Proof. Let $\omega>0$ be a Kähler metric and write $\gamma=\theta+i\partial\bar{\partial}u$ whith $\theta$ smoth and $u$ continuous. There exist a sequence $\gamma_{j}=\theta+i\partial\bar{\partial}u_{j}$, whith $u_{j}\in C^{\infty}(X)$, such that $u_{j}\downarrow u$ uniformly and $0<\gamma_{j}+\varepsilon_{j}\omega$ whith $0<\varepsilon_{j}\downarrow 0$. We consider also a regularizing family $(f_{j})_{j}\subset C^{\infty}(X)$, $f_{j}>0$ of $f$ in $L\log^{n+\delta}L(X)$. We can assume

$$\int\limits_{X}(\gamma_{j}+\varepsilon_{j}\omega)^{n}=\int\limits_{X}f_{j}\,\Omega\,,$$

otherwise we multiply $f_{j}$ by a constant $c_{j}>0$ which converges to $1$ by the normalising condition $\int_{X}\gamma^{n}=\int_{X}f\,\Omega$. We distinguish two cases.

Case $\lambda=0$. By Yau’s solution of the Calabi conjecture there exists a unique family $(\psi_{j})_{j}$, $\psi_{j}\in{\cal P}^{0}_{\gamma_{j}+\varepsilon_{j}\omega}$ of smooth solutions of the complex Monge-Ampère equations

$$(\gamma_{j}+\varepsilon_{j}\omega+i\partial\bar{\partial}\psi_{j})^{n}=f_{j}\,\Omega\,.$$

The hypothesis (C1) and (C2a) of statement (C) of theorem 3 are obviously satisfied for the family $(\gamma_{j}+\varepsilon_{j})_{j}$. We deduce that the constant $C_{1}=C_{1}(\delta\,,\,\gamma_{j}+\varepsilon_{j}\omega\,,\,\Omega)>0$ in the statement of theorem 3,A does not blow up as $j\rightarrow+\infty$. Moreover the uniform estimate

$\displaystyle\|f_{j}\|_{L\log^{n+\delta}L(X)}\leq C^{\prime}\|f\|_{L\log^{n+\delta}L(X)}=:K\,,$

(4.1)

holds for all $j$. (See [Ra-Re] page 364 or [Iw-Ma], theorem 4.12.2, page 79.) Thus by theorem 3, A we obtain the uniform estimate $\|\psi_{j}\|_{C^{0}(X)}\leq C$. On the other hand we have a priori a uniform estimate $\int_{X}-\psi_{j}\,\Omega\leq C^{\prime}$ since the local potentials of the family $(\gamma_{j}+\varepsilon_{j})_{j}$ stay uniformly bounded. Thus, by elementary properties of plurisubharmonic functions, (see [Dem1], chapter 1) there exists a $L^{1}$-convergent subsequence $(\psi_{j})_{j}$ (which by abuse of notations we denote in the same way). We can apply theorem 3, B to the complex Monge-Ampère equation in consideration since we dispose of the estimate (4.1) and the constant $C_{2}=C_{2}(\delta\,,\,\gamma_{j}+\varepsilon_{j}\omega\,,\,\Omega,K)>0$ in the relative statement is uniformly bounded in $j$ thanks to the same considerations concerning the constant $C_{1}$. We infer that the sequence $(\psi_{j})_{j}$ is a Cauchy sequence in the $C^{0}$-topology, thus convergent to some $\psi\in{\cal P}_{\gamma}\cap C^{0}(X)$. This implies the convergence of the weak limits

$$(\gamma+i\partial\bar{\partial}\psi)^{n}=\lim_{j\rightarrow+\infty}(\gamma_{j}+\varepsilon_{j}\omega+i\partial\bar{\partial}\psi_{j})^{n}=\lim_{j\rightarrow+\infty}f_{j}\,\Omega=f\,\Omega\,,$$

thus $\psi$ is the required solution of our degenerate complex Monge-Ampère equation. We normalise the solution $\psi$ whith the condition $\max_{X}\psi=0$. We prove now the uniqueness of the solutions in the case $\lambda=0$. Let $\varphi\in\hat{\cal P}^{0}_{\gamma}$ be another solution. Then the identity $\gamma_{\varphi}^{n}=\gamma_{\psi}^{n}$ implies $\varphi\in\check{\cal P}^{0}_{\gamma}$ by claim 2 in the proof of theorem 5. Let $\varphi_{\varepsilon}$, $\psi_{\varepsilon}$ be as in the statement of corollary 3 and set $u:=\psi-\varphi$, $u_{\varepsilon}:=\psi_{\varepsilon}-\varphi_{\varepsilon}$. Let us also recall the formula

$$\alpha^{k}-\beta^{k}=(\alpha-\beta)\wedge\sum_{l=0}^{k-1}\alpha^{l}\wedge\beta^{k-l-1}\,.$$

