In the whole paper, $X$ will denote a compact Kähler manifold of dimension $n$, $\omega$ a smooth closed form of bidegree $(1,1)$ which is non-negative and big, i.e. the smooth measure $\omega^{n}$ is not identical to zero. For convenience we normalize $\omega$ so that

$$Vol_{\omega}(X):=\int_{X}\omega^{n}=1.$$

$V$ will denote a normal complex space. A resolution of $V$ will be a projective bimeromorphic holomorphic morphism $\pi:X\to V$, $X$ being smooth, such that $\pi:\pi^{-1}(V^{reg})\to V^{reg}$ is an isomorphism. A resolution $\pi$ is a log-resolution iff $\pi^{-1}(V^{sing})$ is a divisor with simple normal crossings. Assume we have a coherent ideal sheaf $\mathcal{I}\subset\mathcal{O}_{V}$. A log resolution of $(V,\mathcal{I})$ is a projective bimeromorphic holomorphic morphism $\pi:X\to V$ $X$ being smooth, such that $\pi:\pi^{-1}(V-Z(\mathcal{I}))^{reg})\to(V-Z(\mathcal{I}))^{reg}$ is an isomorphism with the additional property that the ideal sheaf $\pi^{-1}\mathcal{I}.\mathcal{O}_{X}$ ²² 2 $\pi^{-1}\mathcal{I}.\mathcal{O}_{X}$ is locallly the ideal sheaf of $\mathcal{O}_{X}$ generated by the family of holomorphic functions $(\pi^{*}f_{I})_{I}$, where $(f_{I})_{I}$ are local generators of $\mathcal{I}$. The set $Z(\mathcal{I})$ is the analytic subvariety defined by $\mathcal{I}$. satisfies $\pi^{-1}\mathcal{I}.\mathcal{O}_{X}=\mathcal{O}_{X}(-\sum\gamma_{E}E)\subset\mathcal{O}_{X}$ where $\gamma_{E}\in\mathbb{N}$ is a positive integer attached to any exceptional divisor $E$ of $\pi$.

A pair is a pair $(V,\Delta)$ with $V$ a normal complex space and $\Delta$ a $\mathbb{Q}$-Weil divisor $\Delta=\sum_{i}d_{i}E_{i}$ where $0\leq d_{i}\leq 1$ are rational numbers and $(E_{i})_{i}$ is a finite family of pairwise distinct irreducible codimension 1 subvarieties of $V$. A log resolution of a pair is a log resolution of the ideal $\mathcal{I}_{N\Delta}$ where $N$ is an integer such that $Nd_{i}\in\mathbb{N}$. ³³ 3 The MMP is conjectured to work for pairs. This seemingly technical extension of the MMP is known as log-MMP. log-MMP works in dimension $\leq 3$. The philosophy of the log-MMP is to define the canonical divisor of a pair to be $K_{(V,\Delta)}:=K_{V}+\Delta$ and to try and prove the same theorems for pairs and for varieties.

All these flavors of log-resolutions exist by [Hi] if the variety (resp. pair) under consideration is open in (resp. a restriction to an open subset of) a compact variety (pair).

The paper is organized as follows. In section 1 we define, following [GZ 2], the set ${\mathcal{E}}^{1}(X,\omega)$ of $\omega$-psh functions with finite self-energy, and produce weak solutions to complex Monge-Ampère equations $(\omega+dd^{c}\varphi)^{n}=\mu$ in the class ${\mathcal{E}}^{1}(X,\omega)$ (see proposition 1.4). This is our first basic observation: weak solutions are easy to produce in ${\mathcal{E}}^{1}(X,\omega)$.

The continuity of the solutions is established in section 2 (see Theorem 2.1), by using ideas from [K 1,2,3] and [GZ 1]. This, together with propositions 3.1 and 3.3, yields Theorem A. We actually expect the solutions to be Hölder-continuous, as Theorem 3.5 indicates. We indeed establish further regularity results in section 3, especially Theorem 3.6, by using ideas of [Y],[Ts]. This yields Theorem B.

In section 4 we solve Monge-Ampère equations of the type $(\omega+dd^{c}\varphi)^{n}=e^{t\varphi}\mu$, $t>0$ (see Theorems 4.1, 4.4) by a fixed point method. Here again the use of class ${\mathcal{E}}^{1}(X,\omega)$ makes life easier, and allows us to reduce our analysis to previously studied Monge-Ampère equations $(\omega+dd^{c}\varphi)^{n}=\mu^{\prime}$.

In section 5 we recall some basic facts on some of the singularities encountered in the MMP, and in section 6 we explain what sort of measures $\mu$ we need to consider in order to produce Kähler-Einstein metrics. An important observation is lemma 6.4, which shows that it is necessary to restrict to the case of log terminal singularities (see definition 5.3).

Finally in section 7 we show how our results from sections 2,3,4 allow us to produce singular Kähler-Einstein metrics (see Theorems 7.5, 7.8, 7.12). This is where we prove Theorem C.