Hodge numbers of a Fano eightfold of K3 type

Abstract: We construct an explicit semistable degeneration of a Fano eightfold of index three and deduce its Hodge numbers, in particular we show that it has Picard rank one. The Fano variety is of K3 type and it is defined as a connected component of the fixed locus of a suitable antisymplectic involution on a projective variety that is deformation equivalent to the Hilbert scheme of eight points on a K3 surface. We also obtain a description of a projective model of the Hilbert square of a K3 surface of genus eight in terms of secant lines to the surface.