Asymptotic multiplicities and Monge-Amp\`ere masses (with an appendix by S\'ebastien Boucksom)

TL;DR

This paper explores the relationship between asymptotic multiplicities of ideals and Monge-Ampère masses, connecting complex analysis and algebraic geometry, and proves a key continuity property for a new class of functions.

Contribution

It establishes the equivalence between Samuel multiplicity equality and Demailly's strong continuity, and provides an algebraic proof of the equality using intersection theory.

Findings

Equality of asymptotic multiplicities is equivalent to Demailly's strong continuity.

Demailly's strong continuity holds for a new class of plurisubharmonic functions.

An algebraic proof of the multiplicity equality is provided using b-divisors.

Abstract

Ein, Lazarsfeld and Smith asked whether `equality' holds between two Samuel type asymptotic multiplicities for a graded system of zero-dimensional ideals on a smooth complex variety. We find a connection of this question to complex analysis by showing that the `equality' is equivalent to a particular case of Demailly's strong continuity property on the convergence of residual Monge-Amp\`ere masses under approximation of plurisubharmonic functions. On the other hand, in an appendix of this paper, S\'ebastien Boucksom gives an algebraic proof of the `equality' in general, using the intersection theory of b-divisors. We then use these to show that Demailly's strong continuity holds for a new important class of plurisubharmonic functions.