Reduction of singularities in Poisson geometry via weighted blowups

Abstract: Hironaka’s theorem on resolution of singularities via blowups fails in Poisson geometry, since blowups generally destroy Poisson structures. However, there exists a more general operation called a weighted blowup, which may provide a viable alternative. Weighted blowups have proven useful in the recent literature on resolution of singularities; for instance, in the non-Poisson setting, Abramovich-Temkin-Wlodarczyk have shown that fast functorial resolution of singularities can always be achieved by sequentially blowing up the worst singularities via a canonical choice of weighted blowup. We take a similar approach to show that appropriate choices of weighted blowups produce reduction of singularities in low-dimensional Poisson geometry. In joint work with Simon Lapointe, Mykola Matviichuk and Brent Pym, we obtain: (1) a criterion for lifting a Poisson structure to a weighted blowup, (2) a classification of normal forms of Poisson varieties not admitting any weighted blowup, in low-dimension, and (3) a theorem on the extent to which weighted blowups can reduce the singularities of Poisson varieties, in low-dimension.

algebraic geometryanalysis of PDEsalgebraic topologycomplex variablesdifferential geometrygeneral topologygeometric topologyK-theory and homologymetric geometrysymplectic geometry

Audience: researchers in the topic

CRM - Séminaire du CIRGET / Géométrie et Topologie

Series comments: Hybrid seminar of geometry and topology. Laboratory : CIRGET - www.cirget.uqam.ca The homepage of the seminar is www.cirget.uqam.ca/fr/seminaires.html

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