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On symplectic hypersurfaces

A symplectic variety is a normal complex variety X with a holomorphic symplectic form ω on the regular part X reg and with rational Gorenstein sin-gularities. Affine symplectic varieties arise in many different ways such as closures of nilpotent orbits of a complex simple Lie algebra, as Slodowy slices to such nilpotent orbits or as symplectic reductions of holomorphic symplectic manifolds with Hamiltonian actions. Many examples of affine symplectic varieties tend to require large embedding codimensions compared to their dimensions. In this article we treat the rarest case, namely affine symplectic hypersurfaces.