3 - Valuation spaces and multiplier ideals on singular varieties

Summary

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 and Mustaţă. 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.

Dedicated to Robert Lazarsfeld on the occasion of his 60th birthday

1 Introduction

Multiplier ideal sheaves are a fundamental tool both in complex algebraic and complex analytic geometry. They provide a way to approximate a “singularity data,” which can take the form of a (coherent) ideal sheaf, a graded sequence of ideal sheaves, a plurisubharmonic function, a nef b-divisor, etc., by a coherent ideal sheaf satisfying a powerful cohomology vanishing theorem. For the sake of simplicity, we will focus on the case of ideals and graded sequences of ideals in the present paper.

On a smooth (complex) algebraic variety X, the very definition of the multiplier ideal sheaf J (X, a^c) of an ideal sheaf a ⊂ O_X with exponent c > 0 is valuative in nature: a germ f ∈ O_X belongs to J (X, a^c) iff it satisfies

ν(f) > cν(a) − A_X(ν)

for all divisorial valuations ν, i.e., all valuations of the form ν = ord_E (up to a multiplicative constant) with E a prime divisor on a birational model X′, proper over X. Further, it is enough to test these conditions with X′ a fixed log resolution of a (which shows that J (X, a^c) is coherent, as the direct image of a certain coherent fractional ideal sheaf on X′).