The geometry of antisymplectic involutions. I. (Q2118190)

A compact smooth irreducible Kähler manifold \(X\) is called \textit{hyperkähler (HK)} if it is simply connected, and \(H^0(\Omega_X^2)\) is one-dimensional, and spanned by an everywhere non-degenerate holomorphic \(2\)-form. Recall that an involution \(\tau\) on a HK manifold \(X\) is antisymplectic if it acts as \((-1)\) on the space of global holomorphic \(2\)-forms. This article studies the fixed loci Fix\((\tau)\) of antisymplectic involutions \(\tau\) when \(X\) is deformation equivalent to the Hilbert scheme of length \(n\) subschemes of a \(K3\) surface, and the invariant sublattice in the second integral cohomology is spanned by a class of square \(2\). The authors mention two motivations for studying these fixed loci: the first one is to further explore the relations between HK manifolds of \(K3^{[n]}\) type and Fano manifolds of \(K3\) type, the second one is to produce covering families of Lagrangian cycles on HK manifolds. Let \((X,\lambda)\) be a polarized HK manifold of \(K3^{[n]}\) type. Let \(q_X\) be the associated Beauville-Bogomolov-Fujiki form on \(H^2(X;\mathbb{Z})\), and assume that \(q_X(\lambda)=2\). It follows from the Global Torelli Theorem that there exists an involution \(\tau\in\mathrm{Aut}(X)\) such that the invariant lattice \(H^2(\tau)_{+}=\mathbb{Z}\lambda\), and further by \textit{A. Beauville} [Prog. Math. 39, 1--26 (1983; Zbl 0537.53057)] this \(\tau\) is unique. We also recall that the divisibility of the polarization \(\operatorname{div}(\lambda)\) divides \(q_X(\lambda)\), and consequently, under the stated assumption, \(\operatorname{div}(\lambda)\) is either \(1\), or \(2\). The main theorem of this article proves that the number of connected component of the fixed locus Fix\((\tau)\) is equal to div\((\lambda)\). Moreover, when the divisibility is \(2\) (in which case \(4\) divides \(n\)), the article shows that the two connected components can be distinguished by the behavior of a lift of the action of \(\tau\) to the total space the line bundle with class \(\lambda\): over one component, the action on the fibers of \(\lambda\) is trivial, whereas on the other component it acts as multiplication by \((-1)\). The proof is based on a degeneration argument by specializing to the contraction of a smooth HK manifold that is birational to a Lagrangian HK manifold, and the involution on the singular variety comes from an antisymplectic involution on the smooth HK manifold whose \((+1)\)-eigenspace in \(H^2\) has rank \(2\). The authors carry out the study of the fixed loci through an analysis of the fiberwise involutions of Lagrangian HK manifolds.