Thirty years ago, in a celebrated article [Y], S.T. Yau (and independently T. Aubin [A]) solved the Calabi conjecture by studying complex Monge-Ampère equations on a compact Kähler manifold.

Since then, complex Monge-Ampère equations have been extremely useful in Kähler geometry (see for instance [DP]) and in the dynamical study of rational mappings (see [S] and references therein).

Two major developments in the theory of complex Monge-Ampère equations occurred in the last decade. In the local theory, a deeper analysis of the image of the complex Monge-Ampère operator [C], [K 1] has followed the pioneering work of E.Bedford and A.Taylor [BT]. In the global theory a new proof of the ${\mathcal{C}}^{0}$-estimate [K 1,2] has allowed one to treat complex Monge-Ampère equations with more degenerate R.H.S.

In [GZ1], [GZ2], two of us revisited and extended the results of [BT], [C], on complex Monge-Ampère operators to compact Kähler manifolds. In the present article, we use these methods to study complex Monge-Ampère equations with degenerate L.H.S. We first define, in the spirit of [C], [GZ 2], weak solutions to degenerate complex Monge-Ampère equations and then prove, using ideas of [K 1,2], that these solutions are continous:

Furthermore $f\mapsto\varphi$ is a continuous map from $L^{p}(X,\omega^{n})$ to ${\mathcal{C}}^{0}(X)$.

When $\omega$ has algebraic singularities, then $\mu$ can be assumed to have $L^{p}$ density with respect to the Lebesgue measure. With this ${\mathcal{C}}^{0}$-estimate, it is possible to adapt classical ideas of [Y] and [Ts] and prove:

Let $s_{1},...,s_{p}$ (resp. $t_{1},...,t_{q}$) be holomorphic sections of some line bundle. (resp. of some other line bundle). Assume $k,l\in\mathbb{R}_{\geq 0}$ and $F\in{\mathcal{C}}^{\infty}(X,\mathbb{R})$ are fixed so that

$$\int_{X}\frac{1}{|t_{1}|^{2l}+\ldots+|t_{q}|^{2l}}\Omega^{n}<\infty\text{ and }\int_{X}\frac{|s_{1}|^{2k}+\ldots+|s_{p}|^{2k}}{|t_{1}|^{2l}+\ldots+|t_{q}|^{2l}}e^{F}\Omega^{n}=\int_{X}\Omega^{n}.$$

Then the unique continuous function $\varphi$ such that $\omega+dd^{c}\varphi\geq 0$ and

$$(\omega+dd^{c}\varphi)^{n}=\frac{|s_{1}|^{2k}+\ldots+|s_{p}|^{2k}}{|t_{1}|^{2l}+\ldots+|t_{q}|^{2l}}e^{F}\Omega^{n},\text{ with }\sup_{X}\varphi=-1,$$

is smooth outside $B=S\cup\cap_{i}\{s_{i}=0\}\cup\cap_{i}\{t_{i}=0\}$.

This result should be compared with [Y], Theorem 8, p. 403. Yau’s result is stronger in many respects, most notably in the absence of any projectivity/rationality assumption and a more precise regularity theory. On the other hand, the condition on the poles of the L.H.S. is less optimal than here. We expect the projectivity/rationality assumptions to be superfluous.

Observe that the condition on the singularity in the L.H.S. is precisely the condition that the singular metric associated to $l(t_{i})$ has a trivial multiplier ideal sheaf, or if $q=1$ that the pair $(X,l(t_{1}))$ is klt (see definition 6.7). The possibility of solving complex Monge-Ampère equations with $L^{p}$-R.H.S. was first established by S.Kolodziej [K 1,2], and the connection with the singularities of the Minimal Model Program (MMP for short) has been a strong incentive to our work.

¿From an algebraic geometer’s perspective, these results may be viewed as a version of [Y] for normal Kähler spaces.

As a by-product, S.T.Yau constructed Kähler-Einstein metrics on smooth canonically polarized manifolds and Ricci-flat metrics on what is now known as Calabi-Yau manifolds. It had been soon realized [Ko] that this also yields Kähler-Einstein metrics on Kähler orbifolds hence on the canonical models of surfaces of general type since they have isolated quotient singularities.

In higher dimension, in spite of the development of the MMP during the 1980s – culminating with [Mo] and the proof of the existence of canonical models for general type 3-folds [Ka] –, there was no satisfying analog of these Kähler-Einstein metrics.

For smooth minimal general type projective manifolds, H.Tsuji proved [Ts] that an appropriate Kähler-Ricci flow starting with an arbitrary Kähler datum exists in infinite time, converges towards a current representing the canonical class which is smooth outside the exceptional locus of the map to the canonical model, and defines a Kähler-Einstein metric there ¹¹ 1 Although his idea was rather compelling, the details of the proof for convergence were somewhat hard to follow.. The conjecture made there that the current has continuous (or even bounded) local potentials partly motivated our work.

The article [Ts] has been revisited in two recent preprints, [CN] and [TZ], we learnt of when finishing the present work, where a very satisfactory proof of convergence towards a current with bounded potentials is given. The independent work [TZ] uses a slightly weaker version of Theorem A and does not give any detail on the proof. On the other hand, the three approaches tend to emphasize different aspects of the problem and seem to complement each other nicely.

In this article we give a more general theory of singular Kähler-Einstein metrics as a consequence of the following result:

Let $(V,\Delta)$ be a projective klt pair such that $K_{V}+\Delta\cong 0$ ($\mathbb{Q}$-linear equivalence of $\mathbb{Q}$-Cartier divisors). Then in every ample class in $NS_{\mathbb{R}}(V)$ there is a unique semi-Kähler current with continuous potentials, which satisfies a global degenerate Monge-Ampère equation on $V$ and defines a Ricci flat metric on $(V-\Delta)^{reg}$.

The precise formulation and Monge-Ampère equations are to be found in Theorems 7.5 and 7.8 below.

Note that we do not assume the singularities to be quotient singularities nor that $X$ has a smooth minimal model (a strong restriction present in [Ts], [TZ], [CN]). On the other hand if $\pi:X\to X_{can}$ is a resolution of singularities then $[\pi^{*}\omega_{KE}]\cong K_{X}+F$ where $F\geq 0$ and $=0$ iff $\pi$ is crepant and $X$ is a smooth minimal model. If the MMP is confirmed to work in higher dimension, then Corollary D extends there as well. In dimension 4, the MMP seems to work [Sho] but few people claim they understand this deep work in its manifold aspects. There is an active scientific project led by A.Corti aiming at making Shokurov’s work more accessible [Cor]. In higher dimension, substantial progress has been recently announced in [HMcK], Shokurov’s ideas being a major ingredient.

One may also try and construct these singular KE metrics unconditionnally in an approach to the finiteness of the canonical ring [Siu 2]. Needless to say, it is a substantially harder task.