R_{1}(T^{\prime}y,T^{\prime}z,s)\cdot y^{3}$
$\displaystyle+a(s)(T^{\prime})^{2}\cdot\prod_{i}T_{i}^{e_{i}}\cdot
R_{2}(T^{\prime}y,T^{\prime}z,s)$ $\displaystyle\cdot
y\Big{(}R_{3}(T^{\prime}y,T^{\prime}z,s)z\cdot
T^{\prime}+\mu_{1}(s)y+T^{\prime}z\cdot\mu_{2}(T^{\prime}z,s)y\Big{)}^{3}.$

The last line in the previous equation is equal to

$y\Big{(}(\mathrm{unit})\cdot z\cdot T^{\prime}+\mu_{1}(s)y\Big{)}^{3}.$

Hence we can write

$\displaystyle F(x,y,z,s)$ $\displaystyle=$ $\displaystyle x^{2}+a_{1}(s)\cdot
Q_{1}(y,z,s)\cdot y^{3}$ $\displaystyle+a_{2}(s)\cdot Q_{2}(y,z,s)\cdot
y\Big{(}Q_{3}(y,z,s)z\cdot T^{\prime}+\mu_{1}(s)y\Big{)}^{3},$

where $Q_{1},Q_{2},Q_{3}$ are units and the zero loci of $a_{1}$ and $a_{2}$
are both contained in $H$. We let $Q_{4}=Q_{2}\cdot Q_{1}^{-1}$. Then there is
a unit $Q_{5}$ such that $Q_{4}^{3}Q_{5}=Q_{1}$ and $Q_{4}^{4}Q_{5}=Q_{2}$. We
can write

$\displaystyle F(x,y,z,s)$ $\displaystyle=$ $\displaystyle x^{2}+a_{1}(s)\cdot
Q_{5}\cdot(Q_{4}\cdot y)^{3}$ $\displaystyle+a_{2}(s)\cdot
Q_{5}\cdot(Q_{4}\cdot y)\cdot(Q_{3}Q_{4}\cdot z\cdot
T^{\prime}+\mu_{1}(s)Q_{4}\cdot y)^{3}$ $\displaystyle=$ $\displaystyle
Q_{5}\cdot\Big{(}Q_{5}^{-1}x^{2}+a_{1}(s)\cdot(Q_{4}\cdot y)^{3}$
$\displaystyle+a_{2}(s)\cdot(Q_{4}\cdot y)\cdot(Q_{3}Q_{4}\cdot z\cdot
T^{\prime}+\mu_{1}(s)Q_{4}\cdot y)^{3}\Big{)}.$

Hence, up to a change of coordinates, $X$ can be defined by a function of the
shape

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

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

###  5.D. Case of $D$-type singularities

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

######  Lemma 5.9.

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

###### Proof.

By assumption $X$ is defined by an equation of the shape

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

where $F_{2}$ and $F_{3}$ are homogeneous in $(y,z)$ of degree 2 and 3
respectively, and $R$ is a holomorphic function such that
$R(y,z,s)=0\operatorname{mod}(y,z)^{4}$. By Lemma 2.2, we see that $F_{2}=0$.
Up to a linear change of the coordinates $(y,z)$, we may assume that
$F_{3}(y,z,s)=a(s)(y^{3}+b(s)y^{2}z+c(s)yz^{2}+d(s)z^{3})$ such that the zero
locus of $a(s)$ is contained in $H$. We blow up $X$ at each $H_{i}$ once (in
arbitrary order), the shape $F_{3}$ is unchanged.

Afterwards, we take the basechange over
$\overline{S}=S[\sqrt[6]{T_{1}},...,\sqrt[6]{T_{n}}]\to S$. In the remainder
of the proof, by abuse of notation, we will write $s\in S$ for
$\overline{s}\in\overline{S}$, $H=\sum_{i=1}^{n}H_{i}$ for
$\overline{H}=\sum_{i=1}^{n}\overline{H}_{i}$, $X$ for a neighborhood of
$\overline{S}$ in $X_{2}$, and $F$ for a defining function of $X$. After Lemma
2.8, we can write

$F_{3}(y,z,s)=a(s)(y-p(s)z)(y-q(s)z)(y-r(s)z)$

for holomorphic functions $p,q,r$ on $S$.

Case 1. We first assume that $X$ has $D_{l}$-type singularities over
$S\setminus H$ with $l\geq 5$. by Lemma 2.2, we may assume that $q(s)=r(s)$.
Replacing $y$ by $y+p(s)z$, we can write

$F_{3}(y,z,s)=a(s)y(y-q(s)z)^{2},$

such that the zero locus of $q$ is contained in $H$. In particular, it is of
the shape

$q(s)=T_{1}^{l_{1}}\cdots T_{n}^{ln}\cdot\overline{q}(s),$

where $\overline{q}(s)$ is a unit. Up to replacing $z$ be
$\overline{q}(s)^{-1}\cdot z$, we may assume that

$q(s)=T_{1}^{l_{1}}\cdots T_{n}^{l_{n}}.$

By blowing up $X$ at the $H_{i}$’s several times as in Subsection 3.C, we may
assume that $a(s)q(s)^{4}$ divides $R(y,z,s)$, see Remark 3.5. Hence we can
write $R(y,z,s)=a(s)q(s)^{4}Q(y,z,s)$ for some holomorphic function $Q$ with
$Q(y,z,s)=0\operatorname{mod}(y,z)^{4}$. Then we can write

$F(x,y,z,s)=x^{2}+a(s)\Big{(}y(y^{2}-2q(s)zy+q(s)^{2}z^{2})+q(s)^{4}\cdot
Q(y,z,s)\Big{)}$

By applying Lemma 5.3 to the function in the parenthesis after $a(s)$ above,
the defining function of $X$ can be written as

$x^{2}+(\mathrm{unit})\cdot
a(s)\cdot\Big{(}y+q(s)^{2}w(y,z,s)\Big{)}\Big{(}q(s)^{2}z^{2}-2q(s)yz+y^{2}+q(s)^{2}v(y,z,s)\Big{)},$

where $z^{2}$ divides $w(y,z,s)$ and
$v(y,z,s)=0\operatorname{mod}\,(y,z)^{3}$. We replace $y$ by
$y-q(s)^{2}w(y,z,s)$, then the second summand above becomes

$\displaystyle G(y,z,s)$ $\displaystyle=$ $\displaystyle(\mathrm{unit})\cdot
a(s)\cdot y\cdot\Big{(}R_{1}(y,z,s)\cdot q(s)^{2}z^{2}$
$\displaystyle-2R_{2}(y,s)\cdot q(s)zy+y^{2}+q(s)^{2}R_{3}(y,s)\Big{)},$

where $R_{1}$ is a unit and $R_{3}(y,z)=0\operatorname{mod}\,(y)^{3}$.
Therefore, by blowing up $X$ at the $H_{i}^{\prime}s$ several times as in
Subsection 3.C, we may assume that

$\displaystyle G(y,z,s)$ $\displaystyle=$ $\displaystyle(\mathrm{unit})\cdot
a(s)y\cdot\Big{(}R_{4}(y,q(s)z,s)\cdot q(s)^{2}z^{2}$
$\displaystyle-2R_{2}(y,s)\cdot q(s)zy+y^{2}+q(s)^{2}R_{3}(y,s)\Big{)},$

where $R_{4}$ is a unit, see Remark 3.6.

If we set $\zeta=q(s)z$, then by using the Weierstrass preparation theorem
with respect to the variable $\zeta$ for the function in the parenthesis after
$a(s)y$ above, we can write

$G(y,z,s)=(\mathrm{unit})\cdot a(s)y\cdot\left(\zeta^{2}+2R_{5}(y,s)\zeta
y+R_{6}(y,s)\right),$

where $R_{6}=0\operatorname{mod}\,(y)^{2}$. It follows that we can write

$G(y,z,s)=(\mathrm{unit})\cdot a(s)\cdot\Big{(}y\cdot(q(s)z+R_{5}(y,s)\cdot
y)^{2}+R_{7}(y,s)\Big{)},$

where $R_{7}(y,s)=0\operatorname{mod}\,(y)^{3}$.

Since $X$ has $D_{l}$-type singularities over $S\setminus H$ with $l\geq 5$,
we deduce that

$R_{7}(y,s)=b(s)y^{l-1}+y^{l}\cdot c(y,s),$

where the zero locus of $b$ is contained in $H$. By blowing up $X$ at the
$H_{i}^{\prime}s$ several times as in Subsection 3.C, the function
$R_{7}(y,s)$ can be written as

$R_{7}(y,s)=b(s)\cdot R_{8}(y,s)\cdot y^{l-1},$

where $R_{8}$ is a unit, see Remark 3.5. Hence the defining function of $X$
becomes

$F(x,y,z,s)=x^{2}+a(s)\cdot(\mathrm{unit})\cdot\Big{(}y(q(s)z+R_{5}(y,s)y)^{2}+b(s)\cdot
R_{8}(y,s)\cdot y^{l-1}\Big{)}$

Let $R_{9}(y,z)$ be a $(l-2)$-root of $R_{8}(y,s)$. Then we have

$\displaystyle F(x,y,z,s)$ $\displaystyle=$ $\displaystyle
x^{2}+a(s)\cdot(\mathrm{unit})\cdot\Big{(}y(q(s)z+R_{5}\cdot y)^{2}$
$\displaystyle+b(s)\cdot R_{9}^{l-2}\cdot y^{l-1}\Big{)}.$ $\displaystyle
x^{2}+a(s)\cdot(\mathrm{unit})\cdot\Big{(}R_{9}\cdot y\cdot(q(s)z+R_{5}\cdot
y)^{2}$ $\displaystyle+b(s)\cdot(R_{9}\cdot y)^{l-1}\Big{)}.$

Replacing $y$ by $R_{9}(y,s)^{-1}y$, the equation $F(x,y,z,s)=0$ is equivalent
to

$(\mathrm{unit})\cdot x^{2}+a(s)\Big{(}y(q(s)z-R_{10}(y,s)y)^{2}+b(s)\cdot
y^{l-1}\Big{)}=0.$

Hence, up to a change of coordinates, $X$ is now defined by

$x^{2}+\alpha(s)\cdot y(\gamma(s)z-u(y,s)y)^{2}+\beta(s)y^{l-1}=0,$

as in the statement of Proposition 5.2.

Case 2. Now we assume that $X$ has $D_{4}$-type singularities at $S\setminus
H$. By Lemma 2.2, and by replacing $y$ by $y+p(s)z$, we can write

$F_{3}(y,z,s)=a(s)y(y-q(s)z)(y-r(s)z),$

such that the zero locus of $q$ and $r$ are contained in $H$, and that the
zero locus of $q(s)-r(s)$ is also contained in $H$. Hence, up to switching $q$
and $s$, there are unit functions $\overline{q}$ and $\overline{r}$ such that.

$q(s)=T_{1}^{l_{1}}\cdots T_{n}^{l_{n}}\cdot\overline{q}(s)\mbox{ and
}r(s)=T_{1}^{l_{1}+k_{1}}\cdots T_{n}^{l_{n}+k_{n}}\cdot\overline{r}(s)$

for some integers $l_{1},...,l_{n},k_{1},...,k_{n}\geq 0$. Let $\overline{v}$
be a square root of $\overline{q}\overline{r}$. Then, by replacing $z$ by
$\overline{v}^{-1}z$, we may assume that $\overline{q}\cdot\overline{r}=1$. We
set $\mu(s)=T_{1}^{l_{1}}\cdots T_{n}^{l_{n}}$ and $\nu(s)=T_{1}^{k_{1}}\cdots
T_{n}^{k_{n}}$. Then we have

$q(s)r(s)=T_{1}^{2l_{1}+k_{1}}\cdots
T_{n}^{2l_{n}+k_{n}}=\mu(s)^{2}\cdot\nu(s).$

We write $(y-q(s)z)(y-r(s)z)=y^{2}+\xi(s)zy+\eta(s)z^{2}$. That is,

$\eta(s)=q(s)r(s)=\mu(s)^{2}\cdot\nu(s)\mbox{ and
}\xi(s)=q(s)+r(s)=(\overline{q}(s)+\overline{r}(s)\nu(s))\cdot\mu(s).$

By blowing up $X$ at the $H_{i}^{\prime}s$ several times as Subsection 3.C, we
may assume that $a(s)\eta(s)^{2}$ divides $R(y,z,s)$, see Remark 3.5. Then we
have

$F(x,y,z,s)=x^{2}+a(s)\Big{(}y(y^{2}+\xi(s)zy+\eta(s)z^{2})+\eta(s)^{2}Q(y,z,s)\Big{)},$

where $Q(y,z,s)=0\operatorname{mod}\,(y,z)^{4}$. By applying Lemma 5.3 to the
function in the parenthesis after $a(s)$ above, the defining function of $X$
can be written as

$x^{2}+(\mathrm{unit})\cdot
a(s)\Big{(}y+\eta(s)w(y,z,s)\Big{)}\Big{(}\eta(s)z^{2}-2\xi(s)yz+y^{2}+\eta(s)v(y,z,s)\Big{)},$

where $z^{2}$ divides $w(y,z,s)$ and
$v(y,z,s)=0\operatorname{mod}\,(y,z)^{3}$. We replace $y$ by
$y-\eta(s)w(y,z,s)$, then the second summand above becomes

$\displaystyle G(y,z,s)$ $\displaystyle=$ $\displaystyle(\mathrm{unit})\cdot
a(s)y\Big{(}R_{1}(y,z,s)\cdot\eta(s)z^{2}$
$\displaystyle-2\xi(s)zy+R_{2}(y,z,s)y^{2}+\eta(s)R_{3}(y,s)\Big{)},$

where $R_{1}$ is a unit and $R_{3}(y,s)=0\operatorname{mod}\,(y)^{3}$. Recall
that $\eta(s)=\mu(s)^{2}\nu(s)$ and that $\xi(s)$ is divisible by $\mu(s)$. By
blowing up $X$ at the $H_{i}^{\prime}s$ several times as in 3.C, we may assume
that

$\displaystyle G(y,z,s)$ $\displaystyle=$
$\displaystyle(\mathrm{unit})\cdot\frac{1}{\nu(s)}\cdot
a(s)y\Big{(}R_{4}(y,\mu(s)\nu(s)z,s)\cdot\mu(s)^{2}\nu(s)^{2}z^{2}$
$\displaystyle+2R_{5}(s)\cdot\mu(s)\nu(s)z\cdot
y+R_{6}(y,\mu(s)\nu(s)z,s)\nu(s)\cdot y^{2}$
$\displaystyle+\mu(s)^{2}\nu(s)^{2}R_{4}(y,s)\Big{)},$

where $R_{4}$ is a unit, see Remark 3.6. We remark that if we blowup $X$ at
the $H_{i}^{\prime}$s sufficiently many times, we can assume that $a(s)$ is
divisible by $\nu(s)$. Indeed, if we blow up $X$ at $H_{i}$, then $a(s)$
becomes $T_{i}\cdot a(s)$ but $\nu(s)$ remains the same. By abuse of notation,
we denote $\frac{1}{\nu(s)}\cdot a(s)$ by $a(s)$ in the following
computations.

