Higher-dimensional analogues of \(K3\) surfaces (Q2918463)

A compact Kähler manifold \(X\) is irreducible hyperkähler (also known as irreducible holomorphic symplectic) if it is simply connected and the space \(H^0(X, \Omega^2_X)\) is generated by a holomorphic symplectic form on \(X\). Hyperkähler manifolds are a natural generalization of \(K3\) surfaces and share some of their properties. In complex dimension \(2\) all hyperkähler manifolds are \(K3\) surfaces. Fujiki exhibited the first higher dimensional example of a hyperkähler manifold, namely \(\mathrm{Hilb}^2(K3)\) [\textit{A. Fujiki}, Prog. Math. 39, 71--250 (1983; Zbl 0549.32018)]. In every complex dimension \(2n>2\) there are at least two deformation classes of hyperkähler manifolds: Hilbert schemes of points on a \(K3\) surface \(\mathrm{Hilb}^n(K3)\) and the generalized Kummer varieties \(K^{n+1}(A)\). These are the two standard serries of examples due to \textit{A. Beauville} [J. Differ. Geom. 18, 755--782 (1983; Zbl 0537.53056)]. There are two known exceptional examples due to \textit{K. G. O'Grady} in dimensions 6 and 10 [J. Algebr. Geom. 12, No. 3, 435--505 (2003; Zbl 1068.53058); J. Reine Angew. Math. 512, 49--117 (1999; Zbl 0928.14029)] which are not deformation equivalent to Beauville's examples. The author presents a program whose goal is to prove that a numerical Hilbert square is deformation equivalent to \(\mathrm{Hilb}^2(K3)\). Let \(X\) be a generic deformation of a numerical Hilbert square with a nontrivial line bundle \(L\) on \(X\) with Beauville-Bogomolov form \(q_X(c_1(L))=2\) and such that \(H^{1,1}_\mathbb{Z} = \mathbb{Z} c_1(L)\). O'Grady studies the map \(f: X \dashrightarrow |L|^\vee\). He proves that either \(f\) is the natural cover of an EPW-sextic or it is birational onto its image [Commun. Contemp. Math. 10, No. 4, 553--608 (2008; Zbl 1216.14040)]. The author conjectures that the only case that holds is the former one. If this is true then his program would be complete, i.e., every numerical Hilbert square would be a deformation of a \(\mathrm{Hilb}^2(K3)\).NEWLINENEWLINEFor the entire collection see [Zbl 1242.14002].