Holomorphic Lagrangian fibrations of toric hyperkahler manifolds

Abstract

For the sake of hyperk{ä}hler SYZ conjecture, finding holomorphic Lagrangian fibrations becomes an important issue. Toric hyperk{ä}hler manifolds are real dimension $4n$ non-compact hyperk{ä}hler manifolds which are quaternion analog of toric varieties. The $n$ dimensional residue circle action on it admitting a hyperk{ä}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.

9 pages, draft version

References in corpus (3)

• Lectures on complex geometry, Calabi-Yau manifolds and toric geometry
• A survey of hypertoric geometry and topology
• Chern-Simons theory, the 1/N expansion, and string theory