This paper is a first step towards developing a systematic study of Pluripotential theory on compact Kähler manifolds. Since the fundamental work of Bedford and Taylor [4],[5], several authors have developed a ”Pluripotential theory” in domains of $\mathbb{C}^{n}$ (or of Stein manifolds). This theory is devoted to the fine study of plurisubharmonic (psh) functions and can be seen as a non-linear generalization of the classical potential theory (in one complex variable), where subharmonic functions and the Laplace operator $\Delta$ are replaced by psh functions and the complex Monge-Ampère operator $(dd^{c})^{n}$. Here $d,d^{c}$ denote the real differential operators $d:=\partial+\overline{\partial}$, $d^{c}:=\frac{i}{2\pi}[\overline{\partial}-\partial]$ so that $dd^{c}=\frac{i}{\pi}\partial\overline{\partial}$ ; the normalization being chosen so that the positive measure $\left(dd^{c}\frac{1}{2}\log[1+||z||^{2}]\right)^{n}$ has total mass $1$ in $\mathbb{C}^{n}$. We refer the reader to [3],[7],[26] for a survey of this local theory.

Our aim here is to develop a global Pluripotential theory in the context of compact Kähler manifolds. It follows from the maximum principle that there are no psh functions (except constants) on a compact complex manifold $X$. However there are usually plenty of positive closed currents of bidegree $(1,1)$ (we refer the reader to [15], chapter 3, for basic facts on positive currents). Given $\omega$ a real closed smooth form of bidegree $(1,1)$ on $X$, we may consider every positive closed current $\omega^{\prime}$ of bidegree $(1,1)$ on $X$ which is cohomologous to $\omega$. When $X$ is Kähler, it follows from the ”$dd^{c}$-lemma” that $\omega^{\prime}$ can be written as $\omega^{\prime}=\omega+dd^{c}\varphi$, where $\varphi$ is a function which is integrable with respect to any smooth volume form on $X$. Such a function $\varphi$ will be called $\omega$-plurisubharmonic ($\omega$-psh for short). It is globally defined on $X$ and locally given as the sum of a psh and a smooth function. We let $PSH(X,\omega)$ denote the set of $\omega$-psh functions. Such functions were introduced by Demailly, who call them quasiplurisubharmonic (qpsh). These are the main objects of study in this article.

There are several motivations to study qpsh functions on compact Kähler manifolds. First of all they arise naturally in complex analytic geometry as positive singular metrics of holomorphic line bundles (see section 4) whose study is central to several important questions of complex algebraic geometry. Solving Monge-Ampère equations associated to $\omega$-psh functions has been used to produce metrics with prescribed singularities (see [13]). It is also related to the existence of canonical metrics in Kähler geometry (see [37]). Important contributions have been made by Kolodziej in this direction [28] using techniques from local Pluripotential theory. $\omega$-psh functions have also been used in [21] to define a notion of $\omega$-polynomial convexity and study the fine approximation of positive currents by rational divisors.

It seems to us appropriate to develop a theory of qpsh functions of its own rather than view these functions as particular cases of the local theory. Indeed there are at least two basic facts which make the theory easier on compact Kähler manifolds, and which therefore make us think that it should allow to obtain more complete results: there is no pluriharmonic functions (except constants) on a compact manifold, hence each $\omega$-psh function $\varphi$ is canonically associated (up to normalization) to its curvature current $\omega_{\varphi}:=\omega+dd^{c}\varphi\geq 0$. This yields compactness properties of subsets of $PSH(X,\omega)$ (see section 1) which are quite useful (e.g. in complex dynamics, see section 5.2). Another observation is that Stokes theorem is of constant use in the local theory, where boundary terms cause painful technicalities. This does not happen in the compact setting since there is no boundary. As an illustration, we obtain transparent proofs of Chern-Levine-Nirenberg type inequalities (see example 1.8 and section 2). We shall develop these ideas in a series of articles. The present one intends to lay down the foundations of the theory, with an emphasis on studying ”intrinsic capacities”.

Let us now describe more precisely the contents of the article.

This paper lies at the border of Complex Analysis and Complex Geometry. We have tried to make it accessible to mathematicians from both sides. This has of course some consequences for the style of presentation. We have included proofs of some results which may be seen as consequences of results from the local pluripotential theory. We have spent some efforts defining, regularizing and approximating positive singular metrics of holomorphic line bundles, although some of these facts may be considered as classical by complex geometers. Altogether we hope the paper is essentially self-contained. Our efforts will not be vain if for instance we have convinced specialists of the (local) pluripotential theory that the right point of view in studying the Lelong class ${\mathcal{L}}(\mathbb{C}^{n})$ of psh functions with logarithmic growth in $\mathbb{C}^{n}$ is to consider qpsh functions on the complex projective space $\mathbb{C}\mathbb{P}^{n}$. We also think this paper should be useful to people working in complex dynamics in several variables where pluripotential theory has become an important tool.