Tropical Lagrangian coamoebae and free resolutions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916541439672320 |
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| author | Kuo, Christopher Williams, Harold |
| author_facet | Kuo, Christopher Williams, Harold |
| contents | We study the coamoebae of Lagrangian submanifolds of $(\mathbb{C}^\times)^n$, specifically how the combinatorics of their degenerations encodes the homological algebra of mirror coherent sheaves. Concretely, to a minimal free resolution $F^\bullet$ of a module $M$ over $\mathbb{C}[z_1^{\pm 1}, \dotsc, z_n^{\pm 1}]$ we associate a simplicial complex $T(F^\bullet) \subset T^n$. We call $T(F^\bullet)$ a tropical Lagrangian coamoeba. We show that the discrete information in $F^\bullet$ can often be recovered from $T(F^\bullet)$, and that more generally $M$ is mirror to a certain constructible sheaf supported on $T(F^\bullet)$. The resulting interplay between coherent sheaves on $(\mathbb{C}^\times)^n$ and simplicial complexes in $T^n$ provides a higher-dimensional generalization of the spectral theory of dimer models in $T^2$, as well as a symplectic counterpart to the theory of brane brick models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_18071 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tropical Lagrangian coamoebae and free resolutions Kuo, Christopher Williams, Harold Algebraic Geometry Combinatorics Symplectic Geometry We study the coamoebae of Lagrangian submanifolds of $(\mathbb{C}^\times)^n$, specifically how the combinatorics of their degenerations encodes the homological algebra of mirror coherent sheaves. Concretely, to a minimal free resolution $F^\bullet$ of a module $M$ over $\mathbb{C}[z_1^{\pm 1}, \dotsc, z_n^{\pm 1}]$ we associate a simplicial complex $T(F^\bullet) \subset T^n$. We call $T(F^\bullet)$ a tropical Lagrangian coamoeba. We show that the discrete information in $F^\bullet$ can often be recovered from $T(F^\bullet)$, and that more generally $M$ is mirror to a certain constructible sheaf supported on $T(F^\bullet)$. The resulting interplay between coherent sheaves on $(\mathbb{C}^\times)^n$ and simplicial complexes in $T^n$ provides a higher-dimensional generalization of the spectral theory of dimer models in $T^2$, as well as a symplectic counterpart to the theory of brane brick models. |
| title | Tropical Lagrangian coamoebae and free resolutions |
| topic | Algebraic Geometry Combinatorics Symplectic Geometry |
| url | https://arxiv.org/abs/2412.18071 |