Siu's analyticity theorem for positive pluriharmonic currents

Abstract

Let $T$ be a positive $\ddc$-closed current of bidimension $(1,1)$ on a projective manifold $X$ of dimension $n.$ We show that for every $c > 0$ the set of points of $X$ where the Lelong number of $T$ is larger or equal to $c$ is an analytic subset of dimension at most $1$ of $X.$ Moreover, the following Siu decomposition holds $$T=\sum_{i\in I} λ_i[V_i] +T_0,$$ where $\{V_i\}_{i\in I}$ is a (possibly empty) finite or countable family of compact analytic curves in $X,$ $λ_i\in\mathbb{R}^+,$ and $T_0$ is a positive $\ddc$-closed current of bidimension $(1,1)$ on $X$ whose Lelong number vanishes outside a finite or countable set. As a consequence, the cohomology class of every positive $\ddc$-closed current of bidimension $(1, 1)$ on $X,$ which does not give mass to any proper analytic set, belongs to the Poincaré dual of the effective cone of $H^{1,1}(X,\mathbb{R}).$