On the nonsymplectic involutions of the Hilbert square of a K3 surface

TL;DR

This paper explores the geometric properties of nonsymplectic involutions on the Hilbert square of K3 surfaces, revealing new involutions and their relation to moduli spaces of polarized irreducible holomorphic symplectic manifolds.

Contribution

It provides a geometric description of new nonsymplectic involutions on Hilbert squares of K3 surfaces, expanding understanding of their moduli and involution structures.

Findings

Identification of new nonsymplectic involutions

Description of involutions via moduli space geometry

Connection between involutions and invariant lattices

Abstract

We investigate the interplay between the moduli spaces of ample <2>-polarized IHS manifolds of type K3^[2] and of IHS manifolds of type K3^[2] with a nonsymplectic involution with invariant lattice of rank one. In particular we geometrically describe some new involutions of the Hilbert square of a K3 surface, whose existence was proven in a previous work of Boissiere, Cattaneo, Nieper-Wisskirchen and Sarti.