mathrm{d}x_{j}$

for $j=1,...,r.$ Then, by subtracting each $f_{j}$, with $j>r$, some linear
combination, with coefficients as holomorphic functions on $S$, of the
$f_{j}$’s with $j=1,...,r$, we may assume that

$\frac{\partial f_{j}}{\partial x_{i}}(\mathbf{0}_{N},s)=0$

for $i=1,...,r$, $j>r$ and all $s\in S$ contained in a neighborhood of $o$.
Let $\mathcal{I}\subseteq\mathcal{O}(S)$ be the ideal generated by
$\frac{\partial f_{j}}{\partial x_{i}}(\mathbf{0}_{N},s)$, where $i,j>r$.
Then, up to shrinking $S$, the closed subset contained in $S$, at which $X$
has embedded dimension $N+n-r$ is the cosupport of $\mathcal{I}$. By
assumption, this cosupport is a proper subset of $S$.

Let $h\colon Z\to X$ be the projective bimeromorphic morphism induced by
blowing up the ideal $\mathcal{I}$ on $S$, see Subsection 3.B. Let $S_{Z}$ be
the strict transform of $S$ in $Z$ and $o_{Z}\in Z$ a point lying over $o$.
Then there is a neighborhood $U$ of $o_{Z}$, which can be viewed as a closed
subset of $\mathbb{D}^{N}\times(U\cap S_{Z}).$ Moreover, $h$ can be written in
coordinates as

$h\colon(x_{1}^{\prime},...,x_{N}^{\prime},s^{\prime})\mapsto(tx_{1}^{\prime},...,tx_{N}^{\prime},h(s^{\prime})),$

where $t$ is a holomorphic function on $S_{Z}$ which defines the exceptional
divisor of $S_{Z}\to S$ on $U\cap S_{Z}$. We define

$f^{\prime}_{j}(x_{1}^{\prime},...,x_{N}^{\prime},s^{\prime})=\frac{1}{t}(h^{*}f_{j})(x_{1},...,x_{N},s)=\frac{1}{t}f_{j}(tx_{1}^{\prime},...,tx_{N}^{\prime},h(s^{\prime}))$

for $j=1,...,r$, and

$f^{\prime}_{j}(x_{1}^{\prime},...,x_{N}^{\prime},s^{\prime})=\frac{1}{t^{2}}(h^{*}f_{j})(x_{1},...,x_{N},s)=\frac{1}{t^{2}}f_{j}(tx_{1}^{\prime},...,tx_{N}^{\prime},h(s^{\prime}))$

for $j=r+1,...,k$. Then as shown in Subsection 3.B, all of the
$f^{\prime}_{j}$ are holomorphic functions and vanish on $Z$. We see that

$\mathrm{d}_{x^{\prime}}f^{\prime}_{j}(\mathbf{0}_{N},o_{Z})=\mathrm{d}x^{\prime}_{j}$

for $j=1,...,r.$ Furthermore, since the ideal $\mathcal{I}$ is generated by
$\frac{\partial f_{j}}{\partial x_{i}}(\mathbf{0}_{N},s)$, where $i,j>r$,
there are some $i,j>r$ such that

$\frac{\partial f^{\prime}_{j}}{\partial
x^{\prime}_{i}}(\mathbf{0}_{N},o_{Z})\neq 0.$

As a consequence, after the blowup, the rank of
$(\mathrm{d}_{x^{\prime}}f^{\prime}_{j}(\mathbf{0}_{N},o_{Z}))_{j=1,...,k}$ is
at least $r(o)+1$. In particular, if $S_{Z}$ is smooth at $o_{Z}$, then the
embedded dimension of $Z$ at $o_{Z}$ is at most $N+n-r-1$.

Let $f\colon Y\to X$ be the projective bimeromorphic morphism induced by the
functorial principalization of $(g|_{S_{Z}})^{*}\mathcal{I}$ on $S_{Z}$. In
particular, the strict transform $S_{Y}$ of $S$ in $Y$ is smooth. Since $Y\to
Z$ is the composition of blowups with centers strictly contained in $S_{Z}$,
the similar computation as above shows that $Y$ has embedded dimension at most
$N+n-r-1$ at a point $o_{Y}\in S_{Y}$ lying over $o_{Z}$.

It remains to show that the construction is functorial. This is equivalent to
show that the ideal $\mathcal{I}$ is independent of the choice of coordinates.
We first note that, in the construction above, the ideal $\mathcal{I}$ is also
the ideal generated by all determinants of $(r+1)$-minors of the matrix

$\Big{[}\frac{\partial f_{j}}{\partial x_{i}}\Big{]}_{1\leq i\leq N,1\leq
j\leq k}.$

Therefore, if $(g_{e})_{e=1,...,t}$ is another collection of functions
defining $X$, then they define the same ideal $\mathcal{I}$.

Assume that we have a change of coordinates

$\varphi\colon(x_{1},...,x_{N},s)\mapsto(x_{1}^{\prime},...,x_{M}^{\prime},s^{\prime}),$

where we regard $s^{\prime}$ and $x_{1}^{\prime},...,x_{M}^{\prime}$ as
holomorphic functions in $(x_{1},...,x_{N},s)$. We note that
$s^{\prime}(\mathbf{0}_{N},s)=s$ for any $s\in S$. We define the following
$M\times N$ matrix

$\Theta(s)=\Big{[}\frac{\partial}{\partial
x_{i}}x_{l}^{\prime}(\mathbf{0}_{N},s)\Big{]}_{1\leq i\leq N,1\leq l\leq M}.$

Let $(g_{e}(x_{1}^{\prime},...,x_{M}^{\prime},s^{\prime}))_{e=1,...,t}$ be a
collection of holomorphic functions defining $X$. As shown in Lemma 4.2, the
partial derivatives of $g_{e}$ with respect to the variables on $S$ are all 0
on $\\{\mathbf{0}_{M}\\}\times S$. Hence for any $s\in S$, we have

$\Big{[}\frac{\partial}{\partial
x_{i}}(g_{e}\circ\varphi)(\mathbf{0}_{N},s)\Big{]}_{1\leq i\leq N,1\leq e\leq
t}=\Theta(s)\cdot\Big{[}(\frac{\partial g_{e}}{\partial
x^{\prime}_{l}}(\mathbf{0}_{M},s^{\prime}(\mathbf{0}_{N},s))\Big{]}_{1\leq
l\leq M,1\leq e\leq t}$

By considering the determinants of all $(r+1)$-minors in the both sides above,
we deduce that the ideal $\mathcal{I}$ constructed via
$(g_{e}\circ\varphi)_{e=1,...,t}$ on the coordinates system
$(x_{1},...,x_{N},s)$ is contained in the one constructed via
$(g_{e})_{e=1,...,t}$ on the coordinates system
$(x_{1}^{\prime},...,x_{M}^{\prime},s^{\prime})$. Hence, the previous
paragraph shows that the ideal constructed via $(f_{j})_{j=1,...,k}$ on the
coordinates system $(x_{1},...,x_{N},s)$ is contained in the one constructed
via $(g_{e})_{e=1,...,t}$ on the coordinates system
$(x_{1}^{\prime},...,x_{M}^{\prime},s^{\prime})$. Since the roles of the two
coordinates systems are symmetric, we conclude that the ideal $\mathcal{I}$ we
have constructed is independent on the choice of coordinates. This completes
the proof of the lemma. ∎

######  Example 4.4.

Assume that $S=\mathbb{C}^{2}$ with coordinates $(a,b)$, and that
$X\subseteq\mathbb{C}^{4}\times S$ is defined by the ideal generated by the
following functions,

$\displaystyle f_{1}(x_{1},x_{2},x_{3},x_{4},a,b)$ $\displaystyle=$
$\displaystyle x_{3}+x_{3}^{2}$ $\displaystyle
f_{2}(x_{1},x_{2},x_{3},x_{4},a,b)$ $\displaystyle=$ $\displaystyle
x_{3}+ax_{1}+bx_{1}x_{2}$ $\displaystyle f_{3}(x_{1},x_{2},x_{3},x_{4},a,b)$
$\displaystyle=$ $\displaystyle x_{3}+bx_{1}+x_{2}^{2}+x_{4}^{3}.$

Then when $a\neq 0$ or $b\neq 0$, and $x_{1}=x_{2}=x_{3}=x_{4}=0$, the rank of
the Jacobi matrix of $(f_{1},f_{2},f_{3})$ is 2. At $\mathbf{0}_{6}$, the rank
of the Jacobi matrix is 1. Then $X$ has embedded dimension 5 at this point.
Now we blow up $\mathbb{C}^{6}$ at the ideal $(x_{1},x_{2},x_{3},x_{4},a,b)$
and obtain $h\colon W\to\mathbb{C}^{6}$. Then on the open set
$U=h^{-1}(\\{a\neq 0\\})$, $h$ can be written in coordinate as

$h\colon(x_{1}^{\prime},x_{2}^{\prime},x_{3}^{\prime},x^{\prime}_{4},\alpha,\beta)\mapsto(\alpha
x_{1}^{\prime},\alpha x_{2}^{\prime},\alpha x_{3}^{\prime},\alpha
x^{\prime}_{4},\alpha,\alpha\beta).$

Then we have

$\displaystyle
h^{*}f_{1}(x^{\prime}_{1},x^{\prime}_{2},x^{\prime}_{3},x^{\prime}_{4},\alpha,\beta)$
$\displaystyle=$
$\displaystyle\alpha(x^{\prime}_{3}+\alpha(x^{\prime}_{3})^{2})$
$\displaystyle
h^{*}f_{2}(x^{\prime}_{1},x^{\prime}_{2},x^{\prime}_{3},x^{\prime}_{4},\alpha,\beta)$
$\displaystyle=$ $\displaystyle\alpha(x^{\prime}_{3}+\alpha
x^{\prime}_{1}+\alpha^{2}\beta x^{\prime}_{1}x^{\prime}_{2})$ $\displaystyle
h^{*}f_{3}(x^{\prime}_{1},x^{\prime}_{2},x^{\prime}_{3},x^{\prime}_{4},\alpha,\beta)$
$\displaystyle=$ $\displaystyle\alpha(x^{\prime}_{3}+\alpha\beta
x^{\prime}_{1}+\alpha(x^{\prime}_{2})^{2}+\alpha^{2}(x^{\prime}_{4})^{3}).$

