Line Bundle Resolutions via the Coherent-Constructible Correspondence

TL;DR

This paper develops explicit minimal resolutions of coherent sheaves on smooth projective toric varieties using a specific collection of line bundles, linking algebraic and topological data through the coherent-constructible correspondence.

Contribution

It introduces a method to construct minimal line bundle resolutions with explicit Betti numbers, extending to subvarieties and connecting to homological mirror symmetry.

Findings

Explicit formulas for Betti numbers of resolutions

Resolutions relate to the topology of stratified real tori

Recovery of cellular resolutions in specific cases

Abstract

We consider a finite collection of line bundles $\Phi$ 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 $\Phi$, 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 $f_{*}\mathcal{O}_{Y}$ of length $k$ by line bundles in $\Phi$ and relate their Betti numbers to the topology of a stratified real torus. Additionally, we recover a (generally non-minimal) cellular resolution of $f_{*}\mathcal{O}_{Y}$ constructed in Hanlon-Hicks-Lazarev. Aspects of our proof run through the Coherent Constructible Correspondence, a form of homological mirror symmetry for toric varieties.