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

TL;DR

This paper introduces a tropical geometric approach to reconstruct the holomorphic tangent bundle of the complex projective plane, utilizing tropical Lagrangian multi-sections and analyzing wall-crossing phenomena.

Contribution

It develops a novel method of reconstructing complex vector bundles from tropical data, specifically applying tropical Lagrangian multi-sections to $ ext{T}_{ ext{P}^2}$.

Findings

Successful reconstruction of $ ext{T}_{ ext{P}^2}$ from tropical data

Identification of wall-crossing phenomena during reconstruction

Extension of tropical techniques to complex vector bundle problems

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.