Algebraic bounds on analytic multiplier ideals

TL;DR

This paper introduces the diminished ideal as a new analytic tool extending asymptotic multiplier ideals to pseudo-effective divisors, and explores its properties and relation to abundant divisors.

Contribution

It constructs the diminished ideal for pseudo-effective divisors, generalizes multiplier ideals, and characterizes abundant divisors through this new ideal.

Findings

The diminished ideal extends asymptotic multiplier ideals to the pseudo-effective boundary.

For most pseudo-effective divisors, the multiplier ideal of the minimal singularity metric is contained in the diminished ideal.

Abundant divisors are characterized by the coincidence of geometric and analytic information via the diminished ideal.

Abstract

Given a pseudo-effective divisor L we construct the diminished ideal of L, a "continuous" extension of the asymptotic multiplier ideal for big divisors to the pseudo-effective boundary. For most pseudo-effective divisors L the multiplier ideal of the metric of minimal singularities of L is contained in the diminished ideal. We also characterize abundant divisors using the diminished ideal, indicating that in this case the geometric and analytic information should coincide.