  We compute the subgroup of the monodromy group of a generalized Kummer
variety associated to equivalences of derived categories of abelian surfaces.
The result was previously announced in arXiv:1201.0031. Mongardi showed that
the subgroup constructed here is in fact the whole monodromy group. As an
application we prove the Hodge conjecture for the generic abelian fourfold of
Weil type with complex multiplication by an arbitrary imaginary quadratic
number field K, but only for polarizations with discriminant 1. The latter
result is inspired by a recent observation of O'Grady that the third
intermediate Jacobians of smooth projective varieties of generalized Kummer
deformation type form complete families of abelian fourfolds of Weil type.
Finally, we prove the surjectivity of the Abel-Jacobi map from the Chow group
of co-dimension two algebraic cycles homologous to zero on every projective
irreducible holomorphic symplectic manifold Y of Kummer type onto the third
intermediate Jacobian of Y, as predicted by the generalized Hodge Conjecture.
