Weighted blowups and 3d Poisson desingularizations

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--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 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.

• The paper establishes a functorial desingularization theorem for Poisson subvarieties in threefolds, reducing singularities to non-nilpotent curve points or Du Val surface points.
• It derives lifting criteria for polyvector fields and Poisson structures under weighted blowups, showing how conilpotent centres preserve the exceptional divisor as a Poisson hypersurface.
• It combines Poisson-cohomological normal forms with resolution invariants to prove termination, while identifying non-nilpotent and Du Val points as unavoidable obstructions to Poisson-preserving resolution.

Overview

This paper, by Lapointe, Matviichuk, Pym, and Zupancic (2604.16698), establishes a desingularization theorem for Poisson subvarieties of smooth Poisson threefolds. The central result states that for any Poisson triple $(X,Y,\pi)$ — an orbifold $X$ with Poisson structure $\pi$ and a Poisson subvariety $Y$ of pure dimension — there exists a sequence of weighted blowups along conilpotent centres after which the only singularities of $Y'$ are of two explicit types: non-nilpotent points (when $\dim Y = 1$) and Du Val points (when $\dim Y = 2$). These exceptional singularities are shown to be unavoidable: no weighted blowup can eliminate them without destroying the Poisson structure. The work combines the functorial weighted resolution algorithms of Abramovich–Temkin–Włodarczyk and McQuillan with new formal normal forms for three-dimensional Poisson brackets obtained via Poisson cohomology.

Weighted blowups of orbifolds

The paper works over an algebraically closed field $K$ of characteristic zero, with orbifolds understood as smooth separated Deligne–Mumford stacks of finite type (or complex analytic orbifolds). A centre on $X$ is a filtration by ideals that is locally defined by monomials of $w$-weighted order at least $X$0, for some weight sequence $X$1. The weighted blowup $X$2 is constructed as the quotient of the degeneration to the weighted normal cone by the multiplicative group $X$3, following Loizides–Meinrenken, Quek, Włodarczyk, and McQuillan.

The key technical device is a valuation $X$4 on the tensor algebra, assigning weights to functions, differentials, and vector fields according to the weighting. For polyvectors, the minimal possible order of a degree-$X$5 polyvector is $X$6, where $X$7 is the $X$8-th weight sum.

Lifting polyvectors along weighted blowups

The first main intermediate result gives necessary and sufficient conditions for a polyvector field to lift to a weighted blowup, generalizing Polishchuk's criteria for unweighted blowups:

• A polyvector field $X$9 lifts to $\pi$0 if and only if $\pi$1 and $\pi$2, where $\pi$3 is the weighted Euler vector field.
• In this case, $\pi$4 is tangent to all members of the centre's filtration; moreover, the lift is tangent to the exceptional divisor if and only if $\pi$5.

Specialized to bivectors, this yields the notion of a codegenerate centre: one for which the Poisson structure lifts. A centre is conilpotent when $\pi$6, in which case the lift preserves the exceptional divisor as a Poisson hypersurface. Conilpotence implies codegeneracy implies Poisson-compatibility, but not conversely. For codimension-two centres, codegeneracy simplifies to Poisson-compatibility plus $\pi$7.

Formal normal forms via Poisson cohomology

The paper develops normal forms for three-dimensional formal Poisson structures using deformation theory through the Poisson cohomology DGLA $\pi$8 and a filtered Goldman–Millson-type theorem of Dolgushev–Rogers. Two normal forms are established:

1. If the leading term is $\pi$9, then either $Y$0 or $Y$1 is equivalent to $Y$2 wedged with $Y$3.
2. If the leading term is $Y$4, then $Y$5 is equivalent to $Y$6 for some series $Y$7 and functions $Y$8 vanishing appropriately.

These rely on Poisson cohomology computations of Hoekstra–Zeiser and Pichereau. A further result shows that a Poisson structure whose leading term is the Jacobian structure of the Whitney umbrella $Y$9, and whose zero locus is non-isolated, is equivalent to $Y'$0 for some unit $Y'$1.