From there we deduce

	$\displaystyle 0$	$\displaystyle=$	$\displaystyle\int\limits_{X}-u(\gamma^{n}_{\psi}-\gamma^{n}_{\varphi})=\lim_{\varepsilon\rightarrow 0^{+}}\int\limits_{X}-u_{\varepsilon}(\gamma^{n}_{\psi}-\gamma^{n}_{\varphi})$		(4.2)
		$\displaystyle=$	$\displaystyle\lim_{\varepsilon\rightarrow 0^{+}}\,\sum_{l=0}^{n-1}\,\int\limits_{X}-u_{\varepsilon}\,i\partial\bar{\partial}u\wedge\gamma^{l}_{\psi}\wedge\gamma^{n-l-1}_{\varphi}$
		$\displaystyle=$	$\displaystyle\lim_{\varepsilon\rightarrow 0^{+}}\,\sum_{l=0}^{n-1}\,\int\limits_{X}i\partial u_{\varepsilon}\wedge\bar{\partial}u\wedge\gamma^{l}_{\psi}\wedge\gamma^{n-l-1}_{\varphi}$
		$\displaystyle=$	$\displaystyle\sum_{l=0}^{n-1}\,\int\limits_{X}i\partial u\wedge\bar{\partial}u\wedge\gamma^{l}_{\psi}\wedge\gamma^{n-l-1}_{\varphi}=:I\,,$

since $\partial u_{\varepsilon}\rightarrow\partial u$ in $L^{2}(X,\gamma^{l}_{\psi}\wedge\gamma^{n-l-1}_{\varphi})$ by corollary 3. Inspired by an idea of S. Blocki [Blo1], we will prove by induction on $k=0,...,n-1$ that

$\displaystyle\int\limits_{X}i\partial u\wedge\bar{\partial}u\wedge\gamma^{r}_{\psi}\wedge\gamma^{s}_{\varphi}\wedge\gamma^{k}=0$

(4.3)

for all $r,s\geq 0$, $r+s=n-k-1$. For $k=0$ this follows from (4.2). So we assume (4.3) for $k-1$ and we prove it for $k$. In fact consider the identity

$$\gamma^{k}=\gamma^{k}_{\psi}-i\partial\bar{\partial}\psi\wedge\sum_{l=0}^{k-1}\gamma^{l}_{\psi}\wedge\gamma^{k-l-1}\quad\mbox{and set }\quad\Xi:=\gamma^{r}_{\psi}\wedge\gamma^{s}_{\varphi}\wedge\sum_{l=0}^{k-1}\gamma^{l}_{\psi}\wedge\gamma^{k-l-1}\,.$$

By applying several times corollary 3 and by integrating by parts we derive

			$\displaystyle\int\limits_{X}i\partial u\wedge\bar{\partial}u\wedge\gamma^{r}_{\psi}\wedge\gamma^{s}_{\varphi}\wedge\gamma^{k}\;=\;\lim_{\varepsilon\rightarrow 0^{+}}\int\limits_{X}i\partial u_{\varepsilon}\wedge\bar{\partial}u\wedge\gamma^{r}_{\psi}\wedge\gamma^{s}_{\varphi}\wedge\gamma^{k}$
		$\displaystyle=$	$\displaystyle\lim_{\varepsilon\rightarrow 0^{+}}\,\left[\;\int\limits_{X}i\partial u_{\varepsilon}\wedge\bar{\partial}(u\gamma^{r+k}_{\psi}\wedge\gamma^{s}_{\varphi})\;-\;\int\limits_{X}i\partial u_{\varepsilon}\wedge\bar{\partial}(u\,i\partial\bar{\partial}\psi\wedge\Xi)\;\right]$
		$\displaystyle=$	$\displaystyle\lim_{\varepsilon\rightarrow 0^{+}}\,\left[\;\int\limits_{X}i\partial u_{\varepsilon}\wedge\bar{\partial}u\wedge\gamma^{r+k}_{\psi}\wedge\gamma^{s}_{\varphi}\;+\;\int\limits_{X}u_{\varepsilon}\,i\partial\bar{\partial}u\wedge i\partial\bar{\partial}\psi\wedge\Xi\;\right]$
		$\displaystyle=$	$\displaystyle\int\limits_{X}i\partial u\wedge\bar{\partial}u\wedge\gamma^{r+k}_{\psi}\wedge\gamma^{s}_{\varphi}\;-\;\lim_{\varepsilon\rightarrow 0^{+}}\,\int\limits_{X}u_{\varepsilon}\,i\partial\bar{\partial}\psi\wedge(\gamma_{\varphi}-\gamma_{\psi})\wedge\Xi$
		$\displaystyle\leq$	$\displaystyle I+\lim_{\varepsilon\rightarrow 0^{+}}\,\int\limits_{X}i\partial u_{\varepsilon}\wedge\bar{\partial}\left[\,\psi\,(\gamma_{\varphi}-\gamma_{\psi})\wedge\Xi\,\right]$

