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

TL;DR

This paper introduces elliptic points in log symplectic manifolds, classifies their local structure involving simple elliptic surface singularities, and applies this to classify certain Poisson brackets on Fano fourfolds.

Contribution

It defines elliptic points in log symplectic structures, proves a local normal form involving elliptic surface singularities, and classifies specific Poisson structures on Fano fourfolds.

Findings

Elliptic points satisfy a transversality condition involving the modular vector field.

Local normal form involves simple elliptic surface singularities $ ilde{E}_6, ilde{E}_7, ilde{E}_8$.

Feigin-Odesskii Poisson structures of type $q_{5,1}$ are the only log symplectic structures with elliptic singularities on projective four-space.

Abstract

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.