Valuation spaces and multiplier ideals on singular varieties

TL;DR

This paper extends the valuative characterization of multiplier ideals from smooth to all normal complex algebraic varieties, introducing new concepts like numerically Cartier divisors and Q-Gorenstein varieties.

Contribution

It generalizes the valuative approach to multiplier ideals to singular varieties and introduces the notion of numerically Q-Gorenstein varieties.

Findings

Valuative characterization of multiplier ideals extended to all normal varieties.

Numerically Q-Cartier divisors coincide with Q-Cartier divisors for rational singularities.

Simplified form of the valuative characterization for numerically Q-Gorenstein varieties.

Abstract

We generalize to all normal complex algebraic varieties the valuative characterization of multiplier ideals due to Boucksom-Favre-Jonsson in the smooth case. To that end, we extend the log discrepancy function to the space of all real valuations, and prove that it satisfies an adequate properness property, building upon previous work by Jonsson-Musta\c{t}\u{a}. We next give an alternative definition of the concept of numerically Cartier divisors previously introduced by the first three authors, and prove that numerically Q-Cartier divisors coincide with Q-Cartier divisors for rational singularities. These ideas naturally lead to the notion of numerically Q-Gorenstein varieties, for which our valuative characterization of multiplier ideals takes a particularly simple form.