On symplectic hypersurfaces
Seminar Room 1, Newton Institute
The Grothendieck-Brieskorn-Slodowy theorem explains a relation between ADE-surface singularities $X$ and simply laced simple Lie algebras $g$ of the same Dynkin type: Let $S$ be a slice in $g$ to the subregular orbit in the nilpotent cone $N$. Then $X$ is isomorphic to $S\cap N$. Moreover, the restriction of the characteristic map $\chi:g\to g//G$ to $S$ is the semiuniversal deformation of $X$. We (j.w. Namikawa and Sorger) show that the theorem remains true for all non-regular nilpotent orbits if one considers Poisson deformations only. The situation is more complicated for non-simply laced Lie algebras.
It is expected that holomorphic symplectic hypersurface singularities are rare. Besides the ubiquitous ADE-singularities we describe a four-dimensional series of examples and one six-dimensional example. They arise from slices to nilpotent orbits in Liealgebras of type $C_n$ and $G_2$.