Weil Classes and Decomposable Abelian Fourfolds

TL;DR

This paper investigates the deformation of certain Hodge classes on products of abelian surfaces to abelian fourfolds of Weil type, identifying the associated quadratic fields and polarizations, and relating some classes to Markman's Cayley classes.

Contribution

It characterizes which Hodge classes deform to Weil type fourfolds, determines the acting quadratic fields and polarizations, and links specific classes to Markman's Cayley classes.

Findings

Identifies conditions for Hodge class deformation to Weil type fourfolds.

Determines the quadratic fields and polarizations of these fourfolds.

Relates some Hodge classes to Markman's Cayley classes.

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.