Weil Classes and Decomposable Abelian Fourfolds

Bert van Geemen

Dipartimento di Matematica, Università di Milano, Via Saldini 50, I-20133 Milano, Italy

Abstract: We determine which codimension two Hodge classes on $J\times J$, where $J$ is a general abelian surface, deform to Hodge classes on a family of abelian fourfolds of Weil type. If a Hodge class deforms, there is in general a unique such family. We show how to determine the imaginary quadratic field acting on the fourfolds of Weil type in this family as well as their polarization. There are Hodge classes that may deform to more than one family. We relate these to Markman's Cayley classes.

Keywords: abelian varieties, Hodge classes.

MSC: 14C30, 14C25, 14K20

Received: May 11, 2022; in final form December 6, 2022; Published online December 13, 2022

Language: English

DOI: 10.3842/SIGMA.2022.097

Bibliographic databases:

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ArXiv: 2108.02087