Secant sheaves and Weil classes

We will describe a general strategy for proving the algebraicity of the Hodge Weil classes on abelian varieties of Weil type. The latter are even dimensional abelian varieties admitting a suitable embedding of a CM number field in their rational endomorphism ring. We will describe the implementation of the strategy for abelian varieties of dimension 4 and 6, and why it implies the Hodge conjecture for abelian varieties of dimension at most 5.