Reconstruction of holomoprhic tangent bundle of complex projective plane via tropical Lagrangian multi-section

Abstract

In this paper, we study the reconstruction problem of the holomorphic tangent bundle of the complex projective plane. We introduce the notion of tropical Lagrangian multi-section and cook up one tropicalizing the Chern connection associated the Fubini-Study metric. Then we perform the reconstruction of the tangent bundle from this tropical Lagrangian multi-section. Walling-crossing phenomenon will occur in the reconstruction process.