Odd dimensional symplectic manifolds by Zhenqi He.

In this thesis, we introduce the odd dimensional symplectic manifolds. In the first half we study the Hodge theory on the basic symplectic manifolds. We can define two cohomology theories on them, the standard basic de Rham cohomology gheory and a basic version of the Koszul-Brylinski-Mathieu 'harmonic' symplectic cohomology theory. Among our main results are a collection of examples for which these cohomology theories don't coincide, and, in fact, for which the usual basic cohomology theory is infinite dimensional and the symplectic cohomology theory is finite dimensional. On the other hand, we prove an odd version of the Mathieu theorem and the do-lemma: the two theories coincide if and only if a basic version of strong Lefschetz property holds. In the second half, we discuss the group actions on odd dimensional symplectic manifolds. In particular, we study the Hamiltonian group actions. Finally we use the Local-Global-Principle to prove a convexity theorem for the Hamiltonian torus actions on odd dimensional symplectic manifolds.

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2010

Author dc:creator

• He, Zhenqi, Ph. D. Massachusetts Institute of Technology

Advisor dc:contributor.advisor

• Victor W. Guillemin.

• Mathematics.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/60189

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/60189