Automorphism groups of Calabi-Yau manifolds 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,

arXiv comment17 printed pages, dedication is added, Proposition(5.1) due to the referee is added, The first part of proof of Proposition (6.1) is corrected and several typos are corrected

References in corpus (4)

• A survey of Torelli and monodromy results for holomorphic-symplectic varieties
• A global Torelli theorem for hyperkahler manifolds
• Bridgeland's stability and the positive cone of the moduli spaces of stable objects on an abelian surface
• A global Torelli theorem for hyperkaehler manifolds (after Verbitsky)

Cited by in corpus (6)

• Bridgeland's stability and the positive cone of the moduli spaces of stable objects on an abelian surface
• Some aspects of explicit birational geometry inspired by complex dynamics
• On the Cone conjecture for Calabi-Yau manifolds with Picard number two
• Movable vs Monodromy Nilpotent Cones of Calabi-Yau Manifolds
• The simplicity of the first spectral radius of a meromorphic map
• Around and beyond the canonical class