Cycles on abelian 2n-folds of Weil type from secant sheaves on abelian n-folds

TL;DR

This paper proves the algebraicity of Weil classes in certain abelian varieties of dimension four and six, advancing the understanding of the Hodge Conjecture for these varieties through degeneration techniques.

Contribution

It establishes the algebraicity of Weil classes for all abelian fourfolds and sixfolds of Weil type with specific discriminants, extending previous results.

Findings

Weil classes are algebraic for all abelian sixfolds of Weil type with discriminant -1.

Weil classes are algebraic for all abelian fourfolds of Weil type, regardless of discriminant.

The results imply the Hodge Conjecture for these classes in the specified abelian varieties.

Abstract

A. Weil identified a 2-dimensional space of rational classes of Hodge type (n,n) in the middle cohomology of every 2n-dimensional abelian variety with a suitable complex multiplication by an imaginary quadratic number field. These abelian varieties are said to be of Weil type and these Hodge classes are known as Weil classes. We prove that the Weil classes are algebraic for all abelian sixfold of Weil type of discriminant -1, for all imaginary quadratic number fields. The algebraicity of the Weil classes follows for all abelian fourfolds of Weil type (for all discriminants and all imaginary quadratic number fields), by a degeneration argument of C. Schoen. The Hodge Conjecture for abelian fourfolds is known to follow from the above result.