Problems on Quiver Varieties

(1) Study the class of hyper-Kähler manifolds which are hyper-Kähler reductions of finite dimensional quaternion vector spaces by products of unitary groups. (Probably it is better to assume that the action is linear.) Besides quiver varieties, hyper-Kähler toric varieties in the sense of Bielawski and Dancer [2] are such examples. When the quotients are nonsingular ? How much of geometric properties of quiver varieties can be generalized to hyper-Kähler manifolds in the class ? (2) Let G be a compact Lie group. Moduli spaces of G-monopoles on R are identified with moduli spaces of Nahm’s equations on the products of intervals, where each vertex of the Dynkin diagram of G gives an interval, and we impose the boundary conditions according to edges of the diagram. This result was shown when G is a classical group and the symmetry is maximally broken [6]. Extend this result to more generaly symmetry breaking cases. Then study moduli spaces of Nahm’s equations associated with arbitrary Dynkin diagram, not necessarily classical, finite type. They are probably isomorphic to moduli spaces of rational curves into partial flag manifolds attached with the corresponding Kac-Moody Lie algebra. Study their geometries. Relate them to the corresponding representation theory and the theory of Gromov-Witten invariants. See [3] for the study in these directions. (3) In [13, 14] I have given an algorithm to compute Betti numbers of quiver varieties using the torus action and virtual Hodge polynomials. But the algorithm has a recursive structure and it is practically hard to perform the computation. We need to study many quiver varieties at the same time. (Each quiver variety corresponds to a weight space of a representation of the Kac-Moody Lie algebra corresponding to the quiver. We fix a representation and need to study all quiver varieties corresponding to all weight spaces. So far, I have not succeeded the computation for two fundamental representations for E8.) It is desirable to give closed formula of Betti numbers. Such formula should exists also for quiver varieties of affine types, where having the algorithm are not a strong statment since we have infinitely many nonempty quiver varieties.