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Hyperkahler manifolds of Jacobian type

In this paper we define the notion of a hyperkähler manifold (potentially) of Jacobian type. If we view hyperkähler 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 hyperkähler 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.