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Properties of Hamiltonian Torus Actions on Closed Symplectic Manifolds

Andrew L. Fanoe

Title:
Properties of Hamiltonian Torus Actions on Closed Symplectic Manifolds
Author(s):
Fanoe, Andrew L.
Thesis Advisor(s):
McDuff, Dusa
Date:
Type:
Theses
Degree:
Ph.D., Columbia University
Department(s):
Mathematics
Persistent URL:
Abstract:
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.
Subject(s):
Mathematics
Item views
824
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Suggested Citation:
Andrew L. Fanoe, , Properties of Hamiltonian Torus Actions on Closed Symplectic Manifolds, Columbia University Academic Commons, .

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