		$\displaystyle=$	$\displaystyle\lim_{\varepsilon\rightarrow 0^{+}}\,\left[\;\int\limits_{X}i\partial u_{\varepsilon}\wedge\bar{\partial}\psi\wedge\gamma_{\varphi}\wedge\Xi\;-\;\int\limits_{X}i\partial u_{\varepsilon}\wedge\bar{\partial}\psi\wedge\gamma_{\psi}\wedge\Xi\;\right]$		(4.4)
		$\displaystyle=$	$\displaystyle\int\limits_{X}i\partial u\wedge\bar{\partial}\psi\wedge\gamma_{\varphi}\wedge\Xi\;-\;\int\limits_{X}i\partial u\wedge\bar{\partial}\psi\wedge\gamma_{\psi}\wedge\Xi\,.$

Set $\chi=\varphi$ or $\chi=\psi$. Then the Cauchy-Schwarz inequality implies

			$\displaystyle\left|\;\int\limits_{X}i\partial u\wedge\bar{\partial}\psi\wedge\gamma_{\chi}\wedge\Xi\;\right|$
		$\displaystyle\leq$	$\displaystyle\left(\;\int\limits_{X}i\partial u\wedge\bar{\partial}u\wedge\gamma_{\chi}\wedge\Xi\;\right)^{1/2}\left(\;\int\limits_{X}i\partial\psi\wedge\bar{\partial}\psi\wedge\gamma_{\chi}\wedge\Xi\;\right)^{1/2}=0\,,$

by the inductive hypothesis. This combined with (4.4) implies (4.3) for $k$. So at the end of the induction we get

$$0=\int\limits_{X}i\partial u\wedge\bar{\partial}u\wedge\gamma^{n-1}\geq\eta^{n-1}\int\limits_{X}i\partial u\wedge\bar{\partial}u\wedge\theta^{n-1}\geq 0\,,$$

which implies $\varphi=\psi$ by elementary properties of plurisubharmonic functions.

Case $\lambda>0$. We start by proving the following lemma, which is a particular case of a more general result due to Yau. (See [Yau], sect. 6, page 376).

Proof. The argument is a simplification, in our particular case, of Yau’s original argument for the proof of thm. 4, sect. 6 in [Yau]. Set $\varphi^{\prime}_{0}:=\varphi^{\prime},\,\varphi^{\prime\prime}_{0}:=\varphi^{\prime\prime}$ and consider the solutions $\varphi^{\prime}_{j},\,\varphi^{\prime\prime}_{j}$ of the complex Monge-Ampère equations given by the iteration

	$\displaystyle(\omega+i\partial\bar{\partial}\varphi^{\prime}_{j})^{n}$	$\displaystyle=$	$\displaystyle e^{h+(\lambda+1)\varphi^{\prime}_{j}-\varphi^{\prime}_{j-1}}\,\omega^{n}\,,$		(4.6)
	$\displaystyle(\omega+i\partial\bar{\partial}\varphi^{\prime\prime}_{j})^{n}$	$\displaystyle=$	$\displaystyle e^{h+(\lambda+1)\varphi^{\prime\prime}_{j}-\varphi^{\prime\prime}_{j-1}}\,\omega^{n}\,.$		(4.7)

Notice that we can solve this equations even if the terms $e^{h-\varphi^{\prime}_{j-1}}$, $e^{h-\varphi^{\prime\prime}_{j-1}}$ are not normalized, see lem. 2 page 378 in [Yau]. Set $k=\lambda+1$ and consider

$$(\omega+i\partial\bar{\partial}\varphi^{\prime}_{1})^{n}=e^{h+k(\varphi^{\prime}_{1}-\varphi^{\prime}_{0})+\lambda\varphi^{\prime}_{0}}\omega^{n}\geq e^{k(\varphi^{\prime}_{1}-\varphi^{\prime}_{0})}e^{h}\omega^{n}=e^{k(\varphi^{\prime}_{1}-\varphi^{\prime}_{0})}(\omega+i\partial\bar{\partial}\varphi^{\prime}_{0})^{n}\,.$$

