Symplectic

Many fundamental results in algebraic geometry ultimately rely on the existence of resolutions of singularities for algebraic varieties (Hironaka's theorem), which is proven by repeatedly blowing up along smooth subvarieties of the singular locus. To apply these results to symplectic/Poisson varieties, we often need a resolution that also carries a Poisson structure. However, such Poisson resolutions need not exist, e.g. because Poisson brackets are not compatible with blowups in general. It turns out that if we give ourselves the flexibility to use more general "weighted" blowups, we can do considerably better. For example, I will explain that we can reduce the singularities of Poisson surfaces in threefolds to those of Du Val (ADE) type, which are close enough to being smooth for many purposes. This talk is based on forthcoming joint work with Simon Lapointe, Mykola Matviichuk and Boris Zupancic.