Higher-dimensional analogues of K3 surfaces

A K\"ahler manifold $X$ is {\it hyperk\"ahler} (HK) if it is simply connected and it carries a holomorphic {\bf symplectic} form whose cohomology class spans $H^{2,0}(X)$. A HK manifold of dimension $2$ is a $K3$ surface. In many respects higher-dimensional HK 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 HK manifolds. One deformation class of dimension $2n$ is that of the Hilbert scheme $S^{[n]}$ 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 $HK$ $4$-fold $4$ such that there is an isomorphism of abelian groups $H^2(X;\ZZ)\overset{\sim}{\to} H^2(K3^{[2]};\ZZ)$ compatible with the polynomials given by $4$-tuple cup-product.