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.

Key Words:

• Calabi–Yau threefolds
• Conifold transitions
• del Pezzo surfaces

2000 Mathematics Subject Classification:

• 14J32
• 14J10
• 14J30

ACKNOWLEDGMENTS

We would like to thank S. Cynk for his constant help. Some ideas in this article grew up during our stay at the EAGER training site at Warwick University in the spring of 2004. We would like to express our gratitude to M. Reid for mathematical inspiration. We thank M. Gross, A. Langer, V. Nikulin, P. Pragacz, and J. Wiśniewski for answering questions. We acknowledge S. Kleiman for help with the redaction.

The authors are scholars of the project which is co-financed from the European Social Fund and national budget in the frame of The Integrated Regional Operational Programme.

Notes

Communicated by S. Kleiman.