In this paper, we study the reconstruction problem of the holomorphic tangent bundle $\mathbb{ T}_{ \mathbb{ P}^2}$ of the complex projective plane $\mathbb{ P}^2$. We introduce the notion of tropical Lagrangian multi-section and cook up one by tropicalizing the Chern connection associated the Fubini-Study metric. Then we perform the reconstruction of $\mathbb{ T}_{ \mathbb{ P}^2}$ from this tropical Lagrangian multi-section. Walling-crossing phenomenon will occur in the reconstruction process.
Categories: math.SG math.AG math.DG