Special Lagrangian fibrations, mirror symmetry and Calabi-Yau double   covers
  The first part of this paper is a review of the Strominger-Yau-Zaslow conjecture in various settings. In particular, we summarize how, given a pair (X,D) consisting of a Kahler manifold and an anticanonical divisor, families of special Lagrangian tori in X-D and weighted counts of holomorphic discs in X can be used to build a Landau-Ginzburg model mirror to X. In the second part we turn to more speculative considerations about Calabi-Yau manifolds with holomorphic involutions and their quotients. Namely, given a hypersurface H representing twice the anticanonical class in a Kahler manifold X, we attempt to relate special Lagrangian fibrations on X-H and on the (Calabi-Yau) double cover of X branched along H; unfortunately, the implications for mirror symmetry are far from clear. 
Denis Auroux