Torus Actions On Symplectic Manifolds
Springer Science & Business Media, 27 sept. 2004 - 325 pages
How I have (re-)written this book The book the reader has in hand was supposed to be a new edition of . 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. There was absolutely no question of just correcting numerous misprints and a few mathematical errors. When I wrote the first edition, in 1989, 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). 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. So that this "new edition" is, indeed, another book.
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4-manifolds associated Assume bilinear form Check coadjoint commutative complex structure components cone consider coordinates Corollary corresponding critical points critical submanifold Deduce defined denoted diffeomorphism differential element equivariant Euler class Example exceptional orbit Exercise fiber fibration fixed points function fundamental vector field G-action gradient spheres graph group G Hamiltonian action Hamiltonian vector field Hence Hermitian Hirzebruch surface homotopy integral invariant IS8N isomorphism Kahler Lemma Lie algebra Lie group line bundle manifold of dimension matrix metric moduli space momentum mapping morphism nondegenerate normal bundle Notice open subset oriented periodic Hamiltonian Pn(C point of index polyhedron principal G-bundle projective space Proof Proposition Prove quotient real number regular levels regular values Remark S^action S1-action Seifert shown in Figure skew-symmetric smooth stabilizer subgroup submanifold subspace symplectic form symplectic manifold symplectic vector space theorem toric manifold torus action trivial vector bundle vector space vertex zero