2026-09-19T16:20:54Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/601892022-01-13T07:54:35Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Victor W. Guillemin.
He, Zhenqi, Ph. D. Massachusetts Institute of Technology
Massachusetts Institute of Technology. Dept. of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2010-12-06T17:35:39Z
2010-12-06T17:35:39Z
2010
2010
http://hdl.handle.net/1721.1/60189
681920423
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010.
Cataloged from PDF version of thesis.
Includes bibliographical references (p. 65-66).
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.
Ph.D.
66 p.
eng
Massachusetts Institute of Technology
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http://dspace.mit.edu/handle/1721.1/7582
Mathematics.
Odd dimensional symplectic manifolds by Zhenqi He.
Odd dimensional symplectic manifold
Thesis
application/pdf
Publication
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<Abstract>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.</Abstract>
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