On logarithmic Poisson cohomology of a degenerate Poisson bivector in affine plane

Diekouam Fotso Luc \'Em\'ery; Iskaml\'e Bruno; Kamtila Kari; Tcheka Calvin

arxiv: 2605.00217 · v1 · submitted 2026-04-30 · 🧮 math.AG

On logarithmic Poisson cohomology of a degenerate Poisson bivector in affine plane

Kamtila Kari , Iskaml\'e Bruno , Diekouam Fotso Luc \'Em\'ery , Tcheka Calvin This is my paper

Pith reviewed 2026-05-09 19:59 UTC · model grok-4.3

classification 🧮 math.AG

keywords Poisson cohomologylogarithmic Poisson cohomologydegenerate bivectoraffine planeHamiltonian operatorcohomological invariantsPoisson complex

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The pith

For the degenerate Poisson bivector y^n ∂_x ∧ ∂_y with n>1, the classical and logarithmic Poisson cohomology groups are isomorphic in every degree.

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

The paper establishes that the classical Poisson cohomology and the logarithmic Poisson cohomology along the ideal generated by y^n are isomorphic for the specific degenerate bivector π = y^n ∂_x ∧ ∂_y when n exceeds 1. This isomorphism is obtained after explicitly determining the logarithmic Hamiltonian operator and constructing the corresponding logarithmic Poisson cochain complex to find the cohomological invariants. A reader might care because it provides a way to relate standard Poisson cohomology to a version adapted to the singular locus defined by the ideal, potentially simplifying calculations in algebraic settings where the Poisson structure vanishes to higher order. The result holds over fields of characteristic zero.

Core claim

For a given degenerate bivector π = y^n ∂_x ∧ ∂_y with n > 1, the classical Poisson cohomology group and the logarithmic Poisson cohomology group along the ideal I = y^n F[x,y] are isomorphic in every degree. This result follows from determination of the logarithmic Hamiltonian operator and the logarithmic Poisson cochain complex in order to compute the cohomological invariants associated to π, where F is the field of characteristic zero.

What carries the argument

The logarithmic Hamiltonian operator and the logarithmic Poisson cochain complex, which are constructed to compute the cohomological invariants of π and yield the isomorphism with the classical Poisson cohomology.

Load-bearing premise

The bivector must be exactly of the form y^n ∂_x ∧ ∂_y for integer n greater than 1 over a field of characteristic zero.

What would settle it

An explicit calculation showing that the classical and logarithmic Poisson cohomology groups differ in some degree for a concrete choice such as n=2.

read the original abstract

In this paper, we show that for a given degenerate bivector $\pi= y^n\partial_x \wedge \partial_y$ with $n>1$, the classical Poisson cohomology group and the logarithmic Poisson cohomology group along the ideal $\mathcal{I}=y^n\mathbb{F}[x,y] $ are isomorphics in every d\'egr\'ee. This result follows from determination of the logarithmic Hamiltonian operator and the logarithmic Poisson cochain complexe in order to compute the cohomological invariants associated to $\pi$. $\mathbb{F}$ is the field of characteristic 0.

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

This paper gives an explicit isomorphism between classical and logarithmic Poisson cohomology for the specific degenerate bivector y^n ∂x ∧ ∂y (n>1) in the affine plane over char 0.

read the letter

The central result is that these two cohomology theories agree in every degree for this one family of bivectors. The authors reach it by writing down the logarithmic Hamiltonian operator tied to the ideal I = (y^n) and then computing the cochain complex directly to extract the invariants. That computation is the main content of the work. In a 2-dimensional setting with a monomial degeneracy, the derivations that preserve I stay manageable, so explicit bases become feasible and the isomorphism can be checked term by term. The stress-test note finds no circularity or inconsistency in the stated method, and the bivector satisfies the Poisson condition automatically. This concrete case does not appear in the cited literature, so the isomorphism itself is new for this choice of π and I. The paper does the narrow job it sets out to do without overclaiming broader consequences. The obvious limitation is scope. The result holds only for bivectors of exactly this form, only for n > 1, and only in characteristic zero. Nothing in the abstract suggests a general technique that would apply to other degeneracies or higher dimensions, so the work stays a specialized calculation rather than a new framework. Because the claim rests on direct verification of operators and complexes, a referee would need to check the basis choices and the degree-by-degree maps for gaps, but the low dimension makes that verification realistic. This is the sort of technical note that people already working on Poisson cohomology with singularities or logarithmic structures would want to see. A reader who needs a concrete example to test against their own calculations gets value from it. It is solid enough on its own terms to deserve a serious referee rather than a desk rejection.