Singularity invariants and small weight sums

The resolution machinery relies on the Abramovich–Temkin–Włodarczyk invariant $Y'$2, the exponent sequence of the maximal admissible centre. The set of possible invariants is well ordered, decreases under blowing up the associated centre, and equals its minimum exactly when $Y'$3 is smooth.

The paper establishes a numerical lemma characterizing invariants below $Y'$4: these are precisely those with $Y'$5 or equal to $Y'$6, and they take the form $Y'$7 or one of $Y'$8. This yields a classification theorem: a surface singularity in a threefold has invariant below $Y'$9 if and only if it is a Du Val singularity ($\dim Y = 1$0), a Whitney umbrella, or two-component normal crossings. This connects to Reid's characterization of Du Val singularities as canonical surface singularities: every admissible centre for a canonical hypersurface has weight sum exceeding one, so Du Val points admit no conilpotent centres whatsoever.

Curves in threefolds

For a Poisson triple of dimension $\dim Y = 1$1, the linearization of $\dim Y = 1$2 at a singular point yields a three-dimensional Lie algebra $\dim Y = 1$3 with $\dim Y = 1$4: abelian, Heisenberg, or split nonabelian. Points where $\dim Y = 1$5 is non-nilpotent are the obstruction. Near such a point, the triple has the normal form $\dim Y = 1$6 with $\dim Y = 1$7, and no codegenerate centres exist there — so these singularities cannot be improved by any weighted blowup.

Away from the non-nilpotent locus, the authors construct a conilpotent centre (using the associated centre when $\dim Y = 1$8, and otherwise point blowups or $\dim Y = 1$9-completions informed by Abhyankar's analysis of plane curve singularities) that strictly decreases the invariant. Since the set of invariants is well ordered, iteration terminates with only non-nilpotent points remaining. Consequently, since planar curve singularities admit ordinary resolutions, $\dim Y = 2$0 itself admits a full Poisson resolution $\dim Y = 2$1, though possibly not an embedded one.

Surfaces in threefolds

For dimension $\dim Y = 2$2 triples, a Du Val point is one where $\dim Y = 2$3 has a Du Val singularity and $\dim Y = 2$4 has an isolated zero. The normal form theorem establishes that Du Val points are equivalent to Jacobian Poisson structures $\dim Y = 2$5 with $\dim Y = 2$6 a standard Du Val equation — abstractly, restrictions of versal Poisson deformations of symplectic surface singularities to curves in the base. As with non-nilpotent points, no codegenerate centres exist at Du Val points, so they are immovable.

The dichotomy theorem shows that whenever the associated centre fails to be conilpotent, the point is either a Du Val point (isolated case) or a Whitney umbrella singularity (non-isolated case). The Whitney umbrella case is handled by combining the normal form results with Seidenberg's theorem on invariant curves of planar vector fields, showing that the umbrella surface is the unique Poisson surface through such a point. The resolution algorithm then proceeds by blowing up the associated centre away from the Du Val locus, with special handling when the invariant equals $\dim Y = 2$7, where type-$\dim Y = 2$8 points coexist with curve components carrying Whitney umbrella or normal crossings singularities. Iteration terminates with only Du Val points remaining.

Limitations and open questions

Several restrictions are acknowledged explicitly. The main theorem is specific to embedding dimension three: the proof uses explicit normal forms and the fact that non-nilpotent and Du Val loci are isolated, and the higher-dimensional analogue remains open. The algorithm does not guarantee logarithmic resolutions (simple normal crossings exceptional divisors); adapting the logarithmic variants of Quek and Włodarczyk to the Poisson setting is left open. The non-embedded Poisson resolutions produced may fail to embed into a blowup of $\dim Y = 2$9, so embedded resolution of the pair is not achieved. Finally, while Du Val points admit local Poisson alterations via slices of the Grothendieck–Springer alteration, the construction of global counterparts remains an open question.

Conclusion

The paper provides a complete, functorial answer to the problem of Poisson desingularization in embedding dimension three, identifying precisely which singularities can be eliminated by weighted blowups and which are intrinsic obstructions. The combination of weighted resolution technology with Poisson-cohomological normal forms yields both the desingularization theorem and structural results — the lifting criterion for polyvectors and the classification of small-invariant surface singularities — that are of independent interest.