A symplectic fourfold

We present a method to construct irreducible symplectic varieties by studying terminalisations of quotient of hyper-Kähler manifolds by non-natural group actions. In particular, we construct irreducible symplectic varieties of dimension $4$ with $b_2=4$ and non-quotient singularities: this provides explicit examples of ISVs for which a global Torelli theorem is not known to hold.