Cubic hypersurfaces and integrable systems

• Autores: Atanas Iliev, Laurent Manivel
• Localización: American journal of mathematics, ISSN 0002-9327, Vol. 130, Nº 6, 2008, págs. 1445-1475
• Idioma: inglés
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• Resumen

  • 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.