Trivializations of moment maps

TL;DR

This paper investigates various trivializations of moment maps in algebraic geometry, establishing local triviality results for reductive group actions and hyperkähler moment maps, with applications to Nakajima quiver varieties.

Contribution

It provides a rigorous proof of local trivializations of moment maps in both algebraic and hyperkähler settings, filling a gap in the literature for Nakajima quiver varieties.

Findings

Moment map is a locally trivial fibration over a regular locus.

Trivialization of hyperkähler moment maps for Nakajima quiver varieties.

Application of Kempf-Ness, Morse theory, and ideas from Nakajima and Kronheimer.

Abstract

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.