We now prove the following theorem.

Proof.
Step I. We first assume the existence of an effective divisor $D$ and $\delta>0$ such that $\{\omega\}-\delta\{D\}$ is a Kähler class. This is certainly the case if $X$ is projective and $\{\omega\}\in H^{1,1}(X,\mathbb{Q})$. So by using the Lelong-Poincaré formula we deduce $0<\omega_{\delta}=\omega-2\pi\delta[D]+\delta\,i\partial\bar{\partial}\log|s|^{2}$ with $\operatorname{div}(s)=D$. By convention we will put $\delta=0$ if $\omega>0$ and by abusing notations we will denote by $D$ the support $|D|$ of the divisor $D$.

IA) Setup of Step I.
We first consider the case $n\geq 2$. We can assume without any lost of generality $N=M=1$. Let $\alpha>0$ be a Kähler metric, let $\varepsilon\in(0,1)$ and let $c_{\varepsilon}$ be a normalizing constant for the integral condition

$\displaystyle e^{c_{\varepsilon}}\,\int\limits_{X}\frac{(|\sigma|^{2}+\varepsilon^{A})^{l}}{(|\tau|^{2}+\varepsilon)^{h}}\;\Omega=\int\limits_{X}(\omega+\varepsilon\alpha)^{n}\,,$

(5.3)

with $A:=(h+1)/l$. The condition (5.1) combined with lemma 7 implies $c_{\varepsilon}\rightarrow 0$, when $\varepsilon\rightarrow 0^{+}$. Consider the standard solutions $\varphi_{\varepsilon}\in C^{\infty}(X)$ of the complex Monge-Ampère equations

$\displaystyle(\omega+\varepsilon\alpha+i\partial\bar{\partial}\varphi_{\varepsilon})^{n}=e^{c_{\varepsilon}}\,\frac{(|\sigma|^{2}+\varepsilon^{A})^{l}}{(|\tau|^{2}+\varepsilon)^{h}}\;e^{\lambda\varphi_{\varepsilon}}\,\Omega\,,$

(5.4)

given by Yau’s solution of the Calabi conjecture. Notice that the integral condition (5.3) implies that a non identicaly zero solution $\varphi_{\varepsilon}$ changes signs in the case $\lambda>0$. By combining lemma 7 with the estimate of theorem 6 we deduce a uniform bound for the oscillations $\operatorname{Osc}(\varphi_{\varepsilon})\leq C$. Set now $\omega_{\delta,\varepsilon}:=\omega_{\delta}+\varepsilon\alpha$ and $\psi_{\varepsilon}:=\varphi_{\varepsilon}-\delta\log|s|^{2}$. Then

$\displaystyle 0<\omega+\varepsilon\alpha+i\partial\bar{\partial}\varphi_{\varepsilon}=\omega_{\delta,\varepsilon}+i\partial\bar{\partial}\psi_{\varepsilon}$

(5.5)

over $X\smallsetminus D$, and the equation (5.4) rewrites as

$\displaystyle(\omega_{\delta,\varepsilon}+i\partial\bar{\partial}\psi_{\varepsilon})^{n}=e^{F^{\varepsilon}+\lambda\delta\log|s|^{2}+\lambda\psi_{\varepsilon}}\,\omega_{\delta,\varepsilon}^{n}$

(5.6)

on $X\smallsetminus D$, with $F^{\varepsilon}:=f^{\varepsilon}+l\cdot a^{\varepsilon}-h\cdot b^{\varepsilon}$, and with

$$f^{\varepsilon}:=c_{\varepsilon}+\log(\Omega/\omega_{\delta,\varepsilon}^{n})\,,\quad a^{\varepsilon}:=\log(|\sigma|^{2}+\varepsilon^{A})\,,\quad b^{\varepsilon}:=\log(|\tau|^{2}+\varepsilon)\,.$$

(Here the supscripts in $\varepsilon$ are indices and not powers.) Consider now the function $\gamma_{1}^{\delta,\varepsilon}:X\rightarrow\mathbb{R}$ defined by the formula

$$\gamma_{1}^{\delta,\varepsilon}(x):=\min_{\xi\in T^{\otimes 2}_{X,x}\smallsetminus 0_{x}}{\cal C}^{\omega_{\delta,\varepsilon}}_{{}_{X,J}}(\xi,\xi)|\xi|_{\omega_{\delta,\varepsilon}}^{-2}.$$

