Primitive contractions of Calabi-Yau threefolds I

Abstract

We construct examples of primitive contractions of Calabi--Yau threefolds with exceptional locus being ${\mathbb{P}}^1\times {\mathbb{P}}^1$, ${\mathbb{P}}^2$, and smooth del Pezzo surfaces of degrees $\le 5$. 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.

23 pages, the numeration of subarticles changed