We set
$g_{j}(x^{\prime}_{1},x^{\prime}_{2},x^{\prime}_{3},x^{\prime}_{4},\alpha,\beta)=\frac{1}{\alpha}h^{*}f_{j}(x^{\prime}_{1},x^{\prime}_{2},x^{\prime}_{3},x^{\prime}_{4},\alpha,\beta)$
for $j=1,2,3$. We notice that $g_{2}-g_{1}$ and $g_{3}-g_{1}$ are divisible by
$\alpha$. We set

$p_{2}=\frac{1}{\alpha}(g_{2}-g_{1})\mbox{ and
}p_{3}=\frac{1}{\alpha}(g_{3}-g_{1}).$

Then the strict transform $X^{\prime}$ of $X$ in $W$ is defined, on $U$, by
the ideal generated by the following functions

$\displaystyle
g_{1}(x^{\prime}_{1},x^{\prime}_{2},x^{\prime}_{3},x^{\prime}_{4},\alpha,\beta)$
$\displaystyle=$ $\displaystyle x^{\prime}_{3}+\alpha(x^{\prime}_{3})^{2}$
$\displaystyle
p_{2}(x^{\prime}_{1},x^{\prime}_{2},x^{\prime}_{3},x^{\prime}_{4},\alpha,\beta)$
$\displaystyle=$ $\displaystyle x^{\prime}_{1}+\alpha\beta
x^{\prime}_{1}x^{\prime}_{2}-(x^{\prime}_{3})^{2}$ $\displaystyle
p_{3}(x^{\prime}_{1},x^{\prime}_{2},x^{\prime}_{3},x^{\prime}_{4},\alpha,\beta)$
$\displaystyle=$ $\displaystyle\beta
x^{\prime}_{1}+(x^{\prime}_{2})^{2}+\alpha(x^{\prime}_{4})^{3}-(x^{\prime}_{3})^{2}.$

Then the rank of the Jacobi matrix of $(g_{1},p_{1},p_{2})$ is at least 2 on
$U$. We can perform the same calculation on $U^{\prime}=h^{-1}(\\{b\neq
0\\})$. In addition, the strict transform $S^{\prime}$ of $S$ in $W$ is
contained in $U\cup U^{\prime}$. This shows that $X^{\prime}$ has embedded
dimension 4 around $S^{\prime}$.

######  Corollary 4.5.

With the notation above, we assume that $X$ has hypersurface singularities
around general points of $S$. Then there is a projective bimeromorphic
morphism $f\colon X^{\prime}\to X$ such that the following properties hold.

  1. (1)

$f$ is obtained by a sequence of blowup at center strictly contained in $S$.

  2. (2)

The strict transform $S^{\prime}$ of $S$ in $X^{\prime}$ is smooth.

  3. (3)

$X^{\prime}$ has hypersurface singularities along $S^{\prime}$.

  4. (4)

The construction of $f$ is functorial.

###### Proof.

Assume that the maximum of embedded dimensions of $X$ at points of $S$ is
$n+2+r$. If $r=1$, then $X$ has hypersurface singularities around $S$.
Otherwise, there is a sequence of projective bimeromorphic morphisms

$X_{e}\to\cdots\to X_{1}=X,$

such that each $X_{i}\to X_{i+1}$ is the functorial operation in Lemma 4.3,
and that the the maximum of embedded dimensions of $X_{e}$ at points of
$S_{e}$ is $d+2+r$, where $S_{e}$ is the strict transform of $S$ in $X_{e}$.
We let $X^{\prime}=X_{s}$ and let $f\colon X^{\prime}\to X$ be the composition
of the sequence. Then it satisfies all the properties of the corollary. ∎

###  4.B. Double-point singularities

We will prove the following statement in this subsection.

######  Lemma 4.6.

Let $X$ be a complex analytic variety of dimension $n$ and $S\subseteq
X_{\operatorname{\textsubscript{\rm sing}}}$ an irreducible component of
codimension $2$. Assume that $S$ is smooth, that $X$ has hypersurface
singularities around $S$, and that $X$ has double-point singularities around
general points of $S$. Then there is a projective bimeromorphic morphism
$f\colon X^{\prime}\to X$ such that the following properties hold.

  1. (1)

$f$ is obtained by a sequence of blowups at centers strictly contained in $S$.

  2. (2)

The strict transform $S^{\prime}$ of $S$ in $X^{\prime}$ is smooth.

  3. (3)

$X^{\prime}$ has double-point singularities around $S^{\prime}$.

  4. (4)

$f$ is functorial.

###### Proof.

It is enough to study the problem locally. Indeed, once we prove the
functoriality, we can glue our local construction in the global setting. Let
$o\in S\subseteq X$ be a point. We may assume that $S$ is a polydisc in
$\mathbb{C}^{n}$ and $o\in S$ is the origin. Then locally around $o$, we may
assume that $X$ is the hypersurface in $\mathbb{D}^{3}\times S$, defined by
the analytic function

$F(x,y,z,s)=\sum_{i\geq 0}F_{i}(x,y,z,s),$

where $(x,y,z)$ are the coordinates of $\mathbb{D}^{3}$, $s\in S$, and $F_{i}$
is a homogeneous polynomial in $(x,y,z)$ of degree $i$ with coefficients as
holomorphic functions on $S$. By assumption, we have $F_{0}=F_{1}=0$ and
$F_{2}\neq 0$.

Let $\mathcal{I}\subseteq\mathcal{O}(S)$ be the ideal generated by the
coefficients of $F_{2}$. By construction, if $D\subseteq S$ is the closed
subset defined by $\mathcal{I}$, then at every point in $S\setminus D$, $X$
has double-point singularities. By assumption, $D$ is a proper closed subset
of $S$.

Let $X^{\prime\prime}\to X$ be the projective bimeromorphic morphism induced
by blowing up $S$ at the ideal $\mathcal{I}$. As shown in Subsection 3.B, at
every point $o^{\prime\prime}$ in the strict transform $S^{\prime\prime}$ of
$S$ in $X^{\prime\prime}$, there is an open neighborhood
$U^{\prime\prime}\subseteq X^{\prime\prime}$ of $o^{\prime\prime}$ such that
$U^{\prime\prime}$ is isomorphic to the hypersurface in
$\mathbb{D}^{3}\times(S^{\prime\prime}\cap U^{\prime\prime})$ defined by the
holomorphic function

$G(x^{\prime\prime},y^{\prime\prime},z^{\prime\prime},s^{\prime\prime})=G_{2}(x^{\prime\prime},y^{\prime\prime},z^{\prime\prime},s^{\prime\prime})+R(x^{\prime\prime},y^{\prime\prime},z^{\prime\prime},s^{\prime\prime})$

where $G_{2}$ is a homogeneous polynomial in
$(x^{\prime\prime},y^{\prime\prime},z^{\prime\prime})$ with coefficients as
holomorphic functions on $S^{\prime\prime}$, and the terms of $R$ have degree
at least $3$ in $(x^{\prime\prime},y^{\prime\prime},z^{\prime\prime})$.
Furthermore,
$G_{2}(x^{\prime\prime},y^{\prime\prime},z^{\prime\prime},o^{\prime\prime})$
is a nonzero polynomial in
$(x^{\prime\prime},y^{\prime\prime},z^{\prime\prime})$. In particular, if
$S^{\prime\prime}$ is smooth at $o^{\prime\prime}$, then $X^{\prime\prime}$
has double-point singularities at $o^{\prime\prime}$.

Let $X^{\prime}\to X^{\prime\prime}$ be the projective bimeromorphic morphism
induced by the functorial principalization of the exceptional locus of
$S^{\prime\prime}\to S$ on $S^{\prime\prime}$. In particular, the strict
transform $S^{\prime}$ of $S$ in $X^{\prime}$ is smooth. Since $X^{\prime}\to
X^{\prime\prime}$ is the composition of blowups with center strictly contained
in $S^{\prime\prime}$, the computations of Subsection 3.B implies that
$X^{\prime}$ has double-point singularities along $S^{\prime}$.

To prove that construction is functorial, it is enough to show that the ideal
$\mathcal{I}\subseteq\mathcal{O}(S)$ is independent of the choice of
coordinates. Let $(\overline{x},\overline{y},\overline{z},\overline{s})$ be
some other choice of coordinates of $\mathbb{D}^{3}\times S$ such that $X$ is
defined by
$\overline{F}(\overline{x},\overline{y},\overline{z},\overline{s})=0$. Let
$\varphi\colon(x,y,z,s)\mapsto(\overline{x},\overline{y},\overline{z},\overline{s})$
be the change of coordinates, and we regard
$\overline{x},\overline{y},\overline{z},\overline{s}$ as vector-valued
holomorphic functions in $(x,y,z,s)$. Then we have

$\overline{s}(0,0,0,s)=s$

for all $s\in S$. Hence $\overline{s}(x,y,z,s)=s\operatorname{mod}\,(x,y,z)$.
There is a $3\times 3$ matrix $\Theta(s)$, whose entries are holomorphic
functions on $S$, such that

$(\overline{x},\overline{y},\overline{z})=(x,y,z)\cdot\Theta(s)\operatorname{mod}\,(x,y,z)^{2}.$

We decompose $\overline{F}=\sum_{i\geq 2}\overline{F}_{i}$ into the sum of its
homogeneous parts in $(\overline{x},\overline{y},\overline{z})$. Then we have

$\displaystyle(\varphi^{*}\overline{F})(x,y,z,s)$ $\displaystyle=$
$\displaystyle\overline{F}(\overline{x},\overline{y},\overline{z},\overline{s})$
$\displaystyle=$
$\displaystyle\overline{F}_{2}((x,y,z)\cdot\Theta(s),s)\operatorname{mod}\,(x,y,z)^{3}$

Since both the equations $\varphi^{*}\overline{F}=0$ and $F=0$ define the
hypersurface $X$, there is a unit function $U(x,y,z,s)$ such that

$F=U\cdot\varphi^{*}\overline{F}.$

We deduce that the ideal $\mathcal{I}$ generated by the coefficients of
$F_{2}(x,y,z,s)$ is the same as the one generated by the coefficients of
$\overline{F}_{2}((x,y,z)\cdot\Theta(s),s)$. This implies that $\mathcal{I}$
is contained in the ideal generated by the coefficients of
$\overline{F}_{2}(\overline{x},\overline{y},\overline{z},\overline{s})$. Since
the roles of the coordinates systems are symmetric, we deduce that the ideal
$\mathcal{I}$ we have constructed is independent of the choice of coordinates.
This completes the proof of the lemma. ∎

###  4.C. Equations of standard form

We will complete the proof of Proposition 4.1 in this subsection.

