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Yunhyung Cho

Publications and source records attributed to Yunhyung Cho.

33 records · Page 2Linked to original sources
arXiv2019

On the combinatorics of string polytopes

For a reduced word ${\bf i}$ of the longest element in the Weyl group of $\mathrm{SL}_{n+1}(\mathbb{C})$, one can associate the string cone $C_{\bf i}$ which parametrizes the dual canonical bases. In this paper, we classify all ${\bf i}$'s such that $C_{\bf i}$ is simplicial. We also prove that for any regular dominant weight $λ$ of $\mathfrak{sl}_{n+1}(\mathbb{C})$, the corresponding string polytope $Δ_{\bf i}(λ)$ is unimodularly equivalent to the Gelfand-Cetlin polytope associated to $λ$ if and only if $C_{\bf i}$ is simplicial. Thus we completely characterize Gelfand-Cetlin type string polytopes in terms of ${\bf i}$.

math.CO↗
arXiv2018

Hard Lefschetz Property for Hamiltonian torus actions on 6-dimensional GKM manifolds

In this paper, we study the hard Lefschetz property of a symplectic manifold which admits a Hamiltonian torus action. More precisely, let $(M,ω)$ be a 6-dimensional compact symplectic manifold with a Hamiltonian $T^2$-action. We will show that if the moment map image of $M$ is a GKM-graph and if the graph is index-increasing, then $(M,ω)$ satisfies the hard Lefschetz property.

math.SG↗
arXiv2018

Classification of six dimensional monotone symplectic manifolds admitting semifree circle actions I

Let $(M,ω_M)$ be a six dimensional closed monotone symplectic manifold admitting an effective semifree Hamiltonian $S^1$-action. We show that if the minimal (or maximal) fixed component of the action is an isolated point, then $(M,ω_M)$ is $S^1$-equivariant symplectomorphic to some Kähler Fano manifold $(X,ω_X, J)$ with a certain holomorphic $\mathbb{C}^*$-action. We also give a complete list of all such Fano manifolds and describe all semifree $\mathbb{C}^*$-actions on them specifically.

math.SG↗
arXiv2018

Lagrangian fibers of Gelfand-Cetlin systems

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 \emph{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 $\mathcal{F}(n)$ ($n \geq 3$) 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 $\mathcal{F}(3)$ is non-displaceable the question of which was raised by Nohara-Ueda who computed its Floer cohomology to be vanishing.

math.SG↗
arXiv2017

On the Chern numbers for pseudo-free circle actions

Let $(M,ψ)$ be a $(2n+1)$-dimensional oriented closed manifold equipped with a pseudo-free $S^1$-action $ψ: S^1 \times M \rightarrow M$. We first define a \textit{local data} $\mathcal{L}(M,ψ)$ of the action $ψ$ which consists of pairs $(C, (p(C) ; \overrightarrow{q}(C)))$ where $C$ is an exceptional orbit, $p(C)$ is the order of isotropy subgroup of $C$, and $\overrightarrow{q}(C) \in (\mathbb{Z}_{p(C)}^{\times})^n$ is a vector whose entries are the weights of the slice representation of $C$. In this paper, we give an explicit formula of the Chern number $\langle c_1(E)^n, [M/S^1] \rangle$ modulo $\mathbb{Z}$ in terms of the local data, where $E = M \times_{S^1} \mathbb{C}$ is the associated complex line orbibundle over $M/S^1$. Also, we illustrate several applications to various problems arising in equivariant symplectic topology.

math.SG↗
arXiv2016

On the f-vectors of Gelfand-Cetlin polytopes

A Gelfand-Cetlin polytope is a convex polytope obtained as an image of certain completely integrable system on a partial flag variety. In this paper, we give an equivalent description of the face structure of a GC-polytope in terms of so called the face structure of a ladder diagram. Using our description, we obtain a partial differential equation whose solution is the exponential generating function of f-vectors of GC-polytopes. This solves the open problem (2) posed by Gusev, Kritchenko, and Timorin in [GKT].

math.CO↗
arXiv2016

Embedded surfaces for symplectic circle actions

The purpose of this article is to characterize symplectic and Hamiltonian circle actions on symplectic manifolds in terms of symplectic embeddings of Riemann surfaces. More precisely, we will show that (1) if $(M,ω)$ admits a Hamiltonian $S^1$-action, then there exists an $S^1$-invariant symplectic $2$-sphere $S$ in $(M,ω)$ such that $\langle c_1(M), [S] \rangle > 0$, and (2) if the action is non-Hamiltonian, then there exists an $S^1$-invariant symplectic $2$-torus $T$ in $(M,ω)$ such that $\langle c_1(M), [T] \rangle = 0$. As applications, we will give a very simple proof of the following well-known theorem which was proved by Atiyah-Bott \cite{AB}, Lupton-Oprea \cite{LO}, and Ono \cite{O2} : suppose that $(M,ω)$ is a smooth closed symplectic manifold satisfying $c_1(TM)=λ\cdot [ω]$ for some $λ\in \R$ and let $G$ be a compact connected Lie group acting effectively on $M$ preserving $ω$. Then (1) if $λ< 0$, then $G$ must be trivial, (2) if $λ=0$, then the $G$-action is non-Hamiltonian, and (3) if $λ> 0$, then the $G$-action is Hamiltonian.

