Toric actions on b symplectic manifolds request pdf. A symplectic toric manifold is a connected symplectic manifold m equipped with an e ective hamiltonian action of a torus tof dimension dimt. Although the book is still centered on convexity theorems, it contains much more. There was absolutely no question of just correcting numerous. Aug 18, 2017 ive recently been reading torus actions on symplectic manifolds aud04 by michele audin. I have hesitated quite a long time before deciding to do the rewriting workthe first edition has been sold out for a few years. Nonk ahler hamiltonian torus actions brad hannigandaley may 10, 2010 abstract we discuss conditions under which a symplectic manifold equipped with a hamiltonian torus action admits an invariant compatible complex structure, and we describe tolmans construction. 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. We study hamiltonian actions on b symplectic manifolds with a focus on the effective case of half the dimension of the manifold.
Symmetries of symplectic manifolds and related topics. First, we will consider counting hamiltonian tn actions on closed, symplectic manifolds m2n. Hamiltonian circle actions on manifolds of dimension 4 this is an extended second edition of the topology of torus actions on symplectic manifolds published in this series in 1991. Torus brations on symplectic four manifolds ivan smith 1. Topology of symplectic torus actions with symplectic orbits. Symplectic actions of 2tori on 4manifolds pages 1 50. A symplectic toric manifold is a compact connected symplectic manifold m equipped with an effective hamiltonian torus t action, such that dim t 1. Eugene lerman, department of mathematics, university of. By including detailed proofs, illuminating examples and figures, and numerous exercises, the author has made this book a suitable text for a graduate course, especially one centered on hamiltonian torus actions and their applications.
The purpose of this paper is to prove a convexity theorem for the image of the moment map of a hamiltonian torus action on a b m symplectic manifold. 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. A symplectic toric manifold of dimension 2nis a compact, connected, symplectic manifold equipped with an e ective hamiltonian action by a torus of dimension n. In a download the topology of torus actions on symplectic of performers, gilneas is pretty legal to chinese british isles. Pelayoyz abstract we give a concise overview of the classi. From the symplectic geometry point of view, a symplectic toric manifold is a symplectic manifold m2n. In this paper we obtain convexity results for the more general case of nontoric hamiltonian torus. Complex structures on 4manifolds with symplectic torus. We use cohomological techniques to show that there is a. Ana cannas da silva, lectures on symplectic geometry, lecture notes in mathematics 1764, springerverlag 2001. We also prove a cobordism result and compute the cohomology of a special class of origami manifolds. Torus actions on symplectic orbispaces article pdf available in proceedings of the american mathematical society 1294. The middle section is a digression on hamiltonian torus actions culminating in a statement. May 24, 20 in this thesis, we will study the properties of certain hamiltonian torus actions on closed symplectic manifolds.
The topology of torus actions on symplectic manifolds book. Torus actions and singularities in symplectic geometry by geo rey stephen scott a dissertation submitted in partial ful llment of the requirements for the degree of doctor of philosophy mathematics in the university of michigan 2014 doctoral committee. A momentum map for a symplectic g action on a symplectic manifold m. Although the book is still centered on convexity theorems, it contains much more results, proofs and examples. Hamiltonian group actions on exact symplectic manifolds with.
Torus actions on symplectic manifolds michele audin. Let t be the 2torus rz2, and let t act on mby trans lations on the left factor of the product. Show that an effective hamiltonian action of a torus tn on a 2ndimensional symplectic manifold gives rise to an integrable system. Torus actions and singularities in symplectic geometry. The main object the book covers is a symplectic manifold with a torus action. Symplectic torus actions with coisotropic principal orbits. A convexity theorem for the moment image of a hamiltonian torus action on a b symplectic manifold was proved in. In this paper we obtain convexity results for the more general case of nontoric hamiltonian torus actions on b symplectic manifolds. Full text of torus actions on symplectic manifolds electronic resource see other formats. The last section deals with geometric quantization itself and is drawn mostly from 5 and 6. Properties of hamiltonian torus actions on closed symplectic manifolds andrew fanoe in this thesis, we will study the properties of certain hamiltonian torus actions on closed symplectic manifolds. The material and references in this extended second edition of the topology of torus actions on symplectic manifolds, published as volume 93 in this series in 1991, have been updated. Torus actions on symplectic manifolds pdf free download. Although the book is still centered on convexity results, it contains more theorems, more and better proofs, more exercises, and many figures.
Convexity for hamiltonian torus actions on bsymplectic manifolds. The category of hamiltonian groupactions on symplectic manifolds is not closed under symplectic reduction. Topology of symplectic torus actions with symplectic orbits j. In order to deal with symplectic actions which are not hamiltonian, we develop new techniques, extending the theory of atiyah, guilleminsternberg, delzant, and benoist. Compact toric b symplectic manifolds have been investigated in 11,14. Convexity of the moment map image for torus actions on bm. Torus actions on symplectic manifolds michele audin download. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Viktor ginzburg, victor guillemin, and yael karshon, moment maps, cobordisms, and hamiltonian group actions. This is an extended second edition of the topology of torus actions on symplectic manifolds published in this series in 1991.
Such action of t on mis free, it has symplectic 2tori as torbits, and the orbit space mt is equal to the 2sphere s2. Let 2n be a compact symplectic toric manifold with momentum map image a delzant polytope m p. Use hamiltonian torus actions to understand betti numbers via morse theory and coho mology rings via various equivariant tricks of symplectic manifolds. In most symmetric and regular cases suc h as pro jectiv e toric v arieties or hamiltonian torus actions on symplectic manifolds the quotien t can b e iden ti ed with a con v ex p olytop e. In mathematics, specifically in symplectic geometry, the momentum map or moment map is a tool associated with a hamiltonian action of a lie group on a symplectic manifold, used to construct conserved quantities for the action. A symplectic form on a manifold m is a closed 2form on. Convexity for hamiltonian torus actions on b symplectic manifolds victor guillemin, eva miranda, ana rita pires, and geoffrey scott abstract.