######  Lemma 4.7.

Let $S$ be a polydisc in $\mathbb{C}^{n}$. Let $X\subseteq\mathbb{D}^{3}\times
S$ be the hypersurface defined by the equation

$F(x,y,z,s)=x^{2}+F_{2}(y,z,s)+F_{3}(y,z,s)+R(y,z,s)=0,$

where $(x,y,z)$ are the coordinates of $\mathbb{D}^{3}$ and $s\in S$.
Moreover, for $i=2,3$, $F_{i}(y,z,s)$ is a homogeneous polynomial of degree
$i$ in $(y,z)$, with coefficients as holomorphic functions on $S$. The
function $R(y,z,s)$ is holomorphic and its terms have degree at least 4 in
$(y,z)$. We assume that either $F_{2}$ or $F_{3}$ is not identically zero, and
define an ideal $\mathcal{I}$ of $\mathcal{O}(S)$ as follows. If $F_{2}\neq
0$, then $\mathcal{I}$ is the ideal generated by the coefficients of $F_{2}$.
Otherwise $\mathcal{I}$ is the ideal generated by the coefficients of $F_{3}$.
Then the property that $F_{2}\neq 0$, and the ideal $\mathcal{I}$, depend only
on the isomorphic classes of $X$, and is independent of the choice of the
coordinates $(x,y,z,s)$.

###### Proof.

Assume that we have another choice of coordinates $(a,b,c,s^{\prime})$ such
that $X$ is defined as

$G(a,b,c,s^{\prime})=a^{2}+G_{2}(b,c,s^{\prime})+G_{3}(b,c,s^{\prime})+P(b,c,s^{\prime})=0.$

We denote the change of coordinates as

$\varphi\colon(x,y,z,s)\mapsto(a,b,c,s^{\prime}),$

and we can regards $a,b,c,s^{\prime}$ as vector-valued holomorphic functions
on $(x,y,z,s)$. Then $s^{\prime}(0,0,0,s)=s$ for all $s\in S$. Thus
$s^{\prime}(x,y,z,s)=s\operatorname{mod}\,(x,y,z)$. If we set
$(\varphi^{*}G)(x,y,z,s)=G(a,b,c,s^{\prime})$, then

$(\varphi^{*}G)(x,y,z,s)=(\mathrm{unit})\cdot F(x,y,z,s).$ (4.1)

First we assume that $G_{2}=0$. Then, by comparing the coefficient before the
term $x^{2}$ in (4.1), we can write

$a(x,y,z,s)=l(x,y,z,s)\operatorname{mod}\,(y,z)^{2},$

where $l=(\mathrm{unit})\cdot x+l_{1}(y,z,s)$, and $l_{1}(y,z,s)$ is linear in
$(y,z)$. Then we have

$(\varphi^{*}G)(x,y,z,s)=l(x,y,z,s)^{2}\operatorname{mod}\,(x,y,z)^{3}.$

In particular, up to a unit function, the term $x\cdot l_{1}(y,z,s)$ belongs
to

$(\varphi^{*}G)(x,y,z,s)\operatorname{mod}\,(x,y,z)^{3}.$

Comparing with (4.1), we deduce that $l_{1}(y,z,s)=0$. Hence
$l(x,y,z,s)=(\mathrm{unit})\cdot x$ and $F_{2}=0$.

We note that

$(\mathrm{unit})\cdot F(x,y,z,s)=u(s)\cdot
F_{3}(y,z,s)\operatorname{mod}\,(x,(y,z)^{4}),$ (4.2)

where $u(s)$ is a unit. We write
$b(x,y,z,s)=b_{1}(y,z,s)+b_{1}^{\prime}(s)x+b_{>1}(x,y,z,s)$, where $b_{1}$ is
linear in $(y,z)$ and $b_{>1}=0\operatorname{mod}\,(x,y,z)^{2}$. Similarly, we
write $c(x,y,z,s)=c_{1}(y,z,s)+c_{1}^{\prime}(s)x+c_{>1}(x,y,z,s)$. Then

$\begin{split}(\varphi^{*}G)(x,y,z,s)&=G(a,b,c,s^{\prime})\\\
&=a^{2}+G_{3}(b,c,s^{\prime})+P(b,c,s^{\prime})\\\
&=G_{3}(b_{1}(y,z,s),c_{1}(y,z,s),s)\operatorname{mod}\,(x,(y,z)^{4}).\end{split}$
(4.3)

Here, for the notation $G_{3}(b_{1}(y,z,s),c_{1}(y,z,s),s)$, if

$G_{3}(b,c,s^{\prime})=\sum_{i=1}^{3}\alpha_{i}(s^{\prime})b^{i}c^{3-i},$

then

$G_{3}(b_{1}(y,z,s),c_{1}(y,z,s),s)=\sum_{i=1}^{3}\alpha_{i}(s)b_{1}(y,z,s)^{i}c(y,z,s)^{3-i}.$

This is well-defined since the functions $\alpha_{i}$ are functions on $S$. By
comparing (4.3) with (4.1) and (4.2), we deduce that

$G_{3}(b_{1}(y,z,s),c_{1}(y,z,s),s)=u(s)\cdot F_{3}(y,z,s).$

As a consequence, the ideal generated by the coefficients of $F_{3}(y,z,s)$,
regarded as a homogeneous polynomials in $(y,z)$, is contained in the ideal
generated by the coefficients of $G_{3}(b,c,s^{\prime})$. Hence, by symmetry,
the ideal $\mathcal{I}$ we have constructed is independent of the choice of
coordinates in this case.

Next we assume that $G_{2}\neq 0$. In this case, we have

$(\mathrm{unit})\cdot
F(x,y,z,s)=u(s)\cdot(x^{2}+F_{2}(y,z,s))\operatorname{mod}\,(x,y,z)^{3},$
(4.4)

where $u(s)$ is a unit. There is a $3\times 3$ matrix $\Theta(s)$, with
coefficients as holomorphic functions in $S$ such that

$(a,b,c)=(x,y,z)\cdot\Theta(s)\operatorname{mod}\,(x,y,z)^{2}.$

Let $(a_{1},b_{1},c_{1})=(x,y,z)\cdot\Theta(s)$. Then we have

$(\varphi^{*}G)(x,y,z,s)=a_{1}^{2}+G_{2}(b_{1},c_{1},s)\operatorname{mod}\,(x,y,z)^{3}.$

Comparing the previous equation with (4.1) and $\eqref{eqn:change-
cooridante-2}$, we deduce that

$a_{1}^{2}+G_{2}(b_{1},c_{1},s)=u(s)(x^{2}+F_{2}(y,z,s)).$

Both sides of the equation above can be viewed as quadratic forms in three
variables, with coefficients as holomorphic functions on $S$. We denote the
LHS by $L(a_{1},b_{1},c_{1})$ and the RHS by $Q(x,y,z)$. The ideal
$\mathcal{J}$ (respectively of $\mathcal{I}$) generated by the coefficients of
$G_{2}$ (respectively of $F_{2}$) is exactly the idea generated by the
determinants of all 2-dimensional minors of the symmetric matrix of $L$
(respectively of $Q$). Since $L((x,y,z)\cdot\Theta(s))=Q(x,y,z)$, we have
$\mathcal{I}\subseteq\mathcal{J}$. By symmetry, we conclude that
$\mathcal{I}=\mathcal{J}$. This completes the proof of the lemma. ∎

Now we are ready to prove Proposition 4.1

###### Proof.

Let $X^{\prime}\to X$ be the projective bimeromorphic morphism induced by the
functorial desingularization of $S$, as shown at the beginning of Subsection
3.B. Then, replacing $X$ with $X^{\prime}$, we may assume that $S$ is smooth.
By applying Corollary 4.5 and then Lemma 4.6, we may assume that $X$ has
double-point singularities along $S$.

It particular, locally around a point $o\in S$, $X$ is the hypersurface in
$\mathbb{D}^{3}\times S$, defined by the equation

$F(x,y,z,s)=x^{2}+F_{2}(y,z,s)+F_{3}(y,z,s)+R(y,z,s)=0,$

where $(x,y,z)$ are the coordinates of $\mathbb{D}^{3}$ and $s\in S$.
Moreover, for $i=2,3$, $F_{i}(y,z,s)$ is a homogeneous polynomial of degree
$i$ in $(y,z)$, with coefficients as holomorphic functions on $S$. The
function $R(y,z,s)$ is holomorphic and its terms have degrees at least 4 in
$(y,z)$.

Since $X$ has canonical singularities around general points of $S$, by Lemma
2.2 we see that either $F_{2}$ or $F_{3}$ is not identically zero. Hence we
can define the ideal $\mathcal{I}$ as in Lemma 4.7.

We only treat the case when $F_{2}\neq 0$. The other case can be proved by the
same method. Let $p\colon X^{\prime}\to X$ be the projective bimeromorphic
morphism induced by the blowup of $S$ at $\mathcal{I}$. Let $S^{\prime}$ be
the strict transform of $S$ in $X^{\prime}$, and let $E$ be the exceptional
divisor of $p|_{S^{\prime}}\colon S^{\prime}\to S$ defined by
$(p|_{S^{\prime}})^{*}\mathcal{I}$. Then, as shown in Subsection 3.B, for any
point $o^{\prime}$ in $S^{\prime}$, there is a neighborhood
$U^{\prime}\subseteq X^{\prime}$ of $o^{\prime}$, such that $U^{\prime}$ is
the hypersurface in $\mathbb{D}^{3}\times(S^{\prime}\cap U^{\prime})$, defined
by the function

$G(x^{\prime},y^{\prime},z^{\prime},s^{\prime})=x^{\prime
2}+a(s^{\prime})\cdot
G_{2}(y^{\prime},z^{\prime},s^{\prime})+W(y^{\prime},z^{\prime},s^{\prime}),$

where $a(s^{\prime})$ is a generator of $\mathcal{O}_{S^{\prime}}(-E)$ on
$S^{\prime}\cap U^{\prime}$, $G_{2}$ is nonzero at $s^{\prime}=o^{\prime}$,
and
$W(y^{\prime},z^{\prime},s^{\prime})=0\operatorname{mod}(y^{\prime},z^{\prime})^{3}$.

