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 $$S \subseteq {{\,\mathrm{Spec}\,}}A$$ such that for every closed point s of S the reduction of the maximal non-lc ideal filtration $${\mathcal {J}}'(X, \Delta , {\mathfrak {a}}^\lambda )$$ 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 locally principal and the non-klt locus is contained in $$V({\mathfrak {a}})$$ .