标题： 超曲面奇点的分类和性质以及转换 显示英文标题

标题： Classification and Properties of Hyperconifold Singularities and Transitions

摘要： 本文是对一类孤立的Gorenstein三维奇点，称为超锥面（hyperconifolds）的详细研究，这些奇点是锥面的有限商。 首先，表明超锥面奇点在光滑紧致Calabi--Yau三维流形的极限中自然出现（特别是当覆盖空间上的群作用发展出固定点时）。 Z_n-超锥面——即商群为循环群的那些——被分类，展示了这些奇点与三维透镜空间L(n,k)之间的一一对应关系，这些透镜空间作为消失圆周出现。 该分类是构造性的，并导致了一个简单的证明，即Z_n-超锥面与n节点流形是对偶的。 随后论证所有因子Z_n-超锥面都有平展的、投射的解析，这导致了光滑紧致Calabi--Yau三维流形之间的转变，这些转变与某些锥面转变对偶。 推导了在这样的超锥面转变下基本群和Hodge数的变化公式。 最后，给出了一些显式例子，以说明如何使用超锥面转变来构造新的Calabi--Yau流形，同时也突出当这些奇点出现在非因子流形中时可能出现的差异。 显示英文摘要

摘要： 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 Z_n-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 Z_n-hyperconifold is mirror to an n-nodal variety. It is then argued that all factorial Z_n-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.