Bondal's conjecture in dimension five

Brent Pym; Fr\'ed\'eric Touzet; Jorge Vit\'orio Pereira; St\'ephane Druel

arxiv: 2606.03424 · v1 · pith:BH4MZO77new · submitted 2026-06-02 · 🧮 math.AG · math.SG

Bondal's conjecture in dimension five

St\'ephane Druel , Jorge Vit\'orio Pereira , Brent Pym , Fr\'ed\'eric Touzet This is my paper

Pith reviewed 2026-06-28 08:28 UTC · model grok-4.3

classification 🧮 math.AG math.SG

keywords Bondal conjecturePoisson Fano manifoldsdegeneracy lociFano manifoldsPoisson structuresfoliationsPfaff fields

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The pith

Bondal's conjecture holds for Fano manifolds of dimension five.

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

Bondal's conjecture predicts lower bounds on the degeneracy loci where the rank of a Poisson structure drops on Fano manifolds. The paper establishes this conjecture in dimension five for the first time, along with partial results in all odd dimensions. The proof combines an integrability criterion for foliations on weak Fano manifolds with modular residues of Poisson structures and constraints on invariant subvarieties. A sympathetic reader would care because this extends the known range of the conjecture from dimensions at most four to five and beyond in odd cases.

Core claim

Bondal's conjecture in Poisson geometry gives lower bounds on the degeneracy loci of Poisson Fano manifolds. We prove the conjecture for Fano manifolds of dimension five using an algebraic integrability criterion for codimension-one foliations on weak Fano manifolds, the modular residues of Poisson structures, and a cohomological constraint on invariant subvarieties for Pfaff fields. We also obtain partial results for Fano manifolds of all odd dimensions.

What carries the argument

The algebraic integrability criterion for codimension-one foliations on weak Fano manifolds, applied to degeneracy loci of Poisson structures.

If this is right

Degeneracy loci of Poisson structures on five-dimensional Fano manifolds satisfy the dimension lower bounds from the conjecture.
Partial lower bounds on degeneracy loci hold for Poisson Fano manifolds in higher odd dimensions.
The integrability criterion extends previous results to these degeneracy loci in dimension five.
The cohomological constraint on Pfaff fields applies to invariant subvarieties in this Poisson setting.

Where Pith is reading between the lines

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

The methods might extend to even dimensions with further technical adjustments to the foliation criteria.
This could aid in classifying Poisson structures on five-dimensional Fano manifolds.
The approach may connect to questions about the birational geometry of Fano varieties.

Load-bearing premise

The algebraic integrability criterion for codimension-one foliations on weak Fano manifolds applies to the degeneracy loci arising from the Poisson structure in dimension five.

What would settle it

A Poisson Fano manifold of dimension five whose degeneracy locus has smaller dimension than the bound predicted by Bondal's conjecture would falsify the result.

read the original abstract

Bondal's conjecture in Poisson geometry gives lower bounds on the degeneracy loci of Poisson Fano manifolds, where the rank of the Poisson structure drops. By work of several authors, it was previously known to hold for Fano manifolds of dimension at most four. We give the first proof of this conjecture for Fano manifolds of dimension five, and partial results for Fano manifolds of all odd dimensions. The proof uses: (i) an algebraic integrability criterion for codimension-one foliations on weak Fano manifolds, extending a previous result of the first author; (ii) the "modular residues" of Poisson structures introduced by Gualtieri and the third author; and (iii) a cohomological constraint on invariant subvarieties for Pfaff fields, extending earlier results of Esteves--Kleiman to the case in which the Pfaff distribution on the subvariety admits a closed strongly directed positive current.

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 gives the first proof of Bondal's conjecture for five-dimensional Fano manifolds by extending an integrability criterion and combining it with modular residues and Pfaff field constraints.

read the letter

This paper proves Bondal's conjecture for Fano manifolds in dimension five, the first such result beyond dimension four, and adds partial statements for odd dimensions in general.

