Harmonic morphisms and moment maps on hyper-K\"ahler manifolds

TL;DR

This paper characterizes when moment maps on hyper-K"ahler manifolds induced by abelian group actions are harmonic morphisms, providing explicit examples and conditions for their conformality.

Contribution

It offers a detailed analysis of the conditions under which hyper-K"ahler moment maps are harmonic morphisms, including explicit formulas and geometric characterizations.

Findings

For dim T=1, the moment map is a harmonic morphism.

Explicit formula for the moment map on the tangent bundle of complex projective space.

Higher-dimensional cases with critical points or nonflat manifolds do not admit horizontally weakly conformal moment maps.

Abstract

We characterise the actions, by holomorphic isometries on a K\"ahler manifold with zero first Betti number, of an abelian Lie group of dim\geq 2, for which the moment map is horizontally weakly conformal (with respect to some Euclidean structure on the Lie algebra of the group). Furthermore, we study the hyper-K\"ahler moment map $\phi$ induced by an abelian Lie group T acting by triholomorphic isometries on a hyper-K\"ahler manifold M, with zero first Betti number, thus obtaining the following: If dim T=1 then $\phi$ is a harmonic morphism. Moreover, we illustrate this on the tangent bundle of the complex projective space equipped with the Calabi hyper-K\"ahler structure, and we obtain an explicit global formula for the map. If dim T\geq 2 and either $\phi$ has critical points, or M is nonflat and dim M=4 dim T then $\phi$ cannot be horizontally weakly conformal.