Morse theory of the moment map for representations of quivers

Megumi Harada; Graeme Wilkin

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arXiv · 0807.4734

Morse theory of the moment map for representations of quivers

Megumi Harada·Graeme Wilkin

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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.

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BibTeX RIS

Megumi Harada, Graeme Wilkin. 2008-07-29. Morse theory of the moment map for representations of quivers. https://arxiv.org/abs/0807.4734

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