  In this paper we define the notion of a hyperk\"ahler manifold (potentially)
of Jacobian type. If we view hyperk\"ahler manifolds as "abelian varieties",
then those of Jacobian type should be viewed as "Jacobian varieties". Under a
minor assumption on the polarization, we show that a very general polarized
hyperk\"ahler fourfold $F$ of $K3^{[2]}$-type is not of Jacobian type. As a
potential application, we conjecture that if a cubic fourfold is rational then
its variety of lines is of Jacobian type. Under some technical assumption, it
is proved that the variety of lines on a rational cubic fourfold is potentially
of Jacobian type. We also prove the Hodge conjecture in degree 4 for a generic
$F$ of $K3^{[2]}$-type.
