asymptotic behaviour of the sections of $kL$. The case of a general pseudo-effective line bundle $L$ is then handled by approximating it by $L+{\varepsilon}A$, where $A$ is ample.\
The methods used in this paper are mostly “transcendental”, since we heavily rely on the theory of currents, but the results we aim at definitely belong to algebraic geometry, and we have thus tried to make the paper legible for more algebraically inclined readers by providing in the first section a rather detailed account of the tools we need afterwards.

Technical preliminaries
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${\partial\overline{\partial}}$-cohomology
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When $X$ is an arbitrary complex manifold, the ${\partial\overline{\partial}}$-lemma of Kähler geometry does not hold, and it is thus better to work with ${\partial\overline{\partial}}$-cohomology. We will just need the $(1,1)$-cohomology space $H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{C}})$, which is defined as the quotient of the space of $d$-closed smooth $(1,1)$-forms modulo the ${\partial\overline{\partial}}$-exact ones. The real structure on the space of forms induces a real structure on $H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{C}})$, and we denote by ${H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{R}})}$ the space of real points.\
The canonical map from $H^{1,1}_{{\partial\overline{\partial}}}(X,{\mathbf{C}})$ to the quotient of the space of $d$-closed $(1,1)$-currents modulo the ${\partial\overline{\partial}}$-exact ones is injective (because, for any degree $0$ current $f$, ${\partial\overline{\partial}}f$ is smooth iff $f$ is), and 