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--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.