The monodromy of generalized Kummer varieties and algebraic cycles on their intermediate Jacobians

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