So $\gamma_{1}^{\delta,\varepsilon}(x)$ is the smallest eigenvalue of the Chern curvature form ${\cal C}^{\omega_{\delta,\varepsilon}}_{{}_{X,J}}(x)$ of the metric $\omega_{\delta,\varepsilon}>0$. It is well known (see [Kat], chap II, sect. 5.1, theorem 5.1, page 107) that the function $\gamma_{1}^{\delta,\varepsilon}$ is continuous. We observe that the family of metrics $(\omega_{\delta,\varepsilon})_{\varepsilon}$ has bounded geometry. In particular for all $\varepsilon\in(0,1)$

$$\gamma_{1}^{\delta,\varepsilon}\geq C_{\delta}\,,\quad|f^{\varepsilon}|\leq K_{0,\delta}\,,\quad\lambda(\omega_{\delta}-\omega)+i\partial\bar{\partial}f^{\varepsilon}\geq-K_{0,\delta}\,\omega_{\delta,\varepsilon}\,.$$

IB) The Laplacian estimate.
This estimate is obtained as a combination of ideas of Yau, Blocki and Tsuji, [Yau], [Blo2], [Ts]. Consider the continuous function $\Lambda_{\varepsilon}:X\rightarrow(0,+\infty)$ given by the maximal eigenvalues of $\omega_{\delta,\varepsilon}+i\partial\bar{\partial}\psi_{\varepsilon}$ with respect to the Kähler metric $\omega_{\delta,\varepsilon}$,

$$\Lambda_{\varepsilon}(x):=\max_{\xi\in T_{X,x}\smallsetminus 0_{x}}(\omega_{\delta,\varepsilon}+i\partial\bar{\partial}\psi_{\varepsilon})(\xi,J\xi)|\xi|_{\omega_{\delta,\varepsilon}}^{-2}\,,$$

i.e. we extend $\Lambda_{\varepsilon}$ over $D$ by continuity, as is permitted by (5.5). Consider also the continuous function over $X\smallsetminus D$,

$$A_{\varepsilon}:=\log\Lambda_{\varepsilon}-k\psi_{\varepsilon}+hb^{\varepsilon}\,,$$

with $0<k:=2(1+h\,K_{0,\delta}/2-K_{1})$ and

$$K_{1}:=\min\{-[\lambda+(1+l+h)K_{0,\delta}/(2n)]\,,\,C_{\delta}\}<-\lambda\,.$$

The singularity of the function $\psi_{\varepsilon}$ imply the existence of a maximum of the function $A_{\varepsilon}$ at a certain point $x_{\varepsilon}\in X\smallsetminus D$. Let $g$ be a smooth real valued function in a neighborhood of $x_{\varepsilon}$ in $X\smallsetminus D$ such that $\omega_{\delta,\varepsilon}=\frac{i}{2}\partial\bar{\partial}g$, and let $u:=g+2\psi_{\varepsilon}$. Then

$$\omega_{\delta,\varepsilon}+i\partial\bar{\partial}\psi_{\varepsilon}=\frac{i}{2}\partial\bar{\partial}u\,.$$

In the following calculations we use the notation $u_{l,\bar{r}}:=\frac{\partial^{2}u}{\partial z_{l}\partial\bar{z}_{r}}$. Let $(z_{1},\ldots,z_{n})$ be $\omega_{\delta,\varepsilon}$-geodesic holomorphic coordinates with center the point $x_{\varepsilon}$ such that the metric $\omega_{\delta,\varepsilon}+i\partial\bar{\partial}\psi_{\varepsilon}$ can be writen in diagonal form in $x_{\varepsilon}$. Explicitly $\omega_{\delta,\varepsilon}=\frac{i}{2}\sum_{l,r}g_{l,\bar{r}}\,dz_{l}\wedge d\bar{z}_{r}$, with

