Local vanishing for F-pure threefolds

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.

• The paper proves that Grauert–Riemenschneider vanishing holds for normal F-pure threefolds in characteristic p>5, extending classic Kodaira vanishing analogues.
• It establishes Steenbrink-type vanishing and a logarithmic extension theorem for sharply F-pure pairs by leveraging sophisticated Cartier operator techniques.
• The study links local freeness of the tangent sheaf to Frobenius liftability, providing new rigidity criteria and enhancing tools for the minimal model program.

Summary of "Local vanishing for F-pure threefolds" (2604.14760)

Introduction and Problem Statement

The paper advances the theory of vanishing theorems in positive characteristic, specifically for $F$-pure threefolds over perfect fields with $\mathrm{char}(k) > 5$. Classical vanishing theorems such as Kodaira vanishing and Grauert–Riemenschneider (GR) vanishing are foundational to higher-dimensional birational geometry in characteristic zero, but their analogues fail in general in positive characteristic. This work rectifies this gap for a significant class: normal $F$-pure threefolds and sharply $F$-pure pairs, focusing on local vanishing statements instrumental for the minimal model program (MMP) in positive characteristic.

Main Results

The paper establishes several vanishing, extension, and rigidity properties for $F$-pure threefold singularities:

1. Grauert–Riemenschneider Vanishing for F-pure Threefolds

For a normal $F$-pure threefold $X$ over a perfect field with $p > 5$, the higher direct images $R^j\pi_*\omega_Y$ vanish for all $j > 0$ and all resolutions $\mathrm{char}(k) > 5$0. This extends previous results that were limited to the more restrictive class of strongly $\mathrm{char}(k) > 5$1-regular singularities or klt threefolds and provides the critical local vanishing analogue of the global Kodaira vanishing for $\mathrm{char}(k) > 5$2-split varieties in positive characteristic.

Claim: If $\mathrm{char}(k) > 5$3 is a normal $\mathrm{char}(k) > 5$4-pure threefold in characteristic $\mathrm{char}(k) > 5$5, then $\mathrm{char}(k) > 5$6 for all $\mathrm{char}(k) > 5$7.

This demonstrates that GR vanishing holds for a wide class of singularities where Kodaira's theorem fails.

2. Steenbrink-type Vanishing for Sharply F-pure Pairs

Using the above, the author proves Steenbrink-type vanishing for three-dimensional sharply $\mathrm{char}(k) > 5$8-pure pairs $\mathrm{char}(k) > 5$9 when $F$0. This states that for all log resolutions and for $F$1, one has $F$2. The proof leverages the Cartier operator formalism developed for singular spaces, along with the established local GR vanishing.

Implication: This provides vanishing for relative log (pluri)forms, extending the machinery of mixed Hodge modules and cyclic covers to positive characteristic and $F$3-singularity settings.

3. Logarithmic Extension Theorem for One-forms

Combining previously established results on extension of reflexive one-forms on log canonical surfaces and the new vanishing results, the author proves a logarithmic extension theorem for one-forms in the three-dimensional sharply $F$4-pure case with $F$5: for any log resolution $F$6, $F$7 is surjective.

4. Lipman–Zariski-type Theorem

The classical Lipman–Zariski conjecture states that a variety with free tangent sheaf is smooth. While this fails in general in positive characteristic, the author proves that for a normal $F$8-Gorenstein $F$9-pure threefold in characteristic $F$0 with isolated singularities, local freeness of the tangent sheaf forces Frobenius liftability.

Key statement: If $F$1 is locally free, then $F$2 is Frobenius liftable.

This implies a rigidity and strong arithmetic constraint on $F$3-pure threefolds with free tangent sheaf.

Technical Approach

The central technique is to navigate the obstacles presented by highly singular and positive characteristic settings using the structure of $F$4-pure and sharply $F$5-pure pairs. The proofs leverage:

• Dlt modifications and the MMP in characteristic $F$6
• Structural theorems relating $F$7-purity to split or surjective behavior under Frobenius and the Cartier operator
• Careful use of Serre vanishing, the relative ampleness of exceptional divisors, and properties of reflexive differentials
• Inductive and local duality arguments to bridge between vanishing for canonical sheaves and for reflexive pluri-forms

A technical highlight is the construction of the necessary vanishing and splitting statements at the level of resolutions and dlt modifications, in a way compatible with Frobenius pushforward and the Cartier operator.

Contradictory and Sharp Claims

The paper includes a discussion on the sharpness of these results:

• The condition $F$8 is essential in the proofs; the vanishing can fail for $F$9.
• There is an explicit threefold of Totaro in characteristic two for which the Lipman–Zariski conjecture fails and GR vanishing fails, yet which is Frobenius liftable and has free tangent sheaf.
• Whether $F$0 is optimal remains an open question.

Theoretical and Practical Implications

The results have significant implications for the structure theory of higher-dimensional algebraic varieties in positive characteristic:

• These vanishing theorems furnish essential tools for the MMP in positive characteristic, allowing extension of many characteristic zero techniques to the $F$1-singularities setting.
• The logarithmic extension for one-forms has consequences for the study of reflexive differentials, the behavior of the cotangent complex, and the deformation theory of singular threefolds.
• The Lipman–Zariski-type result gives a new arithmetic criterion for smoothness-like behavior in families, showing that the combination of $F$2-purity and locally free tangent sheaf drastically restricts the deformation possibilities.

Future Directions

Potential extensions and open directions include:

• Optimal bounds for the characteristic in which vanishing holds (current work restricts to $F$3)
• Vanishing for higher degrees or in the relative setting, particularly regarding log forms of degree one, where the case remains open
• Analogous results for $F$4-pure or lc fourfolds, which requires overcoming new pathologies in dimension four
• Constructing more counterexamples in low characteristic for further boundary behavior analysis

Conclusion

This work provides robust vanishing, extension, and rigidity results for threefolds with $F$5-pure and sharply $F$6-pure singularities in characteristic $F$7. The synthesis of $F$8-singularity theory with classical vanishing theorems yields new tools for the birational geometry of positive characteristic varieties. These results significantly enhance the toolkit for the MMP and for understanding the behavior of differential forms and singularities in positive characteristic.