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Mar 10, 2012 at 17:19 comment added Tim Perutz Scott, my understanding is that the Fukaya category is equally difficult to define for a rigid CY and for a degenerating one. (The coefficients are in a Novikov series ring.) But in the case of a maximal degeneration, one expects a homological mirror symmetry description of the category, and that the Novikov series actually converge.
Mar 2, 2012 at 8:41 comment added S. Carnahan I am also not an expert, but I have heard that the Fukaya category of a CY is also difficult to define away from infinitesimal deformations of singular degenerations. (I'm afraid to speculate in the company of specialists, though...)
Mar 1, 2012 at 16:59 history answered David E Speyer CC BY-SA 3.0