Family Floer mirror space for local SYZ singularities

Abstract

We state a non-archimedean and Floer-theoretic version of the SYZ conjecture and verify it for the Gross's special Lagrangian fibration in any toric Calabi-Yau manifold. By combining both the ideas of Gross-Hacking-Keel and Kontsevich-Soibelman, the dual analytic fibration with singularities is explicitly written down and is proved to be compatible with the family Floer mirror construction. As a surprising outcome, the Maurer-Cartan set of a singular Lagrangian is naturally only a strict subset of the corresponding dual singular fiber. This responds negatively to the previous expectation and leads to new perspectives of SYZ singularities. As extra evidence, we also check some computations for a well-known folklore conjecture.