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See provided URL for inquiries about permission. Metadata Show full item record. 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.

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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.

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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.

Submission history

Description Thesis Ph. Date issued Department Massachusetts Institute of Technology.

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Publisher Massachusetts Institute of Technology. Collections Mathematics - Ph. Torus Actions On Symplectic Manifolds.

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How I have re- written this book The book the reader has in hand was supposed to be a new edition of [14]. I have hesitated quite a long time before deciding to do the re-writing work-the first edition has been sold out for a few years.

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  7. There was absolutely no question of just correcting numerous misprints and a few mathematical errors. When I wrote the first edition, in , the convexity and Duistermaat-Heckman theorems together with the irruption of toric varieties on the scene of symplectic geometry, due to Delzant, around which the book was organized, were still rather recent less than ten years.

    Symplectic torus actions with coisotropic principal orbits

    I myself was rather happy with a small contribution I had made to the subject. I was giving a post-graduate course on all that and, well, these were lecture notes, just lecture notes. By chance, the book turned out to be rather popular: during the years since then, I had the opportunity to meet quite a few people 1 who kindly pretended to have learnt the subject in this book.

    However, the older book does not satisfy at all the idea I have now of what a good book should be.