An inequality for Betti numbers of hyper-Kähler manifolds of dimension 6

Abstract

A compact hyper-Kahler manifold M for which π1(M) = 0 and H2,0(M) = C is said to be simple (irreducible). According to Bogomolov’s celebrated theorem, any compact hyper-Kahler manifold admits a finite covering by a product of the torus and several irreducible hyper-Kahler manifolds [1]. In complex dimensions 4 and higher, two series of hyper-Kahler manifolds are known, the Hilbert schemes of n points over K3 and the generalized Kummer manifolds [2]; in addition, there are two sporadic examples due to O’Grady (see [3], [4]). It has been proved that the moduli spaces of vector bundles with numerical parameters different from those in the examples mentioned above admit no symplectic resolution [5].