Automorphisms and Periods of Cubic Fourfolds

Abstract

We classify the symplectic automorphism groups for cubic fourfolds. The main inputs are the global Torelli theorem for cubic fourfolds and the classification of the fixed-point sublattices of the Leech lattice. Among the highlights of our results, we note that there are 34 possible groups of symplectic automorphisms, with 6 maximal cases. The six maximal cases correspond to 8 non-isomorphic, and isolated in moduli, cubic fourfolds; six of them previously identified by other authors. Finally, the Fermat cubic fourfold has the largest possible order (174, 960) for the automorphism group (non-necessarily symplectic) among all smooth cubic fourfolds.

Automorphism groups of cubic fourfolds and K3 categories

“…Assume that G is not isomorphic to the trivial group 1 or the cyclic group Z/2Z of order two. By Theorem 1.2 in [27], we have rk(S G (X)) ≥ 12. By Lemma 8.2, we obtain the inequality ρ(D X ) ≥ 12.…”

“…By Theorem 1.8 (1) in [27], the Fermat cubic fourfold X is the unique cubic fourfold with a symplectic automorphism f 9 of order 9. By Theorem 8.4, there is the unique K3 surface S such that D X ≃ D b (S).…”

“…The third result in our paper is about the relation between the existence of non-trivial symplectic automorphisms of cubic fourfolds and the existence of associated K3 surfaces (cf. [25], [27]).…”

“…The proof relies on the classification of symplectic automorphism groups of cubic fourfolds in [27] and the lattice theoretic technique in [31] and [29]. Using Theorem 1.…”

“…This is the triple cover of P 4 branched along the Klein cubic threefold in [3]. By Theorem 1.8 ( 5) in [27], the Klein cubic fourfold X is the unique cubic fourfold with a symplectic automorphism f 11 of order 11 and its symplectic automorphism group is the finite simple group L 2 (11) (cf. [3]).…”