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In my previous work (M. Damian, Ann Sci Ec Nom Sup 4ème série 48(1):237–252, 2015) I conjectured that any closed orientable monotone Lagrangian submanifoldLagrangian submanifoldSubmanifoldLagrangian in \({\mathbf {C}}^n\) is the total space of a fibrationSpace over the circleFibration over the circleFibration over the circle. In the present paper I prove that this statement is true in dimension greater than 6 under some topological hypothesis on the universal coverUniversal cover of the Lagrangian. More generally, the proof is valid for any symplectic ambient manifold with vanishing second homology groupHomology groupGrouphomology, provided that the Lagrangian is displaceable through a Hamiltonian isotopyHamiltonian isotopyIsotopyHamiltonian. A generalisation for monotone Lagrangians in \({\mathbf {CP}}^n\) is also proved.

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On the Topology of Monotone Lagrangians of High Dimension

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  • 发表日期：

    2025年

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  • Mihai Damian

摘要

In my previous work (M. Damian, Ann Sci Ec Nom Sup 4ème série 48(1):237–252, 2015) I conjectured that any closed orientable monotone Lagrangian submanifoldLagrangian submanifoldSubmanifoldLagrangian in \({\mathbf {C}}^n\) is the total space of a fibrationSpace over the circleFibration over the circleFibration over the circle. In the present paper I prove that this statement is true in dimension greater than 6 under some topological hypothesis on the universal coverUniversal cover of the Lagrangian. More generally, the proof is valid for any symplectic ambient manifold with vanishing second homology groupHomology groupGrouphomology, provided that the Lagrangian is displaceable through a Hamiltonian isotopyHamiltonian isotopyIsotopyHamiltonian. A generalisation for monotone Lagrangians in \({\mathbf {CP}}^n\) is also proved.