  Morelli's computation of the K-theory of a toric variety X associates a
polyhedrally constructible function on a real vector space to every equivariant
vector bundle E on X. The coherent-constructible correspondence lifts Morelli's
constructible function to a complex of constructible sheaves kappa(E). We show
that certain filtrations of the cohomology of kappa(E) coming from Morse theory
coincide with the Klyachko filtrations of the generic stalk of E. We give
Morse-theoretic (i.e. microlocal) conditions for a complex of constructible
sheaves to correspond to a vector bundle, and to a nef vector bundle.
