Automorphism Groups of Smooth Cubic Fourfolds through Lattice Theory

Abstract

Laza and the second author have classified all possible symplectic automorphism groups of smooth cubic fourfolds. There are $34$ such groups, and by Koike, there are $48$ families of cubic fourfolds with speficied symplectic automorphism group. For $42$ among those families, we complete the classification of all possible automorphism groups. The approach of this paper is mainly lattice theoretic, starting from known results about the period map of cubic fourfolds by Voisin, Hassett, Looijenga and Laza. We use a lattice-enumeration algorithm implemented in OSCAR.