At a maximum point of $\varphi^{\prime}_{1}-\varphi^{\prime}_{0}$ we have the inequality

$$(\omega+i\partial\bar{\partial}\varphi^{\prime}_{0})^{n}\geq(\omega+i\partial\bar{\partial}\varphi^{\prime}_{1})^{n}\,.$$

By plugging this into the previous one, we deduce $\varphi^{\prime}_{1}\leq\varphi^{\prime}_{0}$. We now prove by induction the inequality $\varphi^{\prime}_{j}\leq\varphi^{\prime}_{j-1}$. In fact by dividing $(\ref{MAith1})_{j}$ with $(\ref{MAith1})_{j-1}$ we get

$$\frac{(\omega+i\partial\bar{\partial}\varphi^{\prime}_{j})^{n}}{(\omega+i\partial\bar{\partial}\varphi^{\prime}_{j-1})^{n}}=e^{k(\varphi^{\prime}_{j}-\varphi^{\prime}_{j-1})-(\varphi^{\prime}_{j-1}-\varphi^{\prime}_{j-2})}\geq e^{k(\varphi^{\prime}_{j}-\varphi^{\prime}_{j-1})}\,.$$

At a maximum point of $\varphi^{\prime}_{j}-\varphi^{\prime}_{j-1}$ we again find the inequality

$$(\omega+i\partial\bar{\partial}\varphi^{\prime}_{j})^{n}\leq(\omega+i\partial\bar{\partial}\varphi^{\prime}_{j-1})^{n}\,.$$

Combining this with the previous one we deduce $\varphi^{\prime}_{j}\leq\varphi^{\prime}_{j-1}$. We also prove by induction the inequality $\varphi^{\prime\prime}_{j}\leq\varphi^{\prime}_{j}$, which is true by definition in the case $j=0$. By dividing $(\ref{MAith1})_{j}$ with $(\ref{MAith2})_{j}$ we get

$$\frac{(\omega+i\partial\bar{\partial}\varphi^{\prime}_{j})^{n}}{(\omega+i\partial\bar{\partial}\varphi^{\prime\prime}_{j})^{n}}=e^{k(\varphi^{\prime}_{j}-\varphi^{\prime\prime}_{j})-(\varphi^{\prime}_{j-1}-\varphi^{\prime\prime}_{j-1})}\leq e^{k(\varphi^{\prime}_{j}-\varphi^{\prime\prime}_{j})}\,,$$

by the induction hypothesis $\varphi^{\prime\prime}_{j-1}\leq\varphi^{\prime}_{j-1}$. At a minimum point of $\varphi^{\prime}_{j}-\varphi^{\prime\prime}_{j}$ we get

$$(\omega+i\partial\bar{\partial}\varphi^{\prime}_{j})^{n}\geq(\omega+i\partial\bar{\partial}\varphi^{\prime\prime}_{j})^{n}\,,$$

hence $\varphi^{\prime\prime}_{j}\leq\varphi^{\prime}_{j}$. As a conclusion, we have proved the sequence of inequalities

$\displaystyle\varphi^{\prime\prime}_{0}\leq\varphi^{\prime\prime}_{j-1}\leq\varphi^{\prime\prime}_{j}\leq\varphi^{\prime}_{j}\leq\varphi^{\prime}_{j-1}\leq\varphi^{\prime}_{0}\,.$

(4.8)

These inequalities imply $0<2n+\Delta_{\omega}\varphi^{\prime}_{j}\leq C\,B_{j}$, where $B_{j}>0$ satisfies the uniform estimate

$\displaystyle 0\geq C_{1}\,B_{j}^{\frac{1}{n-1}}-\left(2n+\max_{X}\Delta_{\omega}\varphi^{\prime}_{j-1}\right)B_{j}^{-1}-C_{0}\,,$

(4.9)

and $C_{0},\,C_{1}>0$, which are obtained by applying the maximum principle in a way similar to Yau’s proof of the second order estimate for the solution of the Calabi conjecture [Yau]. (In the case $n=1$ the uniform estimate $0<2n+\Delta_{\omega}\varphi^{\prime}_{j}\leq C^{\prime}$ follows immediately from the inequalities (4.8).) Fix now a constant $C_{3}>0$ such that the inequality

$$C_{1}\,x^{1+\frac{1}{n-1}}\geq(C_{0}+2C_{2})x-C_{3}\,,$$

hold for all $x\geq 0$. This implies by (4.9) the estimate

$$2(2n+\Delta_{\omega}\varphi^{\prime}_{j})\leq 2C_{2}\,B_{j}\leq\left(2n+\max_{X}\Delta_{\omega}\varphi^{\prime}_{j-1}\right)+C_{3}\,,$$

