Jumping numbers of analytic multiplier ideals (with an appendix by S\'ebastien Boucksom)

TL;DR

This paper extends the concept of jumping numbers of multiplier ideals from algebraic to plurisubharmonic functions, revealing that clustering of these numbers is common and providing a detailed characterization in specific cases.

Contribution

It generalizes the study of jumping numbers to plurisubharmonic functions and characterizes when clustering occurs, including explicit examples in dimension 2.

Findings

Clustering of jumping numbers is a frequent phenomenon.

Uncountably many new examples of clustering are constructed.

Periodic and discrete properties do not hold in the general plurisubharmonic case.

Abstract

We extend the study of jumping numbers of multiplier ideals due to Ein-Lazarsfeld-Smith-Varolin from the algebraic case to the case of general plurisubharmonic functions. While many properties from Ein-Lazarsfeld-Smith-Varolin are shown to generalize to the plurisubharmonic case, important properties such as periodicity and discreteness do not hold any more. Previously only two particular examples with a cluster point (i.e. failure of discreteness) of jumping numbers were known, due to Guan-Li and to Ein-Lazarsfeld-Smith-Varolin respectively. We generalize them to all toric plurisubharmonic functions in dimension 2 by characterizing precisely when cluster points of jumping numbers exist and by computing all those cluster points. This characterization suggests that clustering of jumping numbers is a rather frequent phenomenon. In particular, we obtain uncountably many new such examples.