The geometry of antisymplectic involutions, II

Author:

Laure Flapan, Emanuele Macrì, Kieran G. O'Grady, Giulia Saccà

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG)

journal:

Roma01.math.AG

date:

2023-09-04 16:00:00

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.

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