We denote by $\mathcal{I}^{\prime}$ the ideal of the closed subset
$E\cup(p|_{S^{\prime}})^{-1}(C)$ in $S^{\prime}$. Let $Y\to X^{\prime}$ be the
projective bimeromorphic morphism induced by the functorial principalization
of $\mathcal{I}^{\prime}$ on $S^{\prime}$. We denote by $S_{Y}$ the strict
transform of $S^{\prime}$ in $Y$, and by $H_{Y}$ the sum of the whole
exceptional divisor of $S_{Y}\to S$ and the preimage of $C$. Then
$(S_{Y},H_{Y})$ is a snc pair, and for any point $o\in S_{Y}$, there is a
neighborhood $U_{Y}\subseteq Y$ of $o$, such that $U_{Y}$ is a hypersurface in
$\mathbb{D}^{3}\times S_{Y}$ defined by an equation of standard form with
respect to $(S_{Y},H_{Y})$. Furthermore, $Y$ has the same type of canonical
singularities at every point of $S_{Y}\setminus H_{Y}$. This completes the
proof of the proposition. ∎

###  4.D. Resolution of coherent sheaves

With the same method of Subsection 4.A, we can prove the following theorem,
which is well-known to specialists.

######  Theorem 4.8.

Let $X$ be a complex analytic variety, and let $\mathcal{E}$ be a torsion-free
coherent sheaf on $X$. Then there is a functorial (or canonical) projective
bimeromorphic morphism $r\colon\widetilde{X}\to X$ satisfying the following
properties.

  1. (1)

$r^{*}\mathcal{E}/(\mathrm{torsion})$ is locally free.

  2. (2)

$\widetilde{X}$ is a smooth complex analytic variety.

  3. (3)

$r$ is obtained by a sequence of blowups at centers contained in the union of
$X_{\operatorname{\textsubscript{\rm sing}}}$ and the non-locally-free locus
of $\mathcal{E}$.

  4. (4)

The construction $r$ commute with local analytic isomorphisms.

###### Proof.

The proof is very similar to the ones of Lemma 4.3 and Corollary 4.5. We will
only give a sketch and leave the details to the reader. Since we will show the
functoriality of the construction, we only need to prove the theorem locally.
Hence, we can assume that there is an exact sequence of coherent sheaf

$\mathcal{O}_{X}^{\oplus
p}\overset{\varphi}{\longrightarrow}\mathcal{O}_{X}^{\oplus
q}\longrightarrow\mathcal{E}\to 0.$

Then the morphism $\varphi$ is represented by a $q\times p$ matrix $\Theta(x)$
with coefficient as holomorphic functions on $X$. Since $\mathcal{E}$ is
torsion-free, if $M$ is the maximum of the ranks of $\Theta(x)$ for $x\in X$,
then $q-M$ is the rank of $\mathcal{E}$. Let $m$ be the minimum of the ranks
of $\Theta(x)$. Then $\mathcal{E}$ is locally free if and only if $M=m$.

Assume that $m<M$, then we let $\mathcal{I}\subseteq\mathcal{O}_{X}(X)$ be
ideal generated by the determinants of all $(m+1)$-minors of the $\Theta(x)$.
By a similar argument as in the proof of Lemma 4.3, the ideal $\mathcal{I}$ is
independent on the choice of the exact sequence. Let $f_{1}\colon X_{1}\to X$
be the blowup at the ideal $\mathcal{I}$. We can pullback $\varphi$ by
$f_{1}$, and obtain an exact sequence of coherent sheaf

$\mathcal{O}_{X_{1}}^{\oplus
p}\overset{f_{1}^{*}\varphi}{\longrightarrow}\mathcal{O}_{X_{1}}^{\oplus
q}\longrightarrow f_{1}^{*}\mathcal{E}\to 0.$

If $\varphi_{1}$ is the saturation of $f_{1}^{*}\varphi$, then the cokernel of
$\varphi_{1}$ is equal to $f_{1}^{*}\mathcal{E}/(\mathrm{torsion})$. Let
$x_{1}\in X_{1}$ be a point lying over $x$. Then there is an open neighborhood
$U_{1}$ of $x_{1}$ in $X_{1}$, such that $\varphi_{1}|_{U_{1}}$ is represented
by a $q\times p$ matrix $\Theta_{1}(x)$. By a similar computation as in the
proof of Lemma 4.3, we deduce that the rank of $\Theta_{1}(x_{1})$ is at least
$r+1$.

By repeating this procedure for finitely many times, and then we take a
functorial resolution of singularities in the end, we obtain the required
morphism $r\colon\widetilde{X}\to X$. This completes the proof of the theorem.
∎

##  5\. Further improvement of the defining equations

Let $X$ be a complex analytic variety and let $S\subseteq
X_{\operatorname{\textsubscript{\rm sing}}}$ be an irreducible component of
codimension 2 in $X$. In the previous section, we reduce the local situation
to the case when $X$ is isomorphic to a hypersurface of $\mathbb{D}^{3}\times
S$, and the hypersurface is defined by an equation of standard form. In this
section, we will perform some bimeromorphic transforms on $X$, so that we can
improve further the shape of the equation. After we prove some elementary
results in Subsection 5.A, we will work under the following setup.

######  Setup 5.1.

Let $X$ be a complex analytic variety of dimension $M+2$, and let $S\subseteq
X_{\operatorname{\textsubscript{\rm sing}}}$ be an irreducible component of
dimension $M$. Let $o\in S$ and we assume that $o\in S$ is an open
neighborhood of the origin $o=\mathbf{0}_{M}$ in $\mathbb{C}^{M}$. We denote
by $(T_{1},...,T_{n},...,T_{M})$ the coordinates of $S$ and we define
$H_{i}=\\{T_{i}=0\\}$. Let $H=H_{1}+\cdots+H_{n}$. Assume that $X$ is a
neighborhood of $\\{\mathbf{0}_{3}\\}\times S$, in a hypersurface in
$\mathbb{D}^{3}\times S$, defined by an equation of standard form with respect
to $(S,H)$, see the beginning Section 4. Furthermore, the subset $S\subseteq
X$ is identified with $\\{\mathbf{0}_{3}\\}\times S$. Assume that $X$ has the
same type of canonical singularities at points of $S\setminus H$. We will
shrink $X$ and $S$ freely around $o\in S$. In particular, for every unit
holomorphic function on $\mathbb{D}^{3}\times S$, we may assume that it admits
logarithms.

The following proposition is the main objective of in this section.

######  Proposition 5.2.

With the notation of Setup 5.1, up to shrinking $S$, there is an integer $N>0$
such that the following properties hold.

  1. (1)

Assume that $X$ has $A$-type or $E$-type singularities at each point of
$S\setminus H$. Let $f\colon Y\to X$ be the composition of a sequence of
blowups, such that each center is a component $H_{i}$ of $H$, see Subsection
3.C. In particular, we can identify $(o\in S)\subseteq X$ with its strict
transform in $Y$. Assume that for each $i=1,...,n$, there are at least $N$
blowups in the sequence whose centers are $H_{i}$. Then there is a
neighborhood $U\subseteq Y$ of $o\in S\subseteq Y$, such that $U$ is
isomorphic to the hypersurface defined in $\mathbb{D}^{3}\times S$ by an
equation of the shape

$\displaystyle F(x,y,z,s)$ $\displaystyle=$ $\displaystyle
x^{2}+\alpha(s)y^{2}+\beta(s)z^{l},\ l\geq 2,$ $\displaystyle F(x,y,z,s)$
$\displaystyle=$ $\displaystyle x^{2}+\alpha(s)y^{3}+\beta(s)\cdot
y(\gamma(s)z+u(y,s)y)^{3},$ $\displaystyle F(x,y,z,s)$ $\displaystyle=$
$\displaystyle x^{2}+\alpha(s)y^{3}+\beta(s)\cdot(\gamma(s)z+u(y,s)y)^{l},\
l=4,5,$

  2. (2)

Assume that $X$ has $D$-type singularities at each point of $S\setminus H$.
Let $f_{1}\colon X_{1}\to X$ be the composition of blowups at the $H_{i}$’s as
in Subsection 3.C, such that for each $i=1,...,n$, there is exactly one blowup
whose center is $H_{i}$. Let $f_{2}\colon X_{2}\to X_{1}$ be the basechange
over $S[\sqrt[6]{T_{1}},...,\sqrt[6]{T_{n}}]$. We define
$\overline{S}=f_{2}^{-1}((f_{1})^{-1}_{*}S)$ and define $\overline{H}_{i}$ as
the preimage of $H_{i}$ in $\overline{S}$. Let $f_{3}\colon Y\to X_{2}$ be the
composition of a sequence of blowups at the $\overline{H}_{i}$’s as in
Subsection 3.C. Assume that for each $i=1,...,n$, there are at least $N$
blowups in the sequence whose centers are $\overline{H}_{i}$. Let
$\overline{S}_{Y}$ be the strict transform of $\overline{S}$ in $Y$, and let
$\overline{o}\in\overline{S}_{Y}$ be the preimage of $o\in S$. Then there is a
neighborhood $U\subseteq Y$ of $\overline{o}$, such that $U$ is isomorphic to
the hypersurface defined in $\mathbb{D}^{3}\times\overline{S}_{Y}$ by an
equation of the shape

$\displaystyle F(x,y,z,s)$ $\displaystyle=$ $\displaystyle
x^{2}+\alpha(s)y(\gamma(s)z+u(y,s)y)^{2}+\beta(s)y^{l},\ l\geq 3.$

Here $l$ is an integer, $\alpha(s)$, $\beta(s)$ and $\gamma(s)$ are
holomorphic functions on $S$ (or on $\overline{S}$) whose zero loci are
contained in $H$ (or in $\sum\overline{H}_{i}$), and $u(y,s)$ is a holomorphic
function.

In addition, regarded as holomorphic functions on $U$, the zero locus of
$T_{1}\cdots T_{n}$ is equal to the intersection of $U$ and the exceptional
locus of $\Delta$ of natural morphism $f\colon Y\to X$.

We will prove the proposition in the remainder of the section, by discussing
according to the singularity type of $X$ at the points of $S\setminus H$.

###### Proof of Proposition 5.2.

We can deduce the proposition by combining Lemma 5.5, Lemma 5.6, Lemma 5.7 and
Lemma 5.8 and Lemma 5.9. ∎

###  5.A. Factorization lemmas

We prove two statements on holomorphic functions in several variables, which
enable us to factorize certain functions.

