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arxiv: 2502.17148 · v3 · submitted 2025-02-24 · 🧮 math.AG · math.AC

Extending one-forms on F-regular singularities

Tatsuro Kawakami , Kenta Sato This is my paper

Pith reviewed 2026-05-23 02:34 UTC · model grok-4.3

classification 🧮 math.AG math.AC
keywords logarithmic extension theoremone-formsstrongly F-regular singularitiesklt singularitiesCartier operatorspositive characteristicalgebraic geometrysingularities
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The pith

The logarithmic extension theorem for one-forms holds on strongly F-regular singularities.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proves that one-forms extend logarithmically across strongly F-regular singularities. It also proves the extension for three-dimensional klt singularities when the characteristic exceeds 41. The argument reduces the three-dimensional problem to a two-dimensional base case of klt singularities with imperfect residue fields by means of Cartier operators. A sympathetic reader would care because the result enlarges the class of singularities on which differential forms behave regularly in positive characteristic.

Core claim

We prove the logarithmic extension theorem for one-forms on strongly F-regular singularities. We additionally establish the logarithmic extension theorem for one-forms on three-dimensional klt singularities in characteristic p>41 by reducing the problem to the logarithmic extension theorem for two-dimensional klt singularities with imperfect residue fields using a technique based on Cartier operators.

What carries the argument

Reduction technique based on Cartier operators that lowers the three-dimensional klt problem to the two-dimensional case with imperfect residue fields.

If this is right

  • One-forms admit logarithmic extensions on every strongly F-regular singularity in any characteristic.
  • The extension property holds for all three-dimensional klt singularities when p exceeds 41.
  • The Cartier-operator reduction supplies a dimension-lowering method applicable to related extension questions for differential forms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the two-dimensional base case can be strengthened, the three-dimensional result might extend to higher dimensions or weaker singularity classes.
  • The technique may apply to extension problems for other sheaves or forms on F-regular and klt spaces.
  • Explicit computation of one-forms on low-dimensional F-regular examples could test the boundary of the result.

Load-bearing premise

The logarithmic extension theorem for one-forms holds on two-dimensional klt singularities with imperfect residue fields.

What would settle it

A concrete three-dimensional klt singularity in characteristic p>41 together with a one-form that fails to extend logarithmically would disprove the claim.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript proves the logarithmic extension theorem for one-forms on strongly F-regular singularities. It additionally establishes the same theorem for three-dimensional klt singularities in characteristic p > 41 by reducing the problem, via a Cartier-operator argument, to the logarithmic extension theorem on two-dimensional klt singularities with imperfect residue fields.

Significance. If the results hold, the work provides a new extension theorem for differential forms on strongly F-regular singularities, a class that includes many terminal and klt singularities in positive characteristic. The dimension-reduction technique via Cartier operators is a standard tool, and the high-characteristic 3D klt statement would extend existing results once the base case is secured. The paper does not claim machine-checked proofs or parameter-free derivations.

major comments (1)
  1. [Abstract] Abstract: the three-dimensional klt claim is obtained by reducing to the logarithmic extension theorem on two-dimensional klt singularities with imperfect residue fields. The manuscript presents this 2D statement as the target of the reduction rather than proving it internally; without an independent reference or proof of the base case, the 3D result remains conditional.

Simulated Author's Rebuttal

1 responses · 0 unresolved

Thank you for your careful review. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the three-dimensional klt claim is obtained by reducing to the logarithmic extension theorem on two-dimensional klt singularities with imperfect residue fields. The manuscript presents this 2D statement as the target of the reduction rather than proving it internally; without an independent reference or proof of the base case, the 3D result remains conditional.

    Authors: We agree that the 3D klt statement in characteristic p>41 is established only conditionally via the Cartier-operator reduction to the 2D klt case with imperfect residue fields. The manuscript fully proves the logarithmic extension theorem in the strongly F-regular setting, which is the primary result. The 3D claim is presented as an application of the reduction technique, but without an internal proof or citation of the 2D base case the claim is indeed conditional. We will revise the abstract, introduction, and any relevant statements to clarify the conditional nature of the 3D result or to include a proof/reference for the 2D base case as appropriate. revision: yes

Circularity Check

0 steps flagged

No circularity; standard dimension reduction to independent base case

full rationale

The paper reduces the 3D klt case to the 2D klt logarithmic extension theorem with imperfect residue fields via Cartier operators, as stated in the abstract. This is a conventional inductive strategy in algebraic geometry and does not define the 3D result in terms of itself or rename a fitted quantity as a prediction. No self-citations, ansatzes smuggled via prior work, or uniqueness theorems imported from the authors appear in the provided text. The base case is treated as external to the reduction step rather than derived from the target claim. The derivation chain remains self-contained against external benchmarks and does not reduce any central statement to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on the standard definitions of F-regular and klt singularities together with the applicability of Cartier operators for dimension reduction; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Strongly F-regular and klt singularities satisfy their standard definitions and basic properties in positive characteristic.
    The theorems are stated for these classes, so their established properties are presupposed.
  • domain assumption The logarithmic extension theorem holds for two-dimensional klt singularities with imperfect residue fields.
    The three-dimensional result is obtained by reducing to this case.

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    math.AG 2026-04 unverdicted novelty 7.0

    Grauert-Riemenschneider vanishing holds for F-pure threefolds in char p>5, implying Steenbrink vanishing for sharply F-pure pairs and logarithmic extension for one-forms.

Reference graph

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