	$\displaystyle\displaystyle{g_{l,\bar{r}}=\delta_{l,r}-\sum_{j,k}C^{j,\bar{k}}_{r,l}z_{j}\bar{z}_{k}+O(|z|^{3})\,,\quad g_{j,\bar{k},l,\bar{r}}(x_{\varepsilon})=-C^{j,\bar{k}}_{r,l}\,,}$
	$\displaystyle\displaystyle{{\cal C}^{\omega_{\delta,\varepsilon}}_{{}_{X,J}}(x_{\varepsilon})=\sum_{j,k,l,r}C^{j,\bar{k}}_{r,l}\,dz_{j}\otimes dz_{l}\otimes d\bar{z}_{k}\otimes d\bar{z}_{r}\,.}$

and $\frac{i}{2}\partial\bar{\partial}u=\frac{i}{2}\sum_{l}u_{l,\bar{l}}\,dz_{l}\wedge d\bar{z}_{l}$, with $0<u_{1,\bar{1}}\leq...\leq u_{n,\bar{n}}$ at the point $x_{\varepsilon}$. For every $\zeta\in\mathbb{C}^{n}$ we set $g_{\zeta,\bar{\zeta}}:=\sum_{l,r}g_{l,\bar{r}}\,\zeta_{l}\,\bar{\zeta}_{r}$. Then

$$\Lambda_{\varepsilon}(x)=\max_{\xi\in T_{X,x}\smallsetminus 0_{x}}\,\frac{\partial\bar{\partial}u(\xi^{1,0},\xi^{0,1})}{\partial\bar{\partial}g(\xi^{1,0},\xi^{0,1})}=\max_{|\zeta|=1}\,\frac{u_{\zeta,\bar{\zeta}}}{g_{\zeta,\bar{\zeta}}}\,,$$

and so $\Lambda_{\varepsilon}(x_{\varepsilon})=u_{n,\bar{n}}(x_{\varepsilon})$, $\frac{u_{n,\bar{n}}}{g_{n,\bar{n}}}\leq\Lambda_{\varepsilon}$. We also set

$$\tilde{A}_{\varepsilon}:=\log\frac{u_{n,\bar{n}}}{g_{n,\bar{n}}}-k\psi_{\varepsilon}+hb^{\varepsilon}\,.$$

Then $\tilde{A}_{\varepsilon}\leq A_{\varepsilon}$, with $\tilde{A}_{\varepsilon}(x_{\varepsilon})=A_{\varepsilon}(x_{\varepsilon})$. This implies that $\tilde{A}_{\varepsilon}$ also reaches a maximum at $x_{\varepsilon}$, thus $\Delta_{\psi_{\varepsilon}}\tilde{A}_{\varepsilon}(x_{\varepsilon})\leq 0$, where $\Delta_{\psi_{\varepsilon}}$ is the Laplacian respect to the metric $\omega_{\delta,\varepsilon}+i\partial\bar{\partial}\psi_{\varepsilon}$. All the subsequent computations in this part of the proof will be made at point $x_{\varepsilon}$. By the local expressions for the Ricci tensor we obtain

	$\displaystyle\partial^{2}_{n,\bar{n}}\log\operatorname{det}(u_{j,\bar{k}})$	$\displaystyle=$	$\displaystyle\sum_{l,p}\Big(u_{n,\bar{n},l,\bar{p}}-\sum_{s,t}u_{n,l,\bar{s}}\,u^{s,\bar{t}}\,u_{\bar{n},t,\bar{p}}\Big)u^{p,\bar{l}}$
		$\displaystyle=$	$\displaystyle\sum_{p}\frac{u_{n,\bar{n},p,\bar{p}}}{u_{p,\bar{p}}}-\sum_{p,q}\frac{\,|u_{n,p,\bar{q}}|^{2}}{u_{p,\bar{p}}\,u_{q,\bar{q}}}\,,$

and in a similar way $\partial^{2}_{n,\bar{n}}\log\operatorname{det}(g_{j,\bar{k}})=\sum_{p}\,g_{n,\bar{n},p,\bar{p}}$. Then by differentiating with respect to $\partial^{2}_{n,\bar{n}}$ the identity (5.6), which rewrites as

$$\log\operatorname{det}(u_{j,\bar{k}})=F^{\varepsilon}+\lambda\delta\log|s|^{2}+\lambda(u-g)/2+\log\operatorname{det}(g_{j,\bar{k}})\,,$$

we obtain

	$\displaystyle\sum_{p}\frac{u_{n,\bar{n},p,\bar{p}}}{u_{p,\bar{p}}}-\sum_{p,q}\frac{\,|u_{n,p,\bar{q}}|^{2}}{u_{p,\bar{p}}\,u_{q,\bar{q}}}$	$\displaystyle=$	$\displaystyle f^{\varepsilon}_{n,\bar{n}}+\lambda[(\omega_{\delta})_{n,n}-\omega_{n,n}]/2$
		$\displaystyle+$	$\displaystyle la^{\varepsilon}_{n,\bar{n}}-hb^{\varepsilon}_{n,\bar{n}}+\lambda(u_{n,\bar{n}}-1)/2+\sum_{p}\,g_{n,\bar{n},p,\bar{p}}\,.$

