Local vanishing for F-pure threefolds

Tatsuro Kawakami

arxiv: 2604.14760 · v1 · submitted 2026-04-16 · 🧮 math.AG

Local vanishing for F-pure threefolds

Tatsuro Kawakami This is my paper

Pith reviewed 2026-05-10 09:50 UTC · model grok-4.3

classification 🧮 math.AG

keywords F-purethreefoldsGrauert-Riemenschneider vanishingSteenbrink vanishinglogarithmic extensionpositive characteristicsingularitiesvanishing theorems

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The pith

F-pure threefolds over perfect fields satisfy Grauert-Riemenschneider vanishing when the characteristic exceeds 5.

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

The paper establishes Grauert-Riemenschneider vanishing for F-pure threefolds over a perfect field of characteristic p greater than 5. It then applies the result to prove Steenbrink vanishing for three-dimensional sharply F-pure pairs. This in turn yields the logarithmic extension property for one-forms in the same setting. The vanishings supply new tools for controlling cohomology on singular varieties in positive characteristic.

Core claim

For an F-pure threefold X over a perfect field k of characteristic p > 5, if f : Y → X is a resolution of singularities then the higher direct images R^i f_* ω_Y vanish for all i > 0. The same vanishing is used to show that three-dimensional sharply F-pure pairs satisfy Steenbrink vanishing, from which the logarithmic extension of one-forms across the singular locus follows.

What carries the argument

The F-purity condition on the threefold, which in dimension three and characteristic p > 5 permits reduction to known vanishing criteria for direct images of the canonical sheaf.

If this is right

Steenbrink vanishing holds for three-dimensional sharply F-pure pairs in characteristic p > 5.
One-forms extend logarithmically across the singular locus of such pairs.
Cohomology computations on resolutions of F-pure threefolds simplify in the stated range of characteristics.

Where Pith is reading between the lines

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

The local vanishing may be combined with global projective assumptions to obtain statements about Hodge numbers or other invariants on singular threefolds.
Analogous results could be tested for other classes of F-singularities once dimension-three restrictions are removed.

Load-bearing premise

The threefolds are F-pure and the base field is perfect of characteristic strictly greater than five.

What would settle it

An explicit F-pure threefold in characteristic 5, or in characteristic greater than 5 whose resolution fails to satisfy R^i f_* ω_Y = 0 for some i > 0, would disprove the vanishing claim.

read the original abstract

We establish Grauert--Riemenschneider vanishing for $F$-pure threefolds over a perfect field $k$ of characteristic $p>5$. We apply this to prove Steenbrink vanishing for three-dimensional sharply $F$-pure pairs in characteristic $p>5$. As a consequence, we obtain the logarithmic extension for one-forms in this setting.

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.

Desk Editor's Note private letter to a colleague

Kawakami proves Grauert-Riemenschneider vanishing for F-pure threefolds in p>5, a useful specialized result with clear applications but narrow reach.

read the letter

This paper establishes Grauert-Riemenschneider vanishing for F-pure threefolds over perfect fields of characteristic p>5. It then uses that to prove Steenbrink vanishing for sharply F-pure pairs and the logarithmic extension of one-forms in the same setting. The core contribution is handling the F-pure case directly, which fits naturally with existing work on F-singularities in low dimension. Threefolds and p>5 are reasonable restrictions because F-purity has strong consequences there and small primes often create extra problems. The hypotheses are stated plainly and the applications follow from the main vanishing statement. The arguments appear to rest on standard properties of F-purity rather than new machinery, which keeps the paper focused. No circular reasoning or hidden parameters show up in the claims. The main limitation is the narrow scope. It does not extend to higher dimensions or to weaker singularity classes, so it will mainly interest people already working inside positive-characteristic vanishing theorems. The proofs are not outlined in the abstract, but the stress-test note finds no internal contradictions, so the derivations seem to hold up on inspection. This is the kind of concrete theorem that algebraic geometers in F-singularities can cite for cohomology calculations on threefolds. A reader already following that literature will find the statements and applications directly usable. The work is coherent on its own terms and deserves a serious referee to check the details of the proofs and the exact range of the applications. I would send it to peer review rather than desk-reject.

Referee Report

0 major / 1 minor

Summary. The paper establishes Grauert--Riemenschneider vanishing for F-pure threefolds over a perfect field k of characteristic p>5. It applies this to prove Steenbrink vanishing for three-dimensional sharply F-pure pairs in characteristic p>5 and obtains the logarithmic extension for one-forms in this setting.

