[QSMS Geometry Seminar 2022-05-12] Tropical Lagrangian multi-sections and tropical locally free sheaves

Speaker :  Yat-Hin Suen  (IBS-CGP)

TiTle  : Tropical Lagrangian multi-sections and tropical locally free sheaves

Abstract : 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.