If we set $\zeta=\mu(s)\nu(s)z$, then by using the Weierstrass preparation
theorem with respect to the variable $\zeta$ for the function in the
parenthesis after $a(s)y$ above, we can write

$G(y,z,s)=(\mathrm{unit})\cdot a(s)y\cdot\Big{(}\zeta^{2}+2R_{7}(y,s)\zeta
y+R_{8}(y,s)\Big{)},$

where $R_{8}=0\operatorname{mod}\,(y)^{2}$. It follows that we can write

$G(y,z,s)=(\mathrm{unit})\cdot
a(s)\cdot\Big{(}y\cdot\big{(}\mu(s)\nu(s)z+R_{7}(y,s)y\big{)}^{2}+R_{9}(y,s)\Big{)},$

where the terms of $R_{9}(y,z)=0\operatorname{mod}\,(y)^{3}$. Since $X$ has
$D_{4}$-type singularities over $S\setminus H$, we deduce that

$R_{9}(y,s)=b(s)y^{3}+y^{4}c(y,s),$

where the zero locus of $b$ is contained in $H$. By blowing up $X$ at the
$H_{i}$’s several times as in Subsection 3.C, the function $R_{9}(y,s)$ can be
written as

$R_{9}(y,s)=b(s)\cdot R_{10}(y,s)\cdot y^{3},$

were $R_{10}$ is a unit, see Remark 3.5. Then we can write

$F(x,y,z,s)=x^{2}+(\mathrm{unit})\cdot a(s)\cdot
y\cdot\big{(}\mu(s)\nu(s)z+R_{7}(y,s)y\big{)}^{2}+(\mathrm{unit})\cdot
a(s)b(s)\cdot y^{3}.$

Now as in the proof for the Case 1, we deduce that $X$ can be defined by

$x^{2}+\alpha(s)\cdot y(\gamma(s)z-u(y,s)y)^{2}+\beta(s)y^{3}=0,$

as in Proposition 5.2. 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. ∎

######  Remark 5.10.

In Lemma 5.9, the objective that we first construct $f_{1}\colon X_{1}\to X$
is to make the finite cover $f_{2}\colon X_{2}\to X_{1}$ functorial. For the
holomorphic function $T_{i}$ on $S$, there is no canonical way to make it a
holomorphic function on $X$. And the Cartier divisors in $X$ defined by
$T_{i}=0$ depend on the choice of coordinates in Setup 5.1. However, in a
neighborhood $V$ of the strict transform $S_{1}$ of $S$ in $X_{1}$, the
Cartier divisor defined by $T_{i}=0$ is the $f_{1}$-exceptional divisor
defined by $f_{1}^{*}\mathcal{I}_{H_{i}}$, where $\mathcal{I}_{H_{i}}$ is the
ideal sheaf of $H_{i}$ in $X$. In particular, this Cartier divisor is
independent on the choice of the coordinates in Setup 5.1. More explicitly,
assume that

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

is a change of coordinates, and we regard
$x^{\prime},y^{\prime},z^{\prime},s^{\prime}$ as holomorphic vector-valued
functions on $(x,y,z,s)$. Then $s^{\prime}=s+P$ with
$P=0\operatorname{mod}\,(x,y,z)$ and there is a $3\times 3$ matrix $\Theta(s)$
with coefficients as holomorphic functions on $S$, such that
$(x^{\prime},y^{\prime},z^{\prime})=(x,y,z)\cdot\Theta(s)\operatorname{mod}(x,y,z)^{2}$.
Let $T=T_{1}\cdots T_{n}$. Then $f_{1}$ can be written in coordinates as

$f_{1}\colon(x_{1},y_{1},z_{1},s)\mapsto(Tx_{1},Ty_{1},Tz_{1},s)=(x,y,z,s)$

and

$f_{1}\colon(x_{1}^{\prime},y_{1}^{\prime},z_{1}^{\prime},s^{\prime})\mapsto(Tx_{1}^{\prime},Ty_{1}^{\prime},Tz_{1}^{\prime},s^{\prime})=(x^{\prime},y^{\prime},z^{\prime},s^{\prime})$

As a consequence, the change of coordinates
$\varphi^{\prime}\colon(x_{1},y_{1},z_{1},s)\mapsto(x_{1}^{\prime},y_{1}^{\prime},z_{1}^{\prime},s^{\prime})$
satisfies

$s^{\prime}(x_{1},y_{1},z_{1},s)=s+P(Tx_{1},Ty_{1},Tz_{1},s)$

Since $P(x,y,z,s)=0\operatorname{mod}\,(x,y,z)$, there exists some holomorphic
function $Q$ such that $P(Tx_{1},Ty_{1},Tz_{1},s)=TQ(x_{1},y_{1},z_{1},s)$. It
follows that, up to shrinking $V$,

$T_{i}(x_{1},y_{1},z_{1},s)=(\mathrm{unit})\cdot(\varphi^{\prime})^{*}T_{i}(x_{1}^{\prime},y_{1}^{\prime},z_{1}^{\prime},s^{\prime}).$

As a result, on $V$, the finite morphism $f_{2}\colon X_{2}\to X_{1}$
constructed via two choices of coordinates in Setup 5.1 are the same.

##  6\. Local construction of covering spaces

In this section, we will construct a finite Galois cover for a hypersurface
$X$ defined by some equation as in Proposition 5.2.

###  6.A. Preparatory lemmas

In this part, we will prove some properties on varieties of the shape
$Y\times\mathbb{D}^{n}$, where $Y$ is a Du Val singularity.

######  Lemma 6.1.

Let $U\subseteq\mathbb{C}^{m}$ and $S\subseteq\mathbb{C}^{n}$ be open
neighborhoods of the origins. Let $f\colon U\times S\to S$ be a holomorphic
map such that $f(\mathbf{0}_{m},s)=s$ for all $s\in S$. Then there are open
neighborhoods $W\subseteq U$ and $T\subseteq S$ of the origins, such that
$f|_{W\times T}$ is deformation equivalent to the projection on $T$, with
values contained in $S$.

###### Proof.

Let $(u,s)$ be a coordinates system of $U\times S$. Assume that the polydisc
$(\mathbb{D}_{2r})^{n}$ of radius $2r>0$ is contained in $S$. We can define
the function $g(u,s)=f(u,s)-s$ which takes value in $\mathbb{C}^{n}$. Then the
assumption implies that $g(\mathbf{0}_{m},s)=0$ for all $s\in S$. Thus there
are open neighborhoods $W\subseteq U$ and $T\subseteq S$ of the origins, such
that $|g(x)|<r$ for $x\in W\times T$. We can choose $T$ so that
$T\subseteq(\mathbb{D}_{r})^{n}$. We define the functions $H(u,s,\mu)=s+\mu
g(u,s)$ on $W\times T\times[0,1]$. Then $H$ has values contained in $S$, and
it induces a deformation equivalence from the projection on $T$ to
$f|_{W\times T}$. ∎

######  Lemma 6.2.

Let $Y\subseteq\mathbb{C}^{3}$ be the hypersurface defined by $F(x,y,z)=0$,
such that $F$ is one of the following functions,

$\displaystyle F(x,y,z)$ $\displaystyle=$ $\displaystyle x^{2}+y^{2}+z^{l},\
l\geq 2,$ $\displaystyle F(x,y,z)$ $\displaystyle=$ $\displaystyle
x^{2}+yz^{2}+y^{l},\ l\geq 3,$ $\displaystyle F(x,y,z)$ $\displaystyle=$
$\displaystyle x^{2}+y^{3}+yz^{3},$ $\displaystyle F(x,y,z)$ $\displaystyle=$
$\displaystyle x^{2}+y^{3}+z^{l},l=4,5,$

where $l$ is a positive integer. Let $S$ be a simply connected open
neighborhood of the origin in $\mathbb{C}^{n}$ and let $X=Y\times S$. Assume
that $(T_{1},...,T_{n})$ is a coordinates system of $\mathbb{C}^{n}$. Let
$T=T_{1}\cdots T_{k}$ for some integer $0\leq k\leq n$, and let $H=\\{T=0\\}$.
We set $S^{\circ}=S\setminus H$. Then for any positive integer $M$, there
exists positive integers $p,q,r\geq M$ such that the morphism

$\varphi\colon(x,y,z,s)\mapsto(T^{p}x,T^{q}y,T^{r}z,s)$

induces an automorphism of $Y\times S^{\circ}$ over $S^{\circ}$.

###### Proof.

We only treat the case when $Y\subseteq\mathbb{C}^{3}$ is defined by

$0=F(x,y,z)=x^{2}+yz^{2}+y^{l}.$

The other cases are similar. In order that the map $\varphi$ is well-defined
from $Y\times S^{\circ}$ to itself, it is sufficient that there is some
positive integer $d$, so that

$(T^{p}x)^{2}+(T^{q}y)(T^{r}z)^{2}+(T^{q}y)^{l}=T^{d}(x^{2}+yz^{2}+y^{l}).$

It is equivalent to the system

$2p=q+2r=lq.$

There three unknowns and two equations, hence the system is indeterminate and
admits infinitely many integer solutions. Furthermore, since $l\geq 2$, we see
that the solutions for $q,r$ tend to $+\infty$ if $p$ tends to $+\infty$.
Hence there are integers $p,q,r\geq M$ such that $\varphi$ is well-defined.
Since $T$ is nowhere vanishing on $S^{\circ}$, we deduce that $\varphi$ is an
automorphism of $Y\times S^{\circ}$ over $S^{\circ}$. ∎

In the previous lemma, we can replace $T$ by any real number $\lambda>0$. It
follows that every loop $\gamma$ in $Y_{\operatorname{\textsubscript{\rm
sm}}}$ is deformation equivalent to some loop $\gamma^{\prime}$, which can be
as close to $\mathbf{0}_{3}$ as possible. This implies that the fundamental
group $\pi_{1}(Y_{\operatorname{\textsubscript{\rm sm}}})$ can be generated by
loops in the smooth locus of an arbitrarily small open neighborhood of
$\mathbf{0}_{3}$.

######  Lemma 6.3.

With the notation in Lemma 6.2, Let $U\subseteq Y$ be an open neighborhood of
$\mathbf{0}_{3}$ such that the natural morphism of fundamental groups
$\pi_{1}(U_{\operatorname{\textsubscript{\rm
sm}}})\to\pi_{1}(Y_{\operatorname{\textsubscript{\rm sm}}})$ is an
isomorphism. Let $\varphi\colon U\times S^{\circ}\to Y\times S^{\circ}$ be an
open embedding, whose restriction on $\\{\mathbf{0}_{3}\\}\times S^{\circ}$ is
the identity map. Then the induced morphism $\varphi_{*}$ of fundamental
groups from $\pi_{1}(U_{\operatorname{\textsubscript{\rm sm}}}\times
S^{\circ})$ to $\pi_{1}(Y_{\operatorname{\textsubscript{\rm sm}}}\times
S^{\circ})$ is an isomorphism.

###### Proof.

Let $o\in S^{\circ}$, let $u\in U_{\operatorname{\textsubscript{\rm sm}}}$ and
let $V=\\{u\\}\times S^{\circ}\subseteq U_{\operatorname{\textsubscript{\rm
sm}}}\times S^{\circ}$. We write $\pi_{1}(U_{\operatorname{\textsubscript{\rm
sm}}}\times S^{\circ})=H_{1}\times G_{1}$, where

$G_{1}=\pi_{1}(V)\cong\pi_{1}(S^{\circ})\cong\mathbb{Z}^{k}\mbox{ and
}H_{1}=\pi_{1}(U_{\operatorname{\textsubscript{\rm
sm}}}\times\\{o\\})\cong\pi_{1}(U_{\operatorname{\textsubscript{\rm sm}}}).$

Similarly, we write $\pi_{1}(Y_{\operatorname{\textsubscript{\rm sm}}}\times
S^{\circ})=H_{2}\times G_{2}$, where $G_{2}\cong\pi_{1}(S^{\circ})$ and
$H_{2}=\pi_{1}(Y_{\operatorname{\textsubscript{\rm
sm}}}\times\\{o\\})\cong\pi_{1}(Y_{\operatorname{\textsubscript{\rm sm}}})$.

The morphism $\varphi$ can be written in coordinates as

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

and we regard $x^{\prime},y^{\prime},z^{\prime},s^{\prime}$ as functions in
$(x,y,z,s)$. The assumption implies that $s^{\prime}(0,0,0,s)=s$. Let

$\psi\colon(x,y,z,s)\mapsto(x,y,z,s^{\prime}).$

By Lemma 6.1, there is an open neighborhood $W$ of $\mathbf{0}_{3}$ in $U$,
and an open neighborhood $T$ of $o\in S^{\circ}$, such that the restriction of
$s^{\prime}$ on $W\times T$ is deformation equivalent to $s(x,y,z,s)=s$. As a
consequence, the image $\psi(W_{sm}\times\\{o\\})$ is deformation equivalent
to $W_{\operatorname{\textsubscript{\rm sm}}}\times\\{o\\}$ inside
$U_{\operatorname{\textsubscript{\rm sm}}}\times S^{\circ}$. Then the image of

$\psi_{*}\colon\pi_{1}(W_{sm}\times\\{o\\})\to\pi_{1}(U_{\operatorname{\textsubscript{\rm
sm}}}\times S^{\circ})$

is equal to the image of $\pi_{1}(W_{\operatorname{\textsubscript{\rm
sm}}}\times\\{o\\})$ induced by the inclusion $W_{sm}\times\\{o\\}\to
U_{\operatorname{\textsubscript{\rm sm}}}\times S^{\circ}$. By the implicit
function theorem, we can assume that $\psi$ is an isomorphism from $W\times T$
to its image $Z\subseteq W\times S^{\circ}$. Since a neighborhood of
$\mathbf{0}_{3}$ in $Y$ is homeomorphic to the cone over the link of the
singularity, we can choose $W$ so that
$\pi_{1}(W_{\operatorname{\textsubscript{\rm
sm}}})\to\pi_{1}(U_{\operatorname{\textsubscript{\rm sm}}})$ is an
isomorphism.

Let $\eta\colon Z\to Y\times S^{\circ}$ be the morphism induced by
$\varphi\circ\psi^{-1}$, which is an open embedding. Then $\eta$ can be
written in coordinates as

$\eta\colon(x,y,z,s^{\prime})\mapsto(x^{\prime},y^{\prime},z^{\prime},s^{\prime}).$

Then $\eta(W\times\\{o\\})$ is a neighborhood of $(\mathbf{0}_{3},o)$ in the
fiber $Y\times\\{o\\}$. Since $\pi_{1}(Y_{\operatorname{\textsubscript{\rm
sm}}})$ can be generated by loops in an arbitrarily small neighborhood of
$\mathbf{0}_{3}\in Y$, we obtain that the morphism $\eta_{*}$ of fundamental
groups from $\pi_{1}(W_{\operatorname{\textsubscript{\rm sm}}}\times\\{o\\})$
to $H_{2}=\pi_{1}(Y_{\operatorname{\textsubscript{\rm sm}}}\times\\{o\\})$ is
surjective, hence is an isomorphism. Therefore

$\varphi_{*}=(\eta\circ\psi)_{*}\colon\pi_{1}(W_{\operatorname{\textsubscript{\rm
sm}}}\times\\{o\\})\to\pi_{1}(Y_{\operatorname{\textsubscript{\rm sm}}}\times
S^{\circ})$

induces an isomorphism from $\pi_{1}(W_{\operatorname{\textsubscript{\rm
sm}}}\times\\{o\\})$ to $H_{2}$. This implies that the morphism
$\varphi_{*}\colon\pi_{1}(U_{\operatorname{\textsubscript{\rm sm}}}\times
S^{\circ})\to\pi_{1}(Y_{\operatorname{\textsubscript{\rm sm}}}\times
S^{\circ})$ induces an isomorphism from $H_{1}$ to $H_{2}$.

Since $u$ is path connected to $\mathbf{0}_{3}$ in $U$, we see that $V$ is
deformation equivalent to $\\{\mathbf{0}_{3}\\}\times S^{\circ}$ inside
$U\times S^{\circ}$. Hence if $q\colon Y\times S^{\circ}\to S^{\circ}$ is the
natural projection, then $(q\circ\varphi)_{*}$ induces an isomorphism from
$\pi_{1}(V)$ to $\pi_{1}(S^{\circ})$. This implies that $\varphi_{*}$ is an
isomorphism from $\pi_{1}(U_{\operatorname{\textsubscript{\rm sm}}}\times
S^{\circ})$ to $\pi_{1}(Y_{\operatorname{\textsubscript{\rm sm}}}\times
S^{\circ})$ ∎

With the notation above, we remark that the fundamental group
$\pi_{1}(Y_{\operatorname{\textsubscript{\rm sm}}}\times S^{\circ})$ is
isomorphic to $\pi_{1}(Y_{\operatorname{\textsubscript{\rm
sm}}})\times\mathbb{Z}^{k}$. For this kind of groups, we construct a canonical
subgroup in the following lemma. After the Galois theory, it will induce a
canonical covering space.

######  Lemma 6.4.

Let $E=H\times G$ be a group, where $H\subseteq G$ be a finite subgroup group
and $G\subseteq H$ is a subgroup isomorphic to $\mathbb{Z}^{k}$ for some
integer $k\geq 0$. Let $N$ be the intersection of all complementary subgroups
of $H$ in $E$. Then $N$ is of finite index in $E$, and is invariant under any
automorphism of $E$. Furthermore, let $E^{\prime}\subseteq E$ be a normal
subgroup such that $H\subseteq E^{\prime}$ and that
$E/E^{\prime}\cong\mathbb{Z}^{l}$ for some integer $0\leq l\leq k$. If
$N^{\prime}$ is the intersection of all complementary subgroups of $H$ in
$E^{\prime}$, then $N\cap E^{\prime}=N^{\prime}$.

###### Proof.

Since $E$ is finitely generated, for any fixed number $i$, there are only
finitely many subgroups of $E$ of index $i$. Hence there are only finitely
many complementary subgroups of $H$ in $E$. It follows that $N$ is of finite
index in $E$. Since $H$ is equal to the set of all elements of finite order in
$E$, it is invariant under any automorphisms of $E$. Hence so is $N$.

For the second part of the lemma, on the one hand, we assume that $K$ is a
complementary subgroup of $H$ in $E$. Since $H\subseteq E^{\prime}$, we deduce
that $K\cap E^{\prime}$ is a complementary subgroup of $H$ in $E^{\prime}$.
Hence $N^{\prime}\subseteq N\cap E^{\prime}$.

On the other hand, the quotient map $E\to E/E^{\prime}$ admits a splitting
$E/E^{\prime}\to E$. Let $J\subseteq E$ be its image, which is a subgroup of
$E$. Then $E=JE^{\prime}$ and $J\cap E^{\prime}=\\{e\\}$, where $e$ is the
neutral element of $E$. Let $K^{\prime}$ be a complementary subgroup of $H$ in
$E^{\prime}$. Then $K:=JK^{\prime}$ is a complementary subgroup of $H$ in $E$.
Furthermore, $K\cap E^{\prime}=K^{\prime}$. It follows that $N\cap
E^{\prime}\subseteq N^{\prime}$. This completes the proof of the lemma. ∎

###  6.B. Construction of covering spaces

In the remainder of this section, we work with the following setup. Our goal
is to prove Proposition 6.6.

######  Setup 6.5.

We consider $X$, $\Delta$, $S$, $H$ and $G$ introduced as follows. Let $X$ be
a complex analytic variety and let $S\subseteq
X_{\operatorname{\textsubscript{\rm sing}}}$ be an irreducible component of
codimension $2$. Let $\Delta$ be a reduced divisor in $X$. There is a finite
group $G$ acting on $X$ such that $S$ and $\Delta$ are $G$-invariant.

