Prime Fano threefolds and integrable systems

Abstract

For a general K3 surface S of genus g, with 2 ≤ g ≤ 10, we prove that the intermediate Jacobians of the family of prime Fano threefolds of genus g containing S as a hyperplane section, form generically an algebraic completely integrable Hamiltonian system.

Hilbert schemes of lines and conics and automorphism groups of Fano threefolds

“…To be honest, most of the information provided by Theorem 1.1.1 can be found in the literature (see [AK77], [DR76], [FN89], [Ten74], [Isk80], [Put82], [Mar81], [Ili03], [IM07], [BF13], etc). Our goal was, in a sense, in collecting all the results together, and cleaning things a bit.…”

Integrable systems from intermediate Jacobians of 5-folds

“…Iliev and Manivel [IM1] introduced a symplectic structure on the relative intermediate Jacobian J = J(V/B) → B of the family of prime Fano threefolds V → B of index 1 extending a fixed K3 surface S, in such a way that J → B turns into a Lagrangian fibration. Here, when we say that a 3-fold V extends S, we mean that S is a hyperplane section of V , and B is understood as a moduli (or universal deformation) space of maps S → V with fixed source S, see [B2] for the particular case of K3-Fano pairs and [Ran] for deformations of general maps.…”

“…The approach of [IM1] is based upon the observation of Tyurin that for a K3-Fano pair S ⊂ V with S ∈ | − K V |, the map sending stable sheaves on V to their restrictions to S under certain hypotheses embeds the moduli spaces of sheaves on V as Lagrangian subvarieties of moduli spaces on S. Tyurin [Tyu] originally observed this for stable vector bundles, and Thomas [Th] extended to the ideal sheaves of curves on V , whose restrictions to S are ideal sheaves of zero-dimensional subschemes. The authors of [IM1] apply this to conics in V and thus obtain a Lagrangian immersion F(V ) → S [2] , where F(V ) is the Fano surface of V , parametrizing conics in V , and S [2] is the Fujiki-Beauville symplectic 4-fold [B1], or the Hilbert scheme of 0-dimensional subschemes of S of length 2.…”

An integrable system of K3-Fano flags

“…Thus it is an open part of a torsor under the relative intermediate Jacobian J → F R S . The authors of [12] introduced a Lagrangian structure on J (V/F R S ) for 3 ≤ g ≤ 10 over an open subset of F R S , where S is assumed generic and R = Zρ. Their construction is a generalization of that of [8]: J (V/F R S ) is identified with the relative Picard variety of a family of Lagrangian subvarieties of S [2] , and the relative Picard variety is itself Lagrangian by a result of Donagi-Markman [5].…”

“…The approaches of [1,12] are based upon the observation of Tyurin that for a K3-Fano flag (S, V ), the map sending stable sheaves on V to their restrictions to S under certain hypotheses embeds the moduli spaces of sheaves on V as Lagrangian subvarieties of moduli spaces on S. Tyurin [19] originally stated it for moduli of stable vector bundles, and Thomas [18] extended to the restriction map Hilb t →dt+c (V ) → S [d] defined on sufficiently nice degree-d curves in V ; the authors of [12] reproved the result of Thomas and applied it to conics, that is for d = 2.…”