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Extending one-forms on $F$-regular singularities

Tatsuro Kawakami, Kenta Sato

Abstract

We prove the logarithmic extension theorem for one-forms on strongly $F$-regular singularities. Additionally, we establish the logarithmic extension theorem for one-forms on three-dimensional klt singularities in characteristic $p>41$. To this end, we reduce the problem to the logarithmic extension theorem for two-dimensional klt singularities with imperfect residue fields using a technique based on Cartier operators.

Extending one-forms on $F$-regular singularities

Abstract

We prove the logarithmic extension theorem for one-forms on strongly -regular singularities. Additionally, we establish the logarithmic extension theorem for one-forms on three-dimensional klt singularities in characteristic . To this end, we reduce the problem to the logarithmic extension theorem for two-dimensional klt singularities with imperfect residue fields using a technique based on Cartier operators.

Paper Structure

This paper contains 20 sections, 25 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Klt threefolds in characteristic p>41
  3. Lipman--Zariski type theorem
  4. Strategy of proofs
  5. Preliminaries
  6. Notation and terminology
  7. F-splitting
  8. Quasi-F-injectivity
  9. Cartier operator
  10. Iterations
  11. Reflexive Cartier operators
  12. Extending one-forms
  13. Logarithmic differential sheaf
  14. Definition of C-differentials
  15. Basic properties for C-differentials
  16. ...and 5 more sections

Key Result

Theorem 1

Let $X$ be a strongly $F$-regular variety over a perfect field of positive characteristic. Then $X$ satisfies the logarithmic extension theorem for one-forms, i.e., for any proper birational morphism $f\colon Y\to X$ with reduced $f$-exceptional divisor $E$, the natural restriction is surjective. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (70)

  • Theorem 1: see Theorem \ref{['thm:key']} for a more general statement
  • Theorem 2
  • Theorem 3
  • Theorem 1.1: Kaw4
  • Theorem 4: Theorems \ref{['mainthm:2-dim']} and \ref{['mainthm:2-dim(lc)']}
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • ...and 60 more