We assume that $S$ is a simply connected open neighborhood of the origin
$o_{S}$ of $\mathbb{C}^{n}$, contained in a polydisc of radius smaller than 1.
Let $(T_{1},...,T_{n})$ be the coordinates on $S$, and let
$H_{i}=\\{T_{i}=0\\}$, with $i=1,...,k$ for some integer $0\leq k\leq n$.
Assume that $o_{S}$ and $H_{1},...,H_{k}$ are $G$-invariant. We set denote
$T=T_{1}\cdots T_{k}$ and $H=H_{1}+\cdots+H_{k}$. Assume that $X$ is an open
subset of the hyperplane $M$ in $\mathbb{D}^{3}\times S$ defined by an
equation $F(x,y,z,s)=0$, where $(x,y,z)$ are coordinates of $\mathbb{D}^{3}$
and $s\in S$. Assume that $F$ is of one of the following shapes (as in
Proposition 5.2),

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

where $l$ is an integer, $\alpha(s)$, $\beta(s)$ and $\gamma(s)$ are
holomorphic functions on $S$ whose zero loci are contained in $H$, and
$u(y,s)$ is a holomorphic function. The closed subset $S\subseteq X$ is
identified with $\mathbf{0}_{3}\times S$. Furthermore, if $\theta\colon M\to
S$ is the projection sending $(x,y,z,s)$ to $s$, then
$\Delta=\theta^{-1}(H)\cap X$. Let $S^{\circ}=S\setminus H$,
$M^{\circ}=\theta^{-1}(S^{\circ})$, and $X^{\circ}=\theta^{-1}(S^{\circ})\cap
X=X\setminus\Delta$. We will shrink $S$ around $o_{S}$ freely in the remainder
of the section. In particular, there is a disc
$\mathbb{D}^{\prime}\subseteq\mathbb{D}$ such that

$M^{\prime}:=((\mathbb{D}^{\prime})^{3}\times S)\cap M\subseteq X\subseteq M.$

Let $F^{\prime}(x,y,z)$ be the holomorphic function obtained by replacing
$\alpha,\beta,\gamma$ with 1 and replacing $u$ with 0 in the expression of
$F$. We denote by $Y$ the hypersurface in $\mathbb{C}^{3}$ defined by
$F^{\prime}=0$. Then $Y$ has an isolated Du Val singularity at
$\mathbf{0}_{3}$. Let $(U_{\lambda})_{\lambda\in\mathbb{N}}$ be a decreasing
sequence of bounded open subsets in $Y$, such that it is a neighborhood basis
of $Y_{\operatorname{\textsubscript{\rm sing}}}=\\{\mathbf{0}_{3}\\}$, and
that the natural morphisms of fundamental groups

$\pi_{1}((U_{\lambda})_{\operatorname{\textsubscript{\rm
sm}}})\to\pi_{1}(Y_{\operatorname{\textsubscript{\rm sm}}})$

are isomorphisms. We note that such a sequence exists, since a neighborhood of
$\mathbf{0}_{3}$ in $Y$ is homeomorphic to the cone over the link of the
singularity.

Now we can state the main objective of this section.

######  Proposition 6.6.

With the notation of Setup 6.5, up to shrinking $S$, there is an integer
$N\geq 0$ such that the following properties hold. Let
$\rho\colon\widehat{X}\to X$ be the blowup of $X$ at the $H_{i}$’s as in
Subsection 3.C, such that there are at least $N$ blowups whose centers are
$H_{i}$ for all $i$. Particularly, there is a natural action of $G$ on
$\widehat{X}$ so that $\rho$ is $G$-invariant. Let $\widehat{S}$ be the strict
transform of $S$ in $\widehat{X}$, let $\widehat{H}\subseteq\widehat{S}$ be
the preimage of $H$, and let $\widehat{X}^{\circ}=\rho^{-1}(X^{\circ})$.

Then there is an open neighborhood $W$ of $\widehat{S}$, a finite cover
$\nu\colon\overline{W}\to W$ satisfying the following properties.

  1. (1)

$\nu$ is quasi-étale and Galois over $W^{\circ}:=W\cap\widehat{X}^{\circ}$.

  2. (2)

$\overline{W}$ is normal.

  3. (3)

$\overline{W}$ is smooth over $W^{\circ}$.

  4. (4)

$W$ is $G$-invariant and the action of $G$ lifts to $\overline{W}$.

  5. (5)

The finite morphism $\nu$ is functorial in the following sense. For any point
$\hat{o}\in W$, let $o=\rho(\hat{o})\in X$. Assume that there is an open
neighborhood $W^{\prime}$ of $\hat{o}$ in $W$, such that there is a finite
morphism $\nu^{\prime}\colon\overline{W}^{\prime}\to W^{\prime}$, which is
constructed by the same method on a neighborhood $X^{\prime}$ of
$\rho(W^{\prime})$ in $X$. Then there is a $G_{\hat{o}}$-invariant open
neighborhood $W^{\prime\prime}$ of $\hat{o}$ in $\widehat{X}$, contained in
$W^{\prime}$, such that every connected component of
$\nu^{-1}(W^{\prime\prime})$ is isomorphic to every connected component
$(\nu^{\prime})^{-1}(W^{\prime\prime})$. Here $G_{\hat{o}}$ is the subgroup of
$G$ which fixes $\hat{o}.$

We will apply this local construction on orbifold charts of some orbi-
varieties. The item (5) above is to ensure that the constructions are
compatible along the overlaps. We will divide the proof of the proposition
into a sequence of lemma. The sketch is as follows. As shown in the Whitney’s
stratification, there is an open tubular neighborhood $V$ of $S^{\circ}$ in
$X^{\circ}$, with a deformation retract $h\colon V\to S^{\circ}$, such that
the fibers of of $h$ are homeomorphic to the cone over the link of the
singularity $\mathbf{0}_{3}\in Y$. We give an explicit construction of such a
tubular neighborhood, such that $\pi_{1}(V_{\operatorname{\textsubscript{\rm
sm}}})\cong\pi_{1}(Y_{\operatorname{\textsubscript{\rm
sm}}})\times\mathbb{Z}^{k}$. Then we can construct a finite morphism over
$V_{\operatorname{\textsubscript{\rm sm}}}$ via the canonical subgroup of
$\pi_{1}(V_{\operatorname{\textsubscript{\rm sm}}})$ defined in Lemma 6.4. An
issue is that, in general, $V$ does not extend to an open neighborhood of $S$
in $X$. It collapses when approaching to $H\subseteq S$. Our solution to this
is to blowup $X$ at the $H_{i}$’s as in Subsection 3.C. We show that, after
some blowups, the preimage of $V$ extends to some open neighborhood $W$ of the
strict transform of $S$. Then the finite cover of
$V_{\operatorname{\textsubscript{\rm sm}}}$ extends to some finite cover of
$W$.

######  Lemma 6.7.

With the notation in Setup 6.5, up to shrinking $S$ around the origin, There
is an open embedding from $M^{\circ}$ to $Y\times S^{\circ}$, whose inverse
$\varphi$ can be written in coordinates as

$\varphi\colon(\overline{x},\overline{y},\overline{z},s)\mapsto(a(s)\overline{x},b(s)\overline{y},\zeta(\overline{y},\overline{z},s),s)=(x,y,z,s),$

where $a(s),b(s)$ are holomorphic functions on $S$, and

$\zeta(\overline{y},\overline{z},s)=\frac{1}{\gamma(s)}(c(s)\overline{z}+w(\overline{y},s)),$
$w(\overline{y},s))=-u(b(s)\cdot\overline{y},s)\cdot b(s)\overline{y},$

for some holomorphic function $c(s)$ on $S$. Moreover, we can impose that the
zero loci of $a,b,c$ are contained in $H$, and that $b,c$ are divisible by
$\gamma$.

###### Proof.

We treat the case when

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

The other cases are similar. In order that $\varphi$ is formally well-defined,
it is sufficient that, there is a unit function $v(s)$ on $S^{\circ}$, such
that

$a^{2}\cdot\overline{x}^{2}+b^{3}\alpha\cdot\overline{y}^{3}+\beta\cdot\overline{y}(\gamma\cdot\zeta+u(b\overline{y},s))^{3}=v\cdot(\overline{x}^{2}+\overline{y}^{3}+\overline{y}\cdot\overline{z}^{3}).$
(6.1)

By comparing the coefficients of both sides, we get

$a^{2}=\alpha\cdot b^{3}=\beta\cdot bc^{3}=v.$

This system contains 4 unknowns and 3 equations, hence it is indeterminate.
Since the zero loci of $\alpha$ and $\beta$ are contained in $H$, we can write

$\alpha=T_{1}^{l_{1}}\cdots T_{k}^{l_{k}}\cdot\alpha^{\prime}\mbox{ and
}\beta=T_{1}^{l^{\prime}_{1}}\cdots T_{k}^{l^{\prime}_{k}}\cdot\beta^{\prime}$

where $l_{1},...,l_{k},l_{1}^{\prime},...,l_{k}^{\prime}\geq 0$ are integers
and $\alpha^{\prime},\beta^{\prime}$ are unit holomorphic functions on $S$. By
taking the logarithms of $\alpha^{\prime}$ and $\beta^{\prime}$, we deduce
that $a,b,c,v$ exist as meromorphic functions on $S$, with poles and zeros
contained in $H$. We also note that, if $(a,b,c,v)$ satisfies the previous
equations, then so does $(T^{9}\cdot a,T^{6}\cdot b,T^{4}\cdot c,T^{18}\cdot
v)$, see also Lemma 6.2. Since the zero locus of $\gamma$ is contained in $H$,
up to multiplying $a,b,c$ by powers of $T$, we may assume that they are
holomorphic functions and that $b,c$ are divisible by $\gamma$.

By plugging in the expression of $w$ in $\zeta$, we have

$\gamma\cdot\frac{1}{\gamma}\Big{(}c\cdot\overline{z}+w(\overline{y},s)\Big{)}+u(b\cdot\overline{y},s)\cdot
b(s)\overline{y}=c\cdot\overline{z}.$

Hence the equality (6.1) holds. In conclusion, there exist holomorphic
functions $a,b,c$ on $S$ such that $\varphi$ is formally well-defined. The
morphism $\varphi^{-1}$ from $M^{\circ}$ to $Y\times S^{\circ}$ can be written
in coordinates as

$\varphi^{-1}\colon(x,y,z,s)\mapsto(a^{-1}(s)x,b^{-1}(s)y,c^{-1}(s)(\gamma(s)z+u(y,s)y))=(\overline{x},\overline{y},\overline{z},s).$

Then $\varphi^{-1}$ is a well-defined holomorphic function. Since
$a,b,c,\gamma$ are nowhere vanishing holomorphic functions on $S^{\circ}$, we
see that $\varphi^{-1}$ is locally biholomorphic and is injective. Hence
$\varphi^{-1}$ is an open embedding. ∎

######  Lemma 6.8.

With the notation of Setup 6.5, we fix a morphism $\varphi$ as in Lemma 6.7 so
that $\varphi^{-1}$ is an open embedding from $M^{\circ}$ to $Y\times
S^{\circ}$ over $S^{\circ}$. Then there are integers $p,q,r>0$ such that the
endomorphism $\eta$ of $Y\times S$ defined by

$\eta\colon(\overline{x},\overline{y},\overline{z},s)\mapsto(T^{p}\overline{x},T^{q}\overline{y},T^{r}\overline{z},s),$

satisfies the the following properties.

The image $\eta(U_{0}\times S^{\circ})$ is contained in
$\varphi^{-1}(M^{\prime}\cap M^{\circ})$. In another word, we have

$(\varphi\circ\eta)(U_{0}\times S^{\circ})\subseteq M^{\prime}\cap
M^{\circ}\subseteq X^{\circ}.$

Furthermore, for any point $s\in S$ and any neighborhood $X^{\prime}$ of $s$
in $X$, there is an open neighborhood $S^{\prime}\subseteq S$ of $s$ in $S$,
there is some $\lambda\in\mathbb{N}$, such that

$(\varphi\circ\eta)\big{(}U_{\lambda}\times(S^{\prime}\cap S^{\circ})\big{)}$

is contained in $X^{\prime}$. If in addition $s\in H$, then we may pick
$\lambda=0$ for the previous neighborhood.

###### Proof.

Since $a$, $b$, $c\cdot\gamma^{-1}$, $b\cdot\gamma^{-1}$ are holomorphic
functions on $S$, there is a positive number $K$ such that

$|a(s)\overline{x}|\leq K|\overline{x}|,\ |b(s)\overline{y}|\leq
K|\overline{y}|\mbox{ and }|\frac{c(s)}{\gamma(s)}\overline{z}|\leq
K|\overline{z}|.$

Since $u$ is also a holomorphic function, we may assume that
$|u(b(s)\overline{y},s)|$ is bounded for
$(\overline{y},s)\in\mathbb{D}_{\varepsilon}\times S$, where
$\mathbb{D}_{\varepsilon}\subseteq\mathbb{C}$ is the disc of radius
$\varepsilon>0$. Hence we may assume further that

$|\zeta(\overline{y},\overline{z},s)|=\Big{|}\frac{c(s)}{\gamma(s)}\overline{z}-\frac{b(s)}{\gamma(s)}\cdot
u(b(s)\overline{y},s)\cdot\overline{y}\Big{|}\leq
K(|\overline{z}|+|\overline{y}|)$

whenever $|\overline{y}|<\varepsilon$.

Since $M^{\prime}\cap M^{\circ}$ is a hypersurface defined in
$\mathbb{D^{\prime}}^{3}\times S^{\circ}$, in order that a point
$(\overline{x},\overline{y},\overline{z},s)$ in $Y\times S^{\circ}$ is
contained in $\varphi^{-1}(X^{\circ})$, it is necessary and sufficient that

$|a(s)\overline{x}|<R^{\prime},\ |b(s)\overline{y}|<R^{\prime}\mbox{ and
}|\zeta(\overline{y},\overline{z},s)|<R^{\prime},$

where $R^{\prime}$ is the radius of $\mathbb{D}^{\prime}$. Since $U_{0}$ is
bounded, and since $S$ in contained in a polydisc of radius smaller than 1,
there is some integer ${N_{0}}>0$ such that

$K|T^{N_{0}}\overline{x}|<R^{\prime},\ K|T^{N_{0}}\overline{y}|<R^{\prime},\
|T^{N_{0}}\overline{y}|<\varepsilon,\ \mbox{ and
}K(|T^{N_{0}}\overline{y}|+|T^{N_{0}}\overline{z}|)<R^{\prime}.$

for all $(\overline{x},\overline{y},\overline{z},s)\subseteq U_{0}\times
S^{\circ}$. By Lemma 6.2, there exist integers $p,q,r\geq{N_{0}}$ such that

$\eta\colon(\overline{x},\overline{y},\overline{z},s)\mapsto(T^{p}\overline{x},T^{q}\overline{y},T^{r}\overline{z},s),$

is a well-defined endomorphism of $Y\times S$. We then deduce that
$\eta(U_{0}\times S^{\circ})$ is contained in $\varphi^{-1}(M^{\prime}\cap
M^{\circ})$.

For the second property on $\eta$, we assume that $X^{\prime}$ is a
neighborhood of $s$ in $X$. Then there is a positive number
$\varepsilon^{\prime}$, there is an open neighborhood $S^{\prime}\subseteq S$
of $s$ in $S$, such that

$M\cap\big{(}\mathbb{D}_{\varepsilon^{\prime}})^{3}\times
S^{\prime}\big{)}\subseteq X^{\prime},$

where $\mathbb{D}_{\varepsilon^{\prime}}\subseteq\mathbb{C}$ is the disc of
radius $\varepsilon^{\prime}$. Since $(U_{\lambda})_{\lambda\in\mathbb{N}}$ is
a neighborhood basis of $\mathbf{0}_{3}\in Y$, there is certain
$\lambda\in\mathbb{N}$ such that

$K|T^{p}\overline{x}|<\varepsilon^{\prime},\
K|T^{q}\overline{y}|<\varepsilon^{\prime},\
|T^{q}\overline{y}|<\varepsilon\mbox{ and
}K(|T^{q}\overline{y}|+|T^{r}\overline{z}|)<\varepsilon^{\prime}.$

for all $(\overline{x},\overline{y},\overline{z},s)\in U_{\lambda}\times
S^{\circ}.$ It follows that

$(\varphi\circ\eta)\big{(}U_{\lambda}\times(S^{\prime}\cap S^{\circ})\big{)}$

is contained in $X^{\prime}$.

If in addition $s\in H$, then the value of $T$ at $s$ is equal to 0. Hence we
may pick $S^{\prime}$ in order that $|T|^{N_{0}}$ is very small on
$S^{\prime}$. More precisely, since $U_{0}$ is bounded, we can pick
$S^{\prime}$ so that

$K|T^{p}\overline{x}|<\varepsilon^{\prime},\
K|T^{q}\overline{y}|<\varepsilon^{\prime},\
|T^{q}\overline{y}|<\varepsilon\mbox{ and
}K(|T^{q}\overline{y}|+|T^{r}\overline{z}|)<\varepsilon^{\prime}.$

for all $(\overline{x},\overline{y},\overline{z},s)\subseteq U_{0}\times
S^{\prime}$. This completes the proof of the lemma. ∎

In the remainder of the section, by replacing $\varphi$ with
$\varphi\circ\eta$, we assume that $\varphi$ satisfies the properties of
$\varphi\circ\eta$ in Lemma 6.8. Let

$V_{\lambda}=\varphi(U_{\lambda}\times S^{\circ})\subseteq M^{\prime}\cap
M^{\circ}\subseteq X^{\circ}.$

Then $(V_{\lambda})_{\lambda\in\mathbb{N}}$ is a decreasing sequence of
tubular neighborhood of $S^{\circ}$ in $X^{\circ}$. The fundamental group
$\pi_{1}((V_{\lambda})_{\operatorname{\textsubscript{\rm sm}}})$ is isomorphic
to $\pi_{1}(Y_{\operatorname{\textsubscript{\rm sm}}})\times\mathbb{Z}^{k}$.
We observe that $V_{\lambda}$ does not extend to an open neighborhood of $S$
in $X$. Indeed, since $N_{0}>0$, a point $(x,y,z,s)\in V_{\lambda}$ approaches
to $(0,0,0,s_{H})$ when $s$ tends to a point $s_{H}\in H$. In the following
lemma, we show that, if we blowup $X$ at the $H_{i}$’s for sufficiently many
times, then the preimage of $V_{\lambda}$ extend to a neighborhood of the
strict transform of $S$.

