jats:titleAjats:scbstract</jats:sc> </jats:title>jats:pWe give a geometric interpretation for superconformal quantum mechanics defined on a hyper-Kähler cone which has an equivariant symplectic resolution. BPS states are identified with certain twisted Dolbeault cohomology classes on the resolved space and their index degeneracies can also be related to the Euler characteristic computed in equivariant sheaf cohomology. In the special case of the Hilbert scheme of jats:italicK</jats:italic> points on ℂjats:sup2</jats:sup>, we obtain a rigorous estimate for the exponential growth of the index degeneracies of BPS states as jats:italicK</jats:italic> → ∞. This growth serves as a toy model for our recently proposed duality between a seven dimensional black hole and superconformal quantum mechanics.</jats:p>