  We study various trivializations of moment maps. First in the general
framework of a reductive group $G$ acting on a smooth affine variety. We prove
that the moment map is a locally trivial fibration over a regular locus of the
center of the Lie algebra of $H$ a maximal compact subgroup of $G$. The
construction relies on Kempf-Ness theory and Morse theory of the square norm of
the moment map studied by Kirwan, Ness-Mumford and Sjamaar. Then we apply it
together with ideas from Nakajima and Kronheimer to trivialize the hyperkaehler
moment map for Nakajima quiver varieties. Notice this trivialization result
about quiver varieties was known and used by experts such as Nakajima and
Maffei but we could not locate a proof in the literature.