math.SG↗
arXiv2016

Hard Lefschetz property of symplectic structures on compact Kaehler manifolds

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 McDuff as follows. According to symplectic Hodge theory, any symplectic form $ω$ on a smooth manifold $M$ defines \textit{symplectic harmonic forms} on $M$. In \cite{Yan}, Khesin and McDuff posed a question whether there exists a path of symplectic forms $\{ω_t \}$ such that the dimension $h^k_{hr}(M,ω)$ of the space of \textit{symplectic harmonic $k$-forms} varies along $t$. By \cite{Yan} and \cite{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. \cite{Dr}).

math.SG↗
arXiv2015

Unimodality of the Betti numbers for Hamiltonian circle actions with index-increasing moment maps

The unimodality conjecture posed by Tolman in the conference `Moment maps in Various Geometry" in 2005 states that if (M,w) is a 2n-dimensional smooth compact symplectic manifold equipped with a Hamiltonian circle action with only isolated fixed points, then the sequence of Betti numbers is unimodal. Recently, the author and M. Kim proved that the unimodality holds in eight-dimensional cases by using equivariant cohomology theory. In this paper, we generalize the idea in \cite{CK} to an arbitrary dimensional case. Also, we prove the conjecture in arbitrary dimension with an assumption that a moment map "index-increasing."

math.SG↗
arXiv2013

Hamiltonian circle action with self-indexing moment map

Let $(M,ω)$ be a $2n$-dimensional smooth compact symplectic manifold equipped with a Hamiltonian circle action with only isolated fixed points and let $μ: M \rightarrow \R$ be a corresponding moment map. Let $Λ_{2k}$ be the set of all fixed points of index $2k$. In this paper, we will show that if $μ$ is constant on $Λ_{2k}$ for each $k$, then $(M,ω)$ satisfies the hard Lefschetz property. In particular, if $(M,ω)$ admits a self-indexing moment map, i.e. $μ(p) = 2k$ for every $p \in Λ_{2k}$ and $k=0,1,\cdots,n,$ then $(M,ω)$ satisfies the hard Lefschetz property.

math.SG↗
arXiv2013

Semifree Hamiltonian circle actions on 6-dimensional symplectic manifolds with non-isolated fixed point set

Let $(M, ω)$ be a 6-dimensional closed symplectic manifold with a symplectic $S^1$-action with $M^{S^1} \neq \emptyset$ and $\dim M^{S^1} \leq 2$. Assume that $ω$ is integral with a generalized moment map $μ$. We first prove that the action is Hamiltonian if and only if $b_2^+(M_{\red})=1$, where $M_{\red}$ is any reduced space with respect to $μ$. It means that if the action is non-Hamiltonian, then $b_2^+(M_{\red}) \geq 2$. Secondly, we focus on the case when the action is semifree and Hamiltonian. We prove that if $M^{S^1}$ consists of surfaces, then the number $k$ of fixed surfaces with positive genera is at most four. In particular, if the extremal fixed surfaces are spheres, then $k$ is at most one. Finally, we prove that $k \neq 2$ and we construct some examples of 6-dimensional semifree Hamiltonian $S^1$-manifolds such that $M^{S^1}$ contains $k$ surfaces of positive genera for $k = 0$ and 4. Examples with $k=1$ and 3 were given in \cite{L2}.

math.SG↗
arXiv2013

An example of circle actions on symplectic Calabi-Yau manifolds with non-empty fixed points

Let $(X,σ,J)$ be a compact Kähler Calabi-Yau manifold equipped with a symplectic circle action. By Frankel's theorem \cite{F}, the action on $X$ is non-Hamiltonian and $X$ does not have any fixed point. In this paper, we will show that a symplectic circle action on a compact non-Kähler symplectic Calabi-Yau manifold may have a fixed point. More precisely, we will show that the symplectic $S^1$-manifold constructed by D. McDuff \cite{McD} has the vanishing first Chern class. This manifold has the Betti numbers $b_1 = 3$, $b_2 = 8$, and $b_3 = 12$. In particular, it does not admit any Kähler structure.

math.SG↗
arXiv2011

Log-concavity of complexity one Hamiltonian torus actions

Let $(M,ω)$ be a closed $2n$-dimensional symplectic manifold equipped with a Hamiltonian $T^{n-1}$-action. Then Atiyah-Guillemin-Sternberg convexity theorem implies that the image of the moment map is an $(n-1)$-dimensional convex polytope. In this paper, we show that the density function of the Duistermaat-Heckman measure is log-concave on the image of the moment map.

math.SG↗