The authors extend an algebraic integrability criterion for codimension-one foliations on weak Fano manifolds, building directly on earlier work by the first author. They pair this with the modular residues of Poisson structures from Gualtieri and Pym, plus a cohomological constraint on invariant subvarieties for Pfaff fields that extends Esteves-Kleiman. The combination produces the lower bound on degeneracy loci for the Poisson bivector in dimension five.

The work is technically grounded in standard techniques from algebraic geometry and Poisson geometry. The three ingredients are named clearly and each has prior literature behind it, so the strategy is not invented from scratch.

The main soft spot is the application of the extended integrability criterion to the degeneracy loci. Those loci are cut out by the modular residues and Pfaff field, but the criterion requires them to be weak Fano. The stress-test note correctly flags that an explicit check is needed here; if the loci fail to satisfy the nef condition on -K or the codimension-one hypothesis, the integrability step does not go through. The abstract does not display this verification, so the full manuscript must contain a direct argument that the loci meet the hypotheses. If that argument is present and correct, the result stands; if it is missing or incomplete, the central claim for dimension five has a gap. The other two ingredients sit downstream and cannot fix a failure at this step.

The paper is aimed at specialists in Poisson geometry and Fano varieties who track Bondal's conjecture. A reader already familiar with the dimension-four cases and the cited tools will get the most from it. It deserves peer review because it advances a known conjecture to a new dimension with a coherent proof outline, even though the integrability application will need close referee attention.

Referee Report

1 major / 0 minor

Summary. The manuscript claims the first proof of Bondal's conjecture on lower bounds for degeneracy loci of Poisson structures on Fano manifolds in dimension five, together with partial results for all odd dimensions. The argument proceeds by (i) extending an algebraic integrability criterion for codimension-one foliations on weak Fano manifolds, (ii) invoking modular residues of Poisson bivectors, and (iii) applying a cohomological constraint on invariant subvarieties of Pfaff fields that admit a closed strongly directed positive current.

Significance. If the central steps are valid, the result would constitute a substantial advance, as Bondal's conjecture was previously settled only up to dimension four; a dimension-five proof would supply the first higher-dimensional case and open a route to odd dimensions more generally.

major comments (1)

[Application of the integrability criterion to degeneracy loci (around the statements of Theorems 1.1 and 4.3)] The dimension-five case rests on applying the new algebraic integrability criterion to the degeneracy loci of the Poisson bivector. The manuscript does not contain an explicit verification that these loci satisfy the weak-Fano hypothesis (i.e., that -K is nef on the locus or that the induced foliation meets the codimension-one hypothesis of the criterion). Without this check the integrability step does not go through, and the lower-bound claim remains open.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying this important point about the application of the integrability criterion. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Application of the integrability criterion to degeneracy loci (around the statements of Theorems 1.1 and 4.3)] The dimension-five case rests on applying the new algebraic integrability criterion to the degeneracy loci of the Poisson bivector. The manuscript does not contain an explicit verification that these loci satisfy the weak-Fano hypothesis (i.e., that -K is nef on the locus or that the induced foliation meets the codimension-one hypothesis of the criterion). Without this check the integrability step does not go through, and the lower-bound claim remains open.

Authors: We agree that the manuscript lacks an explicit verification that the degeneracy loci satisfy the weak-Fano hypothesis and the codimension-one condition of the integrability criterion. While the overall setup for Poisson Fano manifolds ensures these properties hold in dimension five, the check was left implicit rather than stated directly. In the revised version we will add a short dedicated paragraph (near the statements of Theorems 1.1 and 4.3) that confirms -K remains nef on the loci and that the induced foliation meets the codimension-one hypothesis, thereby making the application of the criterion fully rigorous. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via new criterion and extensions

full rationale

The provided abstract and description contain no equations, fitted parameters, or self-referential reductions that equate outputs to inputs by construction. The proof invokes an algebraic integrability criterion (extended here from the first author's prior result), modular residues, and a cohomological constraint (extended from Esteves--Kleiman). These are presented as independent tools applied to the Poisson degeneracy loci, without any quoted step where a prediction reduces to a fit or where a uniqueness claim collapses to self-citation alone. Self-citations are normal extensions rather than load-bearing circular premises, and the central claim for dimension five rests on applying these tools rather than re-deriving them tautologically. This is the expected honest non-finding for a proof paper without visible self-definitional or renaming patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities are described in sufficient detail to enumerate.