Combining this with the inequality $\Delta_{\psi_{\varepsilon}}\tilde{A}_{\varepsilon}(x_{\varepsilon})\leq 0$, we get

	$\displaystyle 0$	$\displaystyle\geq$	$\displaystyle\sum_{p}\frac{\tilde{A}_{p,\bar{p}}}{u_{p,\bar{p}}}=\sum_{p}\left(\frac{u_{n,\bar{n},p,\bar{p}}}{u_{p,\bar{p}}\,u_{n,\bar{n}}}-\frac{\,|u_{n,\bar{n},p}|^{2}}{u_{p,\bar{p}}\,u^{2}_{n,\bar{n}}}+\frac{k/2+hb^{\varepsilon}_{p,\bar{p}}-g_{n,\bar{n},p,\bar{p}}}{u_{p,\bar{p}}}\right)-nk/2$
		$\displaystyle=$	$\displaystyle\sum_{p,q}\frac{\,|u_{n,p,\bar{q}}|^{2}}{u_{p,\bar{p}}\,u_{q,\bar{q}}\,u_{n,\bar{n}}}-\sum_{p}\frac{\,|u_{n,\bar{n},p}|^{2}}{u_{p,\bar{p}}\,u^{2}_{n,\bar{n}}}$
		$\displaystyle+$	$\displaystyle\frac{f_{n,\bar{n}}+\lambda[(\omega_{\delta})_{n,n}-\omega_{n,n}-1]/2+la^{\varepsilon}_{n,\bar{n}}-hb^{\varepsilon}_{n,\bar{n}}}{u_{n,\bar{n}}}$
		$\displaystyle+$	$\displaystyle\sum_{p}\left(\frac{g_{n,\bar{n},p,\bar{p}}}{u_{n,\bar{n}}}+\frac{k/2+hb^{\varepsilon}_{p,\bar{p}}-g_{n,\bar{n},p,\bar{p}}}{u_{p,\bar{p}}}\right)-(nk-\lambda)/2\,.$

We use now (see the Appendix) the existence of smooth $(1,1)$-forms $iT^{a^{\varepsilon}}\geq 0$, $iT^{b^{\varepsilon}}\geq 0$, on $X$ such that

	$\displaystyle-hi\partial\bar{\partial}b^{\varepsilon}$	$\displaystyle\geq$	$\displaystyle-hK_{0,\delta}\,\omega_{\delta,\varepsilon}-hiT^{b^{\varepsilon}}\,,$
	$\displaystyle hi\partial\bar{\partial}b^{\varepsilon}$	$\displaystyle\geq$	$\displaystyle-hK_{0,\delta}\,\omega_{\delta,\varepsilon}+hiT^{b^{\varepsilon}}\,,$
	$\displaystyle li\partial\bar{\partial}a^{\varepsilon}$	$\displaystyle\geq$	$\displaystyle-lK_{0,\delta}\,\omega_{\delta,\varepsilon}\,.$

By plugging these inequalities in the previous computations we get

	$\displaystyle 0\geq-\frac{hT_{n,\bar{n}}^{b^{\varepsilon}}}{u_{n,\bar{n}}}+\sum_{p}\left(\frac{K_{1}-C_{p,p}^{n,\bar{n}}}{u_{n,\bar{n}}}+\frac{-K_{1}+C_{p,p}^{n,\bar{n}}}{u_{p,\bar{p}}}+\frac{1}{u_{p,\bar{p}}}+\frac{hT_{p,\bar{p}}^{b^{\varepsilon}}}{u_{p,\bar{p}}}\right)-(nk-\lambda)/2$
	$\displaystyle\geq\sum_{p}\frac{(C_{p,p}^{n,\bar{n}}-K_{1})(u_{n,\bar{n}}-u_{p,\bar{p}})}{u_{p,\bar{p}}\,u_{n,\bar{n}}}+\sum_{p}\frac{1}{u_{p,\bar{p}}}-(nk-\lambda)/2\,.\qquad\qquad\qquad\qquad\;\,$

