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

TL;DR

This paper introduces a new linear transform for positive closed currents on compact Kähler manifolds that preserves Lelong numbers and leads to self-intersection inequalities, advancing the understanding of complex geometry.

Contribution

The authors construct a continuous linear transform for positive closed currents on compact Kähler manifolds that preserves Lelong numbers, enabling new self-intersection inequalities.

Findings

Constructed a linear transform $\\mathcal{L}_p(T)$ preserving Lelong numbers.

Derived self-intersection inequalities for positive closed currents.

Extended the theory of currents on Kähler manifolds.

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.