The Mumford-Tate group of the general abelian variety is the full symplectic group. It follows that the only Hodge cycles on such a variety are in the Q-algebra generated by the theta divisor.
However, this does not say much about the Hodge conjecture for the general point of a proper closed subvariety of the moduli of abelian varieties.
In joint work with Indranil Biswas we prove this for subvariety consisting of Prym varieties. These are abelian varieties defined as follows. Let X be a double cover of a smooth projective curve Y ; this induces a “norm” map from the Jacobian of X to the Jacobian of Y . The Prym variety P(X → Y ) of this double cover is defined as the connected component of identity in the kernel of the norm map.
Theorem 13.1. Fix non-negative integers g and r. Let X → Y be the general point in the moduli of coverings of a curve Y of genus g which is branched at 2r points. Then the Mumford-Tate group of P(X → Y ) is the full symplectic group.
As a corollary we obtain the Hodge conjecture for this variety.