 Date : 20220512 14:00 ~ 16:00
 Place : 129406 (SNU) or Zoom

Speaker : YatHin Suen (IBSCGP)

TiTle : Tropical Lagrangian multisections and tropical locally free sheaves

Abstract : Based on the SYZ conjecture, it is a folklore that "mirror of Lagrangian multisections are locally free sheaves". However, due to the nontrivial contribution of holomorphic disks and wallcrossing phenomenon, this folklore has only been verified in very few cases. In this talk, I would like to introduce the notion of tropical Lagrangian multisections over any integral affine manifold with singularities $B$ equipped with a polyhedral decomposition $\mathscr{P}$. As the name suggests, tropical Lagrangian multisections are tropical/combinatorial replacement of Lagrangian multisections in the SYZ program. Given a tropical Lagrangian multisection, together with some linear algebra data, we are going to construct a locally free sheaf $\mathcal{E}_0(\mathbb{L})$ on the log CalabiYau space $X_0(B,\mathscr{P})$ associated to $(B,\mathscr{P})$, which plays a key role in the famous GrossSiebert 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 multisections modulo certain nontrivial 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 multisections and locally free shaves using microlocal sheaf theory.