Thresholds of singularities in characteristic zero

Abstract

We study properties of characteristic zero variants of Frobenius thresholds $c^J(\mathfrak a)$ inspired by the work of Epstein-McDonald-R.G. and the second author. Suppose $R$ is an excellent domain of equal characteristic zero with a dualizing complex and $\mathfrak a, J \subseteq R$ are nonzero ideals with $\mathfrak a \subseteq \sqrt{J}$. By using a log resolution of singularities $Y$ of $(R, \mathfrak a)$ as well as the derived global sections of a line bundle on $Y$, we construct two analogs of the Frobenius threshold, the Hironaka-threshold and the Koszul-Hironaka threshold. We show that these two thresholds agree for parameter ideals and are always rational numbers. We prove that these thresholds also share many of the properties of the classical Frobenius threshold, especially for parameter ideals. If $R$ is regular, we show that the set of thresholds coincides with the set of possible multiplier-ideal-jumping numbers. Finally, we show that for parameter ideals in a Kawamata log terminal ring, our thresholds coincide with the limit of the Frobenius thresholds of the mod-$p$-reductions as $p$ goes to infinity. There is even a version of this without that Kawamata log terminal hypothesis.