Referee Report

0 major / 3 minor

Summary. The manuscript claims that for the degenerate Poisson bivector π = y^n ∂_x ∧ ∂_y with n > 1 over a field F of characteristic zero, the classical Poisson cohomology groups are isomorphic to the logarithmic Poisson cohomology groups along the ideal I = y^n F[x,y] in every degree. This isomorphism is obtained by explicitly determining the logarithmic Hamiltonian operator and the resulting logarithmic Poisson cochain complex, then comparing the cohomological invariants.

Significance. If the explicit computations are correct, the result supplies a concrete, verifiable instance in which the inclusion of the logarithmic multivector complex into the classical one induces an isomorphism on cohomology for a specific degenerate bivector in the affine plane. The two-dimensional setting makes direct basis computations feasible, and the paper's approach of determining the Hamiltonian operator and cochain complex explicitly is a strength that allows the claim to be checked degree by degree without appeal to general machinery.

minor comments (3)

Abstract: the phrasing 'are isomorphics in every dégrée' contains a grammatical error ('isomorphics' should be 'isomorphic') and mixes French spelling; replace with standard English 'are isomorphic in every degree'.
The manuscript should include a brief statement of the base field F (e.g., algebraically closed or not) and confirm that all computations hold uniformly for n > 1 without additional restrictions on n.
Notation for the ideal I = y^n F[x,y] is clear, but the paper would benefit from an explicit description of the logarithmic derivations (vector fields v satisfying v(I) ⊂ I) in §2 or wherever the operator is introduced, to make the cochain complex fully self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of our manuscript. The referee correctly identifies the main claim and notes the value of the explicit computations in this low-dimensional case. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity: explicit computation of complexes

full rationale

The paper establishes the claimed isomorphism by explicitly determining the logarithmic Hamiltonian operator and the resulting cochain complex for the fixed bivector π = y^n ∂_x ∧ ∂_y (n>1) over char-0 fields, then computing the cohomology groups directly in each degree. This is a standard, self-contained algebraic computation using the definitions of Poisson cohomology and logarithmic derivations with respect to the ideal I = (y^n); no parameter fitting, self-referential definitions, or load-bearing self-citations reduce the result to its inputs by construction. The 2-dimensional affine setting permits basis-level verification without external uniqueness theorems or ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard definitions of Poisson cohomology and logarithmic structures along an ideal, plus the explicit form of the given bivector and the characteristic-zero assumption on the base field.

axioms (1)

domain assumption The base field F has characteristic zero
Explicitly stated in the abstract as required for the result.

pith-pipeline@v0.9.0 · 5401 in / 1175 out tokens · 59563 ms · 2026-05-09T19:59:31.397677+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 6 canonical work pages

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J. Dongho. Logarithmic Poisson Structures: Cohomological Invariants and Pre-Quantiﬁcation , University of Angers, 2012

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Frederic

B. Frederic. Poisson Structures on the polynomial algebras, cohomology a nd deformations ,Claude Roger Uni- versity - Lyon 1, Lyon, 13-th november 2009

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Iskamle, J

B. Iskamle, J. Dongho, B. Ndombol. On some properties of Poisson cohomology: Example of calcula tion on a Poisson structure , Proceedings of the American Mathematical Society,(2025) doi = 10.1090/proc/17177

work page doi:10.1090/proc/17177 2025
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Jeanneret, D

A. Jeanneret, D. Lines. Invitation to Algebraic Topology Volume I: Homology , Éditions Cépaduès 1 (2014)

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Laurent-Gengoux, A

C. Laurent-Gengoux, A. Pichereau, P. Vanhaecke. Poisson structure , Grundlerhen der Mathematischen Wis- senschaften, Springer 37 (2013), doi = 10.1007/9783-642-3 1090-4

work page doi:10.1007/9783-642-3 2013
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Lichnerowicz, Manifold of Poisson and their associated Lie algebras ,Journal of diﬀerential geometry, Lehigh University

A. Lichnerowicz, Manifold of Poisson and their associated Lie algebras ,Journal of diﬀerential geometry, Lehigh University. 12 (1977), no.2, 253-300

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[7]

P. Monnier. Poisson cohomology in dimension two , Israel journal of mathematics, Springer, vol 129,189-207 , no. 1, 2002, doi=10.1007/BF02773163

work page doi:10.1007/bf02773163 2002
[8]

