We prove that Lagrangian fibrations on projective hyper-Kähler 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 (i.e., 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.
On a conjecture of Matsushita / L. Van Geemen, C. Voisin. - In: INTERNATIONAL MATHEMATICS RESEARCH NOTICES. - ISSN 1073-7928. - 2016:10(2016), pp. 3111-3123. [10.1093/imrn/rnv230]
On a conjecture of Matsushita
L. Van Geemen;
2016
Abstract
We prove that Lagrangian fibrations on projective hyper-Kähler 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 (i.e., 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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