TITLE: How far can one get with the Gross-Siebert program?
QUESTION [15 upvotes]: The Gross-Siebert program is said to be an algebraic analog of the SYZ conjecture and they used toric degeneration to construct a mirror dual of Calabi-Yau varieties. It seems like the singular central fiber, as is mostly given by some combinatoric data, has some sort of "mirror dual" that's easily written down.
My question is, how far can one get with this program? Are the CY varieties arising this way "the" mirror dual of the original CY varieties or just one candidate of its mirror? What are the most recent progress on it?
Also, this seems very different from the homological mirror conjecture saying that the equivalence of the derived category on one and Fukaya category on another. Can anyone here connect the dots for me?

REPLY [3 votes]: For the question how SYZ is related to HMS, I have recently posted a paper on the arXiv (http://arxiv.org/abs/1201.6454) which explains this connection as a sort of Fourier-Mukai transform. There I gave a local explanation. Global situation requires more machinery, but it is a similar picture.