Algebraic topology blowing up dimension lie group torus manifolds manifold. Complex structures on 4manifolds with symplectic torus actions. In recent work with victor guillemin and eva miranda, we explored this question in the context of poisson manifolds which are not too far from being symplectic the so called b symplectic or bpoisson manifolds in the presence of an abelian symmetry group. 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. Torus actions on symplectic manifolds michele audin springer. Torus actions on symplectic manifolds electronic resource.
The category of hamiltonian group actions on symplectic manifolds is not closed under symplectic reduction. Symplectic manifolds and torus actions are investigated, with numerous examples of torus. In the case of a hamiltonian action on a symplectic manifold, a variety of techniques has made computing h g m. The moment map induces an embedding of the quotient of the manifold by the torus action into the dual of the lie algebra of the torus. Torus actions and their applications in topology and. Topology of symplectic torus actions with symplectic.
Properties of hamiltonian torus actions on closed symplectic. A bsymplectic manifold is a manifold m2n together with a symplectic form. An ndimensional convex polytope is called delzant provided. Download the topology of torus actions on symplectic manifolds. In gmps we proved that the moment map image of a b symplectic toric manifold is a convex bpolytope. Torus actions on symplectic manifolds among the group actions torus group action is of special interest. In the most symmetric and regular cases such as projective toric varieties or hamiltonian torus actions on symplectic manifolds the quotient can be identi. Torus actions on symplectic manifolds i a load of hogwash.
Enumeration of rational curves via torus actions maxim kontsevich. Cornell university 2012 the central theme of this work are hamiltonian torus actions on symplectic manifolds. Complex structures on 4 manifolds with symplectic 2 torus actions j. The invariants are invariants of the topology of the manifold, of the torus action, or of the symplectic form.
The central theme of this work are hamiltonian torus actions on symplectic manifolds. The topology of torus actions on symplectic manifolds. 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. We give a concise overview of the classification theory of symplectic manifolds equipped with torus actions for which the orbits are symplectic this is equivalent to the existence of a symplectic principal orbit, and apply this theory to study the structure of the leaf space induced by the action. How i have rewritten this book the book the reader has in hand was supposed to be a new edition of 14. Enumeration of rational curves via torus actions maxim. So hamiltonian actions of tori of maximal dimension are a special case of integrable systems.
Although the book is still centered on convexity theorems, it contains more results, proofs and examples. Michele audin, torus actions on symplectic manifolds, birkhauser, 2004. Symplectic manifolds and torus actions are investigated, with numerous examples of torus actions, for instance on some moduli spaces. I myself was rather happy with a small contribution i had made to the. The rst part concerns the topological constraints placed on a closed four manifold by the existence of. Structure theory for symplectic torus actions 5mm lecture 2. This download the topology of torus actions on symplectic manifolds makes pretty simple with italian steampunk arts and s many of production on the drenai afterschool, with still worn monsters and a school of response. In particular, all such manifolds are bundles with fiber and base equal to projective spaces. First, we will consider counting hamiltonian torus actions on closed, symplectic manifolds m with 2dimensional second cohomology.
At the end, we need to introduce an important notion, symplectic toric manifolds, which is a special kind of symplectic manifolds with certain torus action. Next, we address the related question of which manifolds as above can be endowed. This is an extended second edition of the topology of torus actions on symplectic manifolds published as pm 93 in 1991. As its title says, this book is mainly devoted to the study of torus actions in symplectic geometry. In this paper we study hamiltonian torus actions on symplectic orbifolds, with an emphasis on completely integrable actions. Complex structures on 4manifolds with symplectic 2torus actions j. Algebraic topology blowing up dimension lie group torus manifolds manifold symplectic geometry topology. Introduction this is the third in a series of papers on folded symplectic manifolds. For two dimensional torus actions on closed symplectic fourmanifolds, we reduce the. Torus actions on symplectic manifolds springerlink. Enumeration of rational curves via torus actions maxim kontsevich maxplanckinstitut fu. Symplectic manifolds and torus actions are investigated, with numerous examples of torus actions, for.
Mthat has a particular kind of orderone singularity along a hypersurface z. In particular, we prove a delzanttype theorem that classifies. The work of goreskykottwitzmacpherson gkm describes this ring combinatorially when gis a torus, ra eld, and the action has very speci c form. The book is nicely written, and is a good reference book. Our abstruse statement, in addition to giving unified proofs of previously different theorems, allows us to classify certain group actions on symplectic manifolds. Symplectic actions of 2tori on 4manifolds alvaro pelayo author address. The rst part concerns the topological constraints placed on a closed fourmanifold by the existence of an integrable system. In this thesis, we will study the properties of certain hamiltonian torus actions on closed symplectic manifolds. Most of the material here is included in mich ele audins book torus actions on symplectic manifolds, which i used heavily in preparing these notes. Hamiltonian torus actions in equivariant cohomology and symplectic topology milena pabiniak, ph. Topology of symplectic torus actions with symplectic orbits 61 not have any. Symplectic geometry nicholas proudfoot department of mathematics, university of oregon, eugene, or 97403 these notes are written for a ten week graduate class on symplectic geometry. Buy torus actions on symplectic manifolds progress in mathematics on free shipping on qualified orders. Id like to write down some of the facts i learned from this book.
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