Abstract
We characterise the actions by holomorphic isometries on a Kähler manifold, of an abelian Lie group, admitting a moment map which is horizontally weakly conformal (with respect to some Euclidean structure on the Lie algebra of the group). Furthermore, let \(\varphi \) be a hyper-Kähler moment map of an abelian Lie group T acting by triholomorphic isometries on a hyper-Kähler manifold M. If \(\dim T=1\) then \(\varphi \) is a harmonic morphism. Moreover, we illustrate this on the tangent bundle of the complex projective space equipped with the Calabi hyper-Kähler structure, and we obtain an explicit global formula for the map. If \(\dim T\ge 2\) and either \(\varphi \) has critical points, or M is nonflat and \(\dim M=4\dim T\) then \(\varphi \) cannot be horizontally weakly conformal.
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Benyounes, M., Loubeau, E. & Pantilie, R. Harmonic morphisms and moment maps on hyper-Kähler manifolds. manuscripta math. 153, 373–388 (2017). https://doi.org/10.1007/s00229-016-0894-3
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DOI: https://doi.org/10.1007/s00229-016-0894-3