Abstract

One version of the realizability problem asks, “Which tropical subvarieties can be lifted to algebraic subvarieties?” In his early work on the subject, Mikhalkin exhibited a superabundant tropical curve that could not be lifted to an algebraic variety. However, in 2014, Cheung, Fantini, Park, and Ulirsch showed that all trivalent tropical curves that are non-superabundant possess algebraic lifts.

On the mirror side, every trivalent tropical curve can be lifted to a Lagrangian submanifold! However, these Lagrangian submanifolds may bound holomorphic disks (and therefore be unsuitable for homological mirror symmetry). It is expected that the realizable tropical subvarieties have “unobstructed” Lagrangian submanifold lifts—-in the sense that the counts of holomorphic disks cancel out in homology. In this talk, we’ll show that the non-superabundance condition implies the unobstructedness of the corresponding tropical Lagrangian lift in dimension three.