Symplectic manifolds specialfeaturesofdimensionfour thesymplecticsum some open questions most of what ive discussed so far about the topology of symplectic fourmanifolds applies equally well to the topology of kahler surfaces. Symplectic manifolds are worth studying for their own sake. A metaplectic structure on a symplectic or almost symplectic manifold is in turn lift of the structure group to the metaplectic group. Im neither a physicist nor particularly wellversed in symplectic geometry, so take this answer with a grain of salt.
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, equipped with a closed nondegenerate differential 2form, called the symplectic form. Staneleyreisner rings, torus actions, toric varieties, quasitoric manifolds, momentangle complexes, subspace arrangements abstract. Following work of delzant and karshon, hamiltonian circle and 2. This trick reduces many questions concerning poisson actions to their well known counterparts from the theory of hamiltonian gactions. Buy a brief introduction to symplectic and contact manifolds nankai tracts in mathematics on free shipping on qualified orders. All such actions are not compatible with any symplectic form on the manifold in question. Finally we use the localglobalprinciple to prove a convexity theorem for the hamiltonian torus actions on odd dimensional symplectic manifolds.
New perspectives in differential geometry rome, 1620 november 2015 vicente munoz ucmicmat manifolds which are complex and symplectic but not kahler 1620 nov 2015 1 26. The topology of torus actions on symplectic manifolds. Pdf semifree hamiltonian circle actions on 6dimensional. Symplectic reduction of 4 manifolds with circle actions. A symplectic manifold is, by definition, a smooth connected 2ndimensional manifold equipped with a symplectic form, the latter being simply a closed 2form with the property that its nfold wedge power does not vanish. We introduce the notion of a strict quantization of a symplectic manifold and show its existence under a topological condition. We now turn to the question of the equivalence of two symplectic manifolds or the embedding of one symplectic manifold inside. Buy torus actions on symplectic manifolds progress in mathematics on free shipping on qualified orders. Local symplectic contractions consider a local symplectic contractions. Nonsymplectic 4manifolds with one basic class ronaldfintushelandronaldj. Secondly a symplectic change of coordinates preserve hamiltonian structure see proposition 3. Semifree hamiltonian circle actions on 6dimensional symplectic manifolds with nonisolated fixed point set article pdf available in journal of symplectic geometry 4 may 2010 with 83 reads. Nonsymplectic smooth circle actions on symplectic manifolds.
The short motivation is just that physicists realized that velocity and momentum are important concepts. Symplectic actions of two tori on four manifolds alvaro pelayo. Toric actions on b symplectic manifolds request pdf. The first six sections of these notes contain a description of some of the basic constructions and results on symplectic manifolds and lagrangian submanifolds. Y is called a symplectic contraction if x is a symplectic manifold, y is normal and. In chapters ii and iii, symplectic geometry and hamiltonian group actions are introduced, especially torus actions and actionangle variables. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the. Furthermore, these manifolds admit no symplectic structure with either orientation. Audins polarized symplectic manifolds, which are smooth symplectic manifolds endowed with a morsebott function having only two critical valuesa minimum, which is attained on a. On the existence of symplectic forms on open manifolds maarten mol a thesis submitted to the department of mathematics at utrecht university in partial ful llment of the requirements for the degree of bachelor in mathematics supervisor. Givenamanifoldm,adivergencefunctiond isasmooth,non. Symplectic reduction of 4manifolds with circle actions. Algebraic topology blowing up dimension lie group torus manifolds manifold symplectic geometry topology.
Our interest in symplectic transformation stems from two important facts. My favourite book on symplectic geometry is symplectic invariants and hamiltonian dynamics by hofer and zehnder. Symplectic manifolds a symplectic manifold m is a smooth manifold equipped with a symplectic form, i. Following work of delzant and karshon, hamiltonian circle and 2 torus actions on.
