Classification and Properties of Hyperconifold Singularities and Transitions

This paper is a detailed study of a class of isolated Gorenstein threefold singularities, called hyperconifolds, that are finite quotients of the conifold. First, it is shown that hyperconifold singularities arise naturally in limits of smooth, compact Calabi–Yau threefolds (in particular), when the group action on the covering space develops a fixed point. The Zn-hyperconifolds—those for which the quotient group is cyclic—are classified, demonstrating a one-to-one correspondence between these singularities and three-dimensional lens spaces L(n, k), which occur as the vanishing cycles. The classification is constructive, and leads to a simple proof that a Zn-hyperconifold is mirror to an n-nodal variety. It is then argued that all factorial Zn-hyperconifolds have crepant, projective resolutions, and this gives rise to transitions between smooth compact Calabi–Yau threefolds, which are mirror to certain conifold transitions. Formulae are derived for the change in both fundamental group and Hodge numbers under such hyperconifold transitions. Finally, a number of explicit examples are given, to illustrate how to construct new Calabi–Yau manifolds using hyperconifold transitions, and also to highlight the differences which can occur when these singularities occur in non-factorial varieties