Lagrangian fibers of Gelfand-Cetlin systems

Singular fibres of the Gelfand–Cetlin system on MANI( n )*

“…-Unlike the toric case, where the singularities are all elliptic non-degenerate (in the sense of Vey-Eliasson; see, for example, [8,[10][11][12][13] for non-degenerate singularities), the Gelfand-Cetlin system admits many degenerate singularities. -The degenerate singular fibres of the Gelfand-Cetlin system are very peculiar in the sense that they are all smooth isotropic submanifolds, as will be shown in this paper (see also Cho et al [14], who obtained the same result by different methods), while many degenerate (and non-degenerate) singular fibres of other integrable Hamiltonian systems are singular varieties (see, for example, [7,[15][16][17] for various results about degenerate singularities). -It turns out that the Gelfand-Cetlin systems can be obtained by the method of toric degenerations; see Nishinou et al [18].…”

“…This paper is the result of a project dating back to 2006, during which the singularities of the Gelfand-Cetlin system were studied from the point of view of the general topological theory of integrable Hamiltonian systems and their singularities. Unlike some other papers on the subject, such as [14,18], which are mainly motivated by algebraic geometry, our work is mainly motivated by dynamical systems.…”

A critical point analysis of Landau–Ginzburg potentials with bulk in Gelfand–Cetlin systems

“…In his dissertation, Kogan [23] found an expression of a Schubert class in terms of a certain union of the inverse images of faces in the GC system of a complete flag manifold (see also Kogan and Miller [24]). Due to presence of nontorus fibers in [6], the inverse image of a single face might have boundary so that it does not form a cycle. What he proved is that a certain combination of faces can form a cycle because the boundaries are cancelled out.…”

“…This article is a sequel to Part I of [6]. Here we focus on detecting nondisplaceable Lagrangian fibers of Gelfand-Cetlin (GC) systems.…”

“…Motivated by those works, we recently undertook a systematic study of topological/geometrical types of GC fibers and provided a complete classification thereof in terms of the combinatorial ladder diagram in [6]. In this paper, we discuss nondisplaceable GC fibers on complete flag manifolds equipped with monotone Kirillov-Kostant-Souriau symplectic forms (KKS forms for short).…”