Lagrangian fibers of Gelfand-Cetlin systems

Abstract

The Gelfand-Cetlin system \(\Phi_\lambda : \mathcal{O}_\lambda \rightarrow \mathbb{R}^n\) is a completely integrable system on a partial flag manifold \((\mathcal{O}_\lambda,\omega_\lambda)\) whose image is a convex polytope \(\triangle_\lambda \subset \mathbb{R}^n\). In the first part of this paper, we are concerned with the topology of Gelfand-Cetlin fibers. We first show that every Gelfand-Cetlin fiber is an isotropic submanifold of \((\mathcal{O}_\lambda, \omega_\lambda)\) and it is an iterated bundle where the fiber at each stage is either a point or a product of odd dimensional spheres. Then, we classify all Lagrangian Gelfand-Cetlin fibers. Also, we show that every fiber over a point lying on the relative interior of an \(r\)-dimensional face of \(\triangle_\lambda\) has a trivial \(r\)-dimensional torus factor, i.e., the diffeomorphic type is \((S^1)^r \times Y\) for some smooth manifold \(Y\). Furthermore, we show that a toric degeneration of \((\mathcal{O}_\lambda, \omega_\lambda, \Phi_\lambda)\) into the Gelfand-Cetlin toric variety can be understood as a fiberwise degeneration \((S^1)^r \times Y \rightarrow (S^1)^r\) by contracting \(Y\) to a point. The second part is devoted to detect displaceable and non-displaceable Lagrangian fibers. We discuss a couple of combinatorial and numerical tests for displaceability of fibers. As a byproduct, all non-torus Lagrangian fibers of \(\mathrm{Gr}(2,p)\) for every prime number \(p\) are shown to be displaceable. We then prove that the Gelfand-Cetlin system on every complete flag manifold \(\mathcal{F}(n)\) \((n \geq 3)\) with a monotone \(\omega_\lambda\) carries a continuum of non-displaceable Lagrangian torus fibers and several non-displaceable Lagrangian non-torus fibers. As a special case, the Lagrangian \(S^3\)-fiber in \(\mathcal{F}(3)\) is non-displaceable, the question of which was raised by Nohara-Ueda who computed its Floer cohomology to be zero.