Denote by $(x_{1},...,x_{n})$ the real part of the complex coordinates $(z_{1},...,z_{n})$. Then the inequality $C^{n,\bar{n}}_{p,p}={\cal C}^{\omega_{\delta,\varepsilon}}_{{}_{X,J}}(\frac{\partial}{\partial x_{n}}\otimes\frac{\partial}{\partial x_{p}},\frac{\partial}{\partial x_{n}}\otimes\frac{\partial}{\partial x_{p}})(x_{0})\geq\gamma_{1}^{\delta,\varepsilon}(x_{0})\geq C_{\delta}$ implies

$\displaystyle 0\geq\sum_{p}\frac{1}{u_{p,\bar{p}}}-C_{0}\geq\left(\frac{u_{n,\bar{n}}}{\prod_{p}u_{p,\bar{p}}}\right)^{\frac{1}{n-1}}-C_{0}=e^{\frac{-\lambda\psi_{\varepsilon}-\lambda\delta\log|s|^{2}-F^{\varepsilon}}{n-1}}\,u_{n,\bar{n}}^{\frac{1}{n-1}}-C_{0}\,,$

where $C_{0}>0$ and all the following constants are indipendents of $\varepsilon$. Consider now the function $B_{\varepsilon}:=e^{A_{\varepsilon}}=\Lambda_{\varepsilon}\,e^{-k\psi_{\varepsilon}+hb^{\varepsilon}}$. Then $x_{\varepsilon}$ is also a maximum point for $B_{\varepsilon}$ over $X\smallsetminus D$ and the previous inequality rewrites as

	$\displaystyle 0$	$\displaystyle\geq$	$\displaystyle e^{\frac{(k-\lambda)\psi_{\varepsilon}-\lambda\delta\log|s|^{2}-hb^{\varepsilon}-F^{\varepsilon}}{n-1}(x_{\varepsilon})}\,\,B_{\varepsilon}(x_{\varepsilon})^{\frac{1}{n-1}}-C_{0}$
		$\displaystyle=$	$\displaystyle e^{\frac{(k-\lambda)\varphi_{\varepsilon}-\delta k\log|s|^{2}-la^{\varepsilon}-f^{\varepsilon}}{n-1}(x_{\varepsilon})}\,\,B_{\varepsilon}(x_{\varepsilon})^{\frac{1}{n-1}}-C_{0}\,.$

Then by the inequalities $k-\lambda>0,\;|s|^{2}\leq C$, $a^{\varepsilon}\leq C$ and $|f^{\varepsilon}|\leq K_{0,\delta}$, it follows the estimate

$$0\geq C_{1}\,e^{\frac{(k-\lambda)}{n-1}\min_{X}\varphi_{\varepsilon}}\,\,B_{\varepsilon}(x_{\varepsilon})^{\frac{1}{n-1}}-C_{0}\,.$$

In conclusion we have found over $X\smallsetminus D$ the estimates

	$\displaystyle 0$	$\displaystyle<$	$\displaystyle 2n+\Delta_{\omega_{\delta,\varepsilon}}\varphi_{\varepsilon}-\delta\Delta_{\omega_{\delta,\varepsilon}}\log|s|^{2}=\operatorname{Tr}_{\omega_{\delta,\varepsilon}}(\omega_{\delta,\varepsilon}+i\partial\bar{\partial}\psi_{\varepsilon})$
		$\displaystyle\leq$	$\displaystyle 2n\Lambda_{\varepsilon}\leq 2n\,e^{k\psi_{\varepsilon}-hb^{\varepsilon}}B_{\varepsilon}(x_{\varepsilon})\leq\frac{C_{2}\,e^{k\varphi_{\varepsilon}-(k-\lambda)\min_{X}\varphi_{\varepsilon}}}{|s|^{2\delta k}(|\tau|^{2}+\varepsilon)^{h}}\leq\frac{C_{2}\,e^{k\operatorname{Osc}(\varphi_{\varepsilon})}}{|s|^{2\delta k}\,|\tau|^{2h}}\,.$