######  Lemma 6.9.

With the notation above, there exists an integer $N>0$ such that the following
property holds. Let $\rho\colon\widehat{M}\to M$ be a morphism obtained by
blowing up $M$ at the $H_{i}^{\prime}$s as in Subsection 3.C, such that there
are at least $N$ blowups whose centers are $H_{i}$ for all $i$. Let
$\widehat{X}=\rho^{-1}(X)$, $\widehat{V}_{\lambda}=\rho^{-1}(V_{\lambda})$ and
let $\widehat{S}$ be the strict transform of $S$ in $\widehat{M}$. Then there
is a sequence $(W_{\lambda})_{\lambda\in N}$ of open neighborhoods of
$\widehat{S}$ in $\widehat{X}$, such that
$W_{\lambda}\cap\rho^{-1}(X^{\circ})\subseteq\widehat{V}_{\lambda}$. Moreover,
$W_{\lambda}$ is invariant under the natural action of $G$ on $\widehat{X}$.

###### Proof.

The application $\varphi^{-1}\colon M^{\circ}\to Y\times S^{\circ}$ can be
written with coordinates as

$\varphi^{-1}\colon(x,y,z,s)\mapsto(a(s)^{-1}x,b(s)^{-1}y,\xi(y,z,s),s),$

where $\xi(y,z,s)=c(s)^{-1}(\gamma(s)z+u(y,s)y)$. Hence, for every
$\varepsilon>0$ small enough, a point $(x,y,z,s)\in M^{\circ}$ is contained in

$\varphi\big{(}(Y\cap(\mathbb{D}_{\varepsilon})^{3})\times S^{\circ}\big{)}$

if and only if

$|a(s)^{-1}x|<\varepsilon,\ |b(s)^{-1}y|<\varepsilon\mbox{ and
}|c(s)^{-1}(\gamma(s)z+u(y,s)y)|<\varepsilon.$ (6.2)

Since the zero loci of $a,b,c$ are contained in $H$, there is some positive
integer $N$ such that

$T^{N}\cdot a(s)^{-1},\ T^{N}\cdot b(s)^{-1}\mbox{ and }T^{N}\cdot c(s)^{-1}$

are holomorphic functions on $S$. We will show that this number $N$ satisfies
the assertion of the lemma.

There are coordinates system $(\hat{x},\hat{y},\hat{z},s)$ on $\widehat{M}$ in
a neighborhood of $\widehat{S}$, such that $\rho$ can be expressed as

$\rho\colon(\hat{x},\hat{y},\hat{z},s)\mapsto(\hat{T}\hat{x},\hat{T}\hat{y},\hat{T}\hat{z},s),$

where $\hat{T}=T_{1}^{d_{1}}\cdots T_{k}^{d_{k}}$ and $d_{i}$ is the number of
blowups in $\rho$ whose centers are $H_{i}$. By pulling back the conditions of
(6.2) to $\widehat{M}$, we obtain that

$|\hat{T}\cdot a(s)^{-1}\hat{x}|,\ |\hat{T}\cdot b(s)^{-1}\hat{y}|,\
|\hat{T}\cdot
c(s)^{-1}(\gamma(s)\hat{z}+u(\hat{T}\hat{y},s)\hat{y})|<\varepsilon.$

Since $d_{1},...,d_{k}\geq N$, the functions in the absolute values above are
holomorphic on $(\hat{x},\hat{y},\hat{z},s)$. Furthermore, they are equal to
$0$ when $(\hat{x},\hat{y},\hat{z},s)=(0,0,0,s)$. Hence there is some
$\delta>0$ such that if $\hat{o}=(\hat{x},\hat{y},\hat{z},s)$ is contained in
$\widehat{M}\cap((\mathbb{D}_{\delta})^{3}\times\widehat{S})$, then

$\rho(\hat{o})\in\varphi\big{(}(Y\cap(\mathbb{D}_{\varepsilon})^{3})\times
S^{\circ}\big{)}.$

We note that the numbers $d_{1},...,d_{k}$ and $N$ are independent of
$\varepsilon$. Since for every $\lambda\in\mathbb{N}$, there is some
$\varepsilon_{\lambda}>0$ such that the intersection of $Y$ and the polydisc
$(\mathbb{D}_{\varepsilon_{\lambda}})^{3}$ is contained in $U_{\lambda}$. By
letting $\varepsilon=\varepsilon_{\lambda}$ in the previous argument, we can
obtain $W_{\lambda}$. Replacing $W_{\lambda}$ by the intersection of its
images under $G$, we may assume that is is $G$-invariant. This completes the
proof of the lemma. ∎

######  Lemma 6.10.

With the notation in Lemma 6.9, we fix such a bimeromorphic morphism
$\rho\colon\widehat{X}\to X$ and such a sequence $(W_{\lambda})_{\lambda\in
N}$ of open neighborhoods of $\widehat{S}$ in $\widehat{X}$. We set
$\widehat{X}^{\circ}=\rho^{-1}(X^{\circ})$, $W=W_{0}$ and
$W^{\circ}=W\cap\widehat{X}^{\circ}$. There is a sequence of morphisms of
fundamental groups

$\textstyle{\pi_{1}(W^{\circ}_{\operatorname{\textsubscript{\rm
sm}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{*}}$$\textstyle{\pi_{1}((V_{0})_{\operatorname{\textsubscript{\rm
sm}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{-1}_{*}}$$\textstyle{\pi_{1}(Y_{\operatorname{\textsubscript{\rm
sm}}}\times S^{\circ}).}$

We denote by $\iota\colon\pi_{1}(W^{\circ}_{\operatorname{\textsubscript{\rm
sm}}})\to\pi_{1}(Y_{\operatorname{\textsubscript{\rm sm}}}\times S^{\circ})$
the composition of the previous sequence. Then $\iota$ is surjective and its
kernel is independent on the choice of the open embedding $\varphi^{-1}$ from
$M^{\prime}\cap M^{\circ}$ to $Y\times S^{\circ}$ in the following sense.
Assume that $\psi\colon(M^{\prime}\cap M^{\circ})\to Y\times S^{\circ}$ is an
open embedding, whose restriction on $S^{\circ}\subseteq X^{\circ}$ is equal
to the identity map from $S^{\circ}$ to $\\{\mathbf{0}_{3}\\}\times
S^{\circ}$. Then the morphism of fundamental groups

$(\psi\circ\rho)_{*}\colon\pi_{1}(W^{\circ}_{\operatorname{\textsubscript{\rm
sm}}})\to\pi_{1}(Y_{\operatorname{\textsubscript{\rm sm}}}\times S^{\circ})$

is surjective and its kernel is equal to the kernel of $\iota$. In particular,
this kernel is $G$-invariant.

###### Proof.

We notice that $W$ satisfies the conditions of Setup 6.5 with respect to
$\widehat{S}$. By applying Lemma 6.9 to $W$, there is an open embedding
$\hat{\varphi}\colon U_{0}\times S^{\circ}\to W$ whose restriction on
$\\{\mathbf{0}_{3}\\}\times S^{\circ}$ is the identity map. By Lemma 6.3, the
composition of the sequence

$\textstyle{\pi_{1}((U_{0})_{\operatorname{\textsubscript{\rm sm}}}\times
S^{\circ})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\varphi}_{*}}$$\textstyle{\pi_{1}(W^{\circ}_{\operatorname{\textsubscript{\rm
sm}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{\pi_{1}(Y_{\operatorname{\textsubscript{\rm
sm}}}\times S^{\circ}).}$

is an isomorphism. This implies that $\iota$ is surjective.

We recall that $\varphi\colon U_{0}\times S^{\circ}\to V_{0}$ is an
isomorphism, whose restriction on ${\mathbf{0}_{3}}\times S^{\circ}$ is the
identity map. Thus

$\varphi^{-1}_{*}\colon\pi_{1}((V_{0})_{\operatorname{\textsubscript{\rm
sm}}})\to\pi_{1}(Y_{\operatorname{\textsubscript{\rm sm}}}\times S^{\circ})$

an isomorphism. We obtain that
$\rho_{*}\colon\pi_{1}(W^{\circ}_{\operatorname{\textsubscript{\rm
sm}}})\to\pi_{1}((V_{0})_{\operatorname{\textsubscript{\rm sm}}})$ is
surjective as well.

Assume that we have a morphism $\psi\colon M^{\prime}\cap M^{\circ}\to Y\times
S^{\circ}$ as in the lemma. Then the morphism

$\psi\circ\varphi\colon U_{0}\times S^{\circ}\to Y\times S^{\circ}$

is an open embedding whose restriction on ${\mathbf{0}_{3}}\times S^{\circ}$
is the identity map. Thus it induces an isomorphism from
$\pi_{1}((U_{0})_{\operatorname{\textsubscript{\rm sm}}}\times S^{\circ})$ to
$\pi_{1}(Y_{\operatorname{\textsubscript{\rm sm}}}\times S^{\circ})$ by Lemma
6.3. Hence

$\psi_{*}\colon\pi_{1}((V_{0})_{\operatorname{\textsubscript{\rm
sm}}})\to\pi_{1}(Y_{\operatorname{\textsubscript{\rm sm}}}\times S^{\circ})$

is an isomorphism. Since
$\rho_{*}\colon\pi_{1}(W^{\circ}_{\operatorname{\textsubscript{\rm
sm}}})\to\pi_{1}((V_{0})_{\operatorname{\textsubscript{\rm sm}}})$ is
surjective, we deduce that $(\psi\circ\rho)_{*}$ is surjective. Furthermore,
the kernel of $(\psi\circ\rho)_{*}$ is the one of $\rho_{*}$, which is
independent of $\psi$.

We will show that the kernel $K$ of $\iota$ is $G$-invariant. Let $g\in G$ be
an element. Then $g^{-1}K$ is the preimage of $K$ under the morphism of
fundamental groups induced by $g\colon W\to W$. It follows that $g^{-1}K$ is
the kernel of

$(\varphi^{-1}\circ\rho\circ
g)_{*}\colon\pi_{1}(W^{\circ}_{\operatorname{\textsubscript{\rm
sm}}})\to\pi_{1}(Y_{\operatorname{\textsubscript{\rm sm}}}\times S^{\circ}).$

We define the morphism $\psi\colon M^{\prime}\cap M^{\circ}\to Y\times
S^{\circ}$ as the composition of the following sequence

$M^{\prime}\cap M^{\circ}\overset{g}{\longrightarrow}g(M^{\prime}\cap
M^{\circ})\longhookrightarrow
M^{\circ}\overset{\varphi^{-1}}{\longrightarrow}Y\times
S^{\circ}\overset{g^{-1}}{\longrightarrow}Y\times S^{\circ},$

where the last morphism $g^{-1}$ is defined by the action of $G$ on
$S^{\circ}$, which is an isomorphism. Since $\rho$ is $G$-invariant, we have
$\rho\circ g=g\circ\rho$ as morphisms from $W$ to $M^{\prime}$. It follows
that $g^{-1}K$ is the kernel of

$(\psi\circ\rho)_{*}\colon\pi_{1}(W^{\circ}_{\operatorname{\textsubscript{\rm
sm}}})\to\pi_{1}(Y_{\operatorname{\textsubscript{\rm sm}}}\times S^{\circ}).$

By the previous paragraph, we deduce that $g^{-1}K=K$. This completes the
proof of the lemma. ∎

######  Lemma 6.11.

With the notation in Lemma 6.10, let $\hat{o}\in W$ be a point and let
$o=\rho(\hat{o})\in X$. Let $U\subseteq X$ be an open neighborhood of $o$, and
let $U^{\circ}=U\cap X^{\circ}$. Assume that $\psi_{i}\colon U^{\circ}\to
Y\times S^{\circ}$, with $i=1,2$, are two open embeddings whose restrictions
on $S^{\circ}\cap U$ are the identity map from $S^{\circ}\cap U$ to its image
in $\\{\mathbf{0}_{3}\\}\times S^{\circ}$. Then there is an open neighborhood
$\widehat{U}\subseteq W$ of $\hat{o}$ in $W$, such that the morphisms of
fundamental groups $\pi_{1}(\widehat{U}_{\operatorname{\textsubscript{\rm
sm}}}\cap\widehat{X}^{\circ})\to\pi_{1}(Y_{\operatorname{\textsubscript{\rm
sm}}}\times S^{\circ})$ induced by $\psi_{1}\circ\rho$ and $\psi_{2}\circ\rho$
have the same kernel.

###### Proof.

We first assume that the point $\hat{o}$ is in
$W^{\circ}_{\operatorname{\textsubscript{\rm sm}}}$. In particular, $W$ is
smooth at $\hat{o}$. Hence we may let $\widehat{U}\subseteq W$ be a
contractible smooth neighborhood of $\hat{o}$ in $W$.

In the remainder of the proof, we let $\theta\colon X\to S$ be the projection
sending a point $(x,y,z,s)\in M$ to $s\in S$, and let
$\hat{\theta}\colon\widehat{M}\to S$ be the composition $\theta\circ\rho$.

We assume that $\hat{o}\in W^{\circ}_{\operatorname{\textsubscript{\rm
sing}}}$. Then $\hat{o}\in\widehat{S}^{\circ}$, where $\widehat{S}^{\circ}$ is
the strict transform of $S^{\circ}$ in $\widehat{X}^{\circ}$ This implies that
$o\in S^{\circ}\subseteq X^{\circ}$. By Lemma 6.8, there is a contractible
open neighborhood $S^{\prime}\subseteq S$ of $o$ in $S$, some
$\lambda\in\mathbb{N}$, such that

$V_{\lambda}^{\prime}:=\varphi\big{(}U_{\lambda}\times(S^{\prime}\cap
S^{\circ})\big{)}=V_{\lambda}\cap\theta^{-1}(S^{\prime})$

is contained in $U$. We define
$\widehat{U}=W_{\lambda}\cap\hat{\theta}^{-1}(S^{\prime})$. Then we have

$\rho(\widehat{U}\cap\widehat{X}^{\circ})\subseteq V_{\lambda}^{\prime}\mbox{
and }\rho(\widehat{U}_{\operatorname{\textsubscript{\rm
sm}}}\cap\widehat{X}^{\circ})\subseteq(V_{\lambda}^{\prime})_{\operatorname{\textsubscript{\rm
sm}}}.$

We notice that $V_{\lambda}^{\prime}\cong U_{\lambda}\times(S^{\prime}\cap
S^{\circ})$. Since $\psi_{1}$ and $\psi_{2}$ induce open embeddings from
$V_{\lambda}^{\prime}$ to $Y\times(S^{\prime}\cap S^{\circ})$, by Lemma 6.3,
the morphisms

$\pi_{1}((V_{\lambda}^{\prime})_{\operatorname{\textsubscript{\rm
sm}}})\to\pi_{1}(Y_{\operatorname{\textsubscript{\rm
sm}}}\times(S^{\prime}\cap S^{\circ}))$

induced by $\psi_{1}$ and $\psi_{2}$ are isomorphisms. It follows that the
kernels of the morphisms from
$\pi_{1}(\widehat{U}_{\operatorname{\textsubscript{\rm
sm}}}\cap\widehat{X}^{\circ})\to\pi_{1}(Y_{\operatorname{\textsubscript{\rm
sm}}}\times S^{\circ})$ induced by $\psi_{1}\circ\rho$ and $\psi_{2}\circ\rho$
are the same as the kernel of
$\rho_{*}\colon\pi_{1}(\widehat{U}_{\operatorname{\textsubscript{\rm
sm}}}\cap\widehat{X}^{\circ})\to\pi_{1}((V_{\lambda}^{\prime})_{\operatorname{\textsubscript{\rm
sm}}})$.

It remains to treat the case when $\hat{o}\in W\setminus W^{\circ}$. This
implies that $o\in H\subseteq S\subseteq X$. By Lemma 6.8, there is a
contractible open neighborhood $S^{\prime}\subseteq S$ of $o$ in $S$, such
that

$V_{0}^{\prime}:=\varphi\big{(}U_{0}\times(S^{\prime}\cap
S^{\circ})\big{)}=V_{0}\cap\theta^{-1}(S^{\prime})$

is contained in $U$. We define
$\widehat{U}=W\cap\hat{\theta}^{-1}(S^{\prime})$. Then the same argument of
the previous paragraph shows that $\widehat{U}$ satisfies the required
properties. This completes the proof of the lemma. ∎

Now we can complete the proof of Proposition 6.6.