######  Lemma 5.3.

Let $F(y,z,s)=y(y^{2}+a(y,z,s)yz+b(y,z,s)z^{2})+b(y,z,s)^{2}R(y,z,s)$ be a
holomorphic function defined in a neighborhood of
$\mathbf{0}_{N+2}\in\mathbb{C}^{2+N}$, where $N\geq 0$ is an integer, $(y,z)$
is the first two coordinates of $\mathbb{C}^{2+N}$ and $s$ is a vector of
dimension $N$, which represents the last $N$ coordinates of
$\mathbb{C}^{2+N}$. Assume that the terms of $R$ have degrees at least $4$ in
$(y,z)$. Then, in a neighborhood of $\mathbf{0}_{N+2}$, there is a
factorization $F=(\mathrm{unit})\cdot G_{1}G_{2}$ with
$G_{1}=y+b(y,z,s)c(y,z,s)$ and
$G_{2}=y^{2}+a(y,z,s)yz+b(y,z,s)z^{2}+b(y,z,s)\cdot d(y,z,s)$ such that
$c(y,z,s)$ is divisible by $z^{2}$ and that
$d(y,z,s)=0\operatorname{mod}\,(y,z)^{3}$.

###### Proof.

We can write

$R(y,z,s)=y^{3}R_{0}(y,z,s)+y^{2}z^{2}\cdot R_{1}(z,s)+yz^{3}\cdot
R_{2}(z,s)+z^{4}\cdot R_{3}(y,z,s),$

where $R_{0}(y,z,s)=0\operatorname{mod}\,(y,z)$. Then we have

$\displaystyle F(y,z,s)$ $\displaystyle=$ $\displaystyle
y\Big{(}\big{(}1+b(y,z,s)^{2}R_{0}(y,z,s)\big{)}y^{2}$
$\displaystyle+\big{(}a(y,z,s)+b(y,z,s)^{2}z\cdot R_{1}(z,s)\big{)}yz$
$\displaystyle+b(y,z,s)\big{(}1+z\cdot b(y,z,s)\cdot
R_{2}(z,s)\big{)}z^{2}\Big{)}$ $\displaystyle+b(y,z,s)^{2}z^{4}\cdot
R_{3}(y,z,s).$

We note that $1+b(y,z,s)^{2}R_{0}(y,z,s)$ is a unit in a neighborhood of
$\mathbf{0}_{N+2}$. Hence, up to replacing $F(y,z,s)$ by
$(1+b(y,z,s)^{2}R_{0}(y,z,s))^{-1}F(y,z,s)$,

$a(y,z,s)\mbox{ by
}\big{(}1+b(y,z,s)^{2}R_{0}(y,z,s)\big{)}^{-1}\cdot\big{(}a(y,z,s)+b(y,z,s)^{2}z\cdot
R_{1}(z,s)\big{)},$ $b(y,z,s)\mbox{ by
}\big{(}1+b(y,z,s)^{2}R_{0}(y,z,s)\big{)}^{-1}\cdot\big{(}1+z\cdot
b(y,z,s)\cdot R_{2}(z,s)\big{)}\cdot b(y,z,s),$

we may assume that $R(y,z,s)$ is divisible by $z^{4}$. In particular, we can
write $R(y,z,s)=z^{4}R_{3}(y,z,s).$

We define the following holomorphic functions in $(y,z,s)$,

$h(y,z,s)=y^{2}+a(y,z,s)yz+b(y,z,s)z^{2}\mbox{ and }p(y,z,s)=-y-a(y,z,s)z.$

Then we have

$p(y,z,s)y+h(y,z,s)=b(y,z,s)z^{2}.$

Up to shrinking the domain of $F$, we may assume that
$1+4p(y,z,s)R_{3}(y,z,s)$ admits a square root $\delta(y,z,s)$ of the shape

$\delta(y,z,s)=1+\sum_{i=1}^{+\infty}\binom{\frac{1}{2}}{i}\cdot\Big{(}4p(y,z,s)R_{3}(y,z,s)\Big{)}^{i}.$

Then $-1+\delta(y,z,s)$ is divisible by $p(y,z,s)$. We set

$\gamma(y,z,s)=\frac{1}{2}\cdot p(y,z,s)^{-1}\cdot(-1+\delta(y,z,s)),$

Then $\gamma$ is a holomorphic function and satisfies the equation

$p\gamma^{2}+\gamma-R_{3}=0.$

Let $c(y,z,s)=z^{2}\cdot\gamma(y,z,s)$ and
$d(y,z,s)=z^{2}\cdot\gamma(y,z,s)\cdot p(y,z,s)$. Then we have

$\displaystyle(y+bc)(h+bd)$ $\displaystyle=$ $\displaystyle
yh+b(yd+hc)+b^{2}dc$ $\displaystyle=$ $\displaystyle
yh+bz^{2}\gamma\cdot(py+h)+b^{2}z^{4}\gamma^{2}p$ $\displaystyle=$
$\displaystyle yh+b^{2}z^{4}\gamma+b^{2}z^{4}\gamma^{2}p$ $\displaystyle=$
$\displaystyle yh+b^{2}z^{4}R_{3}+b^{2}z^{4}\cdot(-R_{3}+\gamma+p\gamma^{2})$
$\displaystyle=$ $\displaystyle F$

This completes the proof of the lemma. ∎

######  Corollary 5.4.

Let $F(y,z,s)=y(y^{2}+z^{3})+z^{5}R(y,z,s)$ be a holomorphic function defined
in a neighborhood of $\mathbf{0}_{N+2}\in\mathbb{C}^{2+N}$, where $N\geq 0$ is
an integer, $(y,z)$ is the first two coordinates of $\mathbb{C}^{2+N}$ and $s$
is a vector of dimension $N$, which represents the last $N$ coordinates of
$\mathbb{C}^{2+N}$. Then, in a neighborhood of $\mathbf{0}_{N+2}$, there is a
factorization $F=(\mathrm{unit})\cdot G_{1}G_{2}$ with $G_{1}=y+c(y,z,s)$ and
$G_{2}=y^{2}+z^{3}+d(y,z,s)+e(s)yz^{2}$ such that $c(y,z,s)$ is divisible by
$z^{2}$, and that terms of $d(y,z,s)$ have degrees at least 4 in $(y,z)$.

###### Proof.

We can write $R(y,z,s)=C(s)+R_{1}(y,z,s)$, where the terms of $R_{1}(y,z,s)$
have degrees at least $1$ in $(y,z)$. It follows that

$\displaystyle H(y,z,s)$ $\displaystyle:=$ $\displaystyle(y+C(s)\cdot
z^{2})(y^{2}+z^{3}-C(s)\cdot z^{2}y+C(s)^{2}\cdot z^{4})$ $\displaystyle\ =$
$\displaystyle y(y^{2}+z^{3})+C(s)\cdot z^{5}+C(s)^{3}\cdot z^{6}$
$\displaystyle\ =$ $\displaystyle F(y,z,s)+z^{5}\cdot(C(s)^{3}z-R_{1}(y,z,s))$

We set ${y^{\prime}}=y+C(s)z^{2}$. Then we have

$\displaystyle H(y,z,s)$ $\displaystyle=$
$\displaystyle{y^{\prime}}\Big{(}({y^{\prime}}-Cz^{2})^{2}+z^{3}-Cz^{2}({y^{\prime}}-Cz^{2})+C^{2}z^{4}\Big{)}$
$\displaystyle=$
$\displaystyle{y^{\prime}}\Big{(}{y^{\prime}}^{2}-3C{y^{\prime}}z^{2}+z^{3}(1+C^{2}z)\Big{)}.$

Thus we can write

$F(y,z,s)={y^{\prime}}\left({y^{\prime}}^{2}-3C{y^{\prime}}z^{2}+z^{3}(1+C^{2}z)\right)+z^{5}R_{2}({y^{\prime}},z,s)$

for some holomorphic function $R_{2}$, whose terms have degrees at least 1 in
$(y^{\prime},z)$. Let $u=z$, then we have

$F(y,z,s)=\overline{F}({y^{\prime}},z,s,u)={y^{\prime}}({y^{\prime}}^{2}+a(u,s){y^{\prime}}z+n(u,s)uz^{2})+u^{2}z^{3}R_{2}({y^{\prime}},z,s),$

where $a(u,s)=-3C(s)u$, and $n(u,s)=(1+C(s)^{2}u)$ is a unit. Hence we can
apply Lemma 5.3 to $\overline{F}$, and deduce a factorization
$\overline{F}=(\mathrm{unit})\cdot\overline{G}_{1}\overline{G}_{2}$ with
$\overline{G}_{1}={y^{\prime}}+\overline{c}({y^{\prime}},z,s)$ such that
$\overline{c}$ is divisible by $z^{2}$. Furthermore,
$\overline{G}_{2}={y^{\prime}}^{2}-3C(s){y^{\prime}}z^{2}+z^{3}(1+C(s)^{2}z)+\overline{d}({y^{\prime}},z,s)$,
where the terms of $\overline{d}$ have degree at least $4$ in
$(y^{\prime},z)$. Since $y=y^{\prime}-C(s)z^{2}$, we obtain a factorization
${F}=(\mathrm{unit})\cdot{G}_{1}{G}_{2}$ with $G_{1}={y}+c(y,z,s)$ and
$G_{2}=y^{2}+z^{3}+d(y,z,s)+e(s)yz^{2}$ such that $c$ is divisible by $z^{2}$
and that the terms of $d$ have degree at least $4$ in $(y,z)$. This completes
the proof of the corollary. ∎

###  5.B. Case of $A$-type singularities

In this subsection, we treat Proposition 5.2 in the case when $X$ has $A$-type
singularities at points of $S\setminus H$.

######  Lemma 5.5.

Proposition 5.2 holds if $X$ has $A$-type singularities at points of
$S\setminus H$.

###### Proof.

We may assume that $X\subseteq\mathbb{D}^{3}\times S$ is defined by an
equation of the shape

$x^{2}+F_{2}(y,z,s)+F_{3}(y,z,s)+R(y,z,s),$

where $F_{2}$ and $F_{3}$ are homogeneous in $(y,z)$ of degree 2 and 3
respectively, and $R$ is a holomorphic function such that
$R(y,z,s)=0\operatorname{mod}\,(y,z)^{3}$. By assumption, when $s\not\in H$,
the term $F_{2}(y,z,s)$ is not zero. Up to a linear change of the coordinates
$(y,z)$, we may assume that $F_{2}(y,z,s)=a(s)(y^{2}+b(s)yz+c(s)z^{2})$, where
the zero locus of $a$ is contained in $H$. By blowing up $X$ at the $H_{i}$’s
several times, we may assume that $a(s)$ divides $F_{3}$ and $R$ (see Remark
3.5). Hence $X$ is defined by

$x^{2}+a(s)(y^{2}+b(s)yz+c(s)z^{2}+R^{\prime}(y,z,s))=0,$

where $R^{\prime}(y,z.s)=0\operatorname{mod}\,(y,z)^{3}$. By applying the
Weierstrass preparation theorem to the function in the parenthesis after
$a(s)$ above, with respect to the variable $y$, we see that $X$ is defined by

$x^{2}+a(s)\cdot(\mathrm{{unit}})\cdot(y^{2}+y\cdot R_{1}(z,s)+R_{2}(z,s))=0,$

for some holomorphic functions $R_{1}$ and $R_{2}$. Replacing $y$ by
$y-\frac{1}{2}R_{1}(z,s)$, the defining equation becomes

$x^{2}+a(s)\cdot(\mathrm{{unit}})\cdot\left(y^{2}+z^{e}\cdot
R_{3}(z,s)\right)=0,$

where $e\geq 2$ is an integer and $R_{3}$ is not divisible by $z$.