Sp2n,\mathbbr \hookrightarrow gl2n,\mathbbrstructure. A brief introduction to symplectic and contact manifolds. Click below the file name on the left for viewing or the arrow on the right for downloading. Torus actions on symplectic manifolds michele audin springer. Use hamiltonian torus actions to understand betti numbers via morse theory and coho mology rings via various equivariant tricks of symplectic manifolds. In particular, we prove a delzanttype theorem that classifies. Divergence functions play a central role in information ge. Function theory on symplectic manifolds mathematical. This extends results of atiyah, guilleminsternberg, delzant and benoist. We construct smooth circle actions on symplectic manifolds with non symplectic fixed point sets or cyclic isotropy sets. Symplectic geometry 81 introduction this is an overview of symplectic geometrylthe geometry of symplectic manifolds. Every 2ndimensional symplectic manifold is locally symplectomorphictor2n. Symplectic 4 manifolds with a free circle action 3 as mentioned above, the present paper covers the case where the euler class e2 h2n of the s1 bration p.
Two especially important classes of symplectic manifolds are. When i wrote the first edition, in 1989, the convexity and duistermaatheckman 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. A symplectic manifold is a manifold along with a symplectic form. September 21, 2018 abstract we classify symplectic actions of 2tori on compact, connected symplectic4manifolds, up to equivariant symplectomorphisms. A symplectic form on a manifold m is a closed 2form on. The symplectic form on each of these manifolds is a little simpler than the. We prove that closed symplectic fourmanifolds do not admit any smooth free circle actions with contractible orbits, without assuming that the actions preserve the symplectic forms. Heckman, kostant, and steinberg formulas for symplectic manifolds. Let m2n denote the set of all symplectic manifolds of dimension 2n. The core of the book is chapter iv which is devoted to applications of morse theory to hamiltonian group actions, including convexity theorems. In dimension 2 symplectic contractions are classical and they are minimal resolutions of du val singularities a n, d n, e6, e7, e8. They are quotients of type c2h with h symplectic 4 manifolds as a purely topological problem that may be more accessible than the original classi cation problem for smooth 4 manifolds.
A symplectic structure on a smooth manifold m is a 2form. So hamiltonian actions of tori of maximal dimension are a special case of integrable systems. A symplectic manifold is a 2nmanifold x together with a symplectic. Complex structures on 4manifolds with symplectic 2torus actions j. This is a book on symplectic topology, a rapidly developing field of mathematics which originated as a geometric tool for problems of classical mechanics. We study hamiltonian actions on b symplectic manifolds with a focus on the effective case of half the dimension of the manifold. Strict quantizations of symplectic manifolds springerlink. Since the 1980s, powerful methods such as gromovs pseudoholomorphic curves and morsefloer theory on loop spaces gave rise to the discovery of unexpected symplectic phenomena. We have seen this in section 1 and will be proved later in this section for general symplectic manifolds. Outline manifolds actions theory poisson manifolds. Torus actions on symplectic manifolds springerlink. Manifolds which are complex and symplectic but not kahler.
This condition means that there exists x m such that tx 0 and the restriction of x to tx t x is nondegenerate. In particular, we prove a delzanttype theorem that classifies these manifolds using polytopes that reside. So this nth wedge power of the symplectic form is in fact a top form, whence a volume form, and we not only have orientation but a great. Fixed point sets in the section we will see corollaries of the slice theorem, that will be used later in the proof of the convexity theorem. Next, we address the related question of which manifolds as above can be endowed. Then i introduce symplectic geometry in chapter ii and hamiltonian group actions in chapter iii, especially torus actions and actionangle variables. This is an extended second edition of the topology of torus actions on symplectic manifolds published in this series in 1991. The aim of this book is to present torus actions as a connecting bridge between combinatorial and convex geometry on one side, and commutative and homological algebra, algebraic geometry and topology on the other. Lectures on symplectic geometry fraydoun rezakhanlou departmet of mathematics, uc berkeley september 18, 2018. Topology of symplectic torus actions with symplectic. We prove that one can always exchange the poisson action to the hamiltonian one by an appropriate change of the symplectic structure. Two forms given a twoform on a manifold m, we can evaluate it on a pair of tangent vectors to produce a number. Symplectic 4manifolds with a free circle action 3 as mentioned above, the present paper covers the case where the euler class e2 h2n of the s1 bration p.
