Large Radius Limit and SYZ Fibrations of Hyper-Kähler Manifold

In this paper the relations between the existence of Lagrangian fibration of Hyper-Kähler 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-Kähler manifold N of complex dimension 2n ≥ 4 is at least 5, then there exists an unipotent element T in the mapping class group Γ(N) such that its action on the second cohomology group satisfies (T − id) 6= 0 and (T − id) = 0. A Theorem of Verbitsky implies that the symmetric power Sn(T ) acts on H and it satisfies (Sn(T ) − id) 6= 0 and (Sn(T ) − id) = 0. This fact established the existence of Large Radius Limit for Hyper-Kähler manifolds for polarized algebraic HyperKähler manifolds. Using the theory of vanishing cycles it is proved that if a Hyper-Kähler 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-Kähler 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-Kähler manifolds of fix dimension.