Abstract

The homogeneous nearly Kähler structure on S^3times S^3 is the only known complete 6-dimensional strictly nearly Kähler structure which is invariant under the action of a 3-torus. We investigate the multi-moment maps associated to such an action and compare with the usual moment map of a toric manifold.

Highlights

  • An almost Hermitian manifold (M, g, J ) is nearly Kähler if ∇g J is skew -symmetric

  • The minimum dimension admitting strict nearly Kähler manifolds is 6, where the nearly Kähler condition is equivalent to the existence of an SU3 structure (ω, ) satisfying dω = 3 Re, d Im = −2ω ∧ ω, where ω is the Kähler form, and is a complex volume form [9]

  • We compute ν and σ for the homogeneous nearly Kähler S3 × S3. We find in this case that ν has many qualitative similarities to the toric moment map μ

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Summary

Introduction

An almost Hermitian manifold (M, g, J ) is nearly Kähler if ∇g J is skew -symmetric. We say a nearly Kähler manifold is strict if it is not Kähler. A 2m-dimensional compact -connected Kähler manifold M2m admitting an effective isometric Tm action is toric. Such a manifold could be studied with use of the moment map μ, which is a T3-equivariant map from the manifold to the dual Lie algebra of the torus, t∗. We find that for the action of a 3-torus T on S3 × S3 to descend to a locally homogeneous quotient \(S3 × S3), the finite group must be a subgroup of T It follows that the multi-moment maps on \(S3 × S3) and S3 × S3 have the same image, and the fibers have the same diffeomorphism type.

Torus actions
Multi-moment maps
The multi-moment map for d!

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