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

Tropical Lagrangian multisections and toric vector bundles

“…In [28], the author provided a definition of tropical Lagrangian multisections over integral affine manifolds with singularities whose domain of the branched covering map is a topological manifold. While in [9], the authors gave a definition of tropical Lagrangian multisections over 2-dimensional integral affine manifolds with singularities equipped with polyhedral decomposition, where they also assumed the domain is also a topological manifold equipped with a polyhedral decomposition that is compatible with the covering map. Of course, if we restrict our attention to the case where the affine manifold is ‫ޒ‬ 2 with polyhedral decomposition being a fan , Definition 3.8 extends Definition 3.6 in [9] because we don't assume L is a topological manifold here.…”

“…Payne [25; 26] studied toric vector bundles and their moduli in terms of piecewise linear functions defined on cone complexes. Motivated by the work of Payne, the notion of tropical Lagrangian multisections was first introduced by the author of this paper in [28] and generalized to arbitrary 2-dimensional integral affine manifolds with singularities in a joint work with Chan and Ma [9].…”

From deformation theory to tropical geometry

“…Thus, for the purpose of proving mirror statements or even understanding mirror symmetry, one may bypass the complicated combinatorics of these objects. This was the motivation behind the recent works [4,8,9], where we directly constructed the DGLA (or better, DGBV algebra) that governs smoothing of degenerate Calabi-Yau varieties and prove existence of smoothing without using scattering diagrams.…”