Non-symplectic Indices of Automorphism Groups of Smooth Cubic Fourfolds

Abstract

We study the full automorphism groups of smooth cubic fourfolds with prescribed symplectic automorphism group. Our starting point is the classification of symplectic automorphism groups by Laza and Zheng. We focus on the non-symplectic index, namely, the index of the symplectic automorphism group in the full automorphism group. We prove general restrictions on this index. We also compute bounds by group-theoretic and lattice-theoretic methods. In several cases, we determine all possible indices. For coinvariant lattices of rank 19, we classify all possible pairs consisting of the symplectic automorphism group and the full automorphism group.