On maximality of involutions of hyper-Kähler manifolds and punctual Hilbert schemes of surfaces

Abstract

Given a holomorphic or anti-holomorphic involution on a complex variety, the Smith inequality says that the total $\mathbb{F}_2$-Betti number of the fixed locus is no greater than the total $\mathbb{F}_2$-Betti number of the ambient variety. The involution is called maximal when the equality is achieved. In this paper, we investigate maximality of involutions of compact hyper-Kähler manifolds and of Hilbert schemes of points on surfaces. We obtain both positive and negative results. On one hand, given a smooth projective surface $S$ with $H^1(S, \mathbb{F}_2)=0$ equipped with a holomorphic (resp.~anti-holomorphic) involution $σ$, we establish the following necessary and sufficient condition for the maximality of the induced involution on the $n$th Hilbert scheme of points: the induced involution is maximal if and only if $σ$ is a maximal involution of $S$ and it acts on $H^2(S, \mathbb{Z})$ trivially (resp.~as $-\operatorname{id}$). This generalizes and completes previous partial results of Fu and Kharlamov--R\u asdeaconu. On the other hand, we show that for $n\geq 2$, a hyper-Kähler manifold of K3$^{[n]}$-deformation type admits neither maximal anti-holomorphic involutions (i.e.~real structures), nor maximal holomorphic (symplectic or anti-symplectic) involutions. In other words, such hyper-Kähler manifolds do not admit maximal (AAB), (ABA), (BAA) or (BBB) brane involutions in the sense of Kapustin--Witten.