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On Frobenius liftability of surface singularities

Tatsuro Kawakami, Teppei Takamatsu

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 $p>5$.

On Frobenius liftability of surface singularities

Abstract

We show that a plt surface singularity is -liftable if and only if it is -pure and is not a rational double point of type in characteristic . As a consequence, we prove the logarithmic extension theorem for -pure surface pairs and Bogomolov-Sommese vanishing for globally -split surface pairs. These results were previously known to hold in characteristic .
Paper Structure (23 sections, 28 theorems, 100 equations, 1 table)

This paper contains 23 sections, 28 theorems, 100 equations, 1 table.

Table of Contents

  1. Introduction
  2. $F$-liftability for surface singularities
  3. Applications of Theorem \ref{['Introthm:F-purity and F-liftability']}
  4. Extendability of differential forms for $F$-pure surface pairs
  5. Bogomolov-Sommese vanishing for globally $F$-split surface pairs
  6. Another application
  7. Preliminaries
  8. Notation and terminology
  9. --singularities
  10. $F$-liftability and reflexive Cartier operators
  11. Reflexive Cartier operators
  12. Splitting sections and canonical liftings
  13. Splitting sections associated to $F$-liftings
  14. Canonical liftings associated to splitting sections
  15. Characterization of $F$-liftability via reflexive Cartier operators
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $X$ be a normal variety and $B$ a reduced divisor on $X$ such that $(X, B)$ is locally $F$-liftable. Let $D$ be a $\mathbb{Q}$-Cartier $\mathbb{Z}$-divisor on $X$. Then, for any proper birational morphism $f\colon Y\to X$ from a normal variety $Y$, the restriction map is an isomorphism for all $i\geq 0$, where $E$ is the reduced $f$-exceptional divisor. We refer to Subsection subsection:Notat

Theorems & Definitions (74)

  • Theorem 1.1: Kaw4
  • Theorem 1
  • Theorem 2
  • Remark 1.2
  • Remark 1.3
  • Theorem 3
  • Remark 1.4
  • Definition 2.1
  • Definition 2.2: F-liftability
  • Definition 3.2
  • ...and 64 more