Cubic hypersurfaces and integrable systems

Together with the cubic and quartic threefolds, the cubic fivefolds are the only hypersurfaces of odd dimension bigger than one for which the intermediate Jacobian is a nonzero principally polarized abelian variety (p.p.a.v.). In this paper we show that the family of $21$-dimensional intermediate Jacobians of cubic fivefolds containing a given cubic fourfold $X$ is generically an algebraic integrable system. In the proof we apply an integrability criterion, introduced and used by Donagi and Markman to find a similar integrable system over the family of cubic threefolds in $X$. To enter in the conditions of this criterion, we write down explicitly the symplectic structure, known by Beauville and Donagi, on the family $F(X)$ of lines on the general cubic fourfold $X$, and prove that the family of planes on a cubic fivefold containing $X$ is embedded as a Lagrangian surface in $F(X)$. By a symplectic reduction we deduce that our integrable system induces on the nodal boundary another integrable system, interpreted generically as the family of $20$-dimensional intermediate Jacobians of Fano threefolds of genus four contained in $X$. Along the way we prove an Abel-Jacobi type isomorphism for the Fano surface of conics in the general Fano threefold of genus four, and compute the numerical invariants of this surface.