Classification of equivariant vector bundles over real projective plane

We classify equivariant topological complex vector bundles over real projective plane under a compact Lie group (not necessarily effective) action. It is shown that nonequivariant Chern classes and isotropy representations at (at most) three points are sufficient to classify equivariant vector bundles over real projective plane except one case. To do it, we relate the problem to classification on two-sphere through the covering map because equivariant vector bundles over two-sphere have been already classified

Frankel's theorem in the symplectic category

We prove that if an (n-1)-dimensional torus acts symplectically on a 2n-dimensional manifold, then the action has a fixed point if and only if the action is Hamiltonian. One may regard it as a symplectic version of Frankel theorem. The case of n=2 is the well known theorem of McDuff. From the well known example of McDuff (a six dimensional symplectic non-Hamiltonian circle action with fixed tori) and its products with copies of a two dimensional sphere with the usual rotation, the condition on the dimension of the acting torus is optimal to obtain the result.Comment: 13 pages, includes some figures. Accepted by Transactions AM

Equivariant pointwise clutching maps

In the paper, we introduce the terminology equivariant pointwise clutching map. By using this, we give details on how to glue an equivariant vector bundle over a finite set so as to obtain a new Lie group representation such that the quotient map from the bundle to the representation is equivariant. Then, we investigate the topology of the set of all equivariant pointwise clutching maps with respect to an equivariant vector bundle over a finite set. Results of the paper play a key role in classifying equivariant vector bundles over two-surfaces in other papers.Comment: 20 pages, 3 figure

Classification of equivariant vector bundles over two-torus

We exhaustively classify topological equivariant complex vector bundles over two-torus under a compact Lie group (not necessarily effective) action. It is shown that inequivariant Chern classes and isotropy representations at (at most) six points are sufficient to classify equivariant vector bundles except a few cases. To do it, we calculate homotopy of the set of equivariant clutching maps. And, classification on real projective plane, Klein bottle will appear soonComment: 45page

Unimodality of the Betti numbers for Hamiltonian circle action with isolated fixed points

Let $(M,\omega)$ be an eight-dimensional closed symplectic manifold equipped with a Hamiltonian circle action with only isolated fixed points. In this article, we will show that the Betti numbers of $M$ are unimodal, i.e. $b_0(M) \leq b_2(M) \leq b_4(M)$.Comment: 4page

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

Let $(X,\sigma,J)$ be a compact K\"{a}hler 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-K\"{a}hler 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 K\"{a}hler structure.Comment: 15 page

Crystal \Bla in B(∞)B(\infty) for G2G_2 type Lie Algebra

A previous work gave a combinatorial description of the crystal $B(\infty)$, in terms of certain simple Young tableaux referred to as the marginally large tableaux, for finite dimensional simple Lie algebras. Using this result, we present an explicit description of the crystal $B(\lambda)$, in terms of the marginally large tableaux, for the $G_2$ Lie algebra type. We also provide a new description of B(\la), in terms of Nakajima monomials, that is in natural correspondence with our tableau description.Comment: 13 pages, submitte

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 $(M,\omega)$ 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 $(M,\omega)$ satisfies the hard Lefschetz property.Comment: 25 pages, final version (to appear in J. Symplectic Geom.

Hamiltonian circle action with self-indexing moment map

Let $(M,\omega)$ be a $2n$-dimensional smooth compact symplectic manifold equipped with a Hamiltonian circle action with only isolated fixed points and let $\mu : M \rightarrow \R$ be a corresponding moment map. Let $\Lambda_{2k}$ be the set of all fixed points of index $2k$. In this paper, we will show that if $\mu$ is constant on $\Lambda_{2k}$ for each $k$, then $(M,\omega)$ satisfies the hard Lefschetz property. In particular, if $(M,\omega)$ admits a self-indexing moment map, i.e. $\mu(p) = 2k$ for every $p \in \Lambda_{2k}$ and $k=0,1,\cdots,n,$ then $(M,\omega)$ satisfies the hard Lefschetz property.Comment: 7 pages, 2 figure

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 $(M,\omega)$ admits a Hamiltonian $S^1$-action, then there exists an $S^1$-invariant symplectic $2$-sphere $S$ in $(M,\omega)$ 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 $(M,\omega)$ 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 $(M,\omega)$ is a smooth closed symplectic manifold satisfying $c_1(TM)=\lambda \cdot [\omega]$ for some $\lambda \in \R$ and let $G$ be a compact connected Lie group acting effectively on $M$ preserving $\omega$. Then (1) if $\lambda < 0$, then $G$ must be trivial, (2) if $\lambda=0$, then the $G$-action is non-Hamiltonian, and (3) if $\lambda > 0$, then the $G$-action is Hamiltonian.Comment: 16 page