Date | May 12, 2022 |
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Speaker | Yat-Hin Suen |

Dept. | IBS-CGP |

Room | 129-406 |

Time | 14:00-16:00 |

Based on the SYZ conjecture, it is a folklore that "mirror of Lagrangian multi-sections are locally free sheaves". However, due to the non-trivial contribution of holomorphic disks and wall-crossing phenomenon, this folklore has only been verified in very few cases. In this talk, I would like to introduce the notion of tropical Lagrangian multi-sections over any integral affine manifold with singularities $B$ equipped with a polyhedral decomposition $mathscr{P}$. As the name suggests, tropical Lagrangian multi-sections are tropical/combinatorial replacement of Lagrangian multi-sections in the SYZ program. Given a tropical Lagrangian multi-section, together with some linear algebra data, we are going to construct a locally free sheaf $mathcal{E}_0(mathbb{L})$ on the log Calabi-Yau space $X_0(B,mathscr{P})$ associated to $(B,mathscr{P})$, which plays a key role in the famous Gross-Siebert program. The locally free sheaves arise as such construction are not arbitrary and we call them tropical locally free sheaves. I will also provide the reverse construction and establish a correspondence between isomorphism classes of tropical locally free sheaves and tropical Lagrangian multi-sections modulo certain non-trivial equivalence. I will also provide a combinatorial criterion for the smoothability of the pair $(X_0(B,mathscr{P}),mathcal{E}_0(mathbb{L}))$ in dimension 2 under some additional assumptions on $mathscr{P}$ and $mathbb{L}$. Finally, if time permits, I will discuss how to prove homological mirror symmetry between Lagrangian multi-sections and locally free shaves using microlocal sheaf theory.

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