Morse Theory of the Moment Map for Representations of Quivers

The main results of this manuscript concern the Morse theory associated to the norm-square of a Kähler moment map f = ‖Φ − α‖2 on the space of representations Rep(Q,v) of a quiver; these are the first steps in a larger research program concerning the hyperkähler analogue of the well-known Kirwan surjectivity theorem in symplectic geometry. The first main result is that, although ‖Φ − α‖2 is not necessarily proper, the negative gradient flow with respect to f converges to a critical point of f . Hence we obtain a Morse stratification of Rep(Q,v). We also give explicit descriptions of the critical sets of f in terms of subrepresentations. The second main result concerns the relationship between the analysis and the algebraic geometry: the Morse stratification is equivalent to the algebro-geometric Harder-Narasimhan stratification on Rep(Q,v), and the limit of the negative gradient flow is isomorphic to the associated graded object of the Harder-Narasimhan-Jordan-Hölder filtration of the initial condition. Finally, we explicitly construct local coördinates around the Morse strata, an important tool for future Morsetheoretic applications. The last section contains some immediate applications of these results. First, in the hyperkähler setting of Nakajima quiver varieties, we give a linearized description of the negative normal bundle to the critical sets of f when restricted to the zero set of the complex moment map (under a technical hypothesis on the stability parameter). Second, we prove Kirwan surjectivity theorems in rational cohomology and integral K-theory for moduli spaces of representations of quivers. Finally, we observe that the Morse theory developed in this manuscript immediately generalizes to certain equivariant contexts.