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

TL;DR

This paper classifies all terminalizations of quotients of Hilbert schemes of K3 surfaces or generalized Kummer varieties by symplectic automorphisms, revealing new deformation types of irreducible symplectic varieties.

Contribution

It provides a complete classification of terminalizations for these quotients, including their Betti numbers, fundamental groups, and singularities, expanding the known landscape of symplectic varieties.

Findings

At least nine new deformation types of irreducible symplectic varieties of dimension four.

Terminalizations in the Kummer case have quotient singularities.

Only three smooth terminalizations of K3^{[n]}-type, all previously known.

Abstract

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].