Motivated by the study of Nishinou-Nohara-Ueda on the Floer thoery of Gelfand-Cetlin systems over complex partial flag manifolds, we provide a complete description of the topology of Gelfand-Cetlin fibers. We prove that all fibers are smooth isotropic submanifolds and give a complete description of the fiber to be Lagrangian in terms of combinatorics of Gelfand-Cetlin polytope. Then we study (non-)displaceability of Lagrangian fibers. After a few combinatorial and numercal tests for the displaceability, using the bulk-deformation of Floer cohomology by Schubert cycles, we prove that every full flag manifold F (n) (n ≥ 3) with a monotone Kirillov-Kostant-Souriau symplectic form carries a continuum of non-displaceable Lagrangian tori which degenerates to a non-torus fiber in the Hausdorff limit. In particular, the Lagrangian S 3 -fiber in F (3) is non-displaceable the question of which was raised by Nohara-Ueda who computed its Floer cohomology to be vanishing. CONTENTS 1. Introduction 1 Part 1. The topology of Gelfand-Cetlin fibers 3 2. Introduction to Part 1. 3 3. Gelfand-Cetlin systems 5 4. Ladder diagram and its face structure 9 5. Classification of Lagrangian fibers 12 6. Iterated bundle structures on Gelfand-Cetlin fibers 22 7. Degenerations of fibers to tori 31 Part 2. Non-displaceability of Lagrangian fibers 38 8. Introduction of Part 2 38 9. Criteria for displaceablility of Gelfand-Cetlin fibers 42 10. Lagrangian Floer theory on Gelfand-Cetlin systems 49 11. Decompositions of the gradient of potential function 58 12. Solvability of split leading term equation 70 13. Calculation of potential function deformed by Schubert cycles 78 References 82
This is the second of the series of papers on the classification of six-dimensional closed monotone symplectic manifold admitting a semifree Hamiltonian [Formula: see text]-action. In [Y. Cho, Classification of six-dimensional monotone symplectic manifolds admitting semifree circle actions I, Int. J. Math. 6 1950032], we dealt with the case where at least one of the extremal fixed point is isolated and proved that every such manifold is Kähler Fano. In this paper, we show that if the maximal and the minimal fixed components are both two-dimensional, then the manifold is [Formula: see text]-equivariantly symplectomorphic to some Kähler Fano manifold with a certain holomorphic Hamiltonian [Formula: see text]-action. We also give a complete list of such Fano manifolds together with an explicit description of the [Formula: see text]-actions.
In this paper, we give a new method to construct a compact symplectic manifold which does not satisfy the hard Lefschetz property. Using our method, we construct a simply connected compact Kähler manifold (M, J, ω) and a symplectic form σ on M which does not satisfy the hard Lefschetz property, but is symplectically deformation equivalent to the Kähler form ω. As a consequence, we can give an answer to the question posed by Khesin and Mc-Duff as follows. According to symplectic Hodge theory, any symplectic form ω on a smooth manifold M defines symplectic harmonic forms on M . In [Yan], Khesin and McDuff posed a question whether there exists a path of symplectic forms {ωt} such that the dimension h k hr (M, ω) of the space of symplectic harmonic k-forms varies along t. By [Yan] and [Ma], the hard Lefschetz property holds for (M, ω) if and only if h k hr (M, ω) is equal to the Betti number b k (M ) for all k > 0. Thus our result gives an answer to the question. Also, our construction provides an example of compact Kähler manifold whose Kähler cone is properly contained in the symplectic cone (c.f. [Dr]).Date: October 29, 2018. 2010 Mathematics Subject Classification. 53D20(primary), and 53D05(secondary).
In this paper, we give a formula for the Maslov index of a gradient holomorphic disk, which is a relative version of the Chern number formula of a gradient holomorphic sphere for a Hamiltonian $S^1$-action. Using the formula, we classify all monotone Lagrangian fibers of Gelfand–Cetlin systems on partial flag manifolds.
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