Non-symplectic involutions of irreducible symplectic manifolds of \(K3^{[n]}\)-type (Q303782)

The author considers non-symplectic involutions of irreducible symplectic manifolds of \(K3^{[n]}\)-type. He shows that two symplectic manifolds of \(K3^{[n]}\)-type are deformation equivalent if and only if their stable invariant Kähler cones are equivalent under parallel transport. The author gives a lattice-theoretic description of the deformation types of involutions of type \(M\), where \(M\) is an admissible sublattice of the \(K3^{[n]}\)-lattice, and shows that \(M\) is isometric to an invariant sublattice of a non-symplectic involution. In the last section the author proves that there exists a quasi-projective moduli space of \textit{simple} pairs \((X,i)\) consisting of a symplectic manifold \(X\) of \(K3^{[n]}\)-type together with a non-symplectic involution \(i\), where ''simple'' means that the stable invariant Kähler cone coincides with the invariant Kähler cone.