Tropical Lagrangian multi-sections and tropical locally free sheaves

TL;DR

This paper generalizes tropical Lagrangian multi-sections to higher dimensions and establishes a correspondence with tropical locally free sheaves, expanding the understanding of their geometric and algebraic structures.

Contribution

It introduces a higher-dimensional generalization of tropical Lagrangian multi-sections and constructs a bijective correspondence with tropical locally free sheaves.

Findings

Established a 1-1 correspondence between tropical Lagrangian multi-sections and tropical locally free sheaves.

Extended the notion of tropical Lagrangian multi-sections to arbitrary dimensions.

Provided a construction method linking linear algebra data to tropical sheaves.

Abstract

This article is a continuation of the work "Tropical Lagrangian multi-sections and smoothing of locally free sheaves over degenerated Calabi-Yau surfaces". We generalize the notion of tropical Lagrangian multi-sections to any dimensions. Together with some linear algebra data, we construct a special class of locally free sheaves, called tropical locally free sheaves. We will also provide the reverse construction and show that there is a 1-1 correspondence between isomorphism classes of tropical locally free sheaves and tropical Lagrangian multi-sections modulo certain equivalence.