The deformation theory of a holomorphic Poisson manifold is captured by its Poisson cohomology. The latter can be understood via a natural D-module defined with Pym. When the Poisson structure is "holonomic" (a strong nondegeneracy condition), this D-module is holonomic---hence defines a perverse sheaf. Its structure reflects the geometry of the degeneracy hypersurface. In particular, this partitions its codimension-two singular locus into "smoothable" and "nonsmoothable" components, depending on the Poisson structure.

I will explain how to apply this theory to to understand a toric holomorphic Poisson structure which is generically symplectic (or more generally, a log symplectic structure with a normal crossings hypersurface). The non-toric deformations are given by smoothing the "smoothable" components. In the case of $\mathbb{P}^4$, this allows us to identify 40 new components of the moduli space of Poisson structures, corresponding to very specific smoothings of the hypersurface. In particular, the two types of Feigin--Odesskii structures form two of the components---the ones coming from cyclically symmetric structures. This is joint work in progress with Matviichuk and Pym.

This talk is complemented by the one of Brent Pym.