Line Bundle Resolutions via the Coherent-Constructible Correspondence
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909406078173184 |
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| author | Favero, David Sapronov, Mykola |
| author_facet | Favero, David Sapronov, Mykola |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17873 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Line Bundle Resolutions via the Coherent-Constructible Correspondence Favero, David Sapronov, Mykola Algebraic Geometry 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. |
| title | Line Bundle Resolutions via the Coherent-Constructible Correspondence |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2411.17873 |