Singularity criteria for K-stability of adjoint foliated structures

Theodoros Stylianos Papazachariou

arxiv: 2605.28140 · v1 · pith:JQIK25XEnew · submitted 2026-05-27 · 🧮 math.AG

Singularity criteria for K-stability of adjoint foliated structures

Theodoros Stylianos Papazachariou This is my paper

Pith reviewed 2026-06-29 10:07 UTC · model grok-4.3

classification 🧮 math.AG

keywords K-stabilityadjoint foliated structureslog canonicitymixed Donaldson-Futaki invariantflag idealsfoliationsK-semistabilitysingularity criteria

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The pith

K-semistability of adjoint foliated structures implies log canonicity.

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

The paper establishes singularity criteria for the t-K-stability of adjoint foliated structures. It shows that K-semistability implies log canonicity by extending Odaka's flag ideal characterisation of the mixed Donaldson-Futaki invariant to this setting. It further proves that adjoint Calabi-Yau foliated structures are K-semistable, that klt ones are K-stable, that log canonical adjoint general type foliated structures are K-stable under the canonical polarisation, and that K-semistable adjoint Fano foliated structures are klt. A sympathetic reader cares because these links between stability and singularity types constrain the possible models in foliated algebraic geometry.

Core claim

We prove singularity criteria for the t-K-stability of adjoint foliated structures. We first show that K-semistability of adjoint foliated structures implies log canonicity by extending Odaka's flag ideal characterisation of the mixed Donaldson-Futaki invariant to the adjoint foliated setting. We then prove that adjoint Calabi-Yau foliated structures are K-semistable, and klt ones are K-stable, while log canonical adjoint general type foliated structures are K-stable with respect to the canonical polarisation. We also show that K-semistable adjoint Fano foliated structures are klt. In particular, their ambient varieties are potentially klt and of Fano type.

What carries the argument

Extension of Odaka's flag ideal characterisation of the mixed Donaldson-Futaki invariant to the adjoint foliated setting, used to characterise the invariant and derive the singularity criteria.

If this is right

Adjoint Calabi-Yau foliated structures are K-semistable.
Klt adjoint foliated structures are K-stable.
Log canonical adjoint general type foliated structures are K-stable with respect to the canonical polarisation.
K-semistable adjoint Fano foliated structures are klt, so their ambient varieties are potentially klt and of Fano type.

Where Pith is reading between the lines

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

The criteria may support the construction of moduli spaces for stable adjoint foliated structures by controlling singularities.
Similar extensions of the flag ideal test could apply to non-adjoint foliations or other polarisations.
Explicit checks on toric or projective space examples could test the stability statements directly.

Load-bearing premise

The extension of Odaka's flag ideal characterisation of the mixed Donaldson-Futaki invariant works for adjoint foliated structures without further restrictions on the foliation or the adjoint bundle.

What would settle it

An explicit adjoint foliated structure that is K-semistable yet not log canonical, or a case where the flag ideal method fails to extend as claimed, would falsify the main implication.

read the original abstract

We prove singularity criteria for the $t$-K-stability of adjoint foliated structures. We first show that K-semistability of adjoint foliated structures implies log canonicity by extending Odaka's flag ideal characterisation of the mixed Donaldson--Futaki invariant to the adjoint foliated setting. We then prove that adjoint Calabi--Yau foliated structures are K-semistable, and klt ones are K-stable, while log canonical adjoint general type foliated structures are K-stable with respect to the canonical polarisation. We also show that K-semistable adjoint Fano foliated structures are klt. In particular, their ambient varieties are potentially klt and of Fano type.

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

The paper extends Odaka's flag ideal method to get singularity criteria for K-stability of adjoint foliated structures, but the extension step needs checking for hidden restrictions.

read the letter

The paper extends Odaka's flag ideal characterization of the mixed Donaldson-Futaki invariant to adjoint foliated structures. This produces four statements: K-semistability implies log canonicity, adjoint Calabi-Yau foliated structures are K-semistable, klt ones are K-stable, log canonical adjoint general type ones are K-stable under the canonical polarization, and K-semistable adjoint Fano ones are klt (with ambient varieties potentially klt and of Fano type).

