Projective embeddings and Lagrangian fibrations of Abelian varieties

It is well known that every Abelian variety can be embedded into projective spaces by theta functions and the basis of theta functions are determined by choosing a Lagrangian fibration. In this paper, we prove that the restriction of natural Lagrangian fibrations (moment maps) of projective spaces converge to that of the Abelian variety in "the Gromov-Hausdorff topology''. This is, in some sense, a Lagrangian fibration version of the convergence theorem of G. Tian [6] and S. Zelditch [7] for Kahler metrics.