The dual Lagrangian fibration of known hyper-Kähler manifolds

Abstract

Given a Lagrangian fibration π : X → P n of a compact hyper-Kähler manifold of K3 [n] , Kumn, OG10 or OG6-type, we construct a natural compactification of its dual torus fibration. Specifically, this compactification is given by a quotient of X by Aut • (X/P n ), the group of automorphisms acting trivially on the second cohomology and respecting the Lagrangian fibration. It is a compact hyper-Kähler orbifold with identical period mapping behavior as X.

On fibrations and measures of irrationality of hyper-Kähler manifolds

“…The last result is easy and follows from Beauville's argument in [5] if we know that the group scheme G, which is constructed under some assumptions in [2], [26] and [27], acts on X.…”