Birational geometry of irreducible holomorphic symplectic tenfolds of O'Grady type

Abstract

In this paper, we analyse the birational geometry of O'Grady ten dimensional manifolds, giving a characterization of Kähler classes and lagrangian fibrations. Moreover, we study symplectic compactifications of intermediate jacobian fibrations of smooth cubic fourfolds.

On the period of Li, Pertusi and Zhao's symplectic variety

“…The generality assumption can conjecturally be made more precise by saying that V does not contain a plane or a rational cubic scroll (see [LPZ20, Section 5.7] for the case of a rational cubic scroll -the case of a plane is expected to behave similarly), and in these two cases one expects to find examples of the walls of the Kähler cone described in [MZ16] (in the singular setting) and [MO20] (in the desingularisation).…”

“…By [LPZ20, Theorem 1.3], M λ (V, τ ) is birational to a twisted LSV manifold. By [Sac20, Corollary 3.9] (see also [MO20,Example 3.3]) it is known that if V is very general (in the sense of Hassett), then the twisted LSV manifold is not birational to the untwisted one. Therefore V must be special, i.e.…”

The dual Lagrangian fibration of known hyper-Kähler manifolds

“…For K3 [n] and Kum n -types the computations are done by [Wie16,Wie18]. For OG10 and OG6, the computations are contained in [MO20] and [MR21], respectively. Theorem 3.20 ([Wie16, Wie18], [MO20], [MR21]).…”

“…For OG10 and OG6, the computations are contained in [MO20] and [MR21], respectively. Theorem 3.20 ([Wie16, Wie18], [MO20], [MR21]). Let π : X → B be a Lagrangian fibered compact hyper-Kähler manifold.…”