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A Kaehler Hamiltonian G-manifold is a Kaehler manifold with a Hamiltonian G-action, i.e., a G-action generated by a moment map. In particular, the Killing vector fields which generate the G-action are either holomorphic or antiholomorphic. What is the general explicit expression of this moment map? References would be appreciated.

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    $\begingroup$ Why do you expect the moment map to be defined differently or "more explicitly" in the Kaehler case, compared to the general definition of a moment map of a Hamiltonian G-action on an arbitrary symplectic manifold? The only thing that comes to my mind is that on a Kaehler manifold, due to the relation between the Kaehler metric and the symplectic form, one can probably express the definition of the moment map in terms of the metric instead of the symplectic form. Is this what you are asking for? $\endgroup$
    – B K
    Commented Oct 13, 2017 at 7:40

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I invite you to read these references of Charles-Michel Marle: Symmetries of Hamiltonian Systems on Symplectic and Poisson Manifolds. In "Similarity and Symmetry Methods, Applications in Elasticity and Mechanics of Materials", Jean-François Ganghoffer and Ivaïlo Mladenov, Editors. Lecture notes in Applied and Computational Mechanics 73, Springer 2014, pp. 185-269. https://hal.archives-ouvertes.fr/hal-00940297/file/VarnaArxiv.pdf

From Tools in Symplectic and Poisson Geometry to J.-M. Souriau's theories of Statistical Mechanics and Thermodynamics. Entropy 2016, 18, 370 http://charles-michel.marle.pagesperso-orange.fr/publications/entropy-18-00370.pdf

F. Barbaresco

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