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

Abstract

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 announced by Markman and Mehrotra (2017). Mongardi (2016) 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 with trivial discriminant invariant in ${\mathbb{Q}}^{\ast }/\mathrm{Nm}(K^{\ast })$. 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 ${\mathrm{CH}}^2(Y{)}_0$ of codimension 2 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.