thus

$$2n+\max_{X}\Delta_{\omega}\varphi^{\prime}_{j}\leq 2^{-j}\left(2n+\max_{X}\Delta_{\omega}\varphi^{\prime}_{j-1}\right)+C_{3}\,,$$

by iteration. By taking the derivate in the Green Formula (see [Aub], Th. 4.13 page 108) we get the identity

$$d_{x}\varphi^{\prime}_{j}=-\int\limits_{X}d_{x}G_{\omega}(x,\cdot)\,\Delta_{\omega}\varphi^{\prime}_{j}\,\omega^{n}\,,$$

which implies the estimate $|\nabla_{\omega}\varphi^{\prime}_{j}|_{\omega}\leq C_{\omega}\max_{X}\Delta_{\omega}\varphi^{\prime}_{j}\leq C$. By applying the complex version of the Evans-Krylov theory [Ti2] we deduce the uniform estimate $\|\varphi^{\prime}_{j}\|_{C^{2,\alpha}(X)}\leq C^{\prime}$. This implies that the sequence $(\varphi^{\prime}_{j})_{j}$ converges in the $C^{2,\alpha}$-topology to the unique solution $\varphi$ of the complex Monge-Ampère equation (4.5). Then the conclusion follows from inequalities (4.8). $\Box$

We consider now the unique family $(\psi_{j})_{j}$, $\psi_{j}\in{\cal P}_{\gamma_{j}+\varepsilon_{j}\omega}$ of smooth solutions of the complex Monge-Ampère equations

$$(\gamma_{j}+\varepsilon_{j}\omega+i\partial\bar{\partial}\psi_{j})^{n}=f_{j}\,e^{\lambda\,\psi_{j}}\Omega\,,$$

given by the Aubin-Yau’s solution of the Calabi conjecture. Consider also the solutions $\psi^{\prime}_{j}$, $\psi^{\prime\prime}_{j}$, $\min_{X}\psi^{\prime}_{j}=0=\max_{X}\psi^{\prime\prime}_{j}$ of the complex Monge-Ampère equation

$$(\gamma_{j}+\varepsilon_{j}\omega+i\partial\bar{\partial}\varphi)^{n}=f_{j}\,\Omega\,.$$

By applying lemma 6 we deduce $\psi^{\prime\prime}_{j}\leq\psi_{j}\leq\psi^{\prime}_{j}$ for all $j$. By the same argument in the case $\lambda=0$, we deduce $\|\psi^{\prime}_{j}\|_{C^{0}(X)},\,\|\psi^{\prime\prime}_{j}\|_{C^{0}(X)}\leq C$, thus $\|\psi_{j}\|_{C^{0}(X)}\leq C$ and so

$$\|f_{j}\,e^{\lambda\psi_{j}}\|_{L\log^{n+\delta}L(X)}\leq C^{\prime}e^{\lambda C}\|f\|_{L\log^{n+\delta}L(X)}\,.$$

This fact allows us to apply theorem 3, B as in the case $\lambda=0$ in order to get a sequence of solutions $(\psi_{j})_{j}$ convergent in the uniform topology to some $\psi\in{\cal P}_{\gamma}\cap C^{0}(X)$. This implies the convergence of the weak limits

$$(\gamma+i\partial\bar{\partial}\psi)^{n}=\lim_{j\rightarrow+\infty}(\gamma_{j}+\varepsilon_{j}\omega+i\partial\bar{\partial}\psi_{j})^{n}=\lim_{j\rightarrow+\infty}f_{j}e^{\lambda\psi_{j}}\,\Omega=fe^{\lambda\psi}\,\Omega\,.$$

The integral estimate in the statement of theorem 6 follows immediately from theorem 3, A and from the inequalities $\psi^{\prime\prime}_{j}\leq\psi_{j}\leq\psi^{\prime}_{j}$. We prove now the uniqueness of the solutions. Let $\varphi\in\hat{\cal P}_{\gamma}$ be another solution. The fact that $\varphi\leq C$ implies

$$\|f\,e^{\lambda\varphi}\|_{L\log^{n+\delta}L(X)}\leq e^{\lambda C}\|f\|_{L\log^{n+\delta}L(X)}\,,$$

