The geometry of antisymplectic involutions, I

TL;DR

This paper investigates the fixed points of antisymplectic involutions on certain hyperkähler manifolds, revealing a relationship between the fixed locus components and lattice divisibility.

Contribution

It establishes a precise link between the number of fixed locus components and the divisibility of an ample class in the lattice for -type hyperke4hler manifolds.

Findings

Number of fixed locus components equals the divisibility of the class (1 or 2).

Fixed locus structure depends on the lattice divisibility property.

Results apply to -type hyperke4hler manifolds with specific involutions.

Abstract

We study fixed loci of antisymplectic involutions on projective hyperk\"ahler manifolds of $\mathrm{K3}^{[n]}$-type. When the involution is induced by an ample class of square 2 in the Beauville-Bogomolov-Fujiki lattice, we show that the number of connected components of the fixed locus is equal to the divisibility of the class, which is either 1 or 2.