It does a straightforward job of carrying the existing characterization over and stating the resulting criteria in one place. The work stays close to the prior result instead of introducing new machinery.

The soft spot is the extension itself. The abstract claims the flag ideal construction works in the foliated adjoint setting, but gives no indication of how the foliation's tangent sheaf or the adjoint bundle enters the mixed DF invariant computation. The stress-test concern holds: if the construction requires the foliation to be regular or extra positivity that is not stated, then the implication from semistability to log canonicity does not hold in the claimed generality. The proofs will have to show this step explicitly.

This is for specialists already working on K-stability and foliated pairs in birational geometry. A reader who knows Odaka's work will see the value in the targeted extension. The claims are concrete enough to deserve a serious referee to verify the extension details.

Recommendation: send it to peer review.

Referee Report

1 major / 0 minor

Summary. The manuscript proves singularity criteria for the t-K-stability of adjoint foliated structures. It extends Odaka's flag ideal characterization of the mixed Donaldson-Futaki invariant to the adjoint foliated setting to establish that K-semistability implies log canonicity. It further shows that adjoint Calabi-Yau foliated structures are K-semistable and klt ones are K-stable; log canonical adjoint general type foliated structures are K-stable with respect to the canonical polarization; and K-semistable adjoint Fano foliated structures are klt, implying that their ambient varieties are potentially klt and of Fano type.

Significance. If the extension of the flag ideal characterization holds in the stated generality, the results supply useful singularity criteria that link K-stability conditions with log canonicity and klt singularities for adjoint foliated structures, generalizing Odaka's work from the non-foliated case. The paper explicitly builds on prior characterizations and derives several stability statements for Calabi-Yau, general type, and Fano cases.

major comments (1)

The central implication that K-semistability implies log canonicity rests on the extension of Odaka's flag ideal characterization. The manuscript should explicitly address in the relevant proof section whether the construction of the flag ideal and the computation of the mixed Donaldson-Futaki invariant properly incorporate the foliation's tangent sheaf and possible singularities along the foliation, or whether additional hypotheses (such as regularity of the foliation) are tacitly required; without this verification the generality of the criterion remains open to the concern raised in the stress-test note.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback, which helps strengthen the presentation of our results on singularity criteria for t-K-stability of adjoint foliated structures. We address the major comment below.

read point-by-point responses

Referee: The central implication that K-semistability implies log canonicity rests on the extension of Odaka's flag ideal characterization. The manuscript should explicitly address in the relevant proof section whether the construction of the flag ideal and the computation of the mixed Donaldson-Futaki invariant properly incorporate the foliation's tangent sheaf and possible singularities along the foliation, or whether additional hypotheses (such as regularity of the foliation) are tacitly required; without this verification the generality of the criterion remains open to the concern raised in the stress-test note.

Authors: We appreciate the referee highlighting the need for explicit verification in the proof. In the extension of Odaka's flag ideal characterization (detailed in the proof of the main theorem establishing K-semistability implying log canonicity), the flag ideal is constructed from the adjoint divisor of the foliated structure. This construction directly incorporates the tangent sheaf of the foliation via the adjoint condition in the definition of the mixed Donaldson-Futaki invariant, which uses the foliated canonical class and accounts for singularities along the foliation through the log pair framework. The setup assumes only the standard hypotheses on adjoint foliated structures (saturated subsheaf with the given adjoint divisor); no additional regularity of the foliation is imposed or tacitly required. To make this incorporation fully explicit and address the concern, we will add a clarifying paragraph in the relevant proof section of the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation extends independent external result

full rationale

The paper's core step extends Odaka's flag ideal characterisation of the mixed Donaldson-Futaki invariant (an external prior result by a different author) to the adjoint foliated setting, then derives singularity criteria and stability statements for specific classes (Calabi-Yau, klt, general type, Fano). No quoted equations or steps in the abstract or claimed chain reduce any 'prediction' or stability statement to a fitted quantity or self-defined input inside the paper. The load-bearing implication relies on the validity of the extension of an independent characterisation, not on internal self-citation chains or ansatz smuggling. This is the normal case of a paper building on external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on the background theory of K-stability, log canonicity, and the flag ideal method; the abstract introduces no new free parameters, no new axioms beyond standard algebraic geometry, and no invented entities.