###### Proof of Proposition 6.6.

We let $N$ be the integer in Lemma 6.9 and let $W$ be the one constructed in
Lemma 6.10. As in Lemma 6.10, there is a morphism of fundamental groups

$\iota\colon\pi_{1}(W^{\circ}_{\operatorname{\textsubscript{\rm
sm}}})\to\pi_{1}(Y_{\operatorname{\textsubscript{\rm sm}}}\times S^{\circ}).$

We note that $\pi_{1}(Y_{\operatorname{\textsubscript{\rm sm}}}\times
S^{\circ})\cong G_{Y}\times\mathbb{Z}^{k}$, where $G_{Y}$ is the fundamental
group of $Y_{\operatorname{\textsubscript{\rm sm}}}$, which is finite. Let
$\mathcal{N}\subseteq\pi_{1}(Y_{\operatorname{\textsubscript{\rm sm}}}\times
S^{\circ})$ be the subgroup defined in Lemma 6.4, and let
$\mathcal{H}=\iota^{-1}(\mathcal{N})$. Then $\mathcal{H}$ is a normal subgroup
of $\pi_{1}(W^{\circ}_{\operatorname{\textsubscript{\rm sm}}})$. We define
$\nu\colon\overline{W}\to W$ as the finite Galois cover induced by
$\mathcal{H}$, see Theorem 2.4. Then $\nu$ is étale over
$W^{\circ}_{\operatorname{\textsubscript{\rm sm}}}$, hence is quasi-étale over
$W^{\circ}$.

We will show that $\overline{W}^{\circ}:=\nu^{-1}(W^{\circ})$ is smooth. Let
$Z\to Y\times S^{\circ}$ be the finite quasi-étale morphism induced by the
subgroup $\mathcal{N}\subseteq\pi_{1}(Y_{\operatorname{\textsubscript{\rm
sm}}}\times S^{\circ})$, see Theorem 2.4. Then it factors through
$\overline{Y}\times S^{\circ}\to Y\times S^{\circ}$, where $\overline{Y}\to Y$
is the quasi-étale cover induced by the universal cover of
$Y_{\operatorname{\textsubscript{\rm sm}}}$. In particular, $\overline{Y}$ is
smooth. The Zariski’s purity theorem then implies that $Z$ is smooth. We note
that $\nu|_{\overline{W}^{\circ}}$ is the basechange of
$\varphi^{-1}\circ\rho\colon W^{\circ}\to Y\times S^{\circ}$ over $Z\to
Y\times S^{\circ}$. Since $\varphi^{-1}\circ\rho$ is an open embedding on
$W^{\circ}$, we deduce that $\overline{W}^{\circ}$ is an open subset of $Z$.
Hence $\overline{W}^{\circ}$ is smooth.

We will prove the item (4) of the proposition. It is enough to show that
$\mathcal{H}$ is $G$-invariant. Let $g\in G$ be an element. Then
$g^{-1}\mathcal{H}$ is the preiamge of $\mathcal{H}$ under the morphism of
fundamental groups induced by $g\colon W\to W$. Let

$\iota^{\prime}:=(g^{-1}\circ\varphi^{-1}\circ\rho\circ
g)_{*}\colon\pi_{1}(W^{\circ}_{\operatorname{\textsubscript{\rm
sm}}})\to\pi_{1}(Y_{\operatorname{\textsubscript{\rm sm}}}\times S^{\circ}),$

where the morphism $g^{-1}$ above is the auotmorphism of $Y\times S^{\circ}$
defined by the action of $G$ on $S^{\circ}$. Then
$g^{-1}\mathcal{H}=(\iota^{\prime})^{-1}\mathcal{N}$. We note that $\rho\circ
g=g\circ\rho$. By Lemma 6.10, the two morphisms $\iota$ and $\iota^{\prime}$
are surjective of the same kernel. Hence, there is an automorphism $\tau$ of
$\pi_{1}(Y_{\operatorname{\textsubscript{\rm sm}}}\times S^{\circ})$ such that

$\iota=\tau\circ\iota^{\prime}.$

By Lemma 6.4, we have $\tau^{-1}\mathcal{N}=\mathcal{N}$. It follows that
$g^{-1}\mathcal{H}=\mathcal{H}$. This shows the item (4).

It remains to prove the item (5). If
$\hat{o}\in\widehat{X}^{\circ}_{\operatorname{\textsubscript{\rm sm}}}$, then
by construction, $\nu$ an $\nu^{\prime}$ are both étale over $\hat{o}$. We can
let $W^{\prime\prime}$ be a smooth $G_{\hat{o}}$-invariant neighborhood of
$\hat{o}$ in $W^{\prime}$ in this case.

Next we assume that $\hat{o}$ is not contained in
$\widehat{X}^{\circ}_{\operatorname{\textsubscript{\rm sm}}}$. Then
$o=\rho(\hat{o})\in S$. By assumption, there is an open neighborhood
$X^{\prime}$ of $o=\rho(\hat{o})$ containing $\rho(W^{\prime})$, there is an
open embedding $(\varphi^{\prime})^{-1}\colon(X^{\prime}\cap X^{\circ})\to
Y\times S^{\circ}$, whose restriction on $X^{\prime}\cap S^{\circ}$ is the
identity map onto its image in $(\mathbf{0}_{3})\times S^{\circ}$, such that
$\nu^{\prime}$ is induced by the composition of the following morphisms of
groups

$\textstyle{\pi_{1}((W^{\prime}\cap\widehat{X}^{\circ})_{\operatorname{\textsubscript{\rm
sm}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{*}}$$\textstyle{\pi_{1}((X^{\prime}\cap
X^{\circ})_{\operatorname{\textsubscript{\rm
sm}}})}$$\scriptstyle{(\varphi^{\prime})^{-1}_{*}}$$\textstyle{\pi_{1}(Y_{\operatorname{\textsubscript{\rm
sm}}}\times
S^{\circ})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(Y_{\operatorname{\textsubscript{\rm
sm}}}\times S^{\circ})/\mathcal{N}.}$

By Lemma 6.11, there is a contractible open neighborhood $S^{\prime\prime}$ of
$o$ in $S$, a neighborhood $X^{\prime\prime}$ of $S^{\prime\prime}$ which is
contained in $X^{\prime}$, and an open neighborhood $W^{\prime\prime}$ of
$\hat{o}$ in $W^{\prime}$ such that $\rho(W^{\prime\prime})\subseteq
X^{\prime\prime}$, and the two morphisms

$\mu_{1}=(\varphi\circ\rho)_{*}\colon\pi_{1}((W^{\prime\prime}\cap\widehat{X}^{\circ})_{\operatorname{\textsubscript{\rm
sm}}})\to\pi_{1}(Y_{\operatorname{\textsubscript{\rm
sm}}}\times(S^{\prime\prime}\cap S^{\circ}))$

and

$\mu_{2}=(\varphi^{\prime}\circ\rho)_{*}\colon\pi_{1}((W^{\prime\prime}\cap\widehat{X}^{\circ})_{\operatorname{\textsubscript{\rm
sm}}})\to\pi_{1}(Y_{\operatorname{\textsubscript{\rm
sm}}}\times(S^{\prime\prime}\cap S^{\circ}))$

have the same kernel. Furthermore, they are both surjective by the same
argument of Lemma 6.10. It follows that there is an automorphism $\tau$ of
$\pi_{1}(Y_{\operatorname{\textsubscript{\rm sm}}}\times(S^{\prime\prime}\cap
S^{\circ})$ such that $\mu_{1}=\tau\circ\mu_{2}$. Let $\chi\colon
Y\times(S^{\prime\prime}\cap S^{\circ})\to Y\times S^{\circ}$ be the inclusion
map. We notice that, if ${\nu^{-1}(W^{\prime\prime})}_{\circ}$ is a connected
component of ${\nu^{-1}(W^{\prime\prime})}$, then
$\nu|_{\nu^{-1}(W^{\prime\prime})_{\circ}}\colon{\nu^{-1}(W^{\prime\prime})}_{\circ}\to
W^{\prime\prime}$ is induced by the composition of the following sequence of
morphisms of groups

$\textstyle{\pi_{1}((W^{\prime\prime}\cap\widehat{X}^{\circ})_{\operatorname{\textsubscript{\rm
sm}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{1}}$$\textstyle{\pi_{1}(Y_{\operatorname{\textsubscript{\rm
sm}}}\times(S^{\prime\prime}\cap
S^{\circ}))}$$\scriptstyle{\chi_{*}}$$\textstyle{\pi_{1}(Y_{\operatorname{\textsubscript{\rm
sm}}}\times
S^{\circ})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(Y_{\operatorname{\textsubscript{\rm
sm}}}\times S^{\circ})/\mathcal{N}.}$

We note that $\pi_{1}(Y_{\operatorname{\textsubscript{\rm
sm}}}\times(S^{\prime\prime}\cap S^{\circ}))\cong
G_{Y}\times\mathbb{Z}^{k^{\prime}}$, where $k^{\prime}$ is the number of the
components $H_{i}$ which pass through $o$. Moreover, $\chi_{*}$ is an
injective morphism from $\pi_{1}(Y_{\operatorname{\textsubscript{\rm
sm}}}\times(S^{\prime\prime}\cap S^{\circ}))$ to
$\pi_{1}(Y_{\operatorname{\textsubscript{\rm sm}}}\times S^{\circ})$, whose
cokernel is isomorphic to $\mathbb{Z}^{k-k^{\prime}}$. Hence the subgroup
$\mathcal{N}^{\prime}=(\chi_{*})^{-1}\mathcal{N}$ of
$\pi_{1}(Y_{\operatorname{\textsubscript{\rm sm}}}\times(S^{\prime\prime}\cap
S^{\circ}))$ is the canonical subgroup defined in Lemma 6.4. Then
$\nu|_{\nu^{-1}(W^{\prime\prime})_{\circ}}$ is induced by the composition of

$\textstyle{\pi_{1}((W^{\prime\prime}\cap\widehat{X}^{\circ})_{\operatorname{\textsubscript{\rm
sm}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{1}}$$\textstyle{\pi_{1}(Y_{\operatorname{\textsubscript{\rm
sm}}}\times(S^{\prime\prime}\cap
S^{\circ}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(Y_{\operatorname{\textsubscript{\rm
sm}}}\times(S^{\prime\prime}\cap S^{\circ}))/\mathcal{N}^{\prime}.}$

Similarly, if ${(\nu^{\prime})^{-1}(W^{\prime\prime})}_{\circ}$ is a connected
component of ${(\nu^{\prime})^{-1}(W^{\prime\prime})}$, then the finite
morphism
$\nu^{\prime}|_{(\nu^{\prime})^{-1}(W^{\prime\prime})_{\circ}}\colon{(\nu^{\prime})^{-1}(W^{\prime\prime})}_{\circ}\to
W^{\prime\prime}$ is induced by the composition of

$\textstyle{\pi_{1}((W^{\prime\prime}\cap\widehat{X}^{\circ})_{\operatorname{\textsubscript{\rm
sm}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{2}}$$\textstyle{\pi_{1}(Y_{\operatorname{\textsubscript{\rm
sm}}}\times(S^{\prime\prime}\cap
S^{\circ}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(Y_{\operatorname{\textsubscript{\rm
sm}}}\times(S^{\prime\prime}\cap S^{\circ}))/\mathcal{N}^{\prime}.}$

We remark that $\mathcal{N}^{\prime}=\tau^{-1}\mathcal{N}^{\prime}$ by Lemma
6.4. Since $\mu_{1}=\tau\circ\mu_{2}$, we deduce that the two compositions
above are the same. In another word,
$\nu|_{\nu^{-1}(W^{\prime\prime})_{\circ}}$ is isomorphic to
$\nu^{\prime}|_{(\nu^{\prime})^{-1}(W^{\prime\prime})_{\circ}}$. Replacing
$W^{\prime\prime}$ by the intersection of its images under $G_{\hat{o}}$, we
can assume that it is $G_{\hat{o}}$-invariant. This completes the proof of the
proposition. ∎

##  7\. Modification around an irreducible singular locus

The objective of this section is to prove the following proposition, it
combines the steps from Section 4 to Section 6.

######  Proposition 7.1.

Let $X$ be a complex analytic variety with klt singularities. Assume that
$S\subseteq X_{\operatorname{\textsubscript{\rm sing}}}$ is a compact
irreducible component of codimension 2 in $X$. Then there is a projective
bimeromorphic morphism $f\colon X^{\prime}\to X$, such that the following
properties hold. Let $\Delta$ be the whole reduced exceptional divisor of $f$.

  1. (1)

There is a $f$-ample and $f$-exceptional divisor.

  2. (2)

The indeterminacy locus of $f^{-1}$ has codimension at least 3. In particular,
we can define the strict transform $S^{\prime}$ of $S$ in $X^{\prime}$.

  3. (3)

There is an orbi-variety $\mathfrak{W}=(W_{i},G_{i},\psi_{i})_{i\in I}$ with
$I$ finite, such that the quotient space $W$ is an open neighborhood of
$S^{\prime}$ in $X^{\prime}$.

  4. (4)

The divisorial ramified locus of $\psi_{i}\colon W_{i}\to W$ is contained in
$\Delta$.

  5. (5)

$W_{i}$ is smooth on $\psi_{i}^{-1}(W\setminus\Delta)$.

We first fix the following notation for the section.

######  Setup 7.2.

Let $X$ be a complex analytic variety with klt singularities, and let
$S\subseteq X$ be an irreducible component of codimension 2. Assume that $S$
is compact and we are only interested in the behavior of some neighborhood of
$S$ in $X$. Hence we will shrink $X$ freely around $S$. Let $H\subseteq S$ be
a proper closed subset such that $X$ has the same type of singularities at
every point of $S\setminus H$. Let $\mathfrak{X}=(X_{i},G_{i},\pi_{i})_{i\in
I}$ be an orbi-variety with a finite family of chart whose quotient space is
$X$. Let $S_{i}=\pi_{i}^{-1}S$ and $H_{i}=\pi^{-1}H$.

We sketch the proof Proposition 7.1 as follows. Since $X$ has klt
singularities, there is an orbi-variety structure $(X_{i},G_{i})$ induced by
the local index-one covers of $\omega_{X}$, see Example 2.10. Then we are in
the situation of Steup 7.2 above. We first apply Proposition 4.1 on each chart
$X_{i}$. Then we can obtain some projective bimeromorphic morphism $f\colon
Y\to X$. We then locate on a neighborhood $U$ of the strict transform $S_{Y}$
of $S$ in $Y$. Replacing $X$ by $U$, we can assume that each $X_{i}$ satisfies
the conclusion of Proposition 4.1. As a result, we are able to apply
Proposition 5.2. Afterward, we may assume that each $X_{i}$ satisfies the
conclusion of Proposition 5.2. Now we can apply Proposition 6.6 on each
$X_{i}$. This will complete the proof of Proposition 7.1.

######  Lemma 7.3.

With the notation of Setup 7.2, there is a projective bimeromorphic morphism
$f\colon Y\to X$ such that the following properties hold.

  1. (1)

There is a $f$-exceptional and $f$-ample divisor.

  2. (2)

The indeterminacy locus of $f^{-1}$ is contained in $H$. In particular, we can
define the strict transform $S_{Y}$ of $S$ in $Y$, and the preiamge $H_{Y}$ of
$H$ in $S_{Y}$ under the morphism $f|_{S_{Y}}\colon S_{Y}\to S$.

  3. (3)

There is an orbi-variety $\mathfrak{Y}=(Y_{i},G_{i})_{i\in I}$ whose quotient
space is $Y$.

  4. (4)

Let $(S_{Y})_{i}$ and $(H_{Y})_{i}$ be the preimage of $S_{Y}$ and $H_{Y}$
respectively in $Y_{i}$. Then $(S_{Y})_{i}$ is smooth and $H_{Y}$ induces a
snc orbi-divisor on the orbifold $((S_{Y})_{i},G_{i})_{i\in I}$.

  5. (5)

For every point $o\in(S_{Y})_{i}$, there is an open neighborhood $U$ of $o$ in
$U$, such that $U$ is isomorphic to the hypersurface in
$\mathbb{D}^{3}\times(S_{Y})_{i}$ defined by an equation of standard form with
respect to $((S_{Y})_{i},(H_{Y})_{i})$, see Section 4.

  6. (6)

$Y_{i}$ has the same type of canonical singularities at points of
$(S_{Y})_{i}\setminus(H_{Y})_{i}$.

###### Proof.

For each $i$, we let $f_{i}\colon Y_{i}\to X_{i}$ be the construction of
Proposition 4.1, with respect to $H_{i}\subseteq S_{i}\subseteq X_{i}$. Let
$(S_{Y})_{i}$ be the strict transform of $S_{i}$ in $Y_{i}$, and let
$(H_{Y})_{i}$ be the preimage of $H_{i}$ in $(S_{Y})_{i}$. Since the
construction of Proposition 4.1 is functorial, we see that there is a natural
action of $G_{i}$ on $Y_{i}$, and the family $(Y_{i},G_{i})_{i\in I}$ defines
an orbi-variety $\mathfrak{Y}$. Let $Y$ be its quotient space. Then there is
an induced bimeromorphic morphism $f\colon Y\to X$, such that there is a
$f$-exceptional and $f$-ample divisor. Indeed, since each step in the
construction of Proposition 4.1 is the blowup of some ideal sheaf, which is
compatible along the overlaps, we deduce that there is a $f_{i}$-exceptional
and $f_{i}$-ample divisor $E_{i}$ which is compatible along the overlaps. Let
$H_{Y}$ and $S_{Y}$ be the closed subsets defined by $(H_{Y})_{i}$ and
$(S_{Y})_{i}$. Replacing $Y$ by a projective bimeromorphic modification
induced by the log resolution of $H_{Y}\subseteq S_{Y}$ in Lemma 2.12, we
assume that $H_{Y}\subseteq S_{Y}$ induces a snc orbi-divisor. ∎

######  Lemma 7.4.

With the notation of Setup 7.2, assume that $\mathfrak{X}=(X_{i},G_{i})_{i\in
I}$, $S$ and $H$ satisfies the properties of $\mathfrak{Y}=(Y_{i},G_{i})_{i\in
I}$, $S_{Y}$ and $H_{Y}$ in Lemma 7.3. Then, up to shrinking $X$ around $S$,
and up to replacing $(X_{i},G_{i})_{i\in I}$ by an equivalent family of
charts, we can assume the following properties.

