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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?

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    $\begingroup$ For the limit I suggest you to look at Auroux's paper:www-math.mit.edu/~auroux/papers/slagjdg.pdf, in particular Example 3.1.2. On the other hand, I think your question is not the key point of Vianna's paper. The point of view that holomorphic curves in symplectic manifolds should converge to tropical curves in affine bases is long expected, from the Gross-Siebert point of view of mirror symmetry. Personally I don't think such a point of view is of mathematical interesting because it completely neglects the A-side geometry, which is one of the major applications of mirror symmetry. $\endgroup$
    – YHBKJ
    Oct 6, 2015 at 15:16
  • $\begingroup$ First of all thank you for your reply. I know, that this is not the main point of Vianna's paper, nevertheless it was the part I did not understand. I hoped that maybe there was an easy explanation of this correspondence or a way to see this convergence explicitly in the above example. Otherwise, could you suggest a reference for an explanation of this conjecture, that the holomorphic curves converge to tropical curves? $\endgroup$
    – Sofie
    Oct 7, 2015 at 8:33

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