Surface decompositions of hyper-Kähler manifolds

In recent years, new constructions of complete families of polarized hyper-Kähler manifolds have been found starting from Fano geometry. These hyper-Kähler manifolds also appear as deformations of Hilbert schemes of K3 surfaces or O'Grady manifolds. I will introduce the notion of surface decomposition, which is restrictive and gives a geometric explanation for the Beauville-Fujiki relations, and I will show the existence of such a surface decomposition for the general hyper-Kähler manifold mentioned above. This has implications on the Beauville conjecture.