Density of positive closed currents, a theory of non-generic intersections

TL;DR

This paper develops a new theory of density for positive closed currents on Kähler manifolds, extending existing intersection concepts and enabling solutions to complex dynamics problems involving excess intersections.

Contribution

It introduces a generalized density notion that extends Lelong numbers and intersection theory, allowing for the treatment of excess dimension intersections on Kähler manifolds.

Findings

Constructs cohomology classes representing intersections of currents

Handles cases with excess intersection dimension

Provides foundational calculus for current density

Abstract

We introduce a notion of density which extends both the notion of Lelong number and the theory of intersection for positive closed currents on Kaehler manifolds. For arbitrary finite family of positive closed currents on a compact Kaehler manifold we construct cohomology classes which represent their intersection even when a phenomenon of excess dimension occurs. An example is the case of two algebraic varieties whose intersection has dimension larger than the expected number. The theory allows to solve problems in complex dynamics. Basic calculus on the density of currents is established.