axioms (1)

standard math Standard properties of K-stability, log canonicity, and the mixed Donaldson-Futaki invariant hold in the adjoint foliated setting
The extension of Odaka's characterisation is invoked without re-deriving the background theory.

pith-pipeline@v0.9.1-grok · 5641 in / 1443 out tokens · 38535 ms · 2026-06-29T10:07:38.587061+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages · 1 internal anchor

[1]

Positivity of the moduli part

[Amb+21] F. Ambro, P. Cascini, V. V. Shokurov, and C. Spicer. “Positivity of the moduli part”. In: (2021). arXiv:2111.00423 [math.AG]. [AD13] C. Araujo and S. Druel. “On Fano foliations”. In:Advances in Mathematics238 (2013), pp. 70–118. [Bru15] M. Brunella.Birational Geometry of Foliations. Vol

work page arXiv 2021
[2]

Minimal model program for algebraically integrable adjoint foliated structures

[Cas+24] P. Cascini, J. Han, J. Liu, F. Meng, C. Spicer, R. Svaldi, and L. Xie. “Minimal model program for algebraically integrable adjoint foliated structures”. In: (2024). arXiv:2408.14258 [math.AG]. [Cas+25] P. Cascini, J. Han, J. Liu, F. Meng, C. Spicer, R. Svaldi, and L. Xie. “On finite generation and boundedness of adjoint foliated structures”. In: ...

work page arXiv 2024
[3]

Birational boundedness of stable families

arXiv:2604.24106 [math.AG]. [CS25] P. Cascini and C. Spicer. “On the MMP for rank one foliations on threefolds”. In: Forum of Mathematics, Pi13 (2025), e20. [CDS13] X. Chen, S. Donaldson, and S. Sun. “K¨ ahler–Einstein Metrics and Stability”. In: International Mathematics Research Notices2014.8 (2013), pp. 2119–2125. [Oda12] Y. Odaka. “The Calabi conjectu...

work page internal anchor Pith review Pith/arXiv arXiv 2025

[1] [1]

Positivity of the moduli part

[Amb+21] F. Ambro, P. Cascini, V. V. Shokurov, and C. Spicer. “Positivity of the moduli part”. In: (2021). arXiv:2111.00423 [math.AG]. [AD13] C. Araujo and S. Druel. “On Fano foliations”. In:Advances in Mathematics238 (2013), pp. 70–118. [Bru15] M. Brunella.Birational Geometry of Foliations. Vol

work page arXiv 2021

[2] [2]

Minimal model program for algebraically integrable adjoint foliated structures

[Cas+24] P. Cascini, J. Han, J. Liu, F. Meng, C. Spicer, R. Svaldi, and L. Xie. “Minimal model program for algebraically integrable adjoint foliated structures”. In: (2024). arXiv:2408.14258 [math.AG]. [Cas+25] P. Cascini, J. Han, J. Liu, F. Meng, C. Spicer, R. Svaldi, and L. Xie. “On finite generation and boundedness of adjoint foliated structures”. In: ...

work page arXiv 2024

[3] [3]

Birational boundedness of stable families

arXiv:2604.24106 [math.AG]. [CS25] P. Cascini and C. Spicer. “On the MMP for rank one foliations on threefolds”. In: Forum of Mathematics, Pi13 (2025), e20. [CDS13] X. Chen, S. Donaldson, and S. Sun. “K¨ ahler–Einstein Metrics and Stability”. In: International Mathematics Research Notices2014.8 (2013), pp. 2119–2125. [Oda12] Y. Odaka. “The Calabi conjectu...

work page internal anchor Pith review Pith/arXiv arXiv 2025