2013/09/26 by Rhys Davies, Davies, Rhys · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #FOS: Physical sciences #Geometric and Algebraic Topology #Geometry and complex manifolds #High Energy Physics - Theory (hep-th)
paper · pdf · doi:10.48550/arxiv.1309.6778
openalex publication_date 2013/09/26 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
This paper is a detailed study of a class of isolated Gorenstein threefold\nsingularities, called hyperconifolds, that are finite quotients of the\nconifold. First, it is shown that hyperconifold singularities arise naturally\nin limits of smooth, compact Calabi--Yau threefolds (in particular), when the\ngroup action on the covering space develops a fixed point. The\nZn-hyperconifolds---those for which the quotient group is cyclic---are\nclassified, demonstrating a one-to-one correspondence between these\nsingularities and three-dimensional lens spaces L(n,k), which occur as the\nvanishing cycles. The classification is constructive, and leads to a simple\nproof that a Zn-hyperconifold is mirror to an n-nodal variety. It is then\nargued that all factorial Zn-hyperconifolds have crepant, projective\nresolutions, and this gives rise to transitions between smooth compact\nCalabi--Yau threefolds, which are mirror to certain conifold transitions.\nFormulae are derived for the change in both fundamental group and Hodge numbers\nunder such hyperconifold transitions.\n Finally, a number of explicit examples are given, to illustrate how to\nconstruct new Calabi--Yau manifolds using hyperconifold transitions, and also\nto highlight the differences which can occur when these singularities occur in\nnon-factorial varieties.\n