In this paper, we first provide an explicit description of {\it all} holomorphic discs (``disc instantons'') attached to Lagrangian torus fibers of arbitrary compact toric manifolds, and prove their Fredholm regularity. Using this, we compute Fukaya-Oh-Ohta-Ono's (FOOO's) obstruction (co) chains and the Floer cohomology of Lagrangian torus fibers of Fano toric manifolds. In particular specializing to the formal parameter T^{2}= e^{-1} , our computation verifies the folklore that FOOO's obstruction (co) chains correspond to the Landau-Ginzburg superpotentials under the mirror symmetry correspondence, and also proves the prediction made by K. Hori about the Floer cohomology of Lagrangian torus fibers of Fano toric manifolds. The latter states that the Floer cohomology (for the parameter value T^{2}= e^{-1} ) of all the fibers vanish except at a finite number, the Euler characteristic of the toric manifold, of base points in the momentum polytope that are critical points of the superpotential of the Landau-Ginzburg mirror to the toric manifold. In the latter cases, we also prove that the Floer cohomology of the corresponding fiber is isomorphic to its singular cohomology.

We study Legendrian embeddings of a compact Legendrian submanifold L sitting in a closed contact manifold (M,ξ) whose contact structure is supported by a (contact) open book OB on M. We prove that if OB has Weinstein pages, then there exist a contact structure ξ′ on M, isotopic to ξ and supported by OB, and a contactomorphism f:(M,ξ)→(M,ξ′) such that the image f(L) of any such submanifold can be Legendrian isotoped so that it becomes disjoint from the closure of a page of OB.

We construct positive loops of Legendrian submanifolds in several instances. In particular, we partially recover G. Liu’s result stating that any loose Legendrian admits a positive loop, under some mild topological assumptions on the Legendrian. Moreover, we show contractibility of the constructed loops under an extra topological assumption.