Tropical Lagrangian multi-sections and smoothing of locally free sheaves over degenerate Calabi-Yau surfaces

We introduce the notion of tropical Lagrangian multi-sections over a 2-dimensional integral affine manifold B with singularities, and use them to study the reconstruction problem for higher rank locally free sheaves over Calabi-Yau surfaces. To certain tropical Lagrangian multi-sections L over B, which are explicitly constructed by prescribing local models around the ramification points, we construct locally free sheaves E₀(L,k_s) over the singular projective scheme X₀(B,P,s) associated to B equipped with a polyhedral decomposition P and a gluing data s. We then find combinatorial conditions on such an L under which the sheaf E₀(L,k_s) is simple. This produces explicit examples of smoothable pairs (X₀(B,P,s),E₀(L,k_s)) in dimension 2.