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arxiv: 2604.16698 · v1 · submitted 2026-04-17 · 🧮 math.AG · math.SG

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Weighted blowups and 3d Poisson desingularizations

Simon Lapointe , Mykola Matviichuk , Brent Pym , Boris Zupancic
Authors on Pith no claims yet

Pith reviewed 2026-05-10 06:55 UTC · model grok-4.3

classification 🧮 math.AG math.SG
keywords Poisson subvarietiesweighted blowupsresolution of singularitiesorbifold reductionsDu Val singularitiesPoisson cohomologythreefoldspolyvector fields
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The pith

Weighted blowups reduce singularities of Poisson subvarieties in smooth Poisson threefolds to explicit normal forms.

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

The paper shows that repeated weighted blowups suffice to resolve the singularities of any Poisson subvariety inside a smooth Poisson threefold down to two families of simple local models. One family consists of Du Val surface singularities carrying a Jacobian Poisson structure; the other consists of plane curves inside the zero set of a linear Poisson bracket. A sympathetic reader cares because the construction is functorial, so the reduced models can be used to study global invariants, deformations, and symplectic leaves without losing the Poisson condition. The argument rests on new local normal forms for three-dimensional Poisson brackets obtained via Poisson cohomology, together with general criteria for when a polyvector field lifts across a weighted blowup of an orbifold.

Core claim

We establish existence of functorial orbifold reductions of singularities for Poisson subvarieties in smooth Poisson threefolds. Namely, we show that with enough weighted blowups, one can reduce the singularities of such Poisson subvarieties to certain simple, explicit, local normal forms: Du Val surface singularities where the Poisson structure is locally Jacobian, and plane curves lying in the vanishing locus of a particular linear Poisson structure. The proof combines Abramovich--Temkin--Włodarczyk and McQuillan's recent approach to resolution of singularities for varieties via weighted blowups with some new normal forms for three-dimensional Poisson brackets derived via Poisson Poisson.

What carries the argument

Weighted blowups of orbifolds along suborbifolds, together with necessary and sufficient lifting conditions for polyvector fields and new normal forms for three-dimensional Poisson brackets obtained from Poisson cohomology.

If this is right

  • Any Poisson subvariety in a smooth Poisson threefold admits a functorial sequence of weighted blowups that reduces its singularities to the two listed normal forms.
  • The reduced models are Du Val surfaces with Jacobian Poisson structures or plane curves in linear Poisson vanishing loci.
  • Polyvector fields lift across weighted blowups of orbifolds precisely when certain explicit local conditions on their coefficients hold.
  • Three-dimensional Poisson brackets admit a complete set of local normal forms derived from Poisson cohomology.
  • The orbifold reduction process preserves the Poisson property at every step.

Where Pith is reading between the lines

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

  • The functoriality may allow the reduced normal forms to be glued globally, yielding a canonical partial desingularization of the entire Poisson threefold.
  • The lifting criteria could be checked on concrete examples such as the standard Poisson structure on SL(2) to confirm that the normal forms are attained in practice.
  • If the same weighted-blowup technique extends beyond dimension three, it might supply a uniform resolution method for Poisson subvarieties in higher-dimensional Poisson manifolds.
  • The new cohomology-derived normal forms might simplify computations of Poisson cohomology or deformation theory on the reduced models.

Load-bearing premise

The ambient space is a smooth Poisson threefold, the subvariety is Poisson, and the new normal forms for three-dimensional Poisson brackets derived via Poisson cohomology are valid and sufficient for the reduction.

What would settle it

An explicit Poisson subvariety inside a smooth Poisson threefold whose singularities cannot be reduced to the stated normal forms by any finite sequence of weighted blowups, or a polyvector field that violates the derived lifting criterion on some weighted blowup.

Figures

Figures reproduced from arXiv: 2604.16698 by Boris Zupancic, Brent Pym, Mykola Matviichuk, Simon Lapointe.

Figure 1
Figure 1. Figure 1: Singular Poisson subvarieties Y ⊂ A 3 that cannot be resolved by weighted blowups. these are the three-dimensional Poisson structures given by restricting the versal Poisson deformation of a symplectic surface singularity to a smooth curve in the base of the deformation. Note that these conditions involve both the subvariety Y and the Poisson structure. They can be thought of as a sort of nondegeneracy con… view at source ↗
read the original abstract

We establish existence of functorial orbifold reductions of singularities for Poisson subvarieties in smooth Poisson threefolds. Namely, we show that with enough weighted blowups, one can reduce the singularities of such Poisson subvarieties to certain simple, explicit, local normal forms: Du Val surface singularities where the Poisson structure is locally Jacobian, and plane curves lying in the vanishing locus of a particular linear Poisson structure. The proof combines Abramovich--Temkin--W{\l}odarczyk and McQuillan's recent approach to resolution of singularities for varieties via weighted blowups with some new normal forms for three-dimensional Poisson brackets derived via Poisson cohomology. Along the way, we describe necessary and sufficient conditions for a polyvector field to lift to the weighted blowup of an orbifold along a suborbifold, generalizing criteria of Polishchuk for unweighted blowups of Poisson structures on smooth varieties.

