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.
Wrapped sheaves
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abstract
We construct a sheaf-theoretic analogue of the wrapped Fukaya category in Lagrangian Floer theory, by localizing a category of sheaves microsupported away from some given $\Lambda \subset S^*M$ along continuation maps constructed using the Guillermou-Kashiwara-Schapira sheaf quantization. When $\Lambda$ is a subanalytic singular isotropic, we also construct a comparison map to the category of compact objects in the category of unbounded sheaves microsupported in $\Lambda$, and show that it is an equivalence. The last statement can be seen as a sheaf theoretical incarnation of the sheaf-Fukaya comparison theorem of Ganatra-Pardon-Shende.
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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.