1 Introduction [028R]

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1 Introduction

In a celebrated paper [Yau] Yau solved the Calabi conjecture. As is well known, this problem can be formulated in terms of non degenerate complex Monge-Ampère equations as follows.

Theorem 1

(Yau). Let $X$ be a compact Kähler manifold of complex dimension $n$ and let $\chi$ be a Kähler class. Then for any smooth density $v>0$ on $X$ such that $\int_{X}v=\int_{X}\chi^{n}$ there exists a unique $($ smooth $)$ Kähler metric $\omega\in\chi$ such that $\omega^{n}=v$ .

Another breakthrough concerning the study of complex Monge-Ampère equations has been achieved by Bedford-Taylor [Be-Te]. Their work opened the doors to the study of very degenerate complex Monge-Ampère equations. In fact Kołodziej [Kol] proved the existence of solutions of the equations of type $(\omega+i\partial\bar{\partial}\varphi)^{n}=e^{\lambda\varphi}v$ , $\lambda\geq 0$ with $\omega$ a Kähler metric and $v\geq 0$ a density in $L^{p}$ or in some complicated Orlicz spaces. However in various geometric applications it is necessary to consider $\omega$ which is merely semipositive. This difficulty has been examinated first by Tsuji [Ts]. Tsuji’s technique has been reconsidered in the recent works [Ti-Zha] and [E-G-Z]. In this paper we push further the techniques so far developed and we obtain some very general and sharp results on the existence and uniqueness of degenerate complex Monge-Ampère equations. In order to define the relevant concept of uniqueness of the solutions we need first to introduce the domain of definition of the complex Monge-Ampère operator $MA_{\chi}$ of a pseudoeffective $(1,1)$ -cohomology class $\chi$ and to prove a monotone convergence result. As a consequence of our results (see theorem 6) we derive the following generalization of Yau’s theorem.

Theorem 2

. Let $X$ be a compact Kähler manifold of complex dimension $n$ and let $\chi$ be a big $(1,1)$ -cohomology class admitting a closed positive current with continuous local potentials. Then for any $L\log^{n+\varepsilon}L$ -density $v\geq 0$ , $\varepsilon>0$ on $X$ such that $\int_{X}v=\int_{X}\chi^{n}$ there exists a unique closed positive current $T\in MA_{\chi}$ such that $T^{n}=v$ . Moreover this current possesses continuous local potentials.

We wish to point out that the main examples of Orlicz spaces considered by Kołodziej are contained in some space $L\log^{n+\varepsilon}L$ . In the last section we prove fine regularity properties of the solution of complex Monge-Ampère equations with respect to a given degenerate metric $\omega\geq 0$ and whith right hand side possessing a density carrying complex analytic singularities (see theorem 7). This last type of equation is precisely what is needed in order to construct Kähler-Einstein metrics over irreducible singular Kähler spaces with ample or trivial canonical sheaf. This allows us also to solve generalised equations of the form $\operatorname{Ric}(\omega)=-\lambda\omega+\rho$ , $\lambda\geq 0$ . Quite recently Tian and Kołodziej [Ti-Ko] proved a very particular case of our $L^{\infty}$ -estimate. Their method, which is completely different, is based on an idea developed in [De-Pa]. Our $L^{\infty}$ -estimate allous us to completely solve a Tian’s conjecture stated in [Ti-Ko] (see the remark in the Apendix).