The geometry of antisymplectic involutions, I

Abstract

We study fixed loci of antisymplectic involutions on projective hyperkähler manifolds of 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.

The geometry of antisymplectic involutions, I

“…More precisely, over one component the action on the fibers of λ is trivial while on the other component it is multiplication by (−1). The precise result is Theorem 5.12 and further implications of this will appear in [FM+21].…”

“…In the sequel [FM+21] of the current paper, we study the geometry of these fixed components. In particular, we examine the case of divisibility 2 in more detail and show that one connected component Y of the fixed locus Fix(τ ) is a Fano manifold, generalizing the case of the cubic fourfold in (c).…”

“…In particular, we examine the case of divisibility 2 in more detail and show that one connected component Y of the fixed locus Fix(τ ) is a Fano manifold, generalizing the case of the cubic fourfold in (c). The other component in divisibility 2 and the unique component in divisibility 1 are expected to be of general type, more details are in [FM+21].…”

“…Then we can deduce directly that Def(X, τ ) is smooth of codimension 1 in Def(X), together with the existence of a line bundle L on the universal family with the same properties as in Corollary 2.2. We will expand on this in [FM+21].…”

“…These constructions produce a HK manifold of K3 [n] -type starting from a Fano manifold of K3 type. The results of this paper together with the subsequent paper [FM+21] yield a reverse process in which one starts with a HK manifold X of K3 [n] -type and produces a corresponding Fano manifold arising as a connected component of the fixed locus of an antisymplectic involution on X.…”