Symplectic torus actions with coisotropic principal orbits

Abstract

In this paper we completely classify symplectic actions of a torus T on a compact connected symplectic manifold (M, σ) when some, hence every, principal orbit is a coisotropic submanifold of (M, σ). That is, we construct an explicit model, defined in terms of certain invariants, of the manifold, the torus action and the symplectic form. The invariants are invariants of the topology of the manifold, of the torus action, or of the symplectic form.In order to deal with symplectic actions which are not Hamiltonian, we develop new techniques, extending the theory of Atiyah, Guillemin-Sternberg, Delzant, and Benoist. More specifically, we prove that there is a well-defined notion of constant vector fields on the orbit space M/T . Using a generalization of the Tietze-Nakajima theorem to what we call V -parallel spaces, we obtain that M/T is isomorphic to the Cartesian product of a Delzant polytope with a torus.We then construct special lifts of the constant vector fields on M/T , in terms of which the model of the symplectic manifold with the torus action is defined. * Research stimulated by a KNAW professorship If the effective action of T on (M, σ) is Hamiltonian, then d T = n and the principal orbits are Lagrange submanifolds. Moreover, the image of the momentum mapping is a convex polytope ∆ in the dual space t * of t, where t denotes the Lie algebra of T . ∆ has the special property that at each vertex of ∆ there are precisely n codimension one faces with normals which form a Z-basis of the integral lattice T Z in t, where T Z is defined as the kernel of the exponential mapping from t to T . The classification of Delzant [11] says that for each such polytope ∆ there is a compact connected symplectic manifold with Hamiltonian torus action having ∆ as image of the momentum mapping, and the symplectic manifold with torus action is unique up to equivariant symplectomorphisms. Such polytopes ∆ and corresponding symplectic T -manifolds (M, σ, T ) are called Delzant polytopes and Delzant manifolds in the exposition of this subject by Guillemin [24], after Delzant [11]. Each Delzant manifold has a T -invariant Kähler structure such that the Kähler form is equal to σ.Because critical points of the Hamiltonian function correspond to zeros of the Hamiltonian vector field, a Hamiltonian action on a compact manifold always has fixed points. Therefore the other extreme case of a symplectic torus action with coisotropic principal orbits occurs if the action is free. In this case, M is a principal torus bundle over a torus, hence a nilmanifold for a two-step nilpotent Lie group as described in Palais and Stewart [44]. If the nilpotent Lie group is not commutative, then M does not admit a Kähler structure, cf. Benson and Gordon [5]. For four-dimensional manifolds M, these were the first examples of compact symplectic manifolds without Kähler structure, introduced by Thurston [49]. See the end of Remark 7.6.The general case is a combination of the Hamiltonian case and the free case, in the sense that

Topology of symplectic torus actions with symplectic orbits

“…Benoist [3] proved a symplectic tube theorem for symplectic actions with coisotropic orbits and convexity result in the spirit of the of the Atiyah-Guillemin-Sternberg theorem [3]; Ortega-Ratiu [30] proved a local normal form theorem for symplectic torus actions with coisotropic orbits. These appear to be the most general results prior to the classification of symplectic torus actions with coisotropic principal orbits in Duistermaat-Pelayo [11] and Pelayo [33]. For a concise overview of the classification in [11] and an application to complex and Kähler geometry see [12].…”

“…These appear to be the most general results prior to the classification of symplectic torus actions with coisotropic principal orbits in Duistermaat-Pelayo [11] and Pelayo [33]. For a concise overview of the classification in [11] and an application to complex and Kähler geometry see [12].…”

Actions symplectiques de groupes compacts

“…-Explicit examples where this fibered action is not a product have been discovered by Duistermaat and Pelayo in [2]. I thank them for sending [2] to me.…”

“…I thank them for sending [2] to me. In fact, the aim of [2] is to give a complete list of invariants for all coisotropic actions of tori.…”

“…I thank them for sending [2] to me. In fact, the aim of [2] is to give a complete list of invariants for all coisotropic actions of tori. Such a classification can also be obtained using the cohomological approach of the proof of Proposition 6.17 of [1].…”