Abstract
We prove that Lagrangian fibrations on projective hyper-Kahler 2n-folds with maximal Mumford-Tate group satisfy Matsushita’s conjecture, namely the generic rank of the period map for the fibers of such a fibration is either 0 or maximal (that is n). We establish for this a universal property of the Kuga-Satake variety associated to a K3-type Hodge structure with maximal Mumford-Tate group.
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