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

Local structure of principally polarized stable Lagrangian fibrations