Symplectic actions of 2-tori on 4-manifolds

Abstract

We classify symplectic actions of 2-tori on compact, connected symplectic 4-manifolds, up to equivariant symplectomorphisms. This extends results of Atiyah, Guillemin-Sternberg, Delzant and Benoist. The classification is in terms of a collection of invariants, which are invariants of the topology of the manifold, of the torus action and of the symplectic form. We construct explicit models of such symplectic manifolds with torus actions, defined in terms of these invariants.We also classify, up to equivariant symplectomorphisms, symplectic actions of (2n − 2)dimensional tori on 2n-dimensional symplectic manifolds, when at least one orbit is a (2n−2)--dimensional symplectic submanifold. Then we show that a 2n-dimensional symplectic manifold (M, σ) equipped with a free symplectic action of a (2n − 2)-dimensional torus with at least one (2n − 2)-dimensional symplectic orbit is equivariantly diffeomorphic to M/T × T equipped with the translational action of T . Thus two such symplectic manifolds are equivariantly diffeomorphic if and only if their orbit spaces are surfaces of the same genus. * Partly funded by a Rackham Predoctoral Fellowship (2006-2007) and a Rackham Dissertation Fellowship (2005-2006. He is grateful to professor J.J. Duistermaat for discussions, specifically on sections 2.4.4, 2.5, 2.6, for hospitality on three visits to Utrecht, and for comments on a preliminary version.He thanks professor A. Uribe for conversations on symplectic normal forms, and professor P. Scott for discussions on orbifold theory, and helpful feedback and remarks on Section 4.2. He also has benefited from conversations with professors sat through several talks of the author on the paper and offered feedback. He thanks professor M. The orbit space M/TWe describe the structure of the orbit space M/T , c.f. Definition 2.13. Symplectic form on the T -orbitsWe prove that the symplectic form on every T -orbit of M is given by the same non-degenerate antisymmetric bilinear form.Let X be an element of the Lie algebra t of T , and denote by X M the smooth vector field on M obtained as the infinitesimal action of X on M. Let ω be a smooth differential form, let L v denote the Lie derivative with respect to a vector field v, and let i v ω denote the usual inner product of ω with v. Since the symplectic form σ is T -invariant, we have that d(i X M σ) = L X M σ = 0, where the first equality follows by combining d σ = 0 and the homotopy identityThe following result follows from [12, Lem. 2.1].Lemma 2.1. Let (M, σ) be a compact, connected, symplectic manifold equipped with an effective symplectic action of a torus T for which there is at least one T -orbit which is a dim T -dimensional symplectic submanifold of (M, σ). Then there exists a unique non-degenerate antisymmetric bilinear form σ t : t × t → R on the Lie algebra t of T such thatfor every X, Y ∈ t, and every x ∈ M.Proof. In [12, Lem. 2.1] it was shown that there is a unique antisymmetric bilinear form σ t : t×t → R on the Lie algebra t of T such that expression (2.1) holds for every ...

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].…”

“…At the other extreme of a symplectic Hamiltonian T -action is the case of a symplectic T -action whose principal orbits are symplectic submanifolds of (M, σ ), in which case the action does not have any fixed points and the restriction of the symplectic form to the T -orbits is non-degenerate, which in particular implies that the action is never Hamiltonian. The classification of Pelayo [33], reviewed in the present paper, shows that there are lots of cases where this happens.…”

“…3.2 the particular case when the action is free, and hence the orbit space M/T is a manifold; in this case the classification is more straightforward, see Proposition 3.2. If the action is not free, then the orbit space M/T is a good orbifold (proven in [33]), and the classification of Sect. 3.2 is generalized to this rather more delicate situation in Sect.…”

“…We give a concise review of Pelayo [33, with some modifications in the exposition and present a new fact: Theorem 3.3.…”

“…These local models can also be obtained by applying results of Benoist [3, Proposition 1.9] or Ortega and Ratiu [30,] to the case of a symplectic torus action with symplectic orbits. The proof of Lemma 3.1 in [33] uses these local models.…”