2013/04/18 by Benyounes, M., Loubeau, E., Pantilie, R.
#53C26 #53C43 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1304.5028
We characterise the actions, by holomorphic isometries on a Kähler manifold with zero first Betti number, of an abelian Lie group of dim≥ 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-Kähler moment map ϕ induced by an abelian Lie group T acting by triholomorphic isometries on a hyper-Kähler manifold M, with zero first Betti number, thus obtaining the following: If dim T=1 then ϕ 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≥ 2 and either ϕ has critical points, or M is nonflat and dim M=4 dim T then ϕ cannot be horizontally weakly conformal.