Nakanishi

N. Nakanishi. Poisson cohomology of plane quadratic Poisson structures . Publications of the research institute for mathematical sciences, vol 33, n° 1, 73-89 (1997), doi= 1 0.2977/PRIMS/1195145534

work page arXiv 1997
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Pichereau

A. Pichereau. Poisson (Co)homology and isolated singularities in small di mensions, with an application in deformation theory, Poitiers: theses.fr/2006POIT2354 (2006)

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[10]

Roger and P

C. Roger and P. Vanhaecke. Poisson cohomologyof the aﬃne plane , in preparation
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K. Saito. Theory of logarithmic diﬀerential forms and logarithmic vector ﬁelds , IJournal of the Faculty of Science, Univiversity of Tokyo Sect 1A, Mathematics. 27 (20 24), no. 2, 265-291, doi= 10.15083/00039776

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I. Vaisman. On the geometric quantization of poisson manifolds , Journal of mathematical physics, vol 32(1991), no.12, 3339-3345, doi= 10.1063/1.529447 University of Maroua Email address : kamtilakari@gmail.com University of Maroua Email address : brunoiskamle@gmail.com University of Maroua Email address : diekouamluc@gmail.com University of Dschang Email...

work page doi:10.1063/1.529447 1991

[1] [1]

J. Dongho. Logarithmic Poisson Structures: Cohomological Invariants and Pre-Quantiﬁcation , University of Angers, 2012

2012

[2] [2]

Frederic

B. Frederic. Poisson Structures on the polynomial algebras, cohomology a nd deformations ,Claude Roger Uni- versity - Lyon 1, Lyon, 13-th november 2009

2009

[3] [3]

Iskamle, J

B. Iskamle, J. Dongho, B. Ndombol. On some properties of Poisson cohomology: Example of calcula tion on a Poisson structure , Proceedings of the American Mathematical Society,(2025) doi = 10.1090/proc/17177

work page doi:10.1090/proc/17177 2025

[4] [4]

Jeanneret, D

A. Jeanneret, D. Lines. Invitation to Algebraic Topology Volume I: Homology , Éditions Cépaduès 1 (2014)

2014

[5] [5]

Laurent-Gengoux, A

C. Laurent-Gengoux, A. Pichereau, P. Vanhaecke. Poisson structure , Grundlerhen der Mathematischen Wis- senschaften, Springer 37 (2013), doi = 10.1007/9783-642-3 1090-4

work page doi:10.1007/9783-642-3 2013

[6] [6]

Lichnerowicz, Manifold of Poisson and their associated Lie algebras ,Journal of diﬀerential geometry, Lehigh University

A. Lichnerowicz, Manifold of Poisson and their associated Lie algebras ,Journal of diﬀerential geometry, Lehigh University. 12 (1977), no.2, 253-300

1977

[7] [7]

P. Monnier. Poisson cohomology in dimension two , Israel journal of mathematics, Springer, vol 129,189-207 , no. 1, 2002, doi=10.1007/BF02773163

work page doi:10.1007/bf02773163 2002

[8] [8]

Nakanishi

N. Nakanishi. Poisson cohomology of plane quadratic Poisson structures . Publications of the research institute for mathematical sciences, vol 33, n° 1, 73-89 (1997), doi= 1 0.2977/PRIMS/1195145534

work page arXiv 1997

[9] [9]

Pichereau

A. Pichereau. Poisson (Co)homology and isolated singularities in small di mensions, with an application in deformation theory, Poitiers: theses.fr/2006POIT2354 (2006)

2006

[10] [10]

Roger and P

C. Roger and P. Vanhaecke. Poisson cohomologyof the aﬃne plane , in preparation

[11] [11]

K. Saito. Theory of logarithmic diﬀerential forms and logarithmic vector ﬁelds , IJournal of the Faculty of Science, Univiversity of Tokyo Sect 1A, Mathematics. 27 (20 24), no. 2, 265-291, doi= 10.15083/00039776

work page doi:10.15083/00039776

[12] [12]

I. Vaisman. On the geometric quantization of poisson manifolds , Journal of mathematical physics, vol 32(1991), no.12, 3339-3345, doi= 10.1063/1.529447 University of Maroua Email address : kamtilakari@gmail.com University of Maroua Email address : brunoiskamle@gmail.com University of Maroua Email address : diekouamluc@gmail.com University of Dschang Email...

work page doi:10.1063/1.529447 1991