Reductions of non-lc ideals and non $F$-pure ideals assuming weak ordinarity

Abstract

Assume $X$ is a variety over $\mathbb{C}$, $A\subseteq \mathbb{C}$ is a finitely generated $\mathbb{Z}$-algebra and $X_A$ a model of $X$ (i.e. $X_A{\times }_A\mathbb{C}\cong X$). Assuming the weak ordinarity conjecture we show that there is a dense set $S\subseteq \text{Spec }A$ such that for every closed point $s$ of $S$ the reduction of the maximal non-lc ideal filtration ${\mathcal{J}}^{\prime }(X,\Delta ,{\mathfrak{a}}^{\lambda })$ coincides with the non-$F$-pure ideal filtration $\sigma (X_s,{\Delta }_s,{\mathfrak{a}}_s^{\lambda })$ provided that $(X,\Delta )$ is klt or if $(X,\Delta )$ is log canonical, $\mathfrak{a}$ is principal and the non-klt locus is contained in $\mathfrak{a}$.

11 pages; v2: improved and simplified exposition