which allows us to solve the degenerate complex Monge-Ampère equation

$$(\gamma+i\partial\bar{\partial}u)^{n}=f\,e^{\lambda\varphi}\Omega\,,$$

with $u\in{\cal P}^{0}_{\gamma}\cap C^{0}(X)$. By the uniqueness result in the case $\lambda=0$ we deduce $u=\varphi-\sup_{X}\varphi$, thus $\varphi\in{\cal P}_{\gamma}\cap C^{0}(X)$. By applying the comparison principle we get

$$\int\limits_{\varphi<\psi}\gamma^{n}_{\psi}\,\leq\,\int\limits_{\varphi<\psi}\gamma^{n}_{\varphi}\,=\,\int\limits_{\varphi<\psi}e^{\lambda(\varphi-\psi)}\gamma^{n}_{\psi}\,,$$

which implies $\int_{\varphi<\psi}\gamma^{n}_{\psi}=0$ since $e^{\lambda(\varphi-\psi)}<1$. This implies $\varphi\geq\psi$ $\gamma^{n}_{\psi}$-almost everywhere, thus $\gamma^{n}_{\varphi}=e^{\lambda(\varphi-\psi)}\gamma^{n}_{\psi}\leq\gamma^{n}_{\psi}$ $\gamma^{n}_{\psi}$-almost everywhere. By symmetry we also deduce $\gamma^{n}_{\psi}\leq\gamma^{n}_{\varphi}$ $\gamma^{n}_{\varphi}$-almost everywhere. The fact that $\varphi,\,\psi$ are solutions of our complex Monge-Ampère equation implies that a property holds $\gamma^{n}_{\psi}$-almost everywhere if and only if it holds $\Omega$-almost everywhere and the same for $\gamma^{n}_{\varphi}$. We thus infer $\gamma^{n}_{\psi}=\gamma^{n}_{\varphi}$, which implies $\psi=\varphi$ by the expression of the Monge-Ampère equation. $\Box$

The following lemma gives us an important class of functions for the right hand side of the degenerate complex Monge-Ampère equation.

Proof. We define the coherent complex analytic sheaves ${\cal J}_{\sigma_{j}}:=\sum_{p}{\cal O}_{X}\sigma_{j,p}$ and ${\cal J}_{\tau_{r}}:=\sum_{q}{\cal O}_{X}\tau_{r,q}\subset{\cal O}_{X}$, with $\sigma_{j}=(\sigma_{j,p})_{p}$ , $\tau_{r}=(\tau_{r,q})_{q}$ for some local holomorphic trivializations of the vector bundles $E_{j}$, $F_{r}$. Clearly the definition is independent of the local trivialization, thus this sheaves are globally well defined. By the Hironaka desingularization theorem [Hir] we can find a proper bimeromorphic morphism $\mu:\tilde{X}\rightarrow X$ of compact complex manifolds such that there exists a family $(H_{s})_{s}$, $H_{s}\subset\tilde{X}$ of smooth hypersurfaces with normal crossing in $\tilde{X}$ such that $\mu\left(\bigcup_{s}H_{s}\right)=\bigcup_{j}V(\sigma_{j})\cup\bigcup_{r}V(\tau_{r})$,

$$K_{\tilde{X}}=\mu^{*}K_{X}+\sum_{s}\,a_{s}\,H_{s}\,,\quad a_{s}\in\mathbb{N}\,,$$

and $\mu^{*}{\cal J}_{\sigma_{j}}={\cal O}_{\tilde{X}}\left(-\sum_{s}\,b_{j,s}\,H_{s}\right)$, $\mu^{*}{\cal J}_{\tau_{r}}={\cal O}_{\tilde{X}}\left(-\sum_{s}\,c_{r,s}\,H_{s}\right)$, $b_{j,s}$, $c_{r,s}\in\mathbb{N}$ for all $j,r$. The fact that $\mu$ is a holomorphic map implies $d\mu\otimes_{{}_{\mathbb{R}}}\mathbb{I}_{{}_{\mathbb{C}}}=\partial\mu\oplus\overline{\partial\mu}$, with $\partial\mu\in H^{0}(\tilde{X},\Lambda^{1,0}T^{*}_{\tilde{X}}\otimes\mu^{*}T^{1,0}_{X})$. Thus the divisor of the Jacobian $\Lambda^{n}\partial\mu\in H^{0}(\tilde{X},K_{\tilde{X}}-\mu^{*}K_{X})$ of $\mu$ is by definition $\sum_{s}\,a_{s}\,H_{s}$. On the other hand the invariance of the integral by orientation preserving diffeomorphisms implies

