Metrics with minimal singularities and the Abundance conjecture

TL;DR

This paper explores conditions under which the Abundance conjecture holds for minimal projective klt pairs, focusing on the role of supercanonical currents and their properties, especially when the Euler characteristic is non-zero.

Contribution

It provides a necessary and sufficient condition for the Abundance conjecture based on the asymptotic behavior of multiplier ideals and establishes key properties of supercanonical currents.

Findings

Characterization of the conjecture's validity via multiplier ideals

Structural and regularity properties of supercanonical currents

Indications of their central role in proving the conjecture

Abstract

The Abundance conjecture predicts that on a minimal projective klt pair $(X,\Delta)$, the adjoint divisor $K_X+\Delta$ is semiample. When $\chi(X,\mathcal O_X)\neq0$, we give a necessary and sufficient condition for the conjecture to hold in terms of the asymptotic behaviour of multiplier ideals of currents with minimal singularities of small twists of $K_X+\Delta$. Furthermore, we prove fundamental structural properties as well as regularity and weak convergence behaviour of an important class of currents with minimal singularities: the supercanonical currents. The results of the paper indicate strongly that supercanonical currents are central to the completion of the proof of the Abundance conjecture for minimal klt pairs $(X,\Delta)$ with $\chi(X,\mathcal O_X)\neq0$.