Canonical form of matrix factorizations from Fukaya category of surface
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| Format: | Preprint |
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2024
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| _version_ | 1866911930646528000 |
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| author | Cho, Cheol-Hyun Rho, Kyungmin |
| author_facet | Cho, Cheol-Hyun Rho, Kyungmin |
| contents | This paper concerns homological mirror symmetry for the pair-of-pants surface (A-side) and the non-isolated surface singularity $xyz=0$ (B-side). Burban-Drozd classified indecomposable maximal Cohen-Macaulay modules on the B-side. We prove that higher-multiplicity band-type modules correspond to higher-rank local systems over closed geodesics on the A-side, generalizing our previous work for the multiplicity one case. This provides a geometric interpretation of the representation tameness of the band-type maximal Cohen-Macaulay modules, as every indecomposable object is realized as a geometric object.
We also present an explicit canonical form of matrix factorizations of $xyz$ corresponding to Burban-Drozd's canonical form of band-type maximal Cohen-Macaulay modules. As applications, we give a geometric interpretation of algebraic operations such as AR translation and duality of maximal Cohen-Macaulay modules as well as certain mapping cone operations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_16648 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Canonical form of matrix factorizations from Fukaya category of surface Cho, Cheol-Hyun Rho, Kyungmin Representation Theory Commutative Algebra Symplectic Geometry 53D37, 18G70, 13C14 This paper concerns homological mirror symmetry for the pair-of-pants surface (A-side) and the non-isolated surface singularity $xyz=0$ (B-side). Burban-Drozd classified indecomposable maximal Cohen-Macaulay modules on the B-side. We prove that higher-multiplicity band-type modules correspond to higher-rank local systems over closed geodesics on the A-side, generalizing our previous work for the multiplicity one case. This provides a geometric interpretation of the representation tameness of the band-type maximal Cohen-Macaulay modules, as every indecomposable object is realized as a geometric object. We also present an explicit canonical form of matrix factorizations of $xyz$ corresponding to Burban-Drozd's canonical form of band-type maximal Cohen-Macaulay modules. As applications, we give a geometric interpretation of algebraic operations such as AR translation and duality of maximal Cohen-Macaulay modules as well as certain mapping cone operations. |
| title | Canonical form of matrix factorizations from Fukaya category of surface |
| topic | Representation Theory Commutative Algebra Symplectic Geometry 53D37, 18G70, 13C14 |
| url | https://arxiv.org/abs/2406.16648 |