Families of Lkruville tori on a completely integrable compact complex symplectic manifold are considered as a tool for constructing such manifolds: given a family of n-dimensional tori with degenerations over an n-dimensional. base, find conditions which guarantee the existence of a symplectic structure on this family. such that the generic fibre is,maximal isotropic. This question is studied for families of Jacobians of genus 2 curves in terms of the relative compactified Jacobian arid point Hilbert scheme, The question of possible bases for families of Liouville tori is investigated using results of Fujita-Kawamata-Viehweg'-Kollar on positivity properties of direct images of relative dualizing sheayes. In the case when the base surface is the projective plane, it is proved that the family of Jacobians is Liouville if and only if it is the;Mukai transform of the Fujiki-BeauviUe fourfold built from a hyperelliptic K3 surface.
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