### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 097, 18 pages      arXiv:2108.02087      https://doi.org/10.3842/SIGMA.2022.097

### Weil Classes and Decomposable Abelian Fourfolds

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

Received May 11, 2022, in final form December 06, 2022; Published online December 13, 2022

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.

Key words: abelian varieties; Hodge classes.

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