Multi-moment maps on nearly Kähler six-manifolds

Abstract

We study multi-moment maps induced by a two-torus action on the four homogeneous nearly Kähler six-manifolds. Their explicit expression and stationary orbits are derived. The configuration of fixed-points and one-dimensional orbits is worked out for generic six-manifolds equipped with an $\mathrm{SU}(3)$-structure admitting a two-torus symmetry. Projecting the subspaces obtained to the orbit space yields a trivalent graph. We illustrate this result concretely on the homogeneous nearly Kähler examples.

23 pages, 4 figures. Propositions 3.2, 4.2, 5.2 and Examples 6.2-6.3 added. General exposition improved and some minor mistakes fixed. This is a pre-print of an article published in Geometriae Dedicata. The final version is available at https://doi.org/10.1007/s10711-020-00568-w