In this paper, we introduce the notion of tropical Lagrangian multi-sections over any integral affine manifold B with singularities, and use them to study the reconstruction problem for holomorphic vector bundles over Calabi-Yau surfaces. Given a tropical Lagrangian multi-section L over B with prescribed local models around the ramification points, we construct a locally free sheave E 0 (L) over the projective scheme X0 ( B, P, š) associated to the discrete Legendre transform ( B, P, φ) of (B, P, ϕ), and prove that the pair ( X0 ( B, P, š), E 0 (L)) is smoothable under a combinatorial assumption on L.
Contents1. Introduction 1 Acknowledgment 6 2. The Gross-Siebert program 6 2.1. Affine manifold with singularities and polyhedral decomposition 6 2.2. Toric degenerations 8 2.3. Construction of X0 (B, P) and X0 (B, P, s) 9 3. Tropical Lagrangian multi-sections 10 4. A local model around ramification loci 13 5. Construction of E 0 (L) 14 6. Simplicity and smoothing 19 6.1. Smoothing in rank 2 19 6.2. Smoothing in general rank 21 References 32