Automorphism groups of Calabi-Yau manifolds of Picard number 2

Abstract

We prove that the automorphism group of an odd dimensional Calabi-Yau manifold of Picard number 2 2 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 2 2 . We will also discuss a similar conjectual relation together with the exsistence of a rational curve, expected by the cone conjecture, for a Calabi-Yau threefold of Picard number 2 2 .