Large Radius Limit and SYZ Fibrations of Hyper-Kahler Manifolds

Details Large Radius Limit and SYZ Fibrations of Hyper-Kahler Manifolds Large Radius Limit and SYZ Fibrations of Hyper-Kahler Manifolds

2003-08-22

arxiv.org/abs/math/0308210v2

Mathematics

Symplectic Geometry

two references are added and the names of two mathematicians are corrected

Scientific paper

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 $2n\geq4$ is at least 5, then there exists an unipotent element T in the mapping class group $\Gamma $(N) such that its action on the second cohomology group satisfies $(T-id)^{2}\neq0$ 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.

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