$$\mathcal O_X(K_X)\simeq \omega_X$$ holds when $X$ is normal.

If $f:X\to Y$ is a morphism between Gorenstein schemes, then we put $$\omega_{X/Y}:=\omega_X\otimes f^*\omega^{\otimes -1}_Y.$$

If $f:X\to Y$ is a morphism from a normal scheme $X$ to a normal Gorenstein scheme $Y$, then we put $$K_{X/Y}:=K_X-f^*K_Y.$$

Let us quickly see the definition of singularities of pairs.

\[x-def2.2\] Let $X$ be a normal variety and let $\Delta$ be an effective $\mathbb R$-divisor on $X$ such that $K_X+\Delta$ is $\mathbb R$-Cartier. Let $f:Y\to X$ be a resolution of singularities of $X$ such that ${{\operatorname{Exc}}}(f)\cup f^{-1}_*\Delta$ has a simple normal crossing support, where ${{\operatorname{Exc}}}(f)$ is the exceptional locus of $f$ on $Y$ and $f^{-1}_*\Delta$ is the strict transform of $\Delta$ on $Y$. Then we can write $$K_Y=f^*(K_X+\Delta)+\sum _i a_i E_i$$ with $f_*\left(\sum _i a_i E_i\right)=-\Delta$. We say that $(X, \Delta)$ is [*[log canonical]{}*]{} (resp. [*[klt]{}*]{}) if $a_i\geq -1$ (resp. $a_i>-1$) for every $i$.

If $(X, \Delta)$ is log canonical and there exist a resolution of singularities $f:Y\to X$ as above and a prime divisor $E_i$ on $Y$ with $a_i=-1$, then $f(E_i)$ is called a [*[log canonical center]{}*]{} of $(X, \Delta)$.

\[x-def2.3\] Let $(X, \Delta)$ be a log canonical pair. If there exists a resolution of 