Let $X$ be a closed symplectic manifold equipped with a smooth Lagrangian torus fibration $\pi:X \rightarrow Q$. Assume that $\pi$ admits a Lagrangian section. By work of Kontsevich-Soibelman, one can associate to $(X, \pi)$ a rigid analytic (in the sense of Tate) space $Y$. In recent works, Abouzaid showed that one can associate to $(X, \pi)$ a gerbe $\beta$ on the rigid analytic space $Y$, classified by a class $[\alpha_X] \in H^2(Y,O^{*})$, and define an $A_{\infty}$-functor from the Fukaya category of $X$ to the derived category of $\beta$-twisted coherent sheaves on $Y$. He shows this to be full and faithful embedding. I have heard it remarked that there are ideas floating around about how to extend a result like this in the presence of singular fibres. Do there exist explicit proposals for how one might attempt to do this?
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Yes, there is a concrete program on how to handle singular fibers in the SYZ fibration and several steps of this program are already completed.
You can watch the videos of Abouzaid lectures on the subject at
https://www.youtube.com/watch?v=1PqIE3YJU0I
or look at some of his slides at
answered Jul 25, 2019 at 14:26