On Frobenius liftability of surface singularities

TL;DR

This paper characterizes Frobenius liftability of plt surface singularities, showing it occurs precisely when the singularity is F-pure and not a specific rational double point in characteristic 5, with implications for vanishing theorems.

Contribution

It provides a complete characterization of Frobenius liftability for plt surface singularities in characteristic 5, extending known results to this case.

Findings

F-liftability iff F-purity and not E8^1 rational double point in characteristic 5

Logarithmic extension theorem for F-pure surface pairs

Bogomolov-Sommese vanishing for globally F-split surface pairs

Abstract

We show that a plt surface singularity $(P\in X,B)$ is $F$-liftable if and only if it is $F$-pure and is not a rational double point of type $E_8^1$ in characteristic $p=5$. As a consequence, we prove the logarithmic extension theorem for $F$-pure surface pairs and Bogomolov-Sommese vanishing for globally $F$-split surface pairs. These results were previously known to hold in characteristic $p>5$.