Holomorphic Lagrangian fibrations of toric hyperkahler manifolds

For the sake of hyperk{"a}hler SYZ conjecture, finding holomorphic Lagrangian fibrations becomes an important issue. Toric hyperk{"a}hler manifolds are real dimension $4n$ non-compact hyperk{"a}hler manifolds which are quaternion analog of toric varieties. The $n$ dimensional residue circle action on it admitting a hyperk{"a}hler moment map. We use the complex part of this moment map to construct a holomorphic Lagrangian fibration with generic fiber diffeomorphic to $({\mathbb{C}}^{\ast }{)}^n$, and study the singular fibers.