singularities $f:Y\to X$ such that the exceptional locus ${{\operatorname{Exc}}}(f)$ of $f$ is a divisor on $Y$, ${{\operatorname{Exc}}}(f)\cup f^{-1}_*\Delta$ has a simple normal crossing support, and $$K_Y=f^*(K_X+\Delta)+\sum _i a_i E_i$$ with $a_i>-1$ for every $f$-exceptional divisor $E_i$, then the pair $(X, \Delta)$ is called a [*[dlt]{}*]{} pair.

The following definitions are very useful in this paper.

\[x-def2.4\] Let $f:X\to Y$ be a dominant morphism between normal varieties and let $D$ be an $\mathbb R$-divisor on $X$. We can write $$D=D_{\mathrm{hor}}+D_{\mathrm{ver}}$$ such that every irreducible component of $D_{\mathrm{hor}}$ (resp.  $D_{\mathrm{ver}}$) is mapped (resp. not mapped) onto $Y$. If $D=D_{\mathrm{hor}}$ (resp. $D=D_{\mathrm{ver}}$), $D$ is said to be [*[horizontal]{}*]{} (resp. [*[vertical]{}*]{}).

\[x-def2.5\] Let $D=\sum _i d_i D_i$ be an $\mathbb R$-divisor on a normal variety $X$, where $D_i$ is a prime divisor on $X$ for every $i$, $D_i\ne D_j$ for $i\ne j$, and $d_i \in 
\mathbb R$ for every $i$. Then we put $$\lfloor D\rfloor =\sum _i \lfloor d_i \rfloor D_i, 
\quad \{D\}=D-\lfloor 
D\rfloor, \quad \text{and} \quad\lceil D\rceil =-\lfloor -D\rfloor.$$ Note that $\lfloor d_i \rfloor$ is the integer which satisfies $d_i-1<\lfloor d_i \rfloor \leq d_i$. We also note that $\lfloor D\rfloor$, $\lceil D\rceil$, and $\{D\}$ are called the [*[round-down]{}*]{}, [*[round-up]{}*]{}, and [*[fractional part]{}*]{} of 