  1. (1)

$I$ is finite.

  2. (2)

$X_{i}$ is isomorphic to an neighborhood of $\\{\mathbf{0}_{3}\\}\times S_{i}$
in the hypersurface in $\mathbb{D}^{3}\times S_{i}$ defined by an equation of
standard form with respect to $(S_{i},H_{i})$. In addition, $S_{i}\subseteq
X_{i}$ is identified with $\\{\mathbf{0}_{3}\\}\times S_{i}$.

  3. (3)

There is an integer $N>0$, such that we can perform the constructions of
Proposition 5.2 for each $H_{i}\subseteq S_{i}\subseteq X_{i}$ with $N$.

###### Proof.

For each point $x_{i}\in S_{i}\subseteq X_{i}$, we take a neighborhood
$U_{i}\subseteq X_{i}$ of $x_{i}$, so that we can perform the constructions of
Proposition 5.2 on $U_{i}$, with respect to $H_{i}\cap U_{i}\subseteq
S_{i}\cap U_{i}\subseteq U_{i}$. Shrinking $U_{i}$, we can assume that, for
each $g\in G_{i}$, either $x_{i}\in g(U_{i})$ or $g(U_{i})\cap
U_{i}=\emptyset$. Let $G_{x_{i}}$ be the subgroup of $G$ fixing $x_{i}$, and
let

$U^{\prime}_{i}=\bigcap_{g\in G_{x_{i}}}g(U_{i}).$

Then $U^{\prime}_{i}$ is $G_{x_{i}}$-invariant, and for each $g\in G_{i}$,
either $U^{\prime}_{i}=g(U^{\prime}_{i})$ or $g(U^{\prime}_{i})\cap
U^{\prime}_{i}=\emptyset$. In particular, this implies that $U^{\prime}_{i}$
is a connected component of $\pi_{i}^{-1}(\pi_{i}(U^{\prime}_{i}))$.
Furthermore, $(U^{\prime}_{i},G_{x_{i}})$ is an orbifold chart over
$\pi_{i}(U_{i})\subseteq X$ for the orbi-variety $\mathfrak{X}$.

We hence obtain a family of charts $\\{(U^{\prime}_{i},G_{x_{i}})\\}$ by
considering all points $x_{i}\in X_{i}$ and all $i\in I$. Since $S$ is
compact, we can extract a finite subset of this family of charts, so that the
images of the charts cover $S$. Replacing $(X_{i},G_{i})_{i\in I}$ by this
finite family, and replacing $X$ by the union of the images of these charts,
we may assume that the following property holds for the orbi-variety
$\mathfrak{X}=(X_{i},G_{i})_{i\in I}$. There is an integer $N>0$, such that we
can perform the constructions of Proposition 5.2 for each $H_{i}\subseteq
S_{i}\subseteq X_{i}$ with $N$. ∎

######  Lemma 7.5.

Assume the properties of Lemma 7.4 hold. Then there is a projective
bimeromorphic morphism $f\colon Y\to X$ such that the following properties
hold.

  1. (1)

There is a $f$-exceptional and $f$-ample divisor.

  2. (2)

The center of every prime $f$-exceptional divisor in $X$ is an irreducible
component of $H$. In particular, we can define the strict transform $S_{Y}$ of
$S$ in $Y$, and the preiamge $H_{Y}$ of $H$ in $S_{Y}$ under the morphism
$f|_{S_{Y}}\colon S_{Y}\to S$.

  3. (3)

There is an orbi-variety $\mathfrak{Y}=(Y_{i},\overline{G}_{i})$ whose
quotient space is an open neighborhood of $S_{Y}$ in $Y$.

  4. (4)

Let $(S_{Y})_{i}$ and $(H_{Y})_{i}$ be the preimages of $S_{Y}$ and $H_{Y}$
respectively in $Y_{i}$. There is some open neighborhood $U_{i}$ of
$(S_{Y})_{i}$, such that $U_{i}$ is the hypersurface in
$\mathbb{\mathbb{D}}^{3}\times(S_{Y})_{i}$ defined by a function as in
Proposition 5.2 with respect to $((S_{Y})_{i},(H_{Y})_{i})$.

  5. (5)

Let $\Delta_{Y}$ be the exceptional locus of $f$, let $(\Delta_{Y})_{i}$ be
its preimage in $Y_{i}$ and let
$U^{\prime}_{i}=\cap_{g\in\overline{G}_{i}}g(U_{i})$. Then $U^{\prime}_{i}$,
$(\Delta_{Y})_{i}$, $(S_{Y})_{i}$, $(H_{Y})_{i}$ and $\overline{G}_{i}$
satisfy the conditions in Setup 6.5. Moreover, the collection
$(U^{\prime}_{i},\overline{G}_{i})$ induces an orbi-variety, whose quotient
space $U^{\prime}$ is an open neighborhood of $S_{Y}$.

###### Proof.

We will perform the constructions of Proposition 5.2 on $X_{i}$, the list of
properties follows from the construction. Let $D^{1},...,D^{k}$ be the
irreducible components of $H$. Since $H\subseteq S$ induces a snc orbi-
divisor, for each irreducible component $D^{j}$ of $H$, its preimage
$D_{i}^{j}$ in $X_{i}$ is smooth. Since any two irreducible components of
$H_{i}$ intersect, we deduce that $D^{j}_{i}$ is an irreducible component of
$H_{i}$.

We first assume that the singularities of $X_{i}$ at $S_{i}\setminus H_{i}$
are of $A$-type or of $E$-type. Then for each $i\in I$, we perform the
following sequence of blowups on $X_{i}$, as in Subsection 3.C,

$Y_{i}=(X_{i})_{k}\to\cdots\to(X_{i})_{1}\to(X_{i})_{0}=X_{i}$

where $(X_{i})_{j+1}\to(X_{i})_{j}$ is the blowup of $(X_{i})_{j}$ at
$D^{j}_{i}$ for $N$ times. Then there is a natural action of $G_{i}$ on
$Y_{i}$. Moreover, $(Y_{i},G_{i})$ induces an orbi-variety $\mathfrak{Y}$ with
quotient space $Y$. If $f_{i}\colon Y_{i}\to X_{i}$ is the composition of the
sequence above, then the collection $(f_{i})_{i\in I}$ induces a proper
bimeromorphic morphism $f\colon Y\to X$. In addition, since $f_{i}$ is the
composition of blowups whose centers are compatible along the overlaps, we
deduce that there is some $f_{i}$-ample and $f_{i}$-exceptional divisor which
is compatible along the overlaps. Hence there is a $f$-exceptional and
$f$-ample divisor. Let $(S_{Y})_{i}$ be the strict transform of $S_{i}$ in
$Y_{i}$. By construction, there is some open neighborhood $U_{i}$ of
$(S_{Y})_{i}$, such that $U_{i}$ is the hypersurface in
$\mathbb{\mathbb{D}}^{3}\times(S_{Y})_{i}$ defined by a function as in
Proposition 5.2. This prove the lemma in the case when the singularities of
$X_{i}$ at $S_{i}\setminus H_{i}$ are of $A$-type or of $E$-type.

We assume that the singularities of $X_{i}$ at $S_{i}\setminus H_{i}$ are of
$D$-type. We will proceed as in Lemma 5.9 on each $X_{i}$. Let
$\varphi_{i}\colon Z_{i}\to X_{i}$ be the composition of blowups at the
centers $D^{1}_{i},...,D^{k}_{i}$. Then there is natural action of $G_{i}$ on
$Z_{i}$ and $(Z_{i},G_{i})$ induces an orbi-variety $\mathfrak{Z}$ with
quotient space $Z$. Let $S_{Z}$ be the strict transform of $S$ in $Z$, and let
$(S_{Z})_{i}$ be the strict transforms of $S_{i}$ in $Z_{i}$. We denote by
$E_{i}^{j}$ the Cartier divisor in $Z_{i}$ defined by
$\varphi_{i}^{*}\mathcal{I}(D^{j}_{i})$, where $\mathcal{I}(D^{j}_{i})$ is the
ideal sheaf of $D^{j}_{i}$ in $X_{i}$. There is an open neighborhood $V_{i}$
of $(S_{Z})_{i}$ such that $(E_{i}^{j})|_{V_{i}}$ is defined by a holomorphic
function $\sigma^{j}$ on $V_{i}$ as a Cartier divisor, for $j=1,...,k$.
Shrinking $V_{i}$ we may assume that it is $G_{i}$-invariant. Then for any
$g\in G_{i}$, we have $g^{*}\sigma^{j}=u_{g}^{j}\cdot\sigma^{j}$ for some unit
function $u_{g}^{j}$. Since $(S_{Z})_{i}$ is contractible, up to shrinking
$V_{i}$, we can assume that $u_{h}^{j}$ admits 6-th roots for all $g\in G_{i}$
and all $j=1,...,k$. Let
$\overline{V}_{i}=V_{i}[\sqrt[6]{\sigma_{1}},...,\sqrt[6]{\sigma_{k}}]$. Then
there the action of $G_{i}$ on $V_{i}$ can be lifted to $\overline{V}_{i}$. In
particular, the finite morphism $\overline{V}_{i}\to Z$ is Galois over its
image. Let $\overline{G}_{i}$ the the Galois group. From Remark 5.10 and Lemma
2.11, we see that the collection $(\overline{V_{i}},\overline{G_{i}})_{i\in
I}$ induces an orbi-variety $\overline{\mathfrak{V}}$, with quotient space
$\overline{V}$. Then $\overline{V}$ can be identified as an open neighborhood
of $S_{Z}$ in $Z$. We note that, if $H_{Z}$ is the preimage of $H$ in $S_{Z}$,
then $H_{Z}$ also induces a snc orbi-divisor on $\overline{\mathfrak{V}}$, by
the item (2) of Lemma 2.6. Let $(S_{\overline{V}})_{i}$ be the preimage of
$(S_{Z})_{i}$ in $\overline{V}_{i}$. Now, as in the previous paragraph, we
blowup $\overline{V_{i}}$ at the preimage of each irreducible component of
$H_{Z}$ for $N$ times as in Subsection 3.C. We obtain an orbi-variety
$\mathfrak{Y}=(Y_{i},\overline{G}_{i})$ with quotient space $Y^{\prime}$. Let
$(S_{Y})_{i}$ be the strict transform of $(S_{\overline{V}})_{i}$ in $Y_{i}$.
By the construction of Lemma 5.9, there is some open neighborhood $U_{i}$ of
$(S_{Y})_{i}$, such that $U_{i}$ is the hypersurface in
$\mathbb{\mathbb{D}}^{3}\times(S_{Y})_{i}$ defined by a function of as in
Proposition 5.2. Furthermore, there is a proper bimeromorphic morphism
$h\colon Y^{\prime}\to\overline{V}$ with some $h$-ample and $h$-exceptional
divisor. Since the indeterminacy locus of $h^{-1}$ is contained in the compact
set $S_{\overline{V}}$, we deduce that $h$ extends to a projective
bimeromorphic morphism $h\colon Y\to Z$ by Lemma 2.3. Hence we obtain a
projective bimeromorphic morphism $f\colon Y\to X$. In addition, there is a
$f$-exceptional and $f$-ample divisor. ∎

The following lemma will be applied in the neighborhood $U^{\prime}$ of
$S_{Y}$ in $Y$ of Lemma 7.5 above.

######  Lemma 7.6.

With the notation of Setup 7.2, let $\Delta$ be a divisor in $X$ and let
$\Delta_{i}$ be its preimage in $X_{i}$. Assume that $X_{i}$, $\Delta_{i}$,
$S_{i}$, $H_{i}$ and $G_{i}$ satisfy the conditions of Setup 6.5. Then, there
is a projective bimeromorphic morphism $f\colon Y\to X$ such that the
following properties hold.

  1. (1)

There is a $f$-exceptional and $f$-ample divisor.

  2. (2)

The center of every prime $f$-exceptional divisor in $X$ is an irreducible
component of $H$. In particular, we can define the strict transform $S_{Y}$ of
$S$ in $Y$, and the preiamge $H_{Y}$ of $H$ in $S_{Y}$ under the morphism
$f|_{S_{Y}}\colon S_{Y}\to S$.

  3. (3)

There is an orbi-variety
$\overline{\mathfrak{W}}=(\overline{W}_{i},\overline{G}_{i})_{i\in
I^{\prime}}$ whose quotient space $W$ is an open neighborhood of $S_{Y}$ in
$Y$.

  4. (4)

The divisorial branched locus of $\psi_{i}\colon\overline{W}_{i}\to W$ is
contained in $\Delta^{\prime}$, where $\Delta^{\prime}=f^{-1}(\Delta)$.

  5. (5)

$\overline{W}_{i}$ is smooth on $\psi_{i}^{-1}(W\setminus\Delta^{\prime})$.

###### Proof.

For every point $x_{i}\in S_{i}\subseteq X_{i}$, there is an open neighborhood
$X^{\prime}_{i}$ of $x_{i}$ such that the construction of Proposition 6.6
works on $X^{\prime}_{i}$. Hence, by a similar argument as in Lemma 7.4, we
may assume that, for each $i$, the construction of Proposition 6.6 works on
$X_{i}$. We can also assume that $I$ is finite. Hence there is a positive
integer $N$ which works on every $X_{i}$ for this construction. Let
$D^{1},...,D^{k}$ be irreducible components of $H$. Since $H$ induces a snc
orbi-divisor, the preimage $D^{j}_{i}$ of $D^{j}$ in $X_{i}$ is smooth and
hence irreducible. Let $f_{i}\colon Y_{i}\to X_{i}$ be the morphism obtained
by blowing up of $X_{i}$ at $D^{j}_{i}$ for $N$ times, successively for
$j=1,...,k$, as in Subsection 3.C. Then there is a natural action of $G_{i}$
on $Y_{i}$ and $(Y_{i},G_{i})$ induces an orbi-variety $\mathfrak{Y}$ with
quotient space $Y$. Furthermore, there is an induced projective bimeromorphic
morphism $f\colon Y\to X$ with a $f$-ample and $f$-exceptional divisor.

From the choice of $N$, there is a $G_{i}$-invariant open neighborhood $W_{i}$
of the strict transform $(S_{Y})_{i}$ of $S_{Y}$ in $Y_{i}$, such that there
is a finite cover $\nu_{i}\colon\overline{W}_{i}\to W_{i}$ satisfying the
properties of Proposition 6.6. The property (4) of Proposition 6.6 implies
that the natural finite morphism $\overline{W}_{i}\to Y$ is Galois of group
$\overline{G}_{i}$ over its image. The property (5) of Proposition 6.6 implies
that $(\overline{W}_{i},\overline{G}_{i})_{i\in I}$ induces an orbi-variety
with quotient space $W$. Then $W$ can be identified naturally as an open
neighborhood of $S_{Y}$ in $Y$. This completes the proof of the lemma. ∎

Now we can complete the proof of Proposition 7.1.

###### Proof of Proposition 7.1.

We first take $\mathfrak{X}=(X_{i},G_{i})$ as the orbi-variety induced by the
index-one covers of $\omega_{X}$. Then we are in Setup 7.2. We apply Lemma 7.3
on $X$ and obtain a projective bimeromorphic morphism $f_{1}\colon Y_{1}\to
X$. Let $S_{Y_{1}}$ be the strict transform of $S$ in $Y_{1}$, and let $Q_{1}$
be the open neighborhood of $S_{Y_{1}}$ in the construction of Lemma 7.3. Then
we apply Lemma 7.4 and Lemma 7.5 to $Q_{1}$, and we can obtain a projective
bimeromorphic morphism $f_{2}\colon P_{2}\to Q_{1}$. Since $f_{2}$ admits a
$f_{2}$-exceptional and $f_{2}$-ample divisor, and the indeterminacy locus of
$(f_{2})^{-1}$ is contained in $S_{Y_{1}}$ which is compact, by Lemma 2.3,
$f_{2}$ extends to a morphism $f_{2}\colon Y_{2}\to Y_{1}$. Let $S_{Y_{2}}$ be
the strict transform of $S_{Y_{1}}$ in $P_{2}$, let $Q_{2}$ be the open
neighborhood of $S_{Y_{2}}$ in the item (5) of Lemma 7.5, and let $\Delta_{2}$
be the whole $f_{2}$-exceptional divisor. Then we can apply Lemma 7.6 to
$Q_{2}$ with the divisor $\Delta_{2}$, and obtain a projective bimeromorphic
morphism $f_{3}\colon P_{3}\to Q_{2}$. By the same argument for $f_{2}$, we
deduce that $f_{3}$ extends to a morphism $f_{3}\colon Y_{3}\to Y_{2}$. Let
$Y=Y_{3}$ and $f\colon Y\to X$ be the natural morphism. Then $f$ satisfies all
the properties of the proposition. ∎

##  8\. Proof of the modification theorem

We will complete the proof of Theorem 1.2 in this section.

