Dedicated to Piotr Pragacz on the occasion of his 60th birthday.Abstract. We construct a new 20-dimensional family of projective 6-dimensional irreducible holomorphic symplectic manifolds. The elements of this family are deformation equivalent with the Hilbert scheme of three points on a K3 surface and are constructed as natural double covers of special codimension 3 subvarieties of the Grassmanian G(3, 6). These codimension 3 subvarieties are defined as Lagrangian degeneracy loci and their construction is parallel to that of EPW sextics, we call them the EPW cubes. As a consequence we prove that the moduli space of polarized IHS sixfolds of K3-type, Beauville-Bogomolov degree 4 and divisibility 2 is unirational.

Abstract. We construct several new examples of Calabi-Yau threefolds with Picard group of rank 1. Each of these examples is obtained by smoothing the image of a primitive contraction with exceptional divisor being a del Pezzo surface of degree 4, 5, 6, 7 or P 1 × P 1 .

Abstract. We study the syzygetic structure of projections of del Pezzo surfaces in order to construct singular Calabi-Yau threefolds. By smoothing those threefolds, we obtain new examples of Calabi-Yau threefolds with Picard group of rank 1. We also give an example of type II primitive contraction whose exceptional divisor is the blow-up of the projective plane at a point.

We construct examples of primitive contractions of Calabi-Yau threefolds with exceptional locus being P 1 ×P 1 , P 2 , and smooth del Pezzo surfaces of degrees ≤ 5. We describe the images of these primitive contractions and find their smoothing families. In particular, we give a method to compute the Hodge numbers of a generic fiber of the smoothing family of each Calabi-Yau threefold with one isolated singularity obtained after a primitive contraction of type II. As an application, we get examples of natural conifold transitions between some families of Calabi-Yau threefolds.

Abstract. We study Kustin-Miller unprojections of Calabi-Yau threefolds. As an application we work out the geometric properties of Calabi-Yau threefolds defined as linear sections of determinantal varieties. We compute their Hodge numbers and describe the morphisms corresponding to the faces of their Kähler-Mori cone.