4,644 research outputs found
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 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 are unimodal, i.e. .Comment: 4page
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 be a
6-dimensional compact symplectic manifold with a Hamiltonian -action. We
will show that if the moment map image of is a GKM-graph and if the graph
is index-increasing, then 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 be a -dimensional smooth compact symplectic manifold
equipped with a Hamiltonian circle action with only isolated fixed points and
let be a corresponding moment map. Let
be the set of all fixed points of index . In this paper, we will show that
if is constant on for each , then
satisfies the hard Lefschetz property. In particular, if admits a
self-indexing moment map, i.e. for every and
then satisfies the hard Lefschetz property.Comment: 7 pages, 2 figure
An example of circle actions on symplectic Calabi-Yau manifolds with non-empty fixed points
Let be a compact K\"{a}hler Calabi-Yau manifold equipped with
a symplectic circle action. By Frankel's theorem \cite{F}, the action on is
non-Hamiltonian and 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 -manifold constructed by D. McDuff \cite{McD} has the
vanishing first Chern class. This manifold has the Betti numbers ,
, and . In particular, it does not admit any K\"{a}hler
structure.Comment: 15 page
Crystal \Bla in for type Lie Algebra
A previous work gave a combinatorial description of the crystal ,
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 , in terms of the
marginally large tableaux, for the 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
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 admits a Hamiltonian
-action, then there exists an -invariant symplectic -sphere in
such that , and (2) if the action
is non-Hamiltonian, then there exists an -invariant symplectic
-torus in such that .
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 is a smooth closed symplectic
manifold satisfying for some
and let be a compact connected Lie group acting effectively on
preserving . Then (1) if , then must be trivial, (2)
if , then the -action is non-Hamiltonian, and (3) if , then the -action is Hamiltonian.Comment: 16 page
- β¦