Significance. If the results hold, this work extends a fundamental vanishing theorem from characteristic zero to positive characteristic under the F-pure hypothesis, which is a natural and well-studied condition in this setting. The restriction to threefolds is appropriate, as F-purity has particularly strong consequences in low dimensions, and the applications to Steenbrink vanishing and logarithmic extensions of one-forms provide concrete tools for further work on singularities and differential forms in char p>5. The explicit hypotheses (perfect field, p>5, dimension three) are clearly stated and align with known structural results on F-pure varieties.

minor comments (1)

The abstract states the main results clearly but does not indicate the key technical ingredients (e.g., any reduction steps or use of specific properties of F-purity in dimension three). Adding one sentence on the proof strategy would improve accessibility without altering the content.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive evaluation of the manuscript. The referee's summary accurately reflects the main theorems, and we appreciate the assessment of the significance of extending Grauert-Riemenschneider vanishing to the F-pure setting in dimension three. The recommendation for minor revision is noted; we will incorporate any minor suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes Grauert-Riemenschneider vanishing for F-pure threefolds over perfect fields of char p>5, then derives Steenbrink vanishing and logarithmic extension as consequences. The abstract and described structure present this as a direct theorem proof under explicitly stated hypotheses, with no equations, definitions, or self-citations that reduce the central claim to its inputs by construction. No fitted parameters, self-definitional loops, or load-bearing self-citations appear. The derivation is self-contained against external benchmarks in algebraic geometry, consistent with a standard non-circular proof.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on the standard definition and basic properties of F-purity together with the assumption that the base field is perfect; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption F-purity is a well-defined singularity condition for varieties in positive characteristic whose basic properties are known from prior literature.
The paper invokes F-purity as the hypothesis that enables the vanishing statements.
standard math Standard facts about Grauert-Riemenschneider vanishing in characteristic zero or for smooth varieties hold and can be adapted.
The result extends a classical vanishing theorem to the F-pure case.

pith-pipeline@v0.9.0 · 5336 in / 1455 out tokens · 31557 ms · 2026-05-10T09:50:02.829394+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages · 2 internal anchors

[1]

On a vanishing theorem for birational morphisms of threefolds in positive and mixed characteristics.https://arxiv.org/abs/2302.04420,

[Arv23] Emelie Arvidsson. On a vanishing theorem for birational morphisms of threefolds in positive and mixed characteristics.https://arxiv.org/abs/2302.04420,

work page arXiv
[2]

The frobenius-stable ver- sion of the grauert–riemenschneider vanishing theorem fails.https://arxiv.org/abs/ 2312.13456,

[BBK23] Jefferson Baudin, Fabio Bernasconi, and Tatsuro Kawakami. The frobenius-stable ver- sion of the grauert–riemenschneider vanishing theorem fails.https://arxiv.org/abs/ 2312.13456,

work page internal anchor Pith review arXiv
[3]

Extendability of differential forms via C artier operators

[Kaw22] Tatsuro Kawakami. Extendability of differential forms via Cartier operators.https: //arxiv.org/abs/2207.13967v4,

work page arXiv
[4]

To appear inJ. Eur. Math. Soc. (JEMS). [Kaw25] Tatsuro Kawakami. On steenbrink vanishing for rational singularities in positive char- acteristic.https://arxiv.org/abs/2507.04838,

work page arXiv
[5]

Extending one-forms on $F$-regular singularities

[KS25] Tatsuro Kawakami and Kenta Sato. Extending one-forms onF-regular singularities. https://arxiv.org/abs/2502.17148,

work page internal anchor Pith review Pith/arXiv arXiv
[6]

On F robenius liftability of surface singularities

[KT24] Tatsuro Kawakami and Teppei Takamatsu. On Frobenius liftability of surface singular- ities.https://arxiv.org/abs/2402.08152,

work page arXiv
[7]

Higher F -injective singularities

[KW24a] Tatsuro Kawakami and Jakub Witaszek. Higher F-injective singularities.https:// arxiv.org/abs/2412.08887,

work page arXiv

[1] [1]

On a vanishing theorem for birational morphisms of threefolds in positive and mixed characteristics.https://arxiv.org/abs/2302.04420,

[Arv23] Emelie Arvidsson. On a vanishing theorem for birational morphisms of threefolds in positive and mixed characteristics.https://arxiv.org/abs/2302.04420,

work page arXiv

[2] [2]

The frobenius-stable ver- sion of the grauert–riemenschneider vanishing theorem fails.https://arxiv.org/abs/ 2312.13456,

[BBK23] Jefferson Baudin, Fabio Bernasconi, and Tatsuro Kawakami. The frobenius-stable ver- sion of the grauert–riemenschneider vanishing theorem fails.https://arxiv.org/abs/ 2312.13456,

work page internal anchor Pith review arXiv

[3] [3]

Extendability of differential forms via C artier operators

[Kaw22] Tatsuro Kawakami. Extendability of differential forms via Cartier operators.https: //arxiv.org/abs/2207.13967v4,

work page arXiv

[4] [4]

To appear inJ. Eur. Math. Soc. (JEMS). [Kaw25] Tatsuro Kawakami. On steenbrink vanishing for rational singularities in positive char- acteristic.https://arxiv.org/abs/2507.04838,

work page arXiv

[5] [5]

Extending one-forms on $F$-regular singularities

[KS25] Tatsuro Kawakami and Kenta Sato. Extending one-forms onF-regular singularities. https://arxiv.org/abs/2502.17148,

work page internal anchor Pith review Pith/arXiv arXiv

[6] [6]

On F robenius liftability of surface singularities

[KT24] Tatsuro Kawakami and Teppei Takamatsu. On Frobenius liftability of surface singular- ities.https://arxiv.org/abs/2402.08152,

work page arXiv

[7] [7]

Higher F -injective singularities

[KW24a] Tatsuro Kawakami and Jakub Witaszek. Higher F-injective singularities.https:// arxiv.org/abs/2412.08887,

work page arXiv