Line Bundle Resolutions via the Coherent-Constructible Correspondence

We consider a finite collection of line bundles $\mathrm{\Phi }$ introduced by Bondal on a smooth, projective toric variety $X$. For any coherent sheaf $F$ on $X$, we construct minimal resolutions of $F$ by line bundles in $\mathrm{\Phi }$, up to twist, with length bounded by the dimension of $X$ and provide explicit formulae for their Betti numbers. For a toric subvariety $Y\subset X$ of codimension $k$, we give a construction of the minimal resolution of $f_{\ast }{\mathcal{O}}_Y$ of length $k$ by line bundles in $\mathrm{\Phi }$ and relate their Betti numbers to the topology of a stratified real torus. Additionally, we recover a (generally non-minimal) cellular resolution of $f_{\ast }{\mathcal{O}}_Y$ constructed in Hanlon-Hicks-Lazarev. Aspects of our proof run through the Coherent Constructible Correspondence, a form of homological mirror symmetry for toric varieties.