Lelong-Skoda transform for compact Kaehler manifolds and self-intersection inequalities

Abstract

Let $X$ be a compact Kaehler manifold of dimension $k$ and $T$ be a positive closed current on $X$ of bidimension $(p,p)$ ($1\leq p < k-1$). We construct a continuous linear transform $\mathcal{L}_p(T)$ of $T$ which is a positive closed current on $X$ of bidimension $(k-1,k-1)$ which has the same Lelong numbers as $T$. We deduce from that construction self-intersection inequalities for positive closed currents of any bidegree.