The last inequality follows from the fact that $\lambda\min_{X}\varphi_{\varepsilon}\leq 0$, since a non identicaly zero solution $\varphi_{\varepsilon}$ changes signs in the case $\lambda>0$. Then using the inequality

$$\left|\delta\Delta_{\omega_{\delta,\varepsilon}}\log|s|^{2}\right|=|\operatorname{Tr}_{\omega_{\delta,\varepsilon}}(\omega-\omega_{\delta})|\leq C$$

over $X\smallsetminus D$ we deduce the singular estimate

$$-C<2n+\Delta_{\omega_{\delta,\varepsilon}}\varphi_{\varepsilon}\leq\frac{C_{2}\,e^{k\operatorname{Osc}(\varphi_{\varepsilon})}}{|s|^{2\delta k}\,|\tau|^{2h}}+C\,.$$

IC) Higher order estimates
An elementary computation yields the singular estimate

			$\displaystyle\kern-30.0ptC_{3}^{-1}\,|s|^{2\delta k(n-1)}\,|\sigma|^{2l}\,|\tau|^{2h(n-2)}\,e^{-kn\operatorname{Osc}(\varphi_{\varepsilon})}\,\omega_{\delta,\varepsilon}$		(5.7)
		$\displaystyle\leq$	$\displaystyle\omega+\varepsilon\alpha+i\partial\bar{\partial}\varphi_{\varepsilon}\leq C_{3}\,|s|^{-2\delta k}\,|\tau|^{-2h}\,e^{k\operatorname{Osc}(\varphi_{\varepsilon})}\,\omega_{\delta,\varepsilon}\,.$

Morover the fact that $\varphi_{\varepsilon}\in{\cal P}_{\omega+\varepsilon\alpha}$ implies

$$2|\partial\bar{\partial}\varphi_{\varepsilon}|_{\omega_{\delta,\varepsilon}}\leq\Delta_{\omega_{\delta,\varepsilon}}\varphi_{\varepsilon}+2\operatorname{Tr}_{\omega_{\delta,\varepsilon}}(\omega+\varepsilon\alpha)\,.$$

We set first

$\displaystyle\Sigma_{\omega}:=\bigcap_{\{\omega\}-\delta\{D\}>0\atop D\geq 0,\;\delta>0}D\,.$

Then by the standard Schauder estimates [Gi-Tru] we find that for any coordinate open set $K\subset X\smallsetminus S^{\prime}$ there are uniform constants $C_{K}>0$ such that

$$\max_{K}|\nabla\varphi_{\varepsilon}|\leq C_{K}\left(\max_{K}\Delta\varphi_{\varepsilon}+\max_{K}|\varphi_{\varepsilon}|\right)\,.$$

Therefore, we can apply the complex version of Evans-Krylov theory [Ti2] on every compact set $K\subset X\smallsetminus S$ to get uniform constants $C_{K,2}>0$ such that $\|\varphi_{\varepsilon}\|_{C^{2,\alpha}(K)}\leq C_{K,2}$. Let now $U\subset X\smallsetminus S$ be an open set and $\xi\in{\cal O}(T^{1,0}_{X})(U)$. By deriving with respect to the complex vector field $\xi$ the complex Monge-Ampère equation (5.4), which we rewrite under the form

$\displaystyle(\omega+\varepsilon\alpha+i\partial\bar{\partial}\varphi_{\varepsilon})^{n}=e^{H_{\varepsilon}+\lambda\varphi_{\varepsilon}}\alpha^{n}\,,$

with

$$H_{\varepsilon}:=c_{\varepsilon}+\log(\Omega/\alpha^{n})+la^{\varepsilon}-hb^{\varepsilon}\,,$$

we obtain (see the proof of formula 11 in [Pal])

$\displaystyle\Delta_{\varphi_{\varepsilon}}(\xi\,.\,\varphi_{\varepsilon})-2\lambda\,\xi\,.\,\varphi_{\varepsilon}=-\operatorname{Tr}_{\varphi_{\varepsilon}}L_{\xi}(\omega+\varepsilon\alpha)+\operatorname{Tr}_{\alpha}L_{\xi}\alpha+2\xi\,.\,H_{\varepsilon}\,,$

