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Published online by Cambridge University Press: 29 May 2025
A Kähler manifold X is hyperkähler if it is simply connected and carries a holomorphic symplectic form whose cohomology class spans H2, 0(X). A hyperkähler manifold of dimension 2 is a K3 surface. In many respects higherdimensional hyperkähler manifolds behave like K3 surfaces: they are the higher dimensional analogues of K3 surfaces of the title. In each dimension greater than 2 there is more than one deformation class of hyperkähler manifolds. One deformation class of dimension 2n is that of the Hilbert scheme where S is a K3 surface. We will present a program which aims to prove that a numerical K3[2] is a deformation of K3[2]—a numerical K3[2] is a hyperkähler 4-fold 4 such that there is an isomorphism of abelian groups compatible with the polynomials given by 4-tuple cup-product.
0. Introduction
K3 surfaces were known classically as complex smooth projective surfaces whose generic hyperplane section is a canonically embedded curve; an example is provided by a smooth quartic surface in. One naturally encounters K3's in the Enriques-Kodaira classification of compact complex surfaces: they are defined to be compact Kähler surfaces with trivial canonical bundle and vanishing first Betti number. Here are a few among the wonderful properties of K3's:
(1) [Kodaira 1964] Any two K3 surfaces are deformation equivalent — thus they are all deformations of a quartic surface.
(2) The Kähler cone of a K3 surface X is described as follows. Let be one Kähler class and be the set of nodal classes
The Kähler cone is given by
(3) [Piatetski-Shapiro and Shafarevich 1971; Burns and Rapoport 1975; Looijenga and Peters 1980/81] Weak and strong global Torelli hold. The weak version states that two K3 surfaces X, Y are isomorphic if and only if there exists an integral isomorphism of Hodge structures which is an isometry (with respect to the intersection forms), the strong version states that f is induced by an isomorphism if and only if it maps effective divisors to effective divisors.
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