	$\displaystyle I_{p}$	$\displaystyle:=$	$\displaystyle\int\limits_{X}\,\prod\limits_{j=1}^{N}|\sigma_{j}|^{2l_{j}p}\cdot\prod\limits_{r=1}^{M}|\tau_{r}|^{-2h_{r}p}\,\Omega$
		$\displaystyle=$	$\displaystyle\int\limits_{\tilde{X}}\,\prod\limits_{j=1}^{N}|\sigma_{j}\circ\mu|^{2l_{j}p}\cdot\prod\limits_{r=1}^{M}|\tau_{r}\circ\mu|^{-2h_{r}p}\,(\Omega\circ\mu)\cdot(\Lambda^{n}\partial\mu)\wedge\overline{(\Lambda^{n}\partial\mu)}\,.$

For any open set $U\subset\tilde{X}$ we denote by $\{s_{1},...,s_{m}\}:=\{s\,|\,H_{s}\cap U\not=\emptyset\}$. For any point in $\tilde{X}$ one can find a coordinate neighborhood $U:=\{|z_{k}|<1\}$ such that $H_{s_{t}}=\{z_{t}=0\}$, $t=1,...,m\leq n$. With respect to this coordinates, we have

$$(\Omega\circ\mu)\cdot(\Lambda^{n}\partial\mu)\wedge\overline{(\Lambda^{n}\partial\mu)}=\rho\prod\limits_{t=1}^{m}|z_{t}|^{2a_{s_{t}}}\,i^{n^{2}}dz_{1}\wedge...\wedge dz_{n}\wedge d\bar{z}_{1}\wedge...\wedge d\bar{z}_{n}\,,$$

$\rho>0$, $\rho\in C^{\infty}(U)$ and $\sigma_{j,p}\circ\mu=f_{j,p}\prod_{t=1}^{m}z_{t}^{b_{j,s_{t}}}$, $\tau_{r,q}\circ\mu=g_{r,q}\prod_{t=1}^{m}z_{t}^{c_{r,s_{t}}}$, $f_{j,p},\,g_{r,q}\in{\cal O}(U)$. Then modulo factors that are bounded away from $0$ and $\infty$, we find

	$\displaystyle|\sigma_{j}\circ\mu|^{2}$	$\displaystyle\thicksim$	$\displaystyle\left(\sum_{p}|f_{j,p}|^{2}\right)\prod_{t=1}^{m}|z_{t}|^{2b_{j,s_{t}}}\,,$
	$\displaystyle|\tau_{r}\circ\mu|^{2}$	$\displaystyle\thicksim$	$\displaystyle\left(\sum_{q}|g_{r,q}|^{2}\right)\prod_{t=1}^{m}|z_{t}|^{2c_{r,s_{t}}}\,,$

with $\sum_{p}|f_{j,p}|^{2}>0$, $\sum_{q}|g_{r,q}|^{2}>0$. The latter nonvanishing property follows from the fact that the terms $|\sum_{s}\,b_{j,s}\,H_{s}|=\mu^{-1}V(\sigma_{j})$, $|\sum_{s}\,c_{r,s}\,H_{s}|=\mu^{-1}V(\tau_{r})$, $(\sigma_{j,p}\circ\mu)_{p}$ correspond to local generators of the sheaf ${\cal O}_{\tilde{X}}\left(\sum_{s}\,b_{j,s}\,H_{s}\right)$ and $(\tau_{r,q}\circ\mu)_{q}$ are local generators of the sheaf ${\cal O}_{\tilde{X}}\left(\sum_{s}\,c_{r,s}\,H_{s}\right)$. We infer

$$\prod\limits_{j=1}^{N}|\sigma_{j}|^{2l_{j}}\cdot\prod\limits_{r=1}^{M}|\tau_{r}|^{-2h_{r}}\thicksim\prod_{t=1}^{m}|z_{t}|^{-2d_{s_{t}}}\,,$$

with $d_{s}:=C_{s}H-B_{s}L$, $L:=\sum_{j}l_{j}$, $H:=\sum_{r}h_{r}$, $B_{s}:=\sum_{j}b_{j,s}$ and $C_{s}:=\sum_{r}c_{r,s}$. Set also $T:=\{t\in\{1,...,m\}\,|\,0<d_{s_{t}}-a_{s_{t}}\}$. The hypothesis $I_{0}<+\infty$ implies $d_{s_{t}}<1+a_{s_{t}}$ for all $t\in T$. Thus there exists $p_{U}>1$ such that $p_{U}d_{s_{t}}<1+a_{s_{t}}$ for all $t\in T$. If $(U_{\alpha})_{\alpha}$ is a finite covering of $\tilde{X}$, with $U_{\alpha}$ as $U$ and $p:=\min_{\alpha}p_{U_{\alpha}}>1$ then $I_{p}<+\infty$. This proves the first claim in the statement of lemma 7. In order to prove the convergence in the $L^{p}$ norm of the functions $G_{\varepsilon}$ we distinguish two cases. In the case where $l_{j}=0$ for all $j$, the claim follows imediately from the monotone convergence theorem. The other possible case is $l_{j}>0$ for all $j$. In this case we set $l:=\min_{j}l_{j}>0$. Then our setting implies $A=(H+1)/l$. For all $\varepsilon,\rho\in(0,1)$ consider the sequence of inequalities

