Fano manifolds of degree ten and EPW sextics

Iliev, Atanas; Manivel, Laurent

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 44 (2011) no. 3, pp. 393-426.

O’Grady a démontré que certaines sextiques spéciales dans ${\mathbb{P}}^5$, les sextiques EPW, admettent pour revêtements doubles des variétés symplectiques holomorphes lisses. Nous proposons une nouvelle approche de ces variétés symplectiques, en montrant qu’elles se construisent à partir des schémas de Hilbert de coniques sur des variétés de Fano de dimension quatre et de degré dix. En guise d’application, nous construisons des familles de surfaces lagrangiennes dans ces variétés symplectiques, puis des systèmes intégrables dont les fibres sont des jacobiennes intermédiaires.

O’Grady showed that certain special sextics in ${\mathbb{P}}^5$ called EPW sextics admit smooth double covers with a holomorphic symplectic structure. We propose another perspective on these symplectic manifolds, by showing that they can be constructed from the Hilbert schemes of conics on Fano fourfolds of degree ten. As applications, we construct families of Lagrangian surfaces in these symplectic fourfolds, and related integrable systems whose fibers are intermediate Jacobians.

Classification : 14C05, 14J35, 14J45, 14M15, 37K20, 53C26
Keywords: holomorphic symplectic manifold, Fano manifold, grassmannian, Hilbert scheme, conic, double cover, lagrangian surface, integrable system
Mot clés : variété symplectique holomorphe, variété de Fano, grassmannienne, schéma de Hilbert, conique, revêtement double, surface lagrangienne, système intégrable

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