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