Properties of Hamiltonian Torus Actions on Closed Symplectic Manifolds
In this thesis, we will study the properties of certain Hamiltonian torus actions on closed symplectic manifolds. First, we will consider counting Hamiltonian torus actions on closed, symplectic manifolds M with 2-dimensional second cohomology. In particular, all such manifolds are bundles with fiber and base equal to projective spaces. We use cohomological techniques to show that there is a unique toric structure if the fiber has a smaller dimension than the base. Furthermore, if the fiber and base are both at least 2-dimensional projective spaces, we show that there is a finite number of toric structures on M that are compatible with some symplectic structure on M. Additionally, we show there is uniqueness in certain other cases, such as the case where M is a monotone symplectic manifold. Finally, we will be interested in the existence of symplectic, non-Hamiltonian circle actions on closed symplectic 6-manifolds. In particular, we will use J-holomorphic curve techniques to show that there are no such actions that satisfy certain fixed point conditions. This lends support to the conjecture that there are no such actions with a non-empty set of isolated fixed points.
AC:P:20455
Academic Commons
Fanoe, Andrew L.
Author
McDuff, Dusa
Thesis advisor
Columbia University. Mathematics
Originator
Theses
English
Mathematics
text
2013
eng
2017-06-08T13:56:48Z
2017-06-08T15:37:49Z
Ph.D.
2
Mathematics
Columbia University
10.7916/D8CF9X9J