A log symplectic manifold is a complex manifold equipped with a complex symplectic form that has simple poles on a hypersurface. The possible singularities of such a hypersurface are heavily constrained. We introduce the notion of an elliptic point o
f a log symplectic structure, which is a singular point at which a natural transversality condition involving the modular vector field is satisfied, and we prove a local normal form for such points that involves the simple elliptic surface singularities $tilde{E}_6,tilde{E}_7$ and $tilde{E}_8$. Our main application is to the classification of Poisson brackets on Fano fourfolds. For example, we show that Feigin and Odesskiis Poisson structures of type $q_{5,1}$ are the only log symplectic structures on projective four-space whose singular points are all elliptic.
We describe the possible noncommutative deformations of complex projective three-space by exhibiting the Calabi--Yau algebras that serve as their homogeneous coordinate rings. We prove that the space parametrizing such deformations has exactly six ir
reducible components, and we give explicit presentations for the generic members of each family in terms of generators and relations. The proof uses deformation quantization to reduce the problem to a similar classification of unimodular quadratic Poisson structures in four dimensions, which we extract from Cerveau and Lins Netos classification of degree-two foliations on projective space. Corresponding to the ``exceptional component in their classification is a quantization of the third symmetric power of the projective line that supports bimodule quantizations of the classical Schwarzenberger bundles.
