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

Abstract

© 2022 Elsevier Inc.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,ks) 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,ks) is simple. This produces explicit examples of smoothable pairs (X0(B,P,s),E0(L,ks)) in dimension 2.