Year: 2024
Type: preprint
Abstract: We consider a finite collection of line bundles $Φ$ 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 $Φ$, 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 $Φ$ 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.
Source: arXiv (Cornell University)
Authors: Favero, David, Sapronov, Mykola
Institution: University of Minnesota
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4404990295