Mirror symmetry for Pfaffian Calabi-Yau 3-folds via conifold transitions

In this note we construct conifold transitions between several Calabi-Yau threefolds given by Pfaffians in weighted projective spaces and Calabi-Yau threefolds appearing as complete intersections in toric varieties. We use the obtained results to predict mirrors following ideas of \cite{BCKS, Batsmalltoricdegen}. In particular we consider the family of Calabi--Yau threefolds of degree 25 in $\mathbb{P}^9$ obtained as a transverse intersection of two Grassmannians in their Plucker embeddings.Comment: 11 pages, minor change

Projections of del Pezzo surfaces and Calabi--Yau threefolds

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.Comment: A table of known Calabi--Yau threefolds with Picard number 1 is added, to appear in Advances in Geometr

Primitive contractions of Calabi-Yau threefolds II

We construct 16 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 6, 7 or $\mathbb{P}^1\times \mathbb{P}^1$.Comment: 20 pages, to appear in JLM

EPW Cubes

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.Comment: minor corrections, 25 pages, to appear in J. Reine Angew. Mat

On IHS fourfolds with b2=23b_2=23

The present work is concerned with the study of four-dimensional irreducible holomorphic symplectic manifolds with second Betti number $23$. We describe their birational geometry and their relations to EPW sextics.Comment: to appear in Michigan Math.