Valuation spaces and multiplier ideals on singular varieties

Abstract

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-Mustaţȃ. 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.

Finitistic test ideals on numerically Q-Gorenstein varieties

“…In this section, we recall the notion of numerically Q-Cartier divisors, introduced in [5] and [6], which is a natural generalization of Q-Cartier divisors.…”

Comparing multiplier ideals to test ideals on numerically ℚ‐Gorenstein varieties

“…A priori, one might need to take into account infinitely many resolutions to determine that we have computed J (X, Z). The condition of numerically Q-Gorenstein singularities, introduced and studied in [3,4], is designed to provide a work-around for this issue. The definitions of numerically Q-Cartier and numerically Q-Gorenstein given here follow [4]; they are equivalent to the definitions of numerically Cartier and numerically Gorenstein given in [3].…”

“…The notion of a numerically Q-Cartier divisor given above and studied in [3,4] is, at first glance, particular to characteristic 0 where resolutions of singularities are known to exist. However, in arbitrary characteristic, one may just as well make use of the regular alterations provided by [8].…”

“…The fact that D in the definition is uniquely determined by the given conditions follows by the Negativity Lemma [9, Lemma 3.39] applied to the Stein factorization of π. Since the pull-back of a Cartier divisor to a dominating regular alteration remains relatively numerically trivial, the existence of a numerical pull-back of D to an arbitrary regular alteration follows exactly as in [4,Proposition 5.3], and one can replace π by an arbitrary regular alteration in the definition. In particular, in characteristic 0 the above definition is equivalent to Definition 5.…”

“…In [3,4], the authors introduce and study a new class of singularities, called numerically Q-Gorenstein singularities, which encompasses the class of Q-Gorenstein singularities. It turns out that the theory of multiplier ideals on normal varieties, as introduced in [5], becomes particularly simple on numerically Q-Gorenstein varieties.…”