Symplectic hypersurfaces

A symplectic hypersurface is an even dimensional hypersurface $X$ in complex affine space $\mathbb{C}^{2n+1}$ that carries a nowhere degenerate holomorphic 2-form. Non-trivial examples are necessarily singular, and well known classical examples are provided by the Kleinian singularities $\mathbb{C}^2/G$ in $\mathbb{C}^3$, where $G$ is a finite subgroup in $SU(2)$. I would like to discuss higher dimensional examples found by Namikawa, Sorger, van Straten and myself, which we believe to exhaust all possibilities.