  We give a geometric interpretation for superconformal quantum mechanics
defined on a hyper-Kahler 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 K points on C2, we obtain a rigorous estimate for
the exponential growth of the index degeneracies of BPS states as K goes to
infinity. This growth serves as a toy model for our recently proposed duality
between a seven dimensional black hole and superconformal quantum mechanics.
