On homological mirror symmetry of toric Calabi-Yau threefolds

We use Lagrangian torus fibrations on the mirror X of a toric Calabi-Yau threefold X' to construct Lagrangian sections and various Lagrangian spheres on X. We then propose an explicit correspondence between the sections and line bundles on X' and between spheres and sheaves supported on the toric divisors of X'. We conjecture that these correspondences induce an embedding of the relevant derived Fukaya category of X inside the derived category of coherent sheaves on X'