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arxiv: 2605.22519 · v1 · pith:HA3HCYOEnew · submitted 2026-05-21 · 🧮 math.DG · math.SG

Algebroid Desingularizable Poisson Structures

Shane Rankin This is my paper

Pith reviewed 2026-05-22 02:09 UTC · model grok-4.3

classification 🧮 math.DG math.SG
keywords Poisson manifoldsLie algebroidssymplectic structuresdesingularizationreductive Lie algebrasnilpotent Lie algebrasPoisson geometry
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The pith

Duals of non-abelian reductive Lie algebras never admit algebroid desingularizable Poisson structures.

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

The paper defines algebroid desingularizable Poisson manifolds as those induced by symplectic Lie algebroids whose anchors are almost injective. This definition captures and generalizes several existing classes of singular Poisson structures. The main result establishes that the duals of real finite-dimensional non-abelian reductive Lie algebras fall outside this class. The work also constructs two infinite families of two-step nilpotent Lie algebras to show that the property holds for some but not all Lie algebras.

Core claim

Algebroid desingularizable Poisson manifolds arise precisely from symplectic Lie algebroids with almost-injective anchors. Duals of real finite-dimensional non-abelian reductive Lie algebras never arise in this way. Two infinite families of two-step nilpotent Lie algebras exist, one family admits such structures while the other does not.

What carries the argument

Symplectic Lie algebroid with almost-injective anchor, which induces the Poisson structure and serves as the desingularization device for the manifold.

If this is right

  • The non-existence result separates reductive Lie algebra duals from other Poisson manifolds that permit desingularization.
  • The two nilpotent families provide concrete test cases that distinguish which Lie algebras allow the structures.
  • Generalized classes such as log-symplectic and b^m-symplectic manifolds inherit the same non-existence restriction on reductive duals.
  • The distinction between the two nilpotent families shows that the property depends on specific algebraic features of the Lie algebra.

Where Pith is reading between the lines

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

  • The result may restrict which Poisson singularities on Lie algebra duals can be resolved via algebroid methods.
  • It suggests a possible algebraic obstruction, tied to reductivity and non-abelian character, that prevents almost-injective anchors.
  • Similar non-existence statements might hold for other families of Lie algebras once the anchor condition is examined.

Load-bearing premise

The intended desingularizable Poisson structures are exactly those coming from symplectic Lie algebroids whose anchors are almost injective.

What would settle it

Exhibit a symplectic Lie algebroid with almost-injective anchor on the dual of a real finite-dimensional non-abelian reductive Lie algebra, or prove that one of the constructed nilpotent families fails to satisfy the almost-injective anchor condition.

read the original abstract

We introduce algebroid desingularizable Poisson manifolds, a class of Poisson manifolds induced by symplectic Lie algebroids with almost-injective anchors, generalizing structures including log-symplectic, $b^m$-symplectic, $E$-symplectic manifolds, and hypersurface algebroids. We show that the dual of real, finite-dimensional, non-abelian, reductive Lie algebras never admit such algebroids. We finish by giving two infinite families of $2$-step nilpotent Lie algebras, one of which is desingularizable, and one of which is not.

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.

Referee Report

2 major / 1 minor

Summary. The paper introduces algebroid desingularizable Poisson manifolds as Poisson structures induced by symplectic Lie algebroids whose anchors are almost injective. This class is presented as a common generalization of log-symplectic, b^m-symplectic, E-symplectic manifolds and hypersurface algebroids. The central results are a non-existence theorem asserting that the dual of any real finite-dimensional non-abelian reductive Lie algebra admits no such structure, together with two infinite families of 2-step nilpotent Lie algebras, one of which is desingularizable and one of which is not.

Significance. If the almost-injective-anchor modeling choice correctly captures the geometric features responsible for desingularization in the motivating examples, the non-existence result cleanly separates reductive from certain nilpotent cases and supplies concrete positive and negative instances. The work would then furnish a useful organizing framework for Poisson desingularization questions and a source of falsifiable predictions for further examples.

