Moduli spaces of irreducible symplectic manifolds

We study the moduli spaces of polarised irreducible symplectic manifolds. By a comparison with locally symmetric varieties of orthogonal type of dimension 20, we show that the moduli space of 2d polarised (split type) symplectic manifolds which are deformation equivalent to degree 2 Hilbert schemes of a K3 surface is of general type if d is at least 12.
Comment: Exposition improved, Reference to work of Debarre and Voisin added, Corollary 1.6 removed