(5.8)

where $\Delta_{\varphi_{\varepsilon}}$ and $\operatorname{Tr}_{\varphi_{\varepsilon}}$ are respectively the Laplacian and the trace operators with respect to the Kähler metric $\omega+\varepsilon\alpha+i\partial\bar{\partial}\varphi_{\varepsilon}>0$. By the uniform estimates (5.7) and $\|\varphi_{\varepsilon}\|_{C^{2,\alpha}(K)}\leq C_{K,2}$ it follows that the operator $\Delta_{\varphi_{\varepsilon}}$ is uniformly elliptic with coefficients uniformly bounded in $C^{\alpha}$-norm at least, over any compact set $K\subset U$. The right hand side of equation (5.8) is also uniformly bounded in $C^{\alpha}$-norm at least. By the standard regularity theory for linear elliptic equations [Gi-Tru] we deduce $\|\xi\,.\,\varphi_{\varepsilon}\|_{{C^{2,\alpha}(K)}}\leq C^{\prime}_{K}$ for all $\varepsilon>0$. By conjugation the same hold for $\bar{\xi}\,.\,\varphi_{\varepsilon}$. Thus we obtain the uniform estimate $\|\varphi_{\varepsilon}\|_{C^{3,\alpha}(K)}\leq C_{K,3}$.
In its turn this estimate implies that the coefficients of the Laplacian $\Delta_{\varphi_{\varepsilon}}$ and the right hand side of equation (5.8) are uniformly bounded in $C^{1,\alpha}$-norm at least. By iteration we get the uniform estimates $\|\varphi_{\varepsilon}\|_{C^{r,\alpha}(K)}\leq C_{K,r}$ for all $\varepsilon>0$ and $r\in\mathbb{N}$. We deduce that the family $(\varphi_{\varepsilon})_{\varepsilon>0}\subset C^{\infty}(X\smallsetminus S)$ is precompact in the smooth topology.
On the other hand the uniqueness result of theorem 6, combined with the arguments which showed the existence of a continuous solution $\varphi$ imply the uniform convergence of the family $\varphi_{\varepsilon}$ towards $\varphi$, thus this convergence is achieved in the $C^{\infty}$ topology over $X\smallsetminus S$. In this way we get smoothness of the solution $\varphi$ over $X\smallsetminus S$. The regularity statement in the case $n=1$ follows immediately from the latter considerations.

Step II.
IIA) Setup of step II.
We start with a few definitions adapted to our situation.

Let now $T$ be a closed positive $(1,1)$-current. There are two ways to associate an ideal sheaf to $T$. One can define the ideal sheaf ${\cal J}(T)\subset{\cal O}_{X}$ of germs $f\in{\cal O}_{x}$ such that $\int|f|^{2}e^{-\varphi}<+\infty$, where $\varphi$ is a local plurisubharmonic potential of $T$ in a neighborhood of $x$. One can also define the ideal sheaf ${\cal J}^{T}\subset{\cal O}_{X}$ of germs $f\in{\cal O}_{x}$ such that $|f|^{2}e^{-\varphi}\leq C$. The sheaf ${\cal J}(T)$ is coherent by a result of Nadel [Nad] and $V({\cal J}^{T})=V({\cal J}^{kT})$ for all $k\in\mathbb{R}_{>0}$.

Consider now a polarized compact Kähler manifold $(X,\omega)$ and a class $\chi\in H^{1,1}(X,\mathbb{R})$. We define the set of Kähler currents

$$\chi_{>0}:=\{T\in\chi\,\mid\,\exists\varepsilon>0\,:\,T\geq\varepsilon\omega\,\}\,.$$

If $\chi$ is nef and big then $\chi_{>0}\not=\emptyset$ by a result in [De-Pa]. By the regularization theorem in [Dem1] we deduce that for all $T\in\chi_{>0}$ and for all integers $k\gg 0$ there exists $T_{k}\in\chi_{>0}$ such that ${\cal J}^{kT}=\overline{{\cal J}(kT)}$ and $\mu^{*}T_{k}=2\pi k^{-1}[D]+\alpha$, with $\alpha$ smooth, for any modification $\mu$ such that $\mu^{*}{\cal J}(kT)={\cal O}(-D)$, with $D$ an effective divisor with normal crossing. Notice that $\mu^{*}\overline{{\cal J}(kT)}={\cal O}(-D)$. We deduce that the subset $\chi^{as}_{>0}\subset\chi_{>0}$ of Kähler currents with analytic singularities is also non empty. By the proof of theorem 3.4 in [De-Pa] we deduce the following generalization of Kodaira’s lemma.