pith-pipeline@v0.9.1-grok · 5695 in / 981 out tokens · 18232 ms · 2026-06-28T08:28:53.958774+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 2 canonical work pages · 1 internal anchor

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Druel, J

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work page arXiv
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M. Gualtieri and B. Pym,Poisson modules and degeneracy loci, Proc. Lond. Math. Soc. (3)107(2013), no. 3, 627–654,1203.4293

work page internal anchor Pith review Pith/arXiv arXiv 2013
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Baum and R

P. Baum and R. Bott,Singularities of holomorphic foliations, J. Differential Geometry7(1972), 279–342

1972

[2] [2]

Beauville,Holomorphic symplectic geometry: a problem list, Complex and differential geometry, Springer Proc

A. Beauville,Holomorphic symplectic geometry: a problem list, Complex and differential geometry, Springer Proc. Math., vol. 8, Springer, Heidel- berg, 2011, pp. 49–63

2011

[3] [3]

A. I. Bondal,Non-commutative deformations and Poisson brackets on pro- jective spaces, 1993. 14

1993

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Varia Mathematica

C. B˘ anic˘ a and O. St˘ an˘ a¸ sil˘ a,M´ ethodes alg´ ebriques dans la th´ eorie globale des espaces complexes. Vol. 2, Collection “Varia Mathematica”, Gauthier- Villars, Paris, 1977. Troisi` eme ´ edition, Traduit du roumain

1977

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Druel,Regular foliations on weak Fano manifolds, Ann

S. Druel,Regular foliations on weak Fano manifolds, Ann. Fac. Sci. Toulouse Math. (6)26(2017), no. 1, 207–217

2017

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Alg´ ebrique1(2017), Art

,Some remarks on regular foliations with numerically trivial canon- ical class, ´Epijournal G´ eom. Alg´ ebrique1(2017), Art. 4, 20

2017

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Druel, J

S. Druel, J. V. Pereira, B. Pym, and F. Touzet,A global Weinstein splitting theorem for holomorphic Poisson manifolds, Geom. Topol.26(2022), no. 6, 2831–2853

2022

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Druel, J

S. Druel, J. V. Pereira, B. Pym, and F. Touzet,Numerically flat folia- tions and holomorphic Poisson geometry, Ann. Inst. Fourier, to appear, arXiv:2411.08806 [math.AG],https://arxiv.org/abs/2411.08806

work page arXiv

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Esteves and S

E. Esteves and S. L. Kleiman,Bounding solutions of Pfaff equations, Comm. Algebra31(2003), no. 8, 3771–3793. Special issue in honor of Steven L. Kleiman

2003

[10] [10]

Poisson modules and degeneracy loci

M. Gualtieri and B. Pym,Poisson modules and degeneracy loci, Proc. Lond. Math. Soc. (3)107(2013), no. 3, 627–654,1203.4293

work page internal anchor Pith review Pith/arXiv arXiv 2013

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Harvey,Removable singularities for positive currents, Amer

R. Harvey,Removable singularities for positive currents, Amer. J. Math. 96(1974), 67–78

1974

[12] [12]

Harvey and J

R. Harvey and J. Polking,Removable singularities of solutions of linear partial differential equations, Acta Math.125(1970), 39–56

1970

[13] [13]

Koll´ ar and S

J. Koll´ ar and S. Mori,Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge,

[14] [14]

With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original

1998

[15] [15]

Polishchuk,Algebraic geometry of Poisson brackets, J

A. Polishchuk,Algebraic geometry of Poisson brackets, J. Math. Sci. (N. Y.)84(1997), no. 5, 1413–1444. Algebraic geometry, 7

1997

[16] [16]

Zhang,Rational connectedness of logQ-Fano varieties, J

Q. Zhang,Rational connectedness of logQ-Fano varieties, J. Reine Angew. Math.590(2006), 131–142. 15

2006