Large Radius Limit and SYZ Fibrations of Hyper-Kahler Manifolds

In this paper the relations between the existence of Lagrangian fibration of Hyper-K"{a}hler manifolds and the existence of the Large Radius Limit is established. It is proved that if the the rank of the second homology group of a Hyper-K"{a}hler manifold N of complex dimension $2n\ge 4$ is at least 5, then there exists an unipotent element T in the mapping class group $\mathrm{\Gamma }$(N) such that its action on the second cohomology group satisfies $(T-id{)}^2\mathrm{\ne }0$ and $(T-id{)}^3=0.$ A Theorem of Verbitsky implies that the symmetric power $S^n(T)$ acts on $H^{2n}$ and it satisfies $(S^{n}% (T)-id)^{2n}\neq0$and$(S^{n}(T)-id)^{2n+1}=0.$ This fact established the existence of Large Radius Limit for Hyper-K"{a}hler manifolds for polarized algebraic Hyper-K"{a}hler manifolds. Using the theory of vanishing cycles it is proved that if a Hyper-K"{a}hler manifold admits a Lagrangian fibration then the rank of the second homology group is greater than or equal to five. It is also proved that the fibre of any Lagrangian fibration of a Hyper-K"{a}hler manifold is homological to a vanishing invariant $2n$ cycle of a maximal unipotent element acting on the middle homology. According to Clemens this vanishing invariant cycle can be realized as a torus. I conjecture that the SYZ conjecture implies finiteness of the topological types of Hyper-K"{a}hler manifolds of fix dimension.