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Abstract: A Lagrangian fibration of a compact hyper-Kähler manifold is a generalization of an elliptic fibration of a K3 surface. It is known that an elliptic fibration of a K3 surface is always "self-dual" in a certain sense. This turns out to be not the case for Lagrangian fibrations of higher-dimensional compact hyper-Kähler manifolds. In this talk, I will propose a construction for the dual Lagrangian fibration of all currently known examples of compact hyper-Kähler manifolds, and try to justify this.