Let $\pi:X\to V$ be a log resolution. Write $K_{X}\cong\pi^{*}K_{V}+\displaystyle\sum a_{E}E$. Since $exc(\pi)$ has simple normal crossings, at every $P\in E=exc(\pi)$ there are local coordinates $(z^{i})_{i=1,...,n}$ such that $E$ is described by the equation $z^{1}\ldots z^{q}=0$. Let $E_{j}$ be the divisor $z_{j}=0$. We have: $\pi^{*}v=\prod_{j=1}^{q}|z^{j}|^{2a_{E_{j}}}d\lambda$ where $d\lambda$ is a Lebesgue measure on $X$, hence the measure $\pi^{*}v$ has finite mass near $P$ iff $\forall j,a_{E_{j}}>-1$. Thus $\int_{U_{1}^{reg}}v<\infty$ iff $\forall E,\ a_{E}>-1$.

Let $f_{1}$ be the density of $\pi^{*}v$ with respect to $d\lambda$. Since $f_{1}$ is comparable to $\prod_{j=1}^{q}|z^{j}|^{2a_{E_{j}}}$ near $P$, it follows that $f_{1}$ belongs actually to $L^{p}(X,d\lambda)$ for some $p>1$ when $X$ is log terminal.

Let $D=1/f$ be the density of $\Omega^{n}$ with respect to $v$. We will see here below that $D$ is bounded but it might have zeroes on $E$, hence $f$ is unbounded in general. However we will show that $f\circ\pi\in L^{\alpha}(X,d\lambda)$ for $\alpha>0$ small enough, hence it follows from Hölder’s inequality (as in the proof of lemma 3.2) that

if $\varepsilon>0$ is small enough.

Fix $x\in V$ and let $i:U_{x}\to\mathbb{C}^{m}$ be a local embedding of a neighborhood $U_{x}$ of $x$. We consider the $\left(\begin{array}[]{c}m\\ n\end{array}\right)$ $n$-forms on $U_{x}^{reg}$ $du^{I}=du^{i_{1}}\wedge\ldots du^{i_{n}}$, where $(u^{i})$ is a set of affine coordinates on $\mathbb{C}^{m}$. Observe that $\Omega^{n}$ is comparable to $\sum_{I}v_{du^{I}}$ ¹¹¹¹ 11 Note that the formula for $v_{\beta}$ makes sense even if $\beta$ is not a local generator.. Since $\beta$ is a local generator at $x$ of $\omega_{V}^{[N]}$, we have $(du^{I})^{N}=f_{I}\beta$ where $f_{I}\in\mathcal{O}_{V,x}$ is the germ of an holomorphic function at $x$. Therefore $\Omega^{n}$ is comparable to $\sum_{I}|f_{I}|^{\frac{2}{N}}v$, hence $D$ is comparable to $[\sum_{I}|f_{I}|^{\frac{2}{N}}]^{-1}$ near $x$.

The functions $(f_{I})$ generate an ideal $\mathcal{I}_{x}\subset\mathcal{O}_{V,x}$. Actually, the construction can be globalized to provide a coherent ideal sheaf $\mathcal{I}\subset\mathcal{O}_{V}$ cosupported on $V^{sing}$.

We may assume [Hi] that $\pi:X\to V$ is a log resolution of $(V,\mathcal{I})$, namely a log resolution of $V$ with the additional property that the ideal sheaf $\pi^{-1}\mathcal{I}.\mathcal{O}_{X}$ which is the ideal sheaf of $\mathcal{O}_{X}$ generated by the family of holomorphic functions $(\pi^{*}f_{I})_{I}$, satisfies $\pi^{-1}\mathcal{I}.\mathcal{O}_{X}=\mathcal{O}_{X}(-\sum Nb_{E}E)\subset\mathcal{O}_{X}$ where $N.b_{E}\in\mathbb{N}$ is a positive multiplicity attached to any exceptional divisor of $\pi$.

In local coordinates near $P\in X$, $\pi^{*}D$ is comparable to $\prod_{j}|z_{j}|^{2b_{E_{j}}}$, hence $\pi^{*}(f_{1}D^{-\varepsilon})\text{ is comparable to }\prod_{j}|z_{j}|^{2(a_{E_{j}}-\varepsilon b_{E_{j}})}.$ It follows that for every relatively compact subset $U_{1}\subset\subset U,\ \ f\in L^{1+\varepsilon}(U_{1},\Omega^{n})$ iff $\forall E,\ \ \pi(E)\cap U\not=\emptyset\Rightarrow a_{E}-\varepsilon b_{E}>-1$. ∎