Mobile Product and Zariski decomposition

TL;DR

This paper explores the relationship between standard and mobile intersection products of pseudo-effective classes on Kähler manifolds, applying this to holomorphic Morse inequalities and analyzing Lelong numbers under modifications.

Contribution

It establishes a connection between cohomology and mobile intersection products, and proves new results on Lelong numbers' behavior under modifications in dimension three.

Findings

Relationship between standard and mobile intersection products clarified

Holomorphic Morse inequalities derived using this relationship

Continuity property of Lelong numbers under modifications proved

Abstract

We explain the relationship between $\alpha_{1}...\alpha_{q}$ (standard cohomology product) and $(\alpha_{1}...\alpha_{q})$ (mobile intersection product) of pseudo-effective classes $\alpha_{1},...,\alpha_{q}$ on a compact K$\ddot{a}$hler manifold. We also show how to use this relationship for proving some holomorphic Morse inequalities. Then we prove a result concerning the direct image of Lelong numbers under a modification in dimension 3, deriving a continuity property for the Lelong numbers of the wedge of $(1,1)-$currents.