Relative non-pluripolar products of currents

TL;DR

This paper introduces a new notion of relative non-pluripolar products of currents on compact Kahler manifolds, establishing key properties like monotonicity, full mass intersection criteria, and convexity of related classes.

Contribution

It defines the relative non-pluripolar product for currents, generalizing known concepts, and proves fundamental properties including monotonicity and convexity, with new results even in classical cases.

Findings

Monotonicity property of relative non-pluripolar products

Necessary condition for full mass intersection via Lelong numbers

Convexity of weighted classes of currents

Abstract

Given a closed positive current T on a compact Kahler manifold X, we introduce the notion of non-pluripolar product relative to T of closed positive (1,1)-currents. We recover the well-known non-pluripolar product when T is the current of integration along X. Our main results are a monotonicity property of relative non-pluripolar products, a necessary condition for currents to be of relative full mass intersection in terms of Lelong numbers, and the convexity of weighted classes of currents of relative full mass intersection. The former two results are new even when T is the current of integration along X.