We determine which codimension two Hodge classes on J×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.
Weil Classes and Decomposable Abelian Fourfolds / L. VAN GEEMEN. - In: SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS. - ISSN 1815-0659. - 18:(2022), pp. 97.1-97.18. [10.3842/sigma.2022.097]
Weil Classes and Decomposable Abelian Fourfolds
L. VAN GEEMEN
2022
Abstract
We determine which codimension two Hodge classes on J×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.
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