  A holomorphic Lagrangian fibration is stable if the characteristic cycles of
the singular fibers are of type $I_m, 1 \leq m <\infty,$ or $A_{\infty}$. We
will give a complete description of the local structure of a stable Lagrangian
fibration when it is principally polarized. In particular, we give an explicit
form of the period map of such a fibration and conversely, for a period map of
the described type, we construct a principally polarized stable Lagrangian
fibration with the given period map. This enables us to give a number of
examples exhibiting interesting behavior of the characteristic cycles.
