My goal is to get better understanding how the projection of holomorphic curves converge to tropical disks.
We are given an almost toric fibration $X\rightarrow B$ with special Lagrangian fibers with respect to some meromorphic 2-form $\Omega$, which has poles along a divisor. The divisor is mapped to the boundary of the base $B$ of the almost toric fibration. Then the projection of holomorphic curves in $X$ under this fibration are expected to converge to tropical curves. Renato Vianna mentions in his paper "on exotic Lagrangian tori in $CP^2$" that this limit can be thought of as a limit of a deformation of almost complex structures. I do not understand what is meant by this limit and I would really appreciate if somebody could share their insights. Does it mean a limit complex affine structure on the base $B$?
Also, I believe that there should be another explanation to this limit. Namely by projecting the holomorphic curves and then taking the limit under the logarithm with respect to $t$, where $t\rightarrow \infty$. Is this correct? If yes, how is it related to the "limit of a complex structure" as mentioned above?