  We introduce a natural nondegeneracy condition for Poisson structures, called
holonomicity, which is closely related to the notion of a log symplectic form.
Holonomic Poisson manifolds are privileged by the fact that their deformation
spaces are as finite-dimensional as one could ever hope: the corresponding
derived deformation complex is a perverse sheaf. We develop some basic
structural features of these manifolds, highlighting the role played by the
divergence of Hamiltonian vector fields. As an application, we establish the
deformation-invariance of certain families of Poisson manifolds defined by
Feigin and Odesskii, along with the "elliptic algebras" that quantize them.