On symplectic invariants associated to zoll manifolds by bich t. Symplectic geometry curvature is a local invariant in riemannian geometry darbouxweinstein theorem let p be any point on a symplectic. Hamiltonian group actions are defined on symplectic manifolds. Another lovely book which has just been reissued as an ams chelsea text is abraham and marsdens book foundations of mechanics which covers a lot of symplectic geometry as well as so much more. We study symplectic actions of the torus t on the symplectic manifold m, such that at least one t orbit is a dim t dimensional symplectic submanifold of m. The slice theorem was presented in yakov kerzhners talk. The study of symplectic manifolds is called symplectic geometry or symplectic topology. We prove that closed symplectic four manifolds do not admit any smooth free circle actions with contractible orbits, without assuming that the actions preserve the symplectic forms. Symplectic manifolds throughout we will assume that m is a c1smooth manifold without boundary unless speci c mention is made to the contrary. In part 11 we will give a quantized version of this formula which generalizes to symplectic manifolds the classical multiplicity formulas of kostant 1161 and steinberg ls.
Semifree hamiltonian circle actions on 6dimensional symplectic manifolds with nonisolated fixed point set article pdf available in journal of symplectic geometry 4. Strict quantizations of symplectic manifolds springerlink advertisement. On the one hand, it is easy to find a nonkahler symplectic manifolds with a. Symplectic manifolds and torus actions are investigated, with numerous examples of torus actions, for instance on some moduli spaces. Assume that a lie group gacts properly on a manifold m. Functor and natural operators on symplectic manifolds. On the existence of symplectic forms on open manifolds. To distinguish symplectic 4manifolds from kahler surfaces and from nonsymplectic 4manifolds. Complex structures on 4manifolds with symplectic torus actions. Torus actions on symplectic manifolds pdf free download. In higher dimensions such actions by symplectomorphisms do exist, and we.
We complete the classification of hamiltonian torus and circle actions on symplectic fourdimensional manifolds. A symplectic form is a closed form that is pointwise nondegenerate. Torus actions on symplectic manifolds among the group actions torus group action is of special interest. Torus actions on symplectic manifolds electronic resource. Contact geometry 5 where we are solving for a vector.
Torus actions and their applications in topology and. In particular, we study the hamiltonian group actions. Symplectic geometry curvature is a local invariant. M d is an exact symplectic manifold, a symplectomorphism of mis called exact with respect to the liouville form if the necessarily closed 1form is exact. We know that any ndimensional oriented manifold will haveanondegeneratenform,soitfollowsthatany2dimensionaloriented manifoldhasanondegeneratetwoformandhencewillbesymplectic. The first examples of exotic 4manifolds with cyclic fundamental group were constructed in 4, 17, 25, 27, 28, 34, 36, 40, 72, and examples of 4manifolds with finite fundamental group can be found in 25, 28. From a language for classical mechanics in the xviii century, symplectic geometry has matured since the 1960s to a rich and central branch of differential geometry and topology. Symplectic geometry and hamiltonian group actions 5mm lecture. A symplectic prolegomenon 3 alternative worldview, one could say that any symplectic manifold is locally as complicated as euclidean space, about which a vast array of questions remain open. Complex structures on 4manifolds with symplectic torus. A skewsymmetric bilinear form on an odddimensional vector space is necessarily degenerate. Torus actions on symplectic manifolds i a load of hogwash. Section 7, on intersections of largrangian submanifolds, is still mostly internal to symplectic geometry, but it contains some applications to machanics and dynamical systems. Heckman, kostant, and steinberg formulas for symplectic.
This trick reduces many questions concerning poisson actions to their well known counterparts from the theory of hamiltonian g actions. Chapter i deals with lie group actions on manifolds. In the second half, we discuss the group actions on odd dimensional symplectic manifolds. Abstract we apply the general theory for symplectic torus actions with symplectic or coisotropic orbits to prove that a 4manifold with a symplectic 2torus action admits an invariant complex structure and give. By the above, a symplectic manifold structure is an integrable sp 2 n. We describe some constructions of symplectic 4 manifolds and invariants to distinguish them. On poisson actions of compactliegroups on symplectic. I myself was rather happy with a small contribution i had made to the. Stern we produce examples ofsimply connected 4manifolds which have up to sign only one class with a nontrivial seibergwitten invariant. This is chapter i, in which i spend some time on s 1actions on surfaces and on 3manifolds, especially on seifert manifolds. Divergence functions play a central role in information geometry.
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