The multi-moment map of the nearly K\"ahler $S^3 \times S^3$

TL;DR

This paper studies the multi-moment map in the context of nearly K"ahler manifolds, focusing on the specific case of a $ ext{S}^3 imes ext{S}^3$ with a torus action, revealing similarities and differences with toric geometry.

Contribution

It characterizes the multi-moment map for nearly K"ahler $ ext{S}^3 imes ext{S}^3$ and compares its properties to those of toric manifolds, highlighting new geometric insights.

Findings

Multi-moment map behaves like a toric moment map in this setting

The general case of the multi-moment map differs from toric moment maps

Provides a detailed analysis of the nearly K"ahler $ ext{S}^3 imes ext{S}^3$ case

Abstract

We describe the multi-moment map associated to an almost Hermitian manifold which admits an action of a torus by holomorphic isometries. We investigate in particular the case of a $\mathbb T^3$ action on the homogeneous nearly K\"ahler $ S^3\times S^3$. We find that the multi-moment map in this case acts more-or-less similarly to the moment map of a toric manifold, while the more general case does not.