[Paper Review] Automorphism groups of Calabi-Yau manifolds of Picard number two

Keiji Oguiso|arXiv (Cornell University)|Jun 8, 2012

Algebraic Geometry and Number Theory27 references17 citations

TL;DR

This paper proves that the automorphism group of an odd-dimensional Calabi-Yau manifold with Picard number two is always finite, contrasting sharply with the behavior of K3 surfaces and hyperkähler manifolds. The result hinges on analyzing the nef cone and the absence of a specific quadratic form relation on the Néron-Severi group, with additional results on birational automorphism groups and cone conjectures for Calabi-Yau threefolds of Picard number two.

ABSTRACT

We prove that the automorphism group of an odd dimensional Calabi-Yau manifold of Picard number two is always a finite group. This makes a sharp contrast to the automorphism groups of K3 surfaces and hyperkähler manifolds and birational automorphism groups, as we shall see. We also clarify the relation between finiteness of the automorphism group (resp. birational automorphism group) and the rationality of the nef cone (resp. movable cone) for a hyperkähler manifold of Picard number two. We will also discuss a similar conjectual relation together with exsistence of rational curve, expected by the cone conjecture, for a Calabi-Yau threefold of Picard number two,

Motivation & Objective

• To determine the structure of the automorphism group for Calabi-Yau manifolds of Picard number two.
• To clarify the relationship between the rationality of the nef and movable cones and the finiteness of automorphism or birational automorphism groups in hyperkähler manifolds of Picard number two.
• To investigate the expected relation between rational curves and cone conjectures for Calabi-Yau threefolds of Picard number two.
• To construct an explicit example of a Calabi-Yau threefold with Picard number two whose birational automorphism group is infinite.

Proposed method

• Using the Néron-Severi group and the intersection form on the Picard lattice to analyze the geometry of the nef cone.
• Applying the Fujiki relation and the Beauville-Bogomolov form to detect the absence of a specific quadratic form structure.
• Employing Markman’s solution to the weak movable cone conjecture and Huybrechts-Verbitsky global Torelli theorems for hyperkähler manifolds.
• Using Kawamata’s theorem on decomposition of birational maps into flops to describe the birational automorphism group.
• Performing explicit matrix calculations on the action of flops on the Néron-Severi group to show infinite order of certain automorphisms.
• Constructing a concrete Calabi-Yau threefold via moduli spaces of stable objects on abelian surfaces to realize irrational movable cones.

Experimental results

Research questions

• RQ1 Is the automorphism group of an odd-dimensional Calabi-Yau manifold of Picard number two always finite?
• RQ2 How does the rationality of the nef cone relate to the finiteness of the automorphism group in hyperkähler manifolds of Picard number two?
• RQ3 Can the movable cone conjecture for Calabi-Yau threefolds of Picard number two be linked to the existence of rational curves?
• RQ4 Does the absence of a quadratic form relation of the form $(x^n)_X = c(q_X(x))^{n/2}$ imply finiteness of the automorphism group in even-dimensional Calabi-Yau manifolds of Picard number two?
• RQ5 Can an explicit Calabi-Yau threefold of Picard number two be constructed with an infinite birational automorphism group?

Key findings

• The automorphism group of an odd-dimensional Calabi-Yau manifold of Picard number two is always finite.
• For even-dimensional Calabi-Yau manifolds of Picard number two, the automorphism group is finite if no real quadratic form $q_X(x)$ satisfies $(x^n)_X = c(q_X(x))^{n/2}$.
• In hyperkähler manifolds of Picard number two, the automorphism group is finite if and only if both boundary rays of the nef cone are rational.
• The birational automorphism group of a hyperkähler manifold of Picard number two is infinite if and only if both boundary rays of the movable cone are irrational.
• An explicit Calabi-Yau threefold of Picard number two with an infinite birational automorphism group is constructed, where the movable cone has irrational boundary rays.
• The action of the birational automorphism group on the nef cone has a finite rational polyhedral fundamental domain, supporting the cone conjecture.