The jumping coefficients of non-Q-Gorenstein multiplier ideals

Abstract

Let $\mathfrak a \subset \mathscr O_X$ be a coherent ideal sheaf on a normal complex variety $X$, and let $c \ge 0$ be a real number. De Fernex and Hacon associated a multiplier ideal sheaf to the pair $(X, \mathfrak a^c)$ which coincides with the usual notion whenever the canonical divisor $K_X$ is $\mathbb Q$-Cartier. We investigate the properties of the jumping numbers associated to these multiplier ideals. We show that the set of jumping numbers of a pair is unbounded, countable and satisfies a certain periodicity property. We then prove that the jumping numbers form a discrete set of real numbers if the locus where $K_X$ fails to be $\mathbb Q$-Cartier is zero-dimensional. It follows that discreteness holds whenever $X$ is a threefold with rational singularities. Furthermore, we show that the jumping numbers are rational and discrete if one removes from $X$ a closed subset $W \subset X$ of codimension at least three, which does not depend on $\mathfrak a$. We also obtain that outside of $W$, the multiplier ideal reduces to the test ideal modulo sufficiently large primes $p \gg 0$.