major comments (2)
  1. [Abstract, §1] Abstract and §1: the non-existence statement for reductive duals is proved only inside the subclass of Poisson structures induced by symplectic Lie algebroids with almost-injective anchors. It is not shown that every conceivable algebroid resolution of the motivating Poisson structures (log-symplectic, b^m, etc.) necessarily satisfies the almost-injective condition; if counter-examples exist outside this subclass, the non-existence claim does not apply to the original geometric objects.
  2. [§4 (nilpotent families)] The two nilpotent families are offered as positive and negative instances inside the same definition. Any mismatch between the almost-injective-anchor class and the intended desingularization phenomena would therefore affect both the existence and non-existence statements symmetrically; a direct comparison of the anchor maps in the two families with the anchors arising in the classical log-symplectic and b^m cases is needed to confirm consistency.
minor comments (1)
  1. [Introduction] Notation for the anchor map and the almost-injective condition should be introduced once and used uniformly; several paragraphs in the introduction repeat the same definition with slightly varying wording.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. The points raised concern the precise scope of our definition and the need for explicit verification against classical examples. We respond to each major comment below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [Abstract, §1] Abstract and §1: the non-existence statement for reductive duals is proved only inside the subclass of Poisson structures induced by symplectic Lie algebroids with almost-injective anchors. It is not shown that every conceivable algebroid resolution of the motivating Poisson structures (log-symplectic, b^m, etc.) necessarily satisfies the almost-injective condition; if counter-examples exist outside this subclass, the non-existence claim does not apply to the original geometric objects.

    Authors: We acknowledge the distinction. The manuscript defines algebroid desingularizable Poisson manifolds precisely as Poisson structures induced by symplectic Lie algebroids whose anchors are almost injective; this condition is chosen because it holds in the motivating examples (log-symplectic, b^m-symplectic, E-symplectic, and hypersurface algebroids). The non-existence theorem is proved strictly inside this class. We do not claim that every possible Lie algebroid resolution of the underlying singularities must have an almost-injective anchor; resolutions outside the class would constitute different desingularization mechanisms and lie beyond the paper's scope. To clarify this, we will add a short paragraph in §1 stating the intended scope of the definition and noting that the almost-injective condition is verified directly for the classical cases that motivate the work. revision: partial

  2. Referee: [§4 (nilpotent families)] The two nilpotent families are offered as positive and negative instances inside the same definition. Any mismatch between the almost-injective-anchor class and the intended desingularization phenomena would therefore affect both the existence and non-existence statements symmetrically; a direct comparison of the anchor maps in the two families with the anchors arising in the classical log-symplectic and b^m cases is needed to confirm consistency.

    Authors: We agree that an explicit comparison strengthens the presentation. Both families in §4 are constructed via symplectic Lie algebroids on the duals; the first family has anchors that are almost injective (corank 1 along a hypersurface), while the second does not. This mirrors the anchor behavior in the standard log-symplectic Lie algebroid (anchor drops rank by 1 on the degeneracy locus) and b^m-symplectic structures. We will revise §4 to include a direct computation of the anchor maps for representative members of each family, together with a side-by-side rank comparison to the classical log-symplectic and b^m cases, confirming that the positive family satisfies the almost-injective condition while the negative family provides a controlled counter-example within the same framework. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard definitions and external Lie algebra facts

full rationale

The paper introduces algebroid desingularizable Poisson manifolds via the definition of structures induced by symplectic Lie algebroids with almost-injective anchors, then proves non-existence results for duals of non-abelian reductive Lie algebras using properties of Lie algebras and algebroid anchors. These steps draw on standard mathematical definitions and theorems external to the paper rather than reducing any claim to a self-definition, fitted input renamed as prediction, or self-citation chain. The two nilpotent families serve as illustrative examples within the same framework without forcing the central non-existence result by construction. The derivation remains self-contained against external benchmarks in differential geometry and Lie theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claims rest on the newly introduced definition and on standard background facts from Lie algebroid theory; no free parameters or invented physical entities are present.

axioms (1)
  • standard math Standard properties of symplectic Lie algebroids and their anchors
    Invoked when defining the new class and when proving non-admission for reductive cases.
invented entities (1)
  • algebroid desingularizable Poisson manifold no independent evidence
    purpose: A unifying class of Poisson manifolds induced by symplectic Lie algebroids with almost-injective anchors
    Newly defined to generalize log-symplectic, b^m-symplectic, E-symplectic and hypersurface algebroid structures.

pith-pipeline@v0.9.0 · 5607 in / 1469 out tokens · 54050 ms · 2026-05-22T02:09:31.090054+00:00 · methodology

discussion (0)

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Reference graph

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