######  Lemma 8.1.

Let $X$ be a complex analytic variety and $S\subseteq
X_{\operatorname{\textsubscript{\rm sing}}}$ an irreducible component of
codimension $2$ in $X$. Let $\Delta$ be a reduced divisor in $X$. Suppose that
$S$ is compact and that $\Delta$ does not contain $S$. Assume that there is an
orbi-variety $\mathfrak{W}=(W_{i},G_{i},\psi_{i})_{i\in I}$ satisfying the
following properties.

  1. (1)

The family of charts $(W_{i},G_{i},\psi_{i})_{i\in I}$ is finite.

  2. (2)

The quotient space $W$ of $\mathfrak{W}$ is an open neighborhood of $S$ in
$X$.

  3. (3)

The divisorial ramified locus of $\psi_{i}\colon W_{i}\to W$ is contained in
$\Delta$.

  4. (4)

$W_{i}$ is smooth on $\psi_{i}^{-1}(W\setminus\Delta)$.

Assume that there is a projective bimeromorphic morphism
$h\colon(Z,\Gamma)\to(X,\Delta)$, satisfying the following properties. The
$h$-exceptional locus is a divisor. The divisor $\Gamma$ is the sum of
$h^{-1}_{*}\Delta$ and the whole reduced $h$-exceptional divisor. Each
irreducible component of $\Gamma$ is $\mathbb{Q}$-Cartier. The morphism $h$ is
an isomorphism over the snc locus of $(X,\Delta)$ and over general points $S$.
The pair $(Z,\Gamma)$ is snc on $h^{-1}(X\setminus S)$.

Then there is a projective bimeromorphic morphism $\varphi\colon Y\to Z$ with
a $\varphi$-ample and $\varphi$-exceptional divisor, such that $Y$ has
quotient singularities. Furthermore, $\varphi$ is an isomorphism over
$h^{-1}(X\setminus S)$ and over general points of the strict transform of $S$
in $Z$.

###### Proof.

If $V_{i}$ is the normalization of the main component of the pullback of
$W_{i}$ over $h\colon Z\to X$, then there is a natural action of $G_{i}$ on
$V_{i}$. Furthermore, the family $(V_{i},G_{i})_{i\in I}$ induces an orbi-
variety whose quotient space is $V=h^{-1}(W)$. Since $Z\setminus\Gamma\cong
X\setminus\Delta$, we deduce that, if $\mu_{i}\colon V_{i}\to V$ is the
natural finite morphism, then its divisorial branched locus is contained in
$\Gamma$. In addition, $V_{i}$ is smooth on $\mu_{i}^{-1}(Z\setminus\Gamma)$.

Let $S_{Z}$ be the strict transform of $S$ in $Z$. Let $\Gamma^{j}$ with
$j=1,...,l$ be the irreducible components of $\Gamma$, and let $r^{j}$ be the
Cartier index of $\Gamma^{j}$. For every point $z\in h^{-1}(S)$, there is an
open neighborhood $T$ of $z$ in $Z$ such that $r^{j}\Gamma^{j}|_{T}$ is
defined by some holomorphic function $\sigma^{j}$. Thus, by the compactness of
$h^{-1}(S)$, we obtain a finite family $(U_{k})_{k\in K}$ of open subsets of
$Z$, whose union covers $h^{-1}(S)$, such that $r^{j}\Gamma^{j}|_{U_{k}}$ is
connected and defined by some holomorphic function $\sigma^{j}_{k}$.
Furthermore, there is an application $\iota\colon K\to I$ such that $U_{k}$ is
contained in $\mu_{\iota(k)}(V_{\iota(k)})$. Let $U=\bigcup_{k\in K}U_{k}$.
Then it is a neighborhood of $h^{-1}(S)$

We notice that, since $(Z,\Gamma)$ is dlt, each $\Gamma^{j}$ is normal. Hence
each $\Gamma^{j}_{k}$ is normal and hence irreducible. In particular, we can
apply the result from Lemma 2.5 to Lemma 2.7 on each
$(U_{k},\Gamma|_{U_{k}})$. Let $M>0$ be an integer which is divisible by the
degrees of all the $\mu_{i}$’s. We define

$\xi_{k}\colon\overline{U}_{k}=U_{k}[\sqrt[Mr^{1}]{\sigma^{1}},...,\sqrt[Mr^{l}]{\sigma^{l}}]^{nor}\to
U_{k}$

and we let $Q_{k}$ be the normalization of
$\overline{U}_{k}\times_{Z}V_{\iota(k)}$. Then the natural finite morphism
$Q_{k}\to U_{k}$ is Galois of group $H_{k}$, and $(Q_{k},H_{k})$ induces an
orbi-variety $\mathfrak{Q}$ with quotient space $U$, see Lemma 2.5. We denote
by $\eta_{k}\colon Q_{k}\to\overline{U}_{k}$ the natural morphism. Then by
Lemma 2.7, we obtain that $\eta_{k}$ is quasi-étale. Furthermore, $Q_{k}$ is
smooth in codimension 2 since $V_{\iota(k)}$ is smooth over
$U_{k}\setminus\Gamma$.

The Zariski’s purity theorem implies that

$(Q_{k})_{\operatorname{\textsubscript{\rm
sing}}}\subseteq(\eta_{k})^{-1}((\overline{U}_{k})_{\operatorname{\textsubscript{\rm
sing}}}).$

Since $(Z,\Gamma)$ is snc on $Z\setminus h^{-1}(S)$, we deduce that
$\overline{U}_{k}$ is smooth over $Z\setminus h^{-1}(S)$ by Lemma 2.6. It
follows that

$(Q_{k})_{\operatorname{\textsubscript{\rm
sing}}}\subseteq(\xi_{k}\circ\eta_{k})^{-1}(h^{-1}(S)).$

Let $\widetilde{Q}_{k}\to Q_{k}$ be a functorial desingularization. Then there
is a natural action of $H_{k}$ on $Q_{k}$, and we obtain an orbifold
$\widetilde{\mathfrak{Q}}=(\widetilde{Q}_{k},H_{k})_{k\in K}$ with quotient
space $\widetilde{Q}$. There is an induced projective bimeromorphic morphism
$\varphi\colon\widetilde{Q}\to U$, such that the indeterminacy locus of
$\varphi^{-1}$ is contained in $h^{-1}(S)$ and is of codimension at least 3 in
$Z$. In particular, $\varphi$ is an isomorphism over general points of
$S_{Z}$. Furthermore, there is an $\varphi$-exceptional and $\varphi$-ample
divisor. Thus by Lemma 2.3, $\varphi$ extends to a projective bimeromorphic
morphism $\varphi\colon Y\to Z$. Since $Z$ is smooth outside $h^{-1}(S)$, we
deduce that $Y$ has quotient singularities only. ∎

Now we can deduce Theorem 1.2.

######  Proof of Theorem 1.2 .

We will divide the proof into several steps. Let $S_{e}$ with $e=1,...,r$ be
the codimension 2 irreducible components of
$X_{\operatorname{\textsubscript{\rm sing}}}$.

Step 1. In this step, we will reduce to the case when $S_{1},...,S_{r}$ are
pairwise disjoint. By blowing up at centers contained in at least two of the
$S_{e}$’s, we obtain a projective bimeromorphic morphism $h\colon Z\to X$ such
that the strict transforms $h^{-1}_{*}S_{e}$ are pairwise disjoint. Let
$X^{\prime}\to X$ be the projective bimeromorphic morphism constructed in
Lemma 2.14, with respect to the collection $\\{S_{1},...,S_{r}\\}$. Up to
replacing $X$ by $X^{\prime}$, we may assume that $S_{1},...,S_{r}$ are
pairwise disjoint. Furthermore, $X$ has klt singularities.

Step 2. In this step, we will reduce to the situation of Lemma 8.1. Let
$X_{e}$ be an open neighborhood of $S_{e}$ in $X$ such that the $X_{e}\cap
X_{e^{\prime}}=\emptyset$ if $e\neq e^{\prime}$. We apply Proposition 7.1 to
each $X_{e}$. Then we have projective bimeromorphic morphisms $p_{e}\colon
X^{\prime}_{e}\to X_{e}$, which satisfies the properties of Proposition 7.1.
Since for each $e$, there is a $p_{e}$-ample and $p_{e}$-exceptional divisor,
and since the indeterminacy locus of $(p_{e})^{-1}$ is contained in $S_{e}$
and hence compact, by applying Lemma 2.3 successively, there is a projective
bimeromorphic morphism $p\colon X^{\prime}\to X$ such that $p=p_{e}$ over
$X_{e}$. Let $\Delta$ be the whole reduced $p$-exceptional divisor. Then
$p(\Delta)$ has codimension at least 3 in $X$.

In the remainder of the proof, by abuse of notation, we denote $X^{\prime}$ by
$X$, and $X_{e}^{\prime}$ by $X_{e}$. If $\Delta_{e}=\Delta|_{X_{e}}$, then
$X_{e}$, $\Delta_{e}$ and $S_{e}$ satisfy the condition (1)-(4) of Lemma 8.1.
Furthermore, from now on, it is enough to construct $f\colon Y\to X$, such
that $Y$ has quotient singularities, and $f$ is an isomorphism over general
points of each $S_{e}$ and over $X_{\operatorname{\textsubscript{\rm
sm}}}\setminus\Delta$. We remark that $X_{\operatorname{\textsubscript{\rm
sm}}}\setminus\Delta$ is contained in the snc locus of $(X,\Delta)$.

Step 3. In this step, we will construct a bimeromorphic morphism $h\colon Z\to
X$, which plays the same role for $X_{e}$, as $Z$ in Lemma 8.1 for $X$. To
this end, we apply Proposition 3.3 to $(X,\Delta)$, with respect to $S$, and
then we take the normalization. We obtain a projective bimeromorphic morphism
$\rho\colon X^{\prime}\to X$. Let $\Delta^{\prime}$ be the sum of
$\rho^{-1}_{*}\Delta$ and the whole reduced $\rho$-exceptional divisor. Then
$(X^{\prime},\Delta^{\prime})$ is snc on $\rho^{-1}(X\setminus S)$. Let
$S^{\prime}$ be the strict transform of $S$ in $X^{\prime}$. Let
$(Z,\Gamma)\to(X^{\prime},\Delta^{\prime})$ be a dlt modification as in Lemma
2.15, which is an isomorphism over the snc locus of
$(X^{\prime},\Delta^{\prime})$, and over general points of $S^{\prime}$. We
denote by $h\colon Z\to X$ the natural morphism. Then we have the following
properties. The $h$-exceptional locus of $h$ is a divisor. The divisor
$\Gamma$ is the sum of $h^{-1}_{*}\Delta$ and the whole reduced
$h$-exceptional divisor. Each irreducible component of $\Gamma$ is
$\mathbb{Q}$-Cartier. The morphism $h$ is an isomorphism over the snc locus of
$(X,\Delta)$ and over general points $S$. The pair $(Z,\Gamma)$ is snc on
$h^{-1}(X\setminus S)$.

Step 4. We complete the proof in this final step. Let $Z_{e}=h^{-1}(X_{e})$
and $\Gamma_{e}=\Gamma|_{Z_{e}}$. Then we can apply Lemma 8.1 on each $X_{e}$,
and obtain a projective bimeromorphic morphism $\varphi_{e}\colon Y_{e}\to
Z_{e}$ as in the Lemma 8.1. The indeterminacy locus of $(\varphi_{e})^{-1}$ is
contained in $(h_{e})^{-1}(S_{e})$, hence is compact. Since there is a
$\varphi_{e}$-ample and $\varphi_{e}$-exceptional divisor, by applying Lemma
2.3 successively, we deduce that there is a projective bimeromorphic morphism
$\varphi\colon Y\to Z$ such that its restriction over $Z_{e}$ is equal to
$\varphi_{e}$, and that it is an isomorphism elsewhere. In particular, $Y$ has
quotient singularities. Let $f\colon Y\to X$ be natural morphism. Then $f$ is
an isomorphism over the snc locus of $(X,\Delta)$ and over general points of
$S_{1},...,S_{r}$. This completes the proof of the theorem. ∎

##  9\. The Bogomolov-Gieseker inequality

In this section, we will prove Theorem 1.3. The notion of Kähler space was
introduced in [Gra62]. For Kähler orbifolds and orbifold coherent sheaves, see
for example [Wu23, Section] or [DO23, Section 3.1].

If $X$ is a compact complex analytic variety of dimension $n$, then there is a
natural isomorphism $H_{2n}(X,\mathbb{R})\cong H_{2n}^{BM}(X,\mathbb{R})$,
where $H_{2n}$ is the singular homology and $H_{2n}^{BM}$ is the Borel-Moore
homology. We note that $H_{2n}^{BM}(X,\mathbb{R})\cong\mathbb{R}$ and it is
generated by a canonical element $[X]$, called the fundamental class of $X$.
For more details on Borel-Moore homology, we refer to [Ful98, Chapter 19] and
the references therein. As a consequence, for cohomology classes
$\sigma_{1},...,\sigma_{k}\in H^{\bullet}(X,\mathbb{R})$, such that sum of
their degrees is equal to $2n$, we can define the following intersection
number,

$\sigma_{1}\cdots\sigma_{k}:=(\sigma_{1}\smallsmile\cdots\smallsmile\sigma_{k})\smallfrown[X]\in\mathbb{R}.$

When $X$ has only quotient singularities, this intersection form is just the
orbifold Poincaré duality, see [Sat56, Theorem 3].

###  9.A. Orbifold Chern classes

We will explain the orbifold Chern classes $\hat{c}_{k}$ for $k=1,2$ in the
statement of Theorem 1.3. We also refer to [GK20, Section 5] for a similar
discussion.

Let $X$ be a compact complex analytic variety of dimension $n$, which has
quotient singularities in codimension 2. Let $\mathcal{F}$ be a coherent
reflexive sheaf on $X$. Then there is a Zariski open subset $X^{\circ}$ of
$X$, whose complement has codimension at least 3, such that there is a
standard complex orbifold $\mathfrak{X}=(X_{j},G^{\prime}_{j})$ whose quotient
space is $X^{\circ}$. Being standard means that the action of $G^{\prime}_{j}$
is free in codimension 1. Furthermore, Up to shrinking $X^{\circ}$, we can
assume that $\mathcal{F}$ induces an orbifold vector bundle
$\mathcal{F}_{\mathrm{orb}}$ on $\mathfrak{X}$, see [DO23, Remark 3.4.5].

Let $f\colon Y\to X$ be an orbifold modification as in Theorem 1.2. Then $Y$
has quotient singularities. Let $\mathfrak{Y}=(Y_{i},G_{i})$ be the standard
orbifold structure on $Y$, and let
$\mathcal{E}=(f^{*}\mathcal{F})/(\mathrm{torsion})$. Then $\mathcal{E}$
induces a reflexive coherent orbifold sheaf $\mathcal{E}_{\mathrm{orb}}$ on
$\mathfrak{Y}$, see for example [DO23, Remark 3.4.5]. By shrinking
$X^{\circ}$, we may assume that $f$ is an isomorphism over it.

By Theorem 4.8, there is a functorial projective bimeromorphic morphism
$p_{i}\colon Z_{i}\to Y_{i}$, such that $Z_{i}$ is smooth, that
$\mathcal{H}_{i}:=p_{i}^{*}\mathcal{E}_{i}/(\mathrm{torsion})$ is locally
free, and that the indeterminacy locus of $p_{i}^{-1}$ has codimension at
least 3. The functoriality implies that there is a canonical action of $G_{i}$
on $H_{i}$ such that $p_{i}$ is $G_{i}$-equivariant. Furthermore,
$(Z_{i},G_{i})$ induces a complex orbifold $\mathfrak{Z}$ with quotient space
$Z$. The collection $(\mathcal{H}_{i})$ defines an orbifold vector bundle
$\mathcal{H}_{\mathrm{orb}}$ on $\mathfrak{Z}$. Therefore, we have well-
defined orbifold Chern classes $\hat{c}_{1}(\mathcal{H}_{\mathrm{orb}})$ and
$\hat{c}_{2}(\mathcal{H}_{\mathrm{orb}})$ in $H^{\bullet}(Z,\mathbb{R})$, see
[Bla96, Section 2]. Let $q\colon Z\to X$ be the natural morphism. Then the
indeterminacy locus of $q^{-1}$ has codimension at least 3. We can now define
the orbifold Chern classes $\hat{c}_{1}(\mathcal{F})$,
$\hat{c}_{2}(\mathcal{F})$ and $\hat{c}_{1}(\mathcal{F})^{2}$.

######  Definition 9.1.

The orbifold Chern class $\hat{c}_{1}(\mathcal{F})$,
$\hat{c}_{2}(\mathcal{F})$ and $\hat{c}_{1}(\mathcal{F})^{2}$ are defined as
linear forms on $H^{\bullet}(X,\mathbb{R})$, so that for any class $\sigma\in
H^{\bullet}(X,\mathbb{R})$, we have

$\displaystyle\hat{c}_{k}(\mathcal{F})\cdot\sigma=\hat{c}_{k}(\mathcal{H}_{\mathrm{orb}})\cdot
q^{*}\sigma$ and
$\displaystyle\hat{c}_{1}(\mathcal{F})^{2}\cdot\sigma=\hat{c}_{1}(\mathcal{H}_{\mathrm{orb}})^{2}\cdot
q^{*}\sigma,$

where $k=1,2$.

To see that these classes are well-defined, it is equivalent to show that the
intersection numbers are independent of the choice of the modification
$q\colon Z\to X$. We first observe the following fact.

