New Calabi-Yau Manifolds with Small Hodge Numbers

It is known that many Calabi-Yau manifolds form a connected web. The question of whether all Calabi–Yau manifolds form a single web depends on the degree of singularity that is permitted for the varieties that connect the distinct families of smooth manifolds. If only conifolds are allowed then, since shrinking two-spheres and three-spheres to points cannot affect the fundamental group, manifolds with different fundamental groups will form disconnected webs. We examine these webs for the tip of the distribution of Calabi–Yau manifolds where the Hodge numbers (h, h) are both small. In the tip of the distribution the quotient manifolds play an important role. We generate via conifold transitions from these quotients a number of new manifolds. These include a manifold with χ = −6 that is an analogue of the Tian-Yau manifold and manifolds with an attractive structure that may prove of interest for string phenomenology. We also examine the relation of some of these manifolds to the remarkable Gross-Popescu manifolds that have Euler number zero. ar X iv :0 80 9. 46 81 v1 [ he pth ] 2 6 Se p 20 08