The geometry of antisymplectic involutions, II

TL;DR

This paper studies fixed loci of antisymplectic involutions on hyper-Kähler manifolds, revealing geometric structures like Fano and general type components, and generalizing known results to higher dimensions.

Contribution

It proves that fixed loci include Fano manifolds of index 3 and components of general type, extending previous work to higher-dimensional hyper-Kähler manifolds.

Findings

One fixed locus component is a Fano manifold of index 3.

Another component in the LLSvS 8-fold is of general type.

Generalizes the structure of fixed loci to higher dimensions.

Abstract

We continue our study of fixed loci of antisymplectic involutions on projective hyper-K\"ahler manifolds of $\mathrm{K3}^{[n]}$-type induced by an ample class of square 2 in the Beauville-Bogomolov-Fujiki lattice. We prove that if the divisibility of the ample class is 2, then one connected component of the fixed locus is a Fano manifold of index 3, thus generalizing to higher dimensions the case of the LLSvS 8-fold associated to a cubic fourfold. We also show that, in the case of the LLSvS 8-fold associated to a cubic fourfold, the second component of the fixed locus is of general type, thus answering a question by Manfred Lehn.