Siu's analyticity theorem for positive pluriharmonic currents

Abstract

Let $T$ be a positive $d{d}^c$-closed current of bidimension $(q,q)$ on a compact Kähler manifold $X$. For every $c>0$, let $E_c(T)$ be the set of points of $X$ where the Lelong number of $T$ is larger or equal to $c$. We show that $E_c(T)$ is an analytic subset of dimension at most $q$ of $X$. Moreover, the following Siu decomposition holds $$T=\sum _{i\in I}{\lambda }_i[V_i]+T_0,$$ where $\{V_i{\}}_{i\in I}$ is a (possibly empty) finite or countable family of compact analytic subsets of dimension $q$ in $X$, ${\lambda }_i\in {\mathbb{R}}^{+}$, and $T_0$ is a positive $d{d}^c$-closed current such that $E_c(T_0)$ is an analytic subset of dimension at most $q-1$ of $X$ for every $c>0$. The proof relies on the duality between the pseudoeffective cone and the movable cone of a compact Kähler manifold, recently obtained by Tosatti (2026), together with a theorem of Vigny (2009) and the theory of density currents for positive $d{d}^c$-closed currents developed by Sibony, the first and third authors.

13 pages. In this version, Duc-Bao Nguyen joins us as a co-author. The full Siu's theorem for positive ddc-closed currents on compact Kähler manifolds is now proved. The new argument is the use of a recent theorem of Tosatti on dual cones in Kähler geometry