Holomorphic Lagrangian fibrations of toric hyperkahler manifolds

TL;DR

This paper constructs holomorphic Lagrangian fibrations on toric hyperkähler manifolds using the complex part of the hyperkähler moment map, advancing understanding of the hyperkähler SYZ conjecture.

Contribution

It introduces a method to produce holomorphic Lagrangian fibrations on toric hyperkähler manifolds via the hyperkähler moment map.

Findings

Constructed a holomorphic Lagrangian fibration with generic fiber $( ext{C}^*)^n$

Analyzed the structure of singular fibers

Provided new insights into hyperkähler SYZ conjecture

Abstract

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 $(\mathbb{C}^*)^n$, and study the singular fibers.