Antisymplectic involutions of holomorphic symplectic manifolds

TL;DR

This paper investigates antisymplectic involutions on holomorphic symplectic manifolds, revealing fixed loci as Lagrangian submanifolds with genus 1 and classifying their Chern numbers in specific cases.

Contribution

It demonstrates that fixed loci of antisymplectic involutions have -genus 1 and classifies Chern numbers for fixed loci when the manifold deforms from a K3 surface.

Findings

Fixed locus -genus is 1.

Classified Chern numbers for fixed loci.

Identified conditions on the dimension of the manifold.

Abstract

Let X be a holomorphic symplectic manifold, of dimension divisible by 4, and s an antisymplectic involution of X . The fixed locus F of s is a Lagrangian submanifold of X ; we show that its \^A-genus is 1. As an application, we determine all possibilities for the Chern numbers of F when X is a deformation of the Hilbert square of a K3 surface.