Smoothing, scattering and a conjecture of Fukaya
In 2002, Fukaya [19] proposed a remarkable explanation of mirror symmetry detailing the Strominger–Yau–Zaslow (SYZ) conjecture [47] by introducing two correspondences: one between the theory of pseudo-holomorphic curves on a Calabi–Yau manifold $\check {X}$ and the multivalued Morse theory on...
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Pi |
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| author | Kwokwai Chan Naichung Conan Leung Ziming Nikolas Ma |
| author_facet | Kwokwai Chan Naichung Conan Leung Ziming Nikolas Ma |
| author_sort | Kwokwai Chan |
| collection | DOAJ |
| description | In 2002, Fukaya [19] proposed a remarkable explanation of mirror symmetry detailing the Strominger–Yau–Zaslow (SYZ) conjecture [47] by introducing two correspondences: one between the theory of pseudo-holomorphic curves on a Calabi–Yau manifold
$\check {X}$
and the multivalued Morse theory on the base
$\check {B}$
of an SYZ fibration
$\check {p}\colon \check {X}\to \check {B}$
, and the other between deformation theory of the mirror X and the same multivalued Morse theory on
$\check {B}$
. In this paper, we prove a reformulation of the main conjecture in Fukaya’s second correspondence, where multivalued 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 [7, 9] to tropicalize the pre-dgBV algebra which governs smoothing of a maximally degenerate Calabi–Yau log variety introduced in [8]. Then a comparison between this tropicalized algebra with the dgBV algebra associated to the deformation theory of the semiflat part
$X_{\mathrm {sf}} \subset X$
allows us to extract consistent scattering diagrams from appropriate Maurer–Cartan solutions. |
| format | Article |
| id | doaj-art-bc145376cbc84562ba82ae113759b1c6 |
| institution | DOAJ |
| issn | 2050-5086 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Pi |
| spelling | doaj-art-bc145376cbc84562ba82ae113759b1c62025-08-20T02:48:30ZengCambridge University PressForum of Mathematics, Pi2050-50862025-01-011310.1017/fmp.2024.32Smoothing, scattering and a conjecture of FukayaKwokwai Chan0https://orcid.org/0000-0003-1113-6758Naichung Conan Leung1https://orcid.org/0000-0002-7278-5295Ziming Nikolas Ma2https://orcid.org/0000-0001-5397-2442Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong;The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong; E-mail:Department of Mathematics, Southern University of Science and Technology, 1088 Xueyuan Avenue, Xili, Shenzhen, 518055, China; E-mail:In 2002, Fukaya [19] proposed a remarkable explanation of mirror symmetry detailing the Strominger–Yau–Zaslow (SYZ) conjecture [47] by introducing two correspondences: one between the theory of pseudo-holomorphic curves on a Calabi–Yau manifold $\check {X}$ and the multivalued Morse theory on the base $\check {B}$ of an SYZ fibration $\check {p}\colon \check {X}\to \check {B}$ , and the other between deformation theory of the mirror X and the same multivalued Morse theory on $\check {B}$ . In this paper, we prove a reformulation of the main conjecture in Fukaya’s second correspondence, where multivalued 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 [7, 9] to tropicalize the pre-dgBV algebra which governs smoothing of a maximally degenerate Calabi–Yau log variety introduced in [8]. Then a comparison between this tropicalized algebra with the dgBV algebra associated to the deformation theory of the semiflat part $X_{\mathrm {sf}} \subset X$ allows us to extract consistent scattering diagrams from appropriate Maurer–Cartan solutions.https://www.cambridge.org/core/product/identifier/S2050508624000325/type/journal_article14J3314D1514J3214T20 |
| spellingShingle | Kwokwai Chan Naichung Conan Leung Ziming Nikolas Ma Smoothing, scattering and a conjecture of Fukaya Forum of Mathematics, Pi 14J33 14D15 14J32 14T20 |
| title | Smoothing, scattering and a conjecture of Fukaya |
| title_full | Smoothing, scattering and a conjecture of Fukaya |
| title_fullStr | Smoothing, scattering and a conjecture of Fukaya |
| title_full_unstemmed | Smoothing, scattering and a conjecture of Fukaya |
| title_short | Smoothing, scattering and a conjecture of Fukaya |
| title_sort | smoothing scattering and a conjecture of fukaya |
| topic | 14J33 14D15 14J32 14T20 |
| url | https://www.cambridge.org/core/product/identifier/S2050508624000325/type/journal_article |
| work_keys_str_mv | AT kwokwaichan smoothingscatteringandaconjectureoffukaya AT naichungconanleung smoothingscatteringandaconjectureoffukaya AT zimingnikolasma smoothingscatteringandaconjectureoffukaya |