			$\displaystyle\int\limits_{X}\,\prod\limits_{j=1}^{N}|\sigma_{j}|^{2l_{j}p}\cdot\prod\limits_{r=1}^{M}(|\tau_{r}|^{2}+\varepsilon)^{-h_{r}p}\,\Omega\;\leq\;\int\limits_{X}G_{\varepsilon}^{p}\,\Omega$
	$\displaystyle=$		$\displaystyle\int\limits_{\bigcup_{t=1}^{N}\{\rho|\sigma_{t}|^{2}\,<\,\varepsilon^{A}\}}G_{\varepsilon}^{p}\,\Omega\;+\;\int\limits_{\bigcap_{t=1}^{N}\{\rho|\sigma_{t}|^{2}\,\geq\,\varepsilon^{A}\}}G_{\varepsilon}^{p}\,\Omega$
	$\displaystyle\leq$		$\displaystyle\sum_{t=1}^{N}\;\int\limits_{\{\rho|\sigma_{t}|^{2}\,<\,\varepsilon^{A}\}}G_{\varepsilon}^{p}\,\Omega\;+\;\int\limits_{\bigcap_{t=1}^{N}\{\rho|\sigma_{t}|^{2}\,\geq\,\varepsilon^{A}\}}G_{\varepsilon}^{p}\,\Omega$
	$\displaystyle\leq$		$\displaystyle\sum_{t=1}^{N}\;\int\limits_{\{\rho|\sigma_{t}|^{2}\,<\,\varepsilon^{A}\}}\varepsilon^{-Hp}\,\prod\limits_{j=1}^{N}(|\sigma_{j}|^{2}+\varepsilon^{A})^{l_{j}p}\,\Omega$
	$\displaystyle+$		$\displaystyle\int\limits_{\bigcap_{t=1}^{N}\{\rho|\sigma_{t}|^{2}\,\geq\,\varepsilon^{A}\}}\,\prod\limits_{j=1}^{N}(1+\rho)^{l_{j}p}|\sigma_{j}|^{2l_{j}p}\cdot\prod\limits_{r=1}^{M}(|\tau_{r}|^{2}+\varepsilon)^{-h_{r}p}\,\Omega$
	$\displaystyle\leq$		$\displaystyle\sum_{t=1}^{N}(1+\rho^{-1})^{Lp}\,\varepsilon^{(Al-H)p}\int\limits_{\{\rho|\sigma_{t}|^{2}\,<\,\varepsilon^{A}\}}\,\prod\limits_{j\not=t}(|\sigma_{j}|^{2}+\varepsilon^{A})^{l_{j}p}\,\Omega$
	$\displaystyle+$		$\displaystyle(1+\rho)^{Lp}\int\limits_{X}\,\prod\limits_{j=1}^{N}|\sigma_{j}|^{2l_{j}p}\cdot\prod\limits_{r=1}^{M}|\tau_{r}|^{-2h_{r}p}\,\Omega$
	$\displaystyle\leq$		$\displaystyle(1+\rho^{-1})^{Lp}\,\varepsilon^{p}C+(1+\rho)^{Lp}\int\limits_{X}G^{p}_{0}\,\Omega\,,$

where $C>0$ is a constant uniform in $\varepsilon$. Thus by letting $\varepsilon\rightarrow 0^{+}$ and by applying the increasing monotone convergence theorem to the first integral in the previous inequalities, we obtain

$$\int\limits_{X}G_{0}^{p}\,\Omega\leq\liminf_{\varepsilon\rightarrow 0^{+}}\int\limits_{X}G_{\varepsilon}^{p}\,\Omega\leq\limsup_{\varepsilon\rightarrow 0^{+}}\int\limits_{X}G_{\varepsilon}^{p}\,\Omega\leq(1+\rho)^{Lp}\int\limits_{X}G_{0}^{p}\,\Omega\,.$$

Then the conclusion follows by letting $\rho\rightarrow 0^{+}$ and by the fact that $G_{\varepsilon}$ converges pointwise almost everywhere to $G_{0}$ as $\varepsilon\rightarrow 0^{+}$. $\Box$