Automorphisms and periods of cubic fourfolds

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 symplec...

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Bibliographic Details
Published inMathematische Zeitschrift Vol. 300; no. 2; pp. 1455 - 1507
Main Authors Laza, Radu, Zheng, Zhiwei
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2022
Springer Nature B.V
Subjects
Automorphisms
Fixed points (mathematics)
Lattices (mathematics)
Mathematics
Mathematics and Statistics
Subgroups
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Summary: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.
ISSN:0025-5874
1432-1823
DOI:10.1007/s00209-021-02810-x