Morse theory and toric vector bundles

Treumann, David

doi:10.1090/tran/6511

Morse theory and toric vector bundles
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Trans. Amer. Math. Soc. 369 (2017), 1-29 Request permission

Abstract:

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 $\mathcal {E}$ on $X$. The coherent-constructible correspondence lifts Morelli’s constructible function to a complex of constructible sheaves $\kappa (\mathcal {E})$. We show that certain filtrations of the cohomology of $\kappa (\mathcal {E})$ coming from Morse theory coincide with the Klyachko filtrations of the generic stalk of $\mathcal {E}$. 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.

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