Completely integrable projective symplectic 4-dimensional varieties

Families of Liouville 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 fiber is maximal isotropic. This question is studied for families of Jacobians of genus 2 curves in terms of the relative compactified Jacobian and point Hilbert scheme. The question on possible bases of families of Liouville tori is investigated in using Fujita–Kawamata–Viehweg–Kollár results on positivity properties of direct images of relative dualizing sheaves. In the case when the base surface is the projective plane, it is proved that the family of Jacobians is Liouville iff it is the Mukai transform of the Fujiki–Beauville 4-fold built from a hyperelliptic K3 surface.
Bibliography: 44 titles.