Contact isotopies in the coherent-constructible correspondence
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908353880391680 |
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| author | Bose, Jishnu Williams, Harold |
| author_facet | Bose, Jishnu Williams, Harold |
| contents | The coherent-constructible correspondence is a realization of toric mirror symmetry in which the A-side is modeled by constructible sheaves on $T^n$. This paper provides a geometric realization of the mirror Picard group action in this correspondence, characterizing it in terms of quantized contact isotopies and providing a sheaf-theoretic counterpart to work of Hanlon in the Fukaya-Seidel setting. Given a toric Cartier divisor $D$, we consider a family of homogeneous Hamiltonians $H_\varepsilon$ on $T^* T^n$. Their flows act on sheaves via a family of kernels $K_\varepsilon$ on $T^n \times T^n$. The nearby cycles kernel $K_0$ corresponds heuristically to the Hamiltonian flow of the non-differentiable function $\lim_{\varepsilon \to 0} H_\varepsilon$, which is the pullback of the support function of $D$ along the cofiber projection. We show that the action of $K_0$ coincides with the convolution action of the associated twisted polytope sheaf, hence mirrors the action of $\mathcal{O}(D)$ on coherent sheaves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_05012 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Contact isotopies in the coherent-constructible correspondence Bose, Jishnu Williams, Harold Algebraic Geometry Symplectic Geometry The coherent-constructible correspondence is a realization of toric mirror symmetry in which the A-side is modeled by constructible sheaves on $T^n$. This paper provides a geometric realization of the mirror Picard group action in this correspondence, characterizing it in terms of quantized contact isotopies and providing a sheaf-theoretic counterpart to work of Hanlon in the Fukaya-Seidel setting. Given a toric Cartier divisor $D$, we consider a family of homogeneous Hamiltonians $H_\varepsilon$ on $T^* T^n$. Their flows act on sheaves via a family of kernels $K_\varepsilon$ on $T^n \times T^n$. The nearby cycles kernel $K_0$ corresponds heuristically to the Hamiltonian flow of the non-differentiable function $\lim_{\varepsilon \to 0} H_\varepsilon$, which is the pullback of the support function of $D$ along the cofiber projection. We show that the action of $K_0$ coincides with the convolution action of the associated twisted polytope sheaf, hence mirrors the action of $\mathcal{O}(D)$ on coherent sheaves. |
| title | Contact isotopies in the coherent-constructible correspondence |
| topic | Algebraic Geometry Symplectic Geometry |
| url | https://arxiv.org/abs/2505.05012 |