相关论文: Displacing Lagrangians in the manifolds of full fl…
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…
We show that in many examples the non-displaceability of Lagrangian submanifolds by Hamiltonian isotopy can be proved via Lagrangian Floer cohomology with non-unitary line bundle. The examples include all monotone Lagrangian torus fibers in…
A Gelfand-Cetlin system is a completely integrable system defined on a partial flag manifold whose image is a rational convex polytope called a Gelfand-Cetlin polytope. Motivated by the study of Nishinou-Nohara-Ueda on the Floer theory of…
We prove that the two-step flag variety $\mathcal{F}\ell(1,n;n+1)$ carries a non-displaceable and non-monotone Lagrangian Gelfand--Zeitlin fiber diffeomorphic to $S^3 \times T^{2n-4}$ and a continuum family of non-displaceable Lagrangian…
Using the bulk-deformation of Floer cohomology by Schubert cycles and non-Archimedean analysis of Fukaya--Oh--Ohta--Ono's bulk-deformed potential function, we prove that every complete flag manifold $\mathrm{Fl}(n)$ ($n \geq 3$) with a…
In this note we analyze displaceability of pre-Lagrangian toric fibers in contact toric manifolds. While every symplectic toric manifold contains at least one non-displaceable Lagrangian toric fiber and infinitely many displaceable ones, we…
In this paper we introduce a new way of displacing Lagrangian fibers in toric symplectic manifolds, a generalization of McDuff's original method of probes. Extended probes are formed by deflecting one probe by another auxiliary probe. Using…
(Non-)displaceability of fibers of integrable systems has been an important problem in symplectic geometry. In this paper, for a large class of classical Liouville integrable systems containing the Lagrangian top, the Kovalevskaya top and…
In this paper we construct and classify Lagrangian T^3-fibrations on non compact symplectic manifolds with singular fibres of prescribed topological type. This contributes to the understanding of the structure of the singular fibres that…
In this paper, we show that every singular fiber of the Gelfand--Cetlin system on coadjoint orbits of unitary groups is a smooth isotropic submanifold which is diffeomorphic to a $2$-stage quotient of a compact Lie group by free actions of…
We consider symplectic fibrations as in Guillemin-Lerman-Sternberg, and derive a spectral sequence to compute the Floer cohomology of certain fibered Lagrangians sitting inside a compact symplectic fibration with small monotone fibers and a…
In this paper, we give a formula for the Maslov index of a gradient holomorphic disc, 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…
This article explores special Lagrangian submanifolds in $\mathbb{CP}^3$, viewed as a nearly K\"ahler manifold, from two different perspectives. Intrinsically, using a moving frame set-up, and extrinsically, using $\mathrm{SU}(2)$…
We consider open symplectic manifolds which admit dilations (in the sense previously introduced by Solomon and the author). We obtain restrictions on collections of Lagrangian submanifolds which are pairwise disjoint (or pairwise…
Let n>3, and let L be a Lagrangian embedding of an n-disk into the cotangent bundle of n-dimensional Euclidean space that agrees with the cotangent fiber over a non-zero point x outside a compact set. Assume that L is disjoint from the…
We show that the Lagrangian torus in the cotangent bundles of the 2-sphere obtained by applying the geodesic flow to the unit circle in a fibre is not displaceable by computing its Lagrangian Floer homology. The computation is based on a…
This note studies the geometric structure of monotone moment polytopes (the duals of smooth Fano polytopes) using probes. The latter are line segments that enter the polytope at an interior point of a facet and whose direction is integrally…
In the framework of Special Bohr - Sommerfeld geometry it was established that an ample divisor in compact algebraic variety can define almost canonically certain real submanifold which is lagrangian with respect to the corresponding Kahler…
We study special Lagrangian fibrations of $\mathrm{SU}(3)$-manifolds, not necessarily torsion-free. In the case where the fiber is a unimodular Lie group $G$, we decompose such $\mathrm{SU}(3)$-structures into triples of solder 1-forms,…
We study the following rigidity problem in symplectic geometry:can one displace a Lagrangian submanifold from a hypersurface? We relate this to the Arnold Chord Conjecture, and introduce a refined question about the existence of relative…