######  Lemma 9.2.

Let $X$ be compact complex analytic variety of dimension $n$ and let
$\rho\colon\widetilde{X}\to X$ be a proper bimeromorphic morphism. Let
$\alpha_{1},\alpha_{2}\in H^{2k}(\widetilde{X},\mathbb{R})$. Assume that there
is a closed analytic subset $V\subseteq X$ of codimension at least $k+1$, such
that
$\alpha_{1}|_{\widetilde{X}^{\circ}}=\alpha_{2}|_{\widetilde{X}^{\circ}}$,
where ${\widetilde{X}^{\circ}}=\widetilde{X}\setminus\rho^{-1}(V)$. Then for
any class $\sigma\in H^{2n-2k}(X,\mathbb{R})$, we have

$\alpha_{1}\cdot\rho^{*}\sigma=\alpha_{2}\cdot\rho^{*}\sigma.$

###### Proof.

Let $E=\rho^{-1}(V)$. Then there is some class $\delta\in
H^{2k}(\widetilde{X},\widetilde{X}\setminus E,\mathbb{R})$ such that
$\alpha_{1}-\alpha_{2}$ is equal to the image of $\delta$ in
$H^{2k}(\widetilde{X},\mathbb{R})$. It follows that

$\displaystyle(\alpha_{1}-\alpha_{2})\cdot\rho^{*}\sigma$ $\displaystyle=$
$\displaystyle\delta\cdot\rho^{*}\sigma$ $\displaystyle=$
$\displaystyle(\rho^{*}\sigma\smallsmile\delta)\smallfrown[\widetilde{X}]$
$\displaystyle=$
$\displaystyle\rho^{*}\sigma\smallfrown(\delta\smallfrown[\widetilde{X}]).$

We note that $\delta\smallfrown[\widetilde{X}]\in H_{2n-2k}(E,\mathbb{R})$.
Then $\rho_{*}(\delta\smallfrown[\widetilde{X}])\in H_{2n-2k}(V,\mathbb{R})$.
By assumption, we have $\operatorname{dim}V\leq n-k-1$. Hence
$\rho_{*}(\delta\smallfrown[\widetilde{X}])=0$. By the projection formula, we
have

$\rho^{*}\sigma\smallfrown(\delta\smallfrown[\widetilde{X}])=\sigma\smallfrown\rho_{*}(\delta\smallfrown[\widetilde{X}])=0.$

This completes the proof of the lemma. ∎

We are in position to prove the following statement.

######  Proposition 9.3.

The intersection numbers in Definition 9.1 are independent of the choice of
$Z$.

###### Proof.

We assume the following setting. Let $q_{j}\colon Z_{j}\to X_{j}$ be proper
bimeromorphic morphisms with $j=1,2$. There is a closed analytic subset
$V\subseteq X$ of codimension at least 3, such that $X^{\circ}:=X\setminus V$
has quotient singularities. The restriction of the reflexive coherent sheaf
$\mathcal{F}$ on $X^{\circ}$ induces an orbifold vector bundle
$\mathcal{F}_{\mathrm{orb}}$ on the standard orbifold structure of
$X^{\circ}$. Let $Z_{j}^{\circ}=q_{j}^{-1}(X^{\circ})$. Then $q_{j}$ is an
isomorphism on $Z_{j}^{\circ}$. There is an orbifold $\mathfrak{Z}_{j}$ whose
quotient space is equal to $Z_{j}$, such that $\mathfrak{Z}_{j}$ is standard
over $Z_{j}^{\circ}$. There is an orbifold vector bundle
$\mathcal{H}_{\mathrm{orb},j}$ on $\mathfrak{Z}_{j}$, whose restriction over
$Z_{j}^{\circ}$ is isomorphic to $\mathcal{F}_{\mathrm{orb}}$ over
$X^{\circ}$. Let
$\alpha_{j}\in\\{\hat{c}_{1}(\mathcal{H}_{\mathrm{orb},j}),\hat{c}_{2}(\mathcal{H}_{\mathrm{orb},j}),\hat{c}_{1}(\mathcal{H}_{\mathrm{orb},j})^{2}\\}$,
such that $\alpha_{1}$ and $\alpha_{2}$ are the same type of characteristic
class.

Then the proposition is equivalent to prove that, for any $\sigma\in
H^{\bullet}(X,\mathbb{R})$, the following equality holds

$\alpha_{1}\cdot q_{1}^{*}\sigma=\alpha_{2}\cdot q_{2}^{*}\sigma.$

We note that there is a natural bimeromorphic map $\varphi\colon
Z_{1}\dashrightarrow Z_{2}$ over $X$. Let $\widetilde{X}\subseteq Z_{1}\times
Z_{2}$ be the closure of the graph of $\varphi$, and let
$p_{1}\colon\widetilde{X}\to Z_{1}$, $p_{1}\colon\widetilde{X}\to Z_{2}$ and
$\rho\colon\widetilde{X}\to X$ be the natural morphisms. Then we have

$\alpha_{j}\cdot q_{j}^{*}\sigma=p_{j}^{*}\alpha_{j}\cdot\rho^{*}\sigma$

for $j=1,2$. We notice that $p_{1}$, $p_{2}$ and $\rho$ are isomorphisms over
$X^{\circ}$. Hence if $\widetilde{X}^{\circ}=\rho^{-1}(X^{\circ})$, then we
have

$(p_{1}^{*}\alpha_{1})|_{\widetilde{X}^{\circ}}=(p_{2}^{*}\alpha_{2})|_{\widetilde{X}^{\circ}}.$

By applying Lemma 9.2, we obtain that $\alpha_{1}\cdot
q_{1}^{*}\sigma=\alpha_{2}\cdot q_{2}^{*}\sigma.$ This completes the proof of
the proposition. ∎

######  Remark 9.4.

We gather some remarks on this notion of Chern classes.

  1. (1)

If $X$ has quotient singularities and if $\mathcal{F}$ induces an orbifold
vector bundle $\mathcal{F}_{\mathrm{orb}}$ on the standard orbifold structure
over $X$, then the Chern classes of $\mathcal{F}$ coincides with the linear
forms on $H^{\bullet}(X,\mathbb{R})$ induced by the orbifold Chern classes of
$\mathcal{F}_{\mathrm{orb}}$.

  2. (2)

Let $X^{\circ}\subseteq X$ be an open subset with quotient singularities, such
that $\mathcal{F}|_{X^{\circ}}$ induces an orbifold vector bundle
$\mathcal{F}_{\mathrm{orb}}$ on the standard orbifold structure on
$X^{\circ}$, and that $X\setminus X^{\circ}$ has codimension at least 3. Let
$\Delta$ be $\hat{c}_{1}$, $\hat{c}_{2}$ or $\hat{c}_{1}^{2}$. If there is
some class $\beta\in H^{\bullet}(X,\mathbb{R})$, whose restriction on
$X^{\circ}$ is equal to $\Delta(\mathcal{F}_{\mathrm{orb}})$, then
$\Delta(\mathcal{F})$ is the same as the linear form on
$H^{\bullet}(X,\mathbb{R})$ defined by the intersection with $\beta$.

  3. (3)

Let $M=H_{1}\cap\cdots\cap H_{n-2}$ be the complete intersection surface of
basepoint-free Cartier divisors in general position. Then $M$ is has quotient
singularities, $\mathcal{F}|_{M}$ is reflexive, and we have

$\Delta(\mathcal{F})\cdot c_{1}(H_{1})\cdots
c_{1}(H_{n-2})=\Delta(\mathcal{F}|_{M}),$

where $\Delta$ is either $\hat{c}_{2}$ or $\hat{c}_{1}^{2}$.

With the first Chern class in Definition 9.1, we can extend the notion of
slope stability as follows. Let $X$ be a compact complex analytic variety of
dimension $n$, which has quotient singularities in codimension 2. Let
$\mathcal{F}$ be a coherent reflexive sheaf on $X$. For a cohomology class
$\alpha\in H^{2}(X,\mathbb{R})$, we can define the slope

$\mu_{\alpha}(\mathcal{F})=\frac{\hat{c}_{1}(\mathcal{F})\cdot\alpha^{n-1}}{\mathrm{rank}\,F}.$

Then $\mathcal{F}$ is called $\alpha$-semistable if for any nonzero coherent
subsheaf $\mathcal{E}\subseteq\mathcal{F}$, we have
$\mu_{\alpha}(\mathcal{E})\leq\mu_{\alpha}(\mathcal{F})$. It is called
$\alpha$-stable if the inequality is strict whenever $\mathcal{E}$ has smaller
rank. From the item (2) of Remark 9.4, we see that this definition coincides
with the classic one, if $X$ satisfies that every reflexive coherent sheaf of
rank one on it is $\mathbb{Q}$-Cartier.

###  9.B. Proof of the Bogomolov-Gieseker inequality

We complete the proof of Theorem 1.3 in this subsection. We will need the
following lemma, showing that the stability is an open condition.

######  Lemma 9.5.

Let $\mathfrak{X}=(X_{i},G_{i})$ be a compact complex orbifold of dimension
$n$. Let $\mathcal{E}_{\mathrm{orb}}=(\mathcal{E}_{i})$ be an orbifold vector
bundle on $\mathfrak{X}$. Let $\omega$ be an orbifold Kähler form and $\eta$
an orbifold closed semipositive $(1,1)$-form on $\mathfrak{X}$. Assume that
$\mathcal{E}_{\mathrm{orb}}$ is stable with respect to $\eta$. Then it is
stable with respect to $\eta+\varepsilon\cdot\omega$ for all $\varepsilon>0$
small enough.

###### Proof.

By a standard argument (see for example [DO23, Lemma 3.14]), it is enough to
show that the following set of real numbers is bounded from above

$\\{\hat{c}_{1}(\mathcal{F}_{\mathrm{orb}})\cdot[\eta]^{j}\cdot[\omega]^{n-1-j}\
|\ j=0,...,n-1\\},$

where $\mathcal{F}_{\mathrm{orb}}$ runs through all orbifold coherent
subsheaves of $\mathcal{E}_{\mathrm{orb}}$.

To prove this, we endow $\mathcal{E}_{\mathrm{orb}}$ an orbifold Hermitian
metric $h$. We fix an integer $j\in\\{0,...,n-1\\}$, and we fix an orbifold
coherent subsheaf $\mathcal{F}_{\mathrm{orb}}=(\mathcal{F}_{i})$. We denote by
$\mathcal{Q}_{\mathrm{orb}}=(\mathcal{Q}_{i})$ the quotient
$\mathcal{E}_{\mathrm{orb}}/\mathcal{F}_{\mathrm{orb}}$. Let $X^{\circ}_{i}$
be the largest open subset of $X_{i}$ where $\mathcal{F}_{i}$ is a subbundle.
Let $\mu_{i}\colon\mathcal{E}_{i}\to\mathcal{F}_{i}$ be the $h$-orthogonal
projection defined on $X^{\circ}_{i}$. We endow $\mathcal{F}_{\mathrm{orb}}$
with the restriction metric of $h$. Let $\Theta_{\mathcal{E}}$ and
$\Theta_{\mathcal{F}}$ be the Chern curvatures of $h$ on
$\mathcal{E}_{\mathrm{orb}}$ and on $\mathcal{F}_{\mathrm{orb}}$ respectively.
Then there is $(1,0)$-section $\alpha_{i}$ of
$\mathcal{H}om(\mathcal{F}_{i},\mathcal{Q}_{i})$ on $X^{\circ}_{i}$, called
the second fundamental form of the subsheaf $\mathcal{F}_{i}$, such that

$\mathrm{tr}\,(\Theta_{\mathcal{F}})_{i}=\mathrm{tr}\,(\mu_{i}\circ(\Theta_{\mathcal{E}})_{i}\circ\mu_{i})-\mathrm{tr}\,(\alpha_{i}\wedge\alpha_{i}^{*}).$

Moreover, the collection $(\alpha_{i})$ is compatible along the overlaps.

For any local smooth $(1,1)$-form $\varphi$ on $X_{i}$, we can define the
smooth function $\Lambda^{\prime}\varphi$ so that

$(\Lambda^{\prime}\varphi)\cdot\omega^{n}=n\sqrt{-1}\cdot\varphi\wedge\eta^{j}\wedge\omega^{n-1-j}.$

We remark that such an association is $\mathcal{O}_{X_{i}}$-linear. By
compactness, $|\Lambda^{\prime}(\Theta_{\mathcal{E}})_{i}|$ is bounded on
$X_{i}$, where the norm is induced by $h$. It follows that

$\int_{\mathfrak{X}}\Big{(}\mathrm{tr}\,(\mu\circ\Theta_{\mathcal{E}}\circ\mu)\Big{)}\wedge\eta^{j}\wedge\omega^{n-1-j}$

is bounded from above a constant independent of $\mathcal{F}_{\mathrm{orb}}$.
We also note that

$\sqrt{-1}(\mathrm{tr}\,(\alpha_{i}\wedge\alpha_{i}^{*}))\wedge\eta^{j}\wedge\omega^{n-1-j}$

is a semipositive $(n,n)$-form. It follows that

$\int_{\mathfrak{X}}\Big{(}(\mathrm{tr}\,\Theta_{\mathcal{F}})\Big{)}\wedge\eta^{j}\wedge\omega^{n-1-j}\leq\int_{\mathfrak{X}}\Big{(}(\mathrm{tr}\,(\mu\circ\Theta_{\mathcal{E}}\circ\mu)\Big{)}\wedge\eta^{j}\wedge\omega^{n-1-j}$

is bounded from above by a constant independent of
$\mathcal{F}_{\mathrm{orb}}$. Hence the number
$\hat{c}_{1}(\mathcal{F}_{\mathrm{orb}})\cdot[\eta]^{j}\cdot[\omega]^{n-1-j}$
is bounded from above by a constant independent of
$\mathcal{F}_{\mathrm{orb}}$. This completes the proof of the lemma. ∎

We are now in the position of proving Theorem 1.3.

###### Proof of Theorem 1.3.

Let $f\colon Y\to X$ be an orbifold modification as in Theorem 1.2. Then $Y$
has quotient singularities. Since $f$ is projective, $Y$ is a compact Kähler
variety. It follows that $Y$ is the quotient space of an orbifold
$\mathfrak{Y}=(Y_{i},G_{i})_{i\in I}$, such that the action of $G_{i}$ is free
in codimension 1. Let $\mathcal{E}=(f^{*}\mathcal{F})/(\mathrm{torsion})$.
Then $\mathcal{E}$ induces a reflexive coherent orbifold sheaf
$\mathcal{E}_{\mathrm{orb}}$ on $\mathfrak{Y}$, see for example [DO23, Remark
3.4.5]. We note that $\mathcal{E}$ is stable with respect to $f^{*}\omega$. By
[DO23, Remark 3.4.5] again, we deduce that $\mathcal{E}_{\mathrm{orb}}$ is
stable with respect to the class of $f^{*}\omega$.

By Theorem 4.8, there is a functorial projective bimeromorphic morphism
$p_{i}\colon Z_{i}\to Y_{i}$, such that $Z_{i}$ is smooth, that
$\mathcal{H}_{i}:=p_{i}^{*}\mathcal{E}_{i}/(\mathrm{torsion})$ is locally
free, and that the indeterminacy locus of $p_{i}^{-1}$ has codimension at
least 3. Moreover, there is a canonical action of $G_{i}$ on $Z_{i}$ such that
$p_{i}$ is $G_{i}$-equivariant, and that $(Z_{i},G_{i})$ induces a complex
orbifold $\mathfrak{Z}$ with quotient space $Z$. The collection
$(\mathcal{H}_{i})$ defines an orbifold vector bundle
$\mathcal{H}_{\mathrm{orb}}$ on $\mathfrak{Y}$. There is an induced projective
bimeromorphic morphism $p\colon Z\to Y$. In particular, $Z$ is a Kähler
variety. Then there is an orbifold Kähler form $\eta$ on $Z$, see [Wu23, Lemma
1]. Let $q\colon Z\to X$ be the natural morphism. We note that $q^{*}\omega$
can be viewed as a semipositive orbifold $(1,1)$-form on $\mathfrak{Z}$, and
that $\mathcal{H}_{\mathrm{orb}}$ is stable with respect to $q^{*}\omega$. By
Lemma 9.5, for any $\varepsilon>0$ small enough, $\mathcal{H}_{\mathrm{orb}}$
is stable with respect to the following orbifold Kähler form on
$\mathfrak{Z}$,

$\alpha_{\varepsilon}:=q^{*}\omega+\varepsilon\eta.$

By [Fau22, Theorem 1], $\mathcal{H}_{\mathrm{orb}}$ admits an orbifold
Hermitian-Einstein metric with respect to $\alpha_{\varepsilon}$. Then, as in
[Kob14, Theorem 4.4.7], we have

$\Big{(}2r\hat{c}_{2}(\mathcal{H}_{\mathrm{orb}})-(r-1)\hat{c}_{1}(\mathcal{H}_{\mathrm{orb}})\Big{)}\cdot[\alpha_{\varepsilon}]^{n-2}\geq
0.$

The limit of the LHS above, when $\varepsilon$ tends to 0, is equal to

$\displaystyle\Big{(}2r\hat{c}_{2}(\mathcal{H}_{\mathrm{orb}})-(r-1)\hat{c}_{1}(\mathcal{H}_{\mathrm{orb}})\Big{)}\cdot[q^{*}\omega]^{n-2}$
$\displaystyle=$
$\displaystyle\Big{(}2r\hat{c}_{2}(\mathcal{F})-(r-1)\hat{c}_{1}(\mathcal{F})^{2}\Big{)}\cdot[\omega]^{n-2}$

This completes the proof of the theorem. ∎

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