We define in the end the complex analytic set

$$\Sigma_{\chi}:=\bigcap_{T\in\chi^{as}_{>0}}V({\cal J}^{T})\,.$$

IIB) Application of step I.
Back to our situation, fix $T\in\chi^{as}_{>0}$ and consider a log resolution $\mu:\tilde{X}\rightarrow X$ of the coherent ideal sheaf ${\cal J}^{kT}$. Let $f\geq 0$ be the integrand in the first inegral in (5.1). Then the integral condition (5.1) implies

$$0<\int\limits_{\tilde{X}}\mu^{*}\omega^{n}=\int\limits_{\tilde{X}}(f\Omega\circ\mu)\cdot(\Lambda^{n}\partial\mu)\wedge\overline{(\Lambda^{n}\partial\mu)}=\int\limits_{\tilde{X}}(f\circ\mu)\left|\Lambda^{n}\partial\mu\right|^{2}_{\alpha,\beta}\,\Omega^{\prime}\,$$

where $\alpha,\beta>0$ are hermitian forms respectively over $X$ and $\tilde{X}$,

$$\left|\Lambda^{n}\partial\mu\right|^{2}_{\alpha,\beta}:=\frac{\alpha^{n}\cdot(\Lambda^{n}\partial\mu)\wedge\overline{(\Lambda^{n}\partial\mu)}}{\beta^{n}}\,\qquad\mbox{and}\qquad\Omega^{\prime}=\left(\frac{\Omega}{\alpha^{n}}\circ\mu\right)\beta^{n}>0\,.$$

Therefore by the generalised Kodaira’s lemma 8 we can solve the degenerate complex Monge-Ampère equation

$$(\mu^{*}\omega+i\partial\bar{\partial}\Phi)^{n}=(f\circ\mu)\left|\Lambda^{n}\partial\mu\right|^{2}_{\alpha,\beta}e^{\lambda\Phi}\,\Omega^{\prime}\,,$$

with the method of step I, so as to obtain a solution

$$\Phi\in{\cal P}_{\mu^{*}\omega}\cap C^{0}(\tilde{X})\cap C^{\alpha}(\tilde{X}\smallsetminus(D\cup\mu^{-1}S^{\prime}_{0}))\cap C^{\infty}(\tilde{X}\smallsetminus(D\cup\mu^{-1}S_{0}))\,.$$

Remark in fact that $|\operatorname{div}(\Lambda^{n}\partial\mu)|\subset|D|$ by our definition of a log resolution. Let now $j_{q}:\mu^{-1}(q)\hookrightarrow\tilde{X}$, $q\in V({\cal J}^{T})$ be the inclusion map. By hypothesis $\mu^{-1}(q)$ is compact and connected and obviously $j_{q}^{*}\mu^{*}\omega=0$. Then by the hypothesis $\mu^{*}\omega+i\partial\bar{\partial}\Phi\geq 0$ we deduce $i\partial\bar{\partial}(\Phi\circ j_{q})\geq 0$, which implies that $\Phi$ is constant along the fibers $\mu^{-1}(q)$. Therefore we can define $\varphi:=\pi_{*}\Phi\in{\cal P}_{\omega}\cap C^{0}(X)$. The fact that $\varphi$ is bounded implies that the current $(\omega+i\partial\bar{\partial}\varphi)^{n}$ does not carry any mass on complex analytic sets. Thus $\varphi$ is the solution of the complex Monge-Ampère equation (5.2) with the required regularity over the set $X\smallsetminus V({\cal J}^{T})$. Then the conclusion about the regularity of the solution $\varphi$ follows by letting $T\in\chi^{as}_{>0}$ vary. We finally remark that if $\Sigma_{\omega}$ is empty then the class of $\omega$ is Kähler. In fact chose the volume form $\Omega>0$ such that $\int_{X}\omega^{n}=\int_{X}\Omega$. By the previous arguments we can find a unique solution $\varphi$ of the equation $(\omega+i\partial\bar{\partial}\varphi)^{n}=\Omega>0$, which is smooth, thus $\omega+i\partial\bar{\partial}\varphi>0$ is a Kähler metric. In the case $n=1$ the solution $\varphi$ of the equation $\omega+i\partial\bar{\partial}\varphi=\Omega>0$ is allways smoth. $\Box$