The geometry of antisymplectic involutions, II

Abstract

We continue our study of fixed loci of antisymplectic involutions on projective hyper-Kähler manifolds of ${K3}^{[n]}$-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.

45 pages; v2: Section 2.3 and Section 5 have been rewritten and explanations added. Final version, to appear in J. Ec. Polytech. Math