. Let $X$ be a compact Kähler manifold of complex dimension $n$, let $\omega\geq 0$ be a big closed smooth $(1,1)$-form such that $\{\omega^{n}=0\}$ is a set of zero measure and let $\Omega>0$ be a smooth volume form. Consider also $\sigma_{j}\in H^{0}(X,E_{j})$, $\tau_{r}\in H^{0}(X,F_{r})$, $j=1,...,N$, $r=1,...,M$ be non identically zero holomorphic sections of some holomorphic vector bundles over $X$, such that the intregral condition

$\displaystyle\int\limits_{X}\,\prod\limits_{j=1}^{N}|\sigma_{j}|^{2l_{j}}\cdot\prod\limits_{r=1}^{M}|\tau_{r}|^{-2h_{r}}\,\Omega=\int\limits_{X}\omega^{n}$

(5.1)

holds for some real numbers $l_{j}\geq 0,\,h_{r}\geq 0$. Then there exists a unique solution $\varphi\in\hat{\cal P}_{\omega}$ of the degenerate complex Monge-Ampère equation

$\displaystyle(\omega+i\partial\bar{\partial}\varphi)^{n}=\prod\limits_{j=1}^{N}|\sigma_{j}|^{2l_{j}}\cdot\prod\limits_{r=1}^{M}|\tau_{r}|^{-2h_{r}}\,e^{\lambda\varphi}\,\Omega\,,\quad\lambda\geq 0\,.$

(5.2)

Moreover there exists a complex analytic set $\Sigma_{\omega}\subset X$ depending only on the $(1,1)$-cohomology class of $\omega$ possessing the following properties.

(A). The set $\Sigma_{\omega}$ is empty if and only if the class of $\omega$ is Kähler.

(B). If we define the complex analytic sets

$\displaystyle S^{\prime}_{0}:=\bigcup_{r}\{\tau_{r}=0\}\,,\quad S_{0}:=S^{\prime}_{0}\cup(\bigcup_{j}\{\sigma_{j}=0\})\,,$

$\displaystyle S^{\prime}:=\Sigma_{\omega}\cup S^{\prime}_{0}\,,\quad S:=S^{\prime}\cup(\bigcup_{j}\{\sigma_{j}=0\})\,,$

then in the case $n\geq 2$

$$\varphi\in{\cal P}_{\omega}\cap C^{0}(X)\cap C^{\alpha}(X\smallsetminus S^{\prime})\cap C^{\infty}(X\smallsetminus S)\,,$$

for all $\alpha\in(0,1)$. If $n=1$ then $\varphi\in{\cal P}_{\omega}\cap C^{0}(X)\cap C^{\alpha}(X\smallsetminus S^{\prime}_{0})\cap C^{\infty}(X\smallsetminus S_{0})$ and the class of of $\omega$ is Kähler.