A lower bound on the analytic log-canonical threshold over local fields of positive characteristic
Itay Glazer, Yotam I. Hendel
TL;DR
The paper studies the $F$-analytic log-canonical threshold over local fields of positive characteristic, proving it is always positive and giving explicit uniform lower bounds in the algebro-geometric setting. The authors develop a strategy that reduces to a single function, then uses Weierstrass preparation to express it as a Weierstrass polynomial and derives small-ball estimates for monic and Weierstrass polynomials to deduce a lower bound on $\operatorname{lct}_{F}$. They prove a concrete bound $\operatorname{lct}_{F}(J;x_{0})\geq \frac{1}{d\,D^{m}}$, uniform in $x_{0}$ on smooth $F$-varieties, and show this bound is optimal via an explicit construction. These results underpin later work on integrability of pushforwards in positive characteristic and connect to the broader study of singularity invariants in this setting.
Abstract
Given a local field $F$ of positive characteristic, an $F$-analytic manifold $X$ and an analytic function $f:X\rightarrow F$, the $F$-analytic log-canonical threshold $\mathrm{lct}_{F}(f;x_{0})$ is the supremum over the values $s\geq0$ such that $\left|f\right|_{F}^{-s}$ is integrable near $x_{0}\in X$. We show that $\mathrm{lct}_{F}(f;x_{0})>0$. Moreover, if $f$ is a regular function on a smooth algebraic $F$-variety, we obtain an effective lower bound $\mathrm{lct}_{F}(f;x_{0})>C$, where $C>0$ is explicit and depends only on the complexity class of $X$ and $f$.