Primitive Contractions of Calabi–Yau Threefolds I

Abstract

We construct examples of primitive contractions of Calabi–Yau threefolds with exceptional locus being ℙ1 × ℙ1, ℙ2, 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 familly 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.