Generalized special Lagrangian torus fibration for Calabi-Yau hypersurfaces in toric varieties I

Abstract

In this paper we start the program of constructing generalized special Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric varieties near the large complex limit, with respect to the restriction of a toric metric on the toric variety to the Calabi-Yau hypersurface. The construction is based on the deformation of the standard toric generalized special Lagrangian torus fibration of the large complex limit X-0. In this paper, we will deal with the region near the smooth top dimensional torus fibers of X-0 and its mirror dual situation - the region near the 0-dimensional fibers of X-0.