Morse theory of the moment map for representations of quivers

TL;DR

This paper explores the Morse theory of the norm-square of the moment map on quiver representation spaces, establishing convergence of gradient flows, stratification equivalences, and applications to Nakajima quiver varieties.

Contribution

It demonstrates the convergence of gradient flows, aligns Morse stratification with algebraic geometric stratification, and extends Kirwan surjectivity to non-compact quiver varieties.

Findings

Gradient flow converges on representation spaces.

Morse stratification matches Harder-Narasimhan stratification.

Kirwan surjectivity extends to non-compact settings.

Abstract

The results of this paper concern the Morse theory of the norm-square of the moment map on the space of representations of a quiver. We show that the gradient flow of this function converges, and that the Morse stratification induced by the gradient flow co-incides with the Harder-Narasimhan stratification from algebraic geometry. Moreover, the limit of the gradient flow is isomorphic to the graded object of the Harder-Narasimhan-Jordan-H\"older filtration associated to the initial conditions for the flow. With a view towards applications to Nakajima quiver varieties we construct explicit local co-ordinates around the Morse strata and (under a technical hypothesis on the stability parameter) describe the negative normal space to the critical sets. Finally, we observe that the usual Kirwan surjectivity theorems in rational cohomology and integral K-theory carry over to this non-compact setting, and that these theorems generalize to certain equivariant contexts.