Terminalizations of quotients of compact hyperk\"ahler manifolds by induced symplectic automorphisms

Terminalizations of symplectic quotients are sources of new deformation types of irreducible symplectic varieties. We classify all terminalizations of quotients of Hilbert schemes of K3 surfaces or of generalized Kummer varieties, by finite groups of symplectic automorphisms induced from the underlying K3 or abelian surface. We determine their second Betti number and the fundamental group of their regular locus. In the Kummer case, we prove that the terminalizations have quotient singularities, and determine the singularities of their universal quasi-\'etale cover. In particular, we obtain at least nine new deformation types of irreducible symplectic varieties of dimension four. Finally, we compare our deformation types with those in [FM21; Men22]. The smooth terminalizations are only three and of K$3^{[n]}$-type, and surprisingly they all appeared in different places in the literature [Fuj83; Kaw09; Flo22].