Examples of Hyperkähler Manifolds as Moduli Spaces of Sheaves on K3 Surfaces

A compact Kähler surface X is a K3 surface if it is simply connected and it carries a global homolorphic symplectic form (i.e. the canonical bundle KX ∼= OX). An example is given by the Fermat quartic: consider the degree four polynomial P (X0, ..., X3) = X 4 0 + X 4 1 + X 4 2 + X 4 3 ∈ C[X0, ..., X3]. The vanishing locus S = V (P ) is an irreducible quartic hypersurface in PC, which is simply connected by the Lefschetz Hyperplane Theorem, and has canonical bundle KS = (OP3(−4) ⊗ OP3(4))|S ∼= OS by adjunction. Hence, the surface S is a K3 surface and, by applying the same reasoning verbatim, every irreducible quartic hypersurface in PC is. K3 surfaces play a fundamental role in the classification of algebraic surfaces, hence it is natural to look for generalizations in higher dimensions. The following (beautiful) classification theorem motivates the definition of a hyperkähler manifold (HK):