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 / 3 minor

Summary. The manuscript establishes existence of functorial orbifold reductions of singularities for Poisson subvarieties in smooth Poisson threefolds. With sufficiently many weighted blowups, singularities reduce to Du Val surface singularities equipped with Jacobian Poisson structure or to plane curves lying in the zero locus of a linear Poisson structure. The argument combines the weighted-blowup resolution framework of Abramovich–Temkin–Włodarczyk and McQuillan with new local normal forms for three-dimensional Poisson brackets obtained via Poisson cohomology; it also supplies necessary and sufficient lifting conditions for polyvector fields to weighted blowups of orbifolds, generalizing Polishchuk’s criteria for unweighted blowups.

Significance. If the central claims hold, the work supplies a concrete, functorial desingularization procedure for Poisson subvarieties in dimension three, together with explicit normal forms and a lifting theorem that may be useful beyond the present setting. The combination of weighted-blowup techniques with Poisson-cohomological normal forms is a substantive contribution to both resolution of singularities and Poisson geometry.

major comments (2)
  1. [§4] §4 (Lifting criterion): the statement that the lifting conditions are necessary and sufficient for a polyvector field to lift to the weighted blowup of an orbifold is central to the inductive step, yet the proof sketch relies on an orbifold version of the Polishchuk criterion whose verification for non-trivial stabilizers is only indicated rather than fully expanded; a complete local computation in the presence of a non-trivial group action would strengthen the claim.
  2. [§3.2] §3.2 (Normal forms via Poisson cohomology): the classification of three-dimensional Poisson structures up to the stated normal forms (Du Val Jacobian and linear) is used to terminate the reduction; however, the argument assumes the ambient threefold is smooth and the Poisson structure is holomorphic, and it is not immediately clear whether the cohomology computation covers all cases when the subvariety is not a hypersurface or when the Poisson bivector vanishes to higher order.
minor comments (3)
  1. [§2] The notation for weighted blowups and orbifold charts is introduced in §2 but reused with slight variations in later sections; a consolidated table of notation would improve readability.
  2. Several citations to the resolution literature (Abramovich–Temkin–Włodarczyk, McQuillan) are given without page or theorem numbers; adding precise references would help readers locate the invoked statements.
  3. Figure 1 (schematic of the reduction sequence) is helpful but the labels on the exceptional divisors do not match the local coordinates used in the normal-form statements; consistency between figure and text would clarify the geometry.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive evaluation, and constructive suggestions. We address the two major comments point by point below and will incorporate revisions to improve the manuscript.

read point-by-point responses

  1. Referee: [§4] §4 (Lifting criterion): the statement that the lifting conditions are necessary and sufficient for a polyvector field to lift to the weighted blowup of an orbifold is central to the inductive step, yet the proof sketch relies on an orbifold version of the Polishchuk criterion whose verification for non-trivial stabilizers is only indicated rather than fully expanded; a complete local computation in the presence of a non-trivial group action would strengthen the claim.

    Authors: We agree that expanding the verification strengthens the central lifting criterion. The proof in §4 indicates the orbifold generalization of Polishchuk's criterion but does not fully detail the local computation when stabilizers are non-trivial. In the revised version we will add a complete local computation, working in suitable étale charts and explicitly verifying the necessary and sufficient conditions for the polyvector field to lift under a non-trivial group action. revision: yes

  2. Referee: [§3.2] §3.2 (Normal forms via Poisson cohomology): the classification of three-dimensional Poisson structures up to the stated normal forms (Du Val Jacobian and linear) is used to terminate the reduction; however, the argument assumes the ambient threefold is smooth and the Poisson structure is holomorphic, and it is not immediately clear whether the cohomology computation covers all cases when the subvariety is not a hypersurface or when the Poisson bivector vanishes to higher order.

    Authors: The Poisson-cohomology computations in §3.2 are performed locally on the smooth ambient threefold and classify all holomorphic Poisson bivectors on smooth threefolds up to the stated normal forms; the classification does not depend on the choice of Poisson subvariety. When the bivector vanishes to higher order the resulting structure falls into the linear normal form. We will insert a short clarifying paragraph in §3.2 stating that the local computation is independent of whether the subvariety is a hypersurface and that higher-order vanishing is covered by the linear case. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives its new normal forms for 3D Poisson brackets explicitly via Poisson cohomology inside the manuscript and combines them with resolution techniques from external citations (Abramovich–Temkin–Włodarczyk and McQuillan). The lifting criterion generalizes Polishchuk independently. No load-bearing step reduces by definition, by fitted input, or by self-citation chain to the target result; the existence claim follows from these separate components without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard resolution-of-singularities results via weighted blowups and on the derivation of new normal forms for Poisson brackets; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Abramovich--Temkin--Włodarczyk and McQuillan's weighted blowup resolution techniques apply to the orbifold setting
    Invoked to reduce singularities to the desired normal forms.
  • domain assumption New normal forms for three-dimensional Poisson brackets exist and are derivable via Poisson cohomology
    Central to obtaining the explicit local models for the reduced singularities.

pith-pipeline@v0.9.0 · 5462 in / 1331 out tokens · 56477 ms · 2026-05-10T06:55:45.210490+00:00 · methodology

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Reference graph

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