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 $V({\mathfrak {a}})$.