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

TL;DR

This paper demonstrates, under the weak ordinarity conjecture, that the reductions of non-lc ideals match non-$F$-pure ideals for a dense set of primes, linking complex and positive characteristic singularity invariants.

Contribution

It establishes a connection between non-lc ideals and non-$F$-pure ideals via reductions assuming weak ordinarity, extending the understanding of singularities across characteristics.

Findings

Reductions of non-lc ideals coincide with non-$F$-pure ideals on a dense set of primes.

The result applies to klt pairs and certain lc pairs with principal ideals.

The work relies on the weak ordinarity conjecture to relate complex and positive characteristic invariants.

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 $\mathcal{J}'(X, \Delta, \mathfrak{a}^\lambda)$ 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 $\mathfrak{a}$.