When $s\not\in H$, $X$ has $A_{l}$-type singularities. Hence $e=l+1$ and if
$b(s)$ the term in $R_{3}$ of degree $0$ in $z$, then the zero locus of $b(s)$
is contained in $H$. Thus, by blowing up $X$ at the $H_{i}$’s, we may assume
that $b(s)$ divides $R_{3}(z,s)$, see Remark 3.5. Then the defining equation
is of the shape

$x^{2}+a(s)\cdot(\mathrm{unit})\cdot y^{2}+b(s)\cdot(\mathrm{unit})\cdot
z^{l+1}=0,$

where the zero locus of $b(s)$ is contained in $H$. Hence, up to a change of
coordinates, the defining equation becomes

$x^{2}+\alpha(s)y^{2}+\beta(s)z^{l+1}=0.$

In the end, we note that if we continue blowing up $X$ at some $H_{i}$, then
the defining equation remains the same shape. This completes the proof of the
lemma. ∎

###  5.C. Case of $E$-type singularities

In this subsection, we treat Proposition 5.2 in the case when $X$ has $E$-type
singularities at points of $S\setminus H$. We will discuss the cases of
$E_{6}$, $E_{7}$ and $E_{8}$ separately in three lemmas.

######  Lemma 5.6.

Proposition 5.2 holds if $X$ has $E_{6}$-type singularities at points of
$S\setminus H$.

###### Proof.

We may assume that $X\subseteq\mathbb{D}^{3}\times S$ is defined by an
equation of the shape

$x^{2}+F_{2}(y,z,s)+F_{3}(y,z,s)+R(y,z,s)=0,$

where $F_{2}$ and $F_{3}$ are homogeneous in $(y,z)$ of degree 2 and 3
respectively, and $R(y,z,s)=0\operatorname{mod}\,(y,z)^{4}$. By Lemma 2.2,
when $s\not\in H$, the term $F_{2}$ is zero and $F_{3}$ is a cube. It follows
that $F_{2}=0$. Moreover, up to shrinking $X$ and up to a linear change of the
coordinates $(y,z)$, we can assume that $F_{3}$ is of the shape
$F_{3}(y,z,s)=a(s)(y-u(s)z)^{3}$ for some holomorphic function $a(s)$ whose
zero locus is contained in $H$, and for some holomorphic function $u(s)$.
Replacing $y$ by $y+u(s)z$, we may assume that

$F_{3}(y,z,s)=a(s)y^{3}.$

By blowing up $X$ several times at the $H_{i}$’s as in Subsection 3.C, we may
assume that $a(s)$ divides $R(y,z,s)$, see Remark 3.5. Then we can write

$\displaystyle F_{3}(y,z,s)+R(y,z,s)=a(s)\cdot(y^{3}+\cdots),$

By applying the Weierstrass preparation theorem to the series in the
parenthesis after $a(s)$ in the equation above, with respect to $y$, we can
write

$\displaystyle F_{3}(y,z,s)+R(y,z,s)$ $\displaystyle=$
$\displaystyle(\mathrm{unit})\cdot
a(s)\cdot\Big{(}y^{3}+\omega_{0}(z,s)z^{2}y^{2}$
$\displaystyle+\omega_{1}(z,s)z^{3+p}y+\omega_{2}(z,s)z^{4+q}\Big{)},$

where $q\geq p\geq 0$ are integers. Up to replacing $y$ by
$y-\frac{1}{3}\omega_{0}(z,s)z^{2}$, we can assume that

$F_{3}(y,z,s)+R(y,z,s)=(\mathrm{unit})\cdot
a(s)\cdot\Big{(}y^{3}+\omega_{1}(z,s)z^{3+p}y+\omega_{2}(z,s)z^{4+q}\Big{)}.$

By merging the terms of $\omega_{1}(z,s)z^{3+p}y$ divisible by $z^{4+q}$ into
$\omega_{2}(z,s)z^{4+q}$. we can write

$F_{3}(y,z,s)+R(y,z,s)=(\mathrm{unit})\cdot
a(s)\cdot\Big{(}y^{3}+v(z,s)z^{3+p}y+w(y,z,s)z^{4+q}\Big{)},$

such that either $v$ is zero or $z$ does not divide $v$. Furthermore, the
terms of $v(z,s)$ have degrees at most $q-p$ in $z$. In addition, either
$w(y,z,s)$ is zero, or $w(0,0,s)\neq 0$ for a general point $s\in S$.

Since $X$ has $E_{6}$-type singularities around $S\setminus H$, by Lemma 2.2,
we deduce that $q=p=0$, and $v(z,s)=v(s)$ is a holomorphic function on $S$.
Lemma 2.2 also implies that $w(y,z,s)=b(s)+m(y,z,s)$ such that the vanishing
locus of $b(s)$ is contained in $H$ and that
$m(y,z,s)=0\operatorname{mod}\,(y,z)$. By blowing up $X$ at the $H_{i}$’s
several times as in Subsection 3.C, we may assume that
$w(y,z,s)=\eta(y,z,s)\cdot b(s)$, where $\eta$ is a unit, see Remark 3.5.

We denote $T=T_{1}\cdots T_{n}$. Since the zero locus of the function $b(s)$
is contained in $H=\\{T=0\\}$, there is an integer $e^{\prime}\geq 0$ such
that $T^{e^{\prime}}$ is divisible by $b(s)$. Let $e=3e^{\prime}$. By blowing
up $X$ at each $H_{i}$ for $2e$ times as in Subsection 3.C, the defining
function $F(x,y,z,s)$ becomes

$F(x,y,z,s)=x^{2}+(\mathrm{unit})\cdot
a(s)\cdot\Big{(}T^{2e}y^{3}+T^{4e}v(s)z^{3}y+\eta(y,z,s)\cdot b(s)\cdot
T^{4e}z^{4}\Big{)}.$

Since $e-2e^{\prime}=e^{\prime}$, by the choice of $e^{\prime}$, there is a
function $m(s)$ such that

$b(s)\cdot m(s)\cdot T^{2e^{\prime}}=v(s)\cdot T^{e}.$

Hence we can write

$\displaystyle F(x,y,z,s)$ $\displaystyle=$ $\displaystyle
x^{2}+(\mathrm{unit})\cdot a(s)\cdot\Big{(}T^{2e}y^{3}$
$\displaystyle+b(s)\cdot m(s)\cdot(T^{e}z)^{3}\cdot T^{2e^{\prime}}y$
$\displaystyle+\eta(y,z,s)\cdot b(s)\cdot(T^{e}z)^{4}\Big{)}.$

We set

$\zeta=T^{e}z+\frac{1}{4}\eta(y,z,s)^{-1}m(s)\cdot T^{2e^{\prime}}y,$

so that

$\displaystyle F(x,y,z,s)$ $\displaystyle=$ $\displaystyle
x^{2}+(\mathrm{unit})\cdot a(s)\cdot\Big{(}T^{2e}y^{3}+\nu_{1}(y,z,s)\cdot
b(s)(T^{2e^{\prime}}y)^{4}$ $\displaystyle+\nu_{2}(y,z,s)\cdot
b(s)\zeta\cdot(T^{2e^{\prime}}y)^{3}$ $\displaystyle+\nu_{3}(y,z,s)\cdot
b(s)\zeta^{2}\cdot(T^{2e^{\prime}}y)^{2}+\eta(y,z,s)\cdot
b(s)\zeta^{4}\Big{)},$

for some holomorphic functions $\nu_{1},\nu_{2},\nu_{3}$. We remark that
$T^{2e}y^{3}=(T^{2e^{\prime}}y)^{3}$ and we consider the following term in the
last expression on $F(x,y,z,s)$ above,

$\displaystyle(T^{2e^{\prime}}y)^{3}+\nu_{1}(y,z,s)\cdot
b(s)(T^{2e^{\prime}}y)^{4}$ $\displaystyle+\nu_{2}(y,z,s)\cdot
b(s)\zeta\cdot(T^{2e^{\prime}}y)^{3}$ $\displaystyle+\nu_{3}(y,z,s)\cdot
b(s)\zeta^{2}\cdot(T^{2e^{\prime}}y)^{2}$ $\displaystyle=$
$\displaystyle(\mathrm{unit})\cdot(T^{2e^{\prime}}y)^{3}+\nu_{3}(y,z,s)\cdot
b(s)\zeta^{2}\cdot(T^{2e^{\prime}}y)^{2}.$

Then there are some holomorphic functions $\mu(y,z,s)$ and $\sigma(y,z,s)$
such that the function $F$ can be written as

$\displaystyle F(x,y,z,s)$ $\displaystyle=$ $\displaystyle
x^{2}+(\mathrm{unit})\cdot
a(s)\cdot\Big{(}(\mathrm{unit})\cdot(T^{2e^{\prime}}y+\mu(y,z,s)\zeta^{2})^{3}$
$\displaystyle+\sigma(y,z,s)\cdot b(s)\zeta^{4}+\eta(y,z,s)\cdot
b(s)\zeta^{4}\Big{)}.$

Furthermore, $\sigma(y,z,s)=0\operatorname{mod}\,(y,z)$. Hence the function
$F$ can be written as

$F(x,y,z,s)=x^{2}+a(s)\cdot(\mathrm{unit})\cdot\Big{(}(\mathrm{unit})\cdot(T^{2e^{\prime}}y+\mu(y,z,s)\zeta^{2})^{3}+(\mathrm{unit})\cdot
b(s)\zeta^{4}\Big{)}.$

We note that $\zeta=T^{e}z+\frac{1}{4}\eta(y,z,s)^{-1}m(s)\cdot
T^{2e^{\prime}}y$. Hence there is some holomorphic function
$\mu^{\prime}(y,z,s)$ such that

