Bia{\l}ynicki-Birula Decompositions of Nakajima Quiver Varieties, Quiver Chains and Star-Shaped Quivers
This paper studies the Bia{\l}ynicki--Birula decomposition associated with the natural $\mathbb C^*$-action on Nakajima quiver varieties given by scaling the maps along the reversed arrows of the doubled quiver. Fixed points of this action are described in terms of representations with relations of auxiliary quivers, called quiver chains. Several aspects of the geometry of the resulting Bia{\l}ynicki--Birula decomposition are also investigated, including the dimensions of their attracting fibers and the corresponding motivic decomposition of the Nakajima quiver variety in terms of quiver-chain moduli spaces. Finally, the general framework is specialized to star-shaped quivers, where, for particular dimension vectors, the fixed-locus components are classified and the motivic class of the full Nakajima quiver variety is computed in a suitable localization of the Grothendieck ring of varieties.