Large Radius Limit and SYZ Fibrations of Hyper-Kahler Manifolds

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dc.creatorTodorov, Andrey
dc.date2003-08-22
dc.date2003-08-26
dc.date.accessioned2026-07-07T05:00:33Z
dc.date.available2026-07-07T05:00:33Z
dc.descriptionIn 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 $2n\geq4$ 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 $(T-id)^{2}\neq0$ 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-Kähler manifolds for polarized algebraic Hyper-Kähler 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.
dc.descriptiontwo references are added and the names of two mathematicians are corrected
dc.identifierhttps://arxiv.org/abs/math/0308210
dc.identifierhttp://arxiv.org/abs/math/0308210
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/68364
dc.subjectSymplectic Geometry
dc.subject14D20
dc.titleLarge Radius Limit and SYZ Fibrations of Hyper-Kahler Manifolds
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