Elliptic singularities on log symplectic manifolds and Feigin-Odesskii Poisson brackets

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 of 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 E6, E7 and E8. Our main application is to the classification of Poisson brackets on Fano fourfolds. For example, we show that Feigin and Odesskii's Poisson structures of type q_5,1 are the only log symplectic structures on projective four-space whose singular points are all elliptic.