Abstract:
An SYZ fibration is a fibration of a symplectic manifold $X$ whose fibers are Lagrangian tori. If $Q$ is the base of the fibration, we can associate to each Lagrangian submanifold $L$ a subset of of $Q$ called the SYZ support. This is defined as the points of $Q$ for which the associated Lagrangian torus fiber $F_q$ has non-trivial Lagrangian intersection Floer homology with $L$. For example, when $L=F_q$ is some torus fiber, the support is simply $q$.
We extend this to the setting where $L$ is a Lagrangian submanifold determined by the data of a tropical curve in $Q$. In specific examples we compute this support explicitly, and make some connections to mirror symmetry for toric varieties.