Compact Tori Associated to Hyperkähler Manifolds of Kummer Type

For X a hyperkähler manifold of Kummer type, let J3(X) be the intermediate Jacobian associated to H3(X).We prove that H2(X) can be embedded into H2(J3(X)).We show that there exists a natural smooth quadric Q(X) in the projectivization of H3(X), such that Gauss–Manin parallel transport identifies the set of projectivizations of H2,1(Y), for Y a deformation of X, with an open subset of a linear section of Q+ (X), one component of the variety of maximal linear subspaces of Q(X). We give a new proof of a result of Mongardi restricting the action of monodromy on H2(X). Lastly, we show that if X is projective, then J3(X) is an abelian fourfold of Weil type.