Ricerc@Sapienza

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.