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 $\tilde{E}_6,\tilde{E}_7$
and $\tilde{E}_8$. Our main application is to the classification of Poisson
brackets on Fano fourfolds. For example, we show that Feigin and 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.