  We interpret symplectic geometry as certain sheaf theory by constructing a
sheaf of curved A_\infty algebras which in some sense plays the role of a
"structure sheaf" for symplectic manifolds. An interesting feature of this
"structure sheaf" is that the symplectic form itself is part of its curvature
term. Using this interpretation homological mirror symmetry can be understood
by well-known duality theories in mathematics: Koszul duality or Fourier- Mukai
transform. In this paper we perform the above constructions over a small open
subset inside the smooth locus of a Lagrangian torus fibration. In a subsequent
work we shall use the language of derived geometry to obtain a global theory
over the whole smooth locus. However we do not know how to extend this
construction to the singular locus. As an application of the local theory we
prove a version of homological mirror symmetry between a toric symplectic
manifold and its Landau-Ginzburg mirror.
