Smoothing, scattering, and a conjecture of Fukaya
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| Format: | Preprint |
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2022
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| _version_ | 1866913706819977216 |
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| author | Chan, Kwokwai Leung, Naichung Conan Ma, Ziming Nikolas |
| author_facet | Chan, Kwokwai Leung, Naichung Conan Ma, Ziming Nikolas |
| contents | In 2002, Fukaya proposed a remarkable explanation of mirror symmetry detailing the SYZ conjecture by introducing two correspondences: one between the theory of pseudo-holomorphic curves on a Calabi-Yau manifold $\check{X}$ and the multi-valued Morse theory on the base $\check{B}$ of an SYZ fibration $\check{p}: \check{X}\to \check{B}$, and the other between deformation theory of the mirror $X$ and the same multi-valued Morse theory on $\check{B}$. In this paper, we prove a reformulation of the main conjecture in Fukaya's second correspondence, where multi-valued Morse theory on the base $\check{B}$ is replaced by tropical geometry on the Legendre dual $B$. In the proof, we apply techniques of asymptotic analysis developed in our previous works to tropicalize the pre-dgBV algebra which governs smoothing of a maximally degenerate Calabi-Yau log variety introduced in another of our recent work. Then a comparison between this tropicalized algebra with the dgBV algebra associated to the deformation theory of the semi-flat part $X_{\text{sf}} \subseteq X$ allows us to extract consistent scattering diagrams from appropriate Maurer-Cartan solutions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2205_09926 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Smoothing, scattering, and a conjecture of Fukaya Chan, Kwokwai Leung, Naichung Conan Ma, Ziming Nikolas Algebraic Geometry Symplectic Geometry In 2002, Fukaya proposed a remarkable explanation of mirror symmetry detailing the SYZ conjecture by introducing two correspondences: one between the theory of pseudo-holomorphic curves on a Calabi-Yau manifold $\check{X}$ and the multi-valued Morse theory on the base $\check{B}$ of an SYZ fibration $\check{p}: \check{X}\to \check{B}$, and the other between deformation theory of the mirror $X$ and the same multi-valued Morse theory on $\check{B}$. In this paper, we prove a reformulation of the main conjecture in Fukaya's second correspondence, where multi-valued Morse theory on the base $\check{B}$ is replaced by tropical geometry on the Legendre dual $B$. In the proof, we apply techniques of asymptotic analysis developed in our previous works to tropicalize the pre-dgBV algebra which governs smoothing of a maximally degenerate Calabi-Yau log variety introduced in another of our recent work. Then a comparison between this tropicalized algebra with the dgBV algebra associated to the deformation theory of the semi-flat part $X_{\text{sf}} \subseteq X$ allows us to extract consistent scattering diagrams from appropriate Maurer-Cartan solutions. |
| title | Smoothing, scattering, and a conjecture of Fukaya |
| topic | Algebraic Geometry Symplectic Geometry |
| url | https://arxiv.org/abs/2205.09926 |