[Paper Review] Classification and Properties of Hyperconifold Singularities and Transitions

Rhys Davies|arXiv (Cornell University)|Sep 26, 2013

Algebraic Geometry and Number Theory23 references3 citations

TL;DR

This paper classifies $π_{n}$-hyperconifold singularities as finite quotients of the conifold, establishing a one-to-one correspondence with lens spaces $L(n,k)$, and demonstrates that these singularities admit crepant projective resolutions. The key contribution is a constructive classification proving that $π_{n}$-hyperconifolds are mirror to $n$-nodal Calabi-Yau threefolds, with explicit formulae for changes in Hodge numbers and fundamental group under hyperconifold transitions.

ABSTRACT

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.

Motivation & Objective

• To classify $π_{n}$-hyperconifold singularities as finite quotients of the conifold and establish their topological and mirror symmetry properties.
• To demonstrate that these singularities arise naturally in limits of smooth Calabi-Yau threefolds when a group action develops a fixed point.
• To prove that all factorial $π_{n}$-hyperconifolds admit crepant, projective resolutions, enabling transitions between smooth Calabi-Yau threefolds.
• To derive explicit formulae for changes in Hodge numbers and fundamental group under hyperconifold transitions.
• To provide explicit examples, including non-factorial cases, to illustrate the differences in resolution behavior and topological transitions.

Proposed method

• Construct a classification of $π_{n}$-hyperconifold singularities via group actions on the conifold, showing a one-to-one correspondence with lens spaces $L(n,k)$.
• Use the holomorphic $(3,0)$-form and symmetry analysis to identify allowed group actions that yield isolated singularities.
• Apply the theory of crepant resolutions to show that factorial $π_{n}$-hyperconifolds admit projective crepant resolutions.
• Derive formulae for changes in Hodge numbers $h^{1,1}$ and $h^{2,1}$, and the fundamental group, under hyperconifold transitions.
• Use local ample/Kähler cone analysis to determine the existence of projective resolutions in the local setting.
• Construct explicit examples, including the $π_{5}$-quotient of the quintic and $π_{3}$-hyperconifolds, to illustrate transitions and non-factorial behavior.

Experimental results

Research questions

• RQ1 What is the complete classification of $π_{n}$-hyperconifold singularities as quotients of the conifold?
• RQ2 How do hyperconifold transitions affect the fundamental group and Hodge numbers of Calabi-Yau threefolds?
• RQ3 Under what conditions do $π_{n}$-hyperconifolds admit crepant, projective resolutions?
• RQ4 What distinguishes the topological and geometric behavior of factorial versus non-factorial $π_{n}$-hyperconifolds?
• RQ5 Can hyperconifold transitions be used to construct new Calabi-Yau manifolds with specific Hodge numbers and fundamental groups?

Key findings

• There is a one-to-one correspondence between $π_{n}$-hyperconifold singularities and three-dimensional lens spaces $L(n,k)$, established through a constructive classification.
• A $π_{n}$-hyperconifold is mirror to an $n$-nodal Calabi-Yau threefold, with a simple proof derived from the classification.
• All factorial $π_{n}$-hyperconifolds admit crepant, projective resolutions, enabling smooth transitions between Calabi-Yau threefolds.
• The change in Hodge numbers under a hyperconifold transition is given by $\Delta h^{1,1} = n$ and $\Delta h^{2,1} = 0$, while the fundamental group changes by a factor of $\mathbb{Z}_n$.
• In non-factorial cases, such as the $π_{3}$-hyperconifold, resolutions may not be projective, and the Euler characteristic change is $+24$ for four singularities resolved via $π_{3}$-hyperconifold transitions.
• An exceptional $π_{4}$-hyperconifold singularity exists that does not arise from a free group action limit, due to non-trivial projection onto the $π_{2}$ symmetry factor.