Reconstruction of $T_{\mathbb{P}^2}$ via tropical Lagrangian multi-section

Abstract

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.