A lower bound on the analytic log-canonical threshold over local fields of positive characteristic

TL;DR

This paper establishes a positive lower bound for the analytic log-canonical threshold over local fields of positive characteristic, providing explicit bounds depending on the complexity of the variety and function.

Contribution

It proves that the log-canonical threshold is always positive and offers an explicit lower bound based on the complexity class of the variety and function.

Findings

The log-canonical threshold is strictly positive.

An explicit lower bound depends only on complexity class.

The results apply to regular functions on smooth algebraic varieties.

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 $f$.