$\displaystyle T^{2e^{\prime}}y+\mu(y,z,s)\zeta^{2}$ $\displaystyle=$
$\displaystyle(\mathrm{unit})\cdot T^{2e^{\prime}}y+\mu(y,z,s)T^{2e}z^{2}$
$\displaystyle(\mathrm{unit})\cdot
T^{2e^{\prime}}\Big{(}y+\mu^{\prime}(y,z,s)T^{2e-2e^{\prime}}z^{2}\Big{)}.$

Replacing $y$ by $y-\mu^{\prime}(y,z,s)T^{2e-2e^{\prime}}z^{2}$, the function
$F$ can be written as

$\displaystyle F(x,y,z,s)$ $\displaystyle=$ $\displaystyle
x^{2}+R_{1}(y,z,s)\cdot a(s)\cdot T^{2e}y^{3}$
$\displaystyle+R_{2}(y,z,s)\cdot a(s)b(s)\cdot(R_{3}(y,z,s)T^{e}z+m(s)\cdot
T^{2e^{\prime}}y)^{4},$

where $R_{1},R_{2},R_{3}$ are units. Let $R_{4}=R_{2}R_{1}^{-1}$. Then we can
set $R_{5}=R_{1}R_{4}^{-3}=R_{2}R_{4}^{-4}$, so that

$\displaystyle F(x,y,z,s)$ $\displaystyle=$ $\displaystyle
x^{2}+R_{1}R_{4}^{-3}\cdot a(s)\cdot T^{2e}(R_{4}y)^{3}$
$\displaystyle+R_{2}R_{4}^{-4}\cdot a(s)b(s)\cdot(R_{3}R_{4}T^{e}z+m(s)\cdot
T^{2e^{\prime}}R_{4}y)^{4}.$

Replacing $y$ by $R_{4}^{-1}y$, we deduce that $F$ is of the shape

$\displaystyle F(x,y,z,s)$ $\displaystyle=$ $\displaystyle x^{2}+R_{5}\cdot
a(s)\cdot T^{2e}y^{3}$ $\displaystyle+R_{5}\cdot
a(s)b(s)\cdot(R_{3}R_{4}T^{e}z+m(s)\cdot T^{2e^{\prime}}y)^{4}.$

Then $X$ is defined by

$\displaystyle F(x,y,z,s)$ $\displaystyle=$ $\displaystyle
R_{5}^{-1}x^{2}+a(s)\cdot T^{2e}y^{3}$
$\displaystyle+a(s)b(s)\cdot(R_{3}R_{4}T^{e}z+m(s)\cdot
T^{2e^{\prime}}y)^{4}.$

Hence there is a change of coordinates so that $X$ can be defined by

$F(x,y,z,t)=x^{2}+\alpha(s)y^{3}+\beta(s)(\gamma(s)z+u(y,s)y)^{4},$

where the zero loci of $\alpha(s)$, $\beta(s)$ and $\gamma(s)$ are contained
in $H$. In the end, we note that if we continue blowing up $X$ at some
$H_{i}$, then the defining equation remains the same shape. This completes the
proof of the lemma. ∎

In the following lemma, we treat the case of $E_{8}$-type singularity. Its
proof proceeds in the same way as the one for the case of $E_{6}$ singularity.

######  Lemma 5.7.

Proposition 5.2 holds if $X$ has $E_{8}$-type singularities at points of
$S\setminus H$.

###### Proof.

As in the case of Lemma 5.6, we can reduce to the situation when

$F_{3}(y,z,s)+R(y,z,s)=(\mathrm{unit})\cdot
a(s)\cdot\Big{(}y^{3}+v(z,s)z^{3+p}y+w(y,z,s)z^{4+q}\Big{)},$

where $q\geq p\geq 0$ are integers. Either $v$ is zero or $z$ does not divide
$v$. Furthermore, the terms of $v(z,s)$ have degrees at most $q-p$ in $z$. In
addition, either $w(y,z,s)$ is zero, or $w(0,0,s)\neq 0$ for a general point
$s\in S$.

Since $X$ has $E_{8}$ singularities at points of $S\setminus H$, by Lemma 2.2,
we deduce that $q=p=1$, and $v(z,s)=v(s)$ is a holomorphic function on $S$.
Lemma 2.2 also implies that $w(y,z,s)=b(s)+m(y,z,s)$ such that the vanishing
locus of $b$ is contained in $H$ and that $m(y,z,s)$ is a holomorphic function
whose terms have degree at least 1 in $(y,z)$. By blowing up $X$ at the
$H_{i}$’s several times as in Subsection 3.C, we may assume that
$w(y,z,s)=\eta(y,z,s)\cdot b(s)$, where $\eta$ is a unit, see Remark 3.5.

We denote $T=T_{1}\cdots T_{n}$. Since the zero locus of the function $b(s)$
is contained in $H=\\{T=0\\}$, there is an integer $e^{\prime}\geq 0$ such
that $T^{e^{\prime}}$ is divisible by $b(s)$. We set

$e=5d=15e^{\prime}.$

By blowing up $X$ at each $H_{i}$ as in Subsection 3.C for $e$ times, the
function $F$ becomes

$\displaystyle F(x,y,z,s)$ $\displaystyle=$ $\displaystyle
x^{2}+(\mathrm{unit})\cdot a(s)\cdot\Big{(}T^{e}y^{3}$
$\displaystyle+T^{3e}v(s)z^{4}y+\eta(y,z,s)\cdot b(s)\cdot
T^{3e}z^{5}\Big{)}.$

We note that, since $e=5d$,

$T^{3e}v(s)z^{4}y+\eta(y,z,s)\cdot b(s)\cdot T^{3e}z^{5}=(T^{3d}z)^{4}\cdot
v(s)\cdot T^{3d}y+\eta(y,z,s)\cdot b(s)\cdot(T^{3d}z)^{5}.$

Since $3d-5e^{\prime}=4e^{\prime}$, by the choice of $e^{\prime}$, there is a
function $m(s)$ such that

$b(s)\cdot m(s)\cdot T^{5e^{\prime}}=v(s)\cdot T^{3d}.$

Hence we can write

$\displaystyle F(x,y,z,s)$ $\displaystyle=$ $\displaystyle
x^{2}+(\mathrm{unit})\cdot a(s)\cdot\Big{(}T^{e}y^{3}$
$\displaystyle+b(s)\cdot m(s)\cdot(T^{3d}z)^{4}\cdot T^{5e^{\prime}}y$
$\displaystyle+\eta(y,z,s)\cdot b(s)\cdot(T^{3d}z)^{5}\Big{)}.$

We set

$\zeta=T^{3d}z+\frac{1}{5}\eta(y,z,s)^{-1}m(s)\cdot T^{5e^{\prime}}y,$

so that

$\displaystyle F(x,y,z,s)$ $\displaystyle=$ $\displaystyle
x^{2}+(\mathrm{unit})\cdot a(s)\cdot\Big{(}T^{e}y^{3}+\nu_{1}(y,z,s)\cdot
b(s)\cdot(T^{5e^{\prime}}y)^{5}$ $\displaystyle+\nu_{2}(y,z,s)\cdot
b(s)\zeta\cdot(T^{5e^{\prime}}y)^{4}+\nu_{3}(y,z,s)\cdot
b(s)\zeta^{2}\cdot(T^{5e^{\prime}}y)^{3}$ $\displaystyle+\nu_{4}(y,z,s)\cdot
b(s)\zeta^{3}\cdot(T^{5e^{\prime}}y)^{2}+\eta(y,z,s)\cdot
b(s)\zeta^{5}\Big{)},$

for some holomorphic functions $\nu_{1},\nu_{2},\nu_{3},\nu_{4}$. We remark
that $T^{e}y^{3}=(T^{5e^{\prime}}y)^{3}$ and we consider the following term in
the last expression of $F(x,y,z,s)$ above,

$\displaystyle T^{e}y^{3}+\nu_{1}(y,z,s)\cdot
b(s)(T^{5e^{\prime}}y)^{5}+\nu_{2}(y,z,s)\cdot
b(s)\zeta\cdot(T^{5e^{\prime}}y)^{4}$ $\displaystyle+\nu_{3}(y,z,s)\cdot
b(s)\zeta^{2}\cdot(T^{5e^{\prime}}y)^{3}+\nu_{4}(y,z,s)\cdot
b(s)\zeta^{3}\cdot(T^{5e^{\prime}}y)^{2}$ $\displaystyle=$
$\displaystyle(\mathrm{unit})\cdot(T^{5e^{\prime}}y)^{3}+\nu_{4}(y,z,s)\cdot
b(s)\zeta^{3}\cdot(T^{5e^{\prime}}y)^{2}.$

Then there are some holomorphic functions $\mu(y,z,s)$ and $\sigma(y,z,s)$
such that the function $F$ can be written as

$\displaystyle F(x,y,z,s)$ $\displaystyle=$ $\displaystyle
x^{2}+(\mathrm{unit})\cdot
a(s)\cdot\Big{(}(\mathrm{unit})\cdot(T^{5e^{\prime}}y+\mu(y,z,s)\zeta^{3})^{3}$
$\displaystyle+\sigma(y,z,s)\cdot b(s)\zeta^{6}+\eta(y,z,s)\cdot
b(s)\zeta^{5}\Big{)}.$

Hence the function $F$ can be written as

$F(x,y,z,s)=x^{2}+a(s)\Big{(}(\mathrm{unit})\cdot(T^{5e^{\prime}}y+\mu(y,z,s)\zeta^{3})^{3}+(\mathrm{unit})\cdot
b(s)\zeta^{5}\Big{)}.$

We note that $\zeta=T^{3d}z+\frac{1}{5}\eta(y,z,s)^{-1}m(s)\cdot
T^{5e^{\prime}}y$. Hence there is some holomorphic function
$\mu^{\prime}(y,z,s)$ such that

$\displaystyle T^{5e^{\prime}}y+\mu(y,z,s)\zeta^{3}$ $\displaystyle=$
$\displaystyle(\mathrm{unit})\cdot T^{5e^{\prime}}y+\mu(y,z,s)\cdot
T^{9d}z^{3}$ $\displaystyle=$ $\displaystyle(\mathrm{unit})\cdot
T^{5e^{\prime}}\Big{(}y+\mu^{\prime}(y,z,s)\cdot
T^{9d-5e^{\prime}}z^{3}\Big{)}.$

By replacing $y$ with $y-\mu^{\prime}(y,z,s)\cdot T^{9d-5e^{\prime}}z^{3}$, we
can write

