Birational automorphism groups of projective varieties of Picard number two

TL;DR

This paper extends results on automorphism groups and the Morrison-Kawamata cone conjecture from Calabi-Yau manifolds to broader classes of singular varieties with Picard number two, revealing structural properties of their automorphism groups.

Contribution

It generalizes previous results to singular varieties and klt Calabi-Yau pairs, showing automorphism groups are either almost cyclic or have finitely many components.

Findings

Automorphism groups of certain singular varieties are almost cyclic.

Automorphism groups of these varieties have finitely many connected components.

Extension of cone conjecture results to broader classes of varieties.

Abstract

We slightly extend a result of Oguiso on birational or automorphism groups (resp. of Lazi\'c - Peternell on Morrison-Kawamata cone conjecture) from Calabi-Yau manifolds of Picard number two to arbitrary singular varieties X (resp. to klt Calabi-Yau pairs in broad sense) of Picard number two. When X has only klt singularities and is not a complex torus, we show that either Aut(X) is almost cyclic, or it has only finitely many connected components.