Yu-Shen Lin (Boston U.): From tropical curves to special Lagrangians

Given a tropical curve in ℝⁿ, a central question in tropical geometry is whether it can be lifted to a holomorphic curve in the corresponding toric variety. For instance, every tropical curve in ℝ² admits such holomorphic lifting. However, such lifting may not exist in general when n > 2. On the mirror side, Mikhalkin showed that any tropical curve admits a Lagrangian lifting. In this talk, we will show a refinement of the result of Mikhalkin: Every locally planar tropical curve can be lifted to a special Lagrangian in (ℂ^*)ⁿ via a gluing construction. Moreover, we construct a one-parameter family of special Lagrangians whose Gromov–Hausdorff limit collapses to the given tropical curve in the adiabatic limit. This is joint work with S.-K. Chiu and Y. Li.