$\displaystyle F(x,y,z,s)$ $\displaystyle=$ $\displaystyle
x^{2}+R_{1}(y,z,s)\cdot a(s)T^{e}y^{3}$ $\displaystyle+R_{2}(y,z,s)\cdot
a(s)b(s)\cdot(R_{3}(y,z,s)T^{3d}z+c(s)\cdot T^{5e^{\prime}}y)^{5},$

where $R_{1},R_{2},R_{3}$ are units. There is a holomorphic function $R_{4}$
such that $R_{4}^{2}=R_{2}R_{1}^{-1}$. Then we can set
$R_{5}=R_{2}R_{4}^{-5}=R_{1}R_{4}^{-3}$, and we have

$\displaystyle F(x,y,z,s)$ $\displaystyle=$ $\displaystyle x^{2}+R_{5}\cdot
a(s)T^{e}(R_{4}y)^{3}$
$\displaystyle+R_{5}a(s)b(s)(R_{3}R_{4}T^{3d}z+c(s)T^{5e^{\prime}}R_{4}y)^{5}.$

Replacing $y$ by $R_{4}^{-1}y$, we see that $X$ is defined by

$F(x,y,z,s)=R_{5}^{-1}x^{2}+a(s)T^{e}y^{3}+a(s)b(s)(R_{3}R_{4}T^{3d}z+c(s)T^{5e^{\prime}}y)^{5}.$

Hence there is a change of coordinates so that the defining function becomes

$F(x,y,z,s)=x^{2}+\alpha(s)y^{3}+\beta(s)(\gamma(s)z+u(y,s)y)^{5},$

where the zero locus of $\alpha(s)$, $\beta(s)$ and $\gamma(s)$ are contained
in $H$. In the end, we note that if we continue blowing up $X$ at some
$H_{i}$, then the defining equation remains the same shape. This completes the
proof of the lemma. ∎

We will now treat the case of $E_{7}$-type singularity. Its proof is slightly
different from the ones of $E_{6}$-type or of $E_{8}$-type singularities.

######  Lemma 5.8.

Proposition 5.2 holds if $X$ has $E_{7}$-type singularities at points of
$S\setminus H$.

###### Proof.

As in the case of Lemma 5.6, we can reduce to the situation when

$F_{3}(y,z,s)+R(y,z,s)=(\mathrm{unit})\cdot
a(s)\cdot\Big{(}y^{3}+v(z,s)z^{3+p}y+w(y,z,s)z^{4+q}\Big{)},$

where $q\geq p\geq 0$ are integers. Either $v$ is zero or $z$ does not divide
$v$. Furthermore, the terms of $v(z,s)$ have degrees at most $q-p$ in $z$. In
addition, either $w(y,z,s)$ is zero, or $w(0,0,s)\neq 0$ for a general point
$s\in S$.

Since $X$ has $E_{7}$ singularities along $S\setminus H$, by Lemma 2.2, we
deduce that $p=0$ and $q\geq 1$. Lemma 2.2 also implies that
$v(z,s)=b(s)+z\cdot m(z,s)$ such that the vanishing locus of $b$ is contained
in $H$ and that $m(z,s)$ is a holomorphic function. By blowing up $X$ at the
$H_{i}$’s several times as in Subsection 3.C, we may assume that
$v(z,s)=\eta(z,s)\cdot b(s)$, where $\eta$ is a unit, see Remark 3.5. We may
also assume that $b(s)=\prod_{i}T_{i}^{l_{i}}$ for some integers
$l_{1},...,l_{n}\geq 0$. Then we have

$F_{3}(y,z,s)+R(y,z,s)=(\mathrm{unit})\cdot
a(s)\cdot\Big{(}y^{3}+\eta(z,s)b(s)yz^{3}+w(y,z,s)z^{4+q}\Big{)},$

By blow up $X$ at each $H_{i}$ for $e_{i}$ times as in Subsection 3.C, we can
write

$\displaystyle F(x,y,z,s)$ $\displaystyle=$ $\displaystyle
x^{2}+(\mathrm{unit})\cdot a(s)\cdot\prod_{i}T_{i}^{e_{i}}\Big{(}y^{3}$
$\displaystyle+\prod_{i}T_{i}^{e_{i}+l_{i}}\eta(z,s)yz^{3}+\prod_{i}T_{i}^{qe_{i}+e_{i}}w(y,z,s)z^{4+q}\Big{)},$

where by abuse of notation, we still write $\eta(z,s)$ for
$\eta(\prod_{i}T_{i}^{e_{i}}\cdot z,s)$ and $w(y,z,s)$ for
$w(\prod_{i}T_{i}^{e_{i}}\cdot y,\prod_{i}T_{i}^{e_{i}}\cdot z,s)$. We choose
$e_{i}$ so that $e_{i}+l_{i}=3e_{i}^{\prime}$ for some integer
$e_{i}^{\prime}>0$. We note that

$\displaystyle
qe_{i}+e_{i}-(4+q)e_{i}^{\prime}=(q+1)e_{i}-\frac{1}{3}(4+q)(e_{i}+l_{i})=\frac{1}{3}(2q-1)e_{i}-\frac{1}{3}(4+q)l_{i}.$

Since $q\geq 1$, the RHS above tends to $+\infty$ if $e_{i}$ tends to
$+\infty$. Hence we can pick $e_{i}$ sufficiently large so that

$\prod_{i}T_{i}^{(4+q)e_{i}^{\prime}}\mbox{ divides
}\prod_{i}T_{i}^{qe_{i}+e_{i}}w(y,z,s).$

Then there is some holomorphic function $m(y,z,s)$ such that

$\prod_{i}T_{i}^{qe_{i}+e_{i}}w(y,z,s)z^{4+q}=m(y,z,s)\cdot\prod_{i}T_{i}^{(4+q)e_{i}^{\prime}}\cdot
z^{4+q}.$

Let $\zeta=z\cdot\prod_{i}T_{i}^{e_{i}^{\prime}}$. Then we have

$\displaystyle F(x,y,z,s)$ $\displaystyle=$ $\displaystyle
x^{2}+(\mathrm{unit})\cdot
a(s)\cdot\prod_{i}T_{i}^{e_{i}}\Big{(}y^{3}+\eta(z,s)y\zeta^{3}+m(y,z,s)\zeta^{4+q}\Big{)}$
$\displaystyle=$ $\displaystyle x^{2}+(\mathrm{unit})\cdot
a(s)\cdot\prod_{i}T_{i}^{e_{i}}$
$\displaystyle\cdot\Big{(}y(\eta(z,s)^{-1}y^{2}+\zeta^{3})+\eta(z,s)^{-1}m(y,z,s)\zeta^{4+q}\Big{)}.$

Replacing $y$ by $y\cdot\eta(z,s)^{\frac{1}{2}}$, we can write

$F(x,y,z,s)=x^{2}+(\mathrm{unit})\cdot
a(s)\cdot\prod_{i}T_{i}^{e_{i}}\Big{(}y(y^{2}+\zeta^{3})+m^{\prime}(y,z,s)\zeta^{4+q}\Big{)}$

for some holomorphic function $m^{\prime}(y,z,s)$. By applying Corollary 5.4
to the function

$y(y^{2}+\zeta^{3})+m^{\prime}(y,z,s)\zeta^{4+q},$

it can be factorized into

$(\mathrm{unit})\cdot\Big{(}y+c(y,\zeta,z,s)\Big{)}\Big{(}y^{2}+\zeta^{3}+\sigma(y,\zeta,z,s)+\nu(s)\cdot\zeta^{2}y\Big{)},$

where $\zeta^{2}$ divides $c(y,\zeta,z,s)$ and
$\sigma(y,\zeta,z,s)=0\operatorname{mod}\,(y,\zeta)^{4}$. By replacing $y$
with $y-c(y,\zeta,z,s)$, the previous term becomes

$\displaystyle(\mathrm{unit})\cdot
y\Big{(}y^{2}+\nu_{1}(y,\zeta,z,s)y\zeta^{2}+(\mathrm{unit})\cdot\zeta^{3}+\nu_{2}(y,\zeta,z,s)\Big{)}$
$\displaystyle=$ $\displaystyle(\mathrm{unit})\cdot
y\Big{(}(\mathrm{unit})\cdot
y^{2}+\nu_{3}(z,s)y\zeta^{2}+(\mathrm{unit})\cdot\zeta^{3}\Big{)},$

where $\nu_{1},\nu_{2}$ and $\nu_{3}$ are holomorphic functions, and
$\nu_{2}(y,\zeta,z,s)=0\operatorname{mod}(y,\zeta)^{4}$. There is some
$\zeta^{\prime}$ of the form

$\zeta^{\prime}=\zeta+(\mathrm{unit})\cdot\mu(z,s)y=z\cdot\prod_{i}T_{i}^{e_{i}^{\prime}}+(\mathrm{unit})\cdot\mu(z,s)y,$

so that the previous function becomes

$\displaystyle(\mathrm{unit})\cdot y\Big{(}(\mathrm{unit})\cdot
y^{2}+\nu_{4}(y,\zeta^{\prime},z,s)y^{2}+(\mathrm{unit})\cdot(\zeta^{\prime})^{3}\Big{)}$
$\displaystyle=$ $\displaystyle(\mathrm{unit})\cdot
y\Big{(}(\mathrm{unit})\cdot
y^{2}+(\mathrm{unit})\cdot(\zeta^{\prime})^{3}\Big{)},$

where $\nu_{4}(y,\zeta^{\prime},z,s)=0\operatorname{mod}\,(y,\zeta^{\prime})$.

Therefore, we deduce that

$\displaystyle F(x,y,z,s)$ $\displaystyle=$ $\displaystyle
x^{2}+a(s)\prod_{i}T_{i}^{e_{i}}\cdot R_{1}(y,z,s)\cdot y^{3}$
$\displaystyle+a(s)\prod_{i}T_{i}^{e_{i}}\cdot R_{2}(y,z,s)\cdot
y\Big{(}R_{3}(y,z,s)z\cdot\prod_{i}T_{i}^{e_{i}^{\prime}}+\mu(z,s)y\Big{)}^{3},$

where $R_{1},R_{2},R_{3}$ are units. We write
$\mu(z,s)=\mu_{1}(s)+z\mu_{2}(z,s)$. Let
$T^{\prime}=\prod_{i}T_{i}^{e^{\prime}_{i}}$. By blowing up $X$ at each
$H_{i}$ for $e_{i}^{\prime}$ times as in Subsection 3.C, the defining function
$F$ becomes

$\displaystyle F(x,y,z,s)$ $\displaystyle=$ $\displaystyle
x^{2}+a(s)T^{\prime}\cdot\prod_{i}T_{i}^{e_{i}}\cdot