On smooth projective toric varieties, every coherent sheaf has a minimal line bundle resolution of length at most the dimension, and for toric subvarieties the Betti numbers are compactly supported cohomology groups of mirror-torus strata.
A categorification of Morelli's theorem
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Line Bundle Resolutions via the Coherent-Constructible Correspondence
On smooth projective toric varieties, every coherent sheaf has a minimal line bundle resolution of length at most the dimension, and for toric subvarieties the Betti numbers are compactly supported cohomology groups of mirror-torus strata.