Centered complexity one Hamiltonian torus actions
Transactions of the American Mathematical Society
We consider symplectic manifolds with Hamiltonian torus actions which are "almost but not quite completely integrable": the dimension of the torus is one less than half the dimension of the manifold. We provide a complete set of invariants for such spaces when they are "centered" and the moment map is proper. In particular, this classifies the preimages under the moment map of all sufficiently small open sets, which is an important step towards global classification. As an application, we
... uct a full packing of each of the Grassmannians Gr + (2, R 5 ) and Gr + (2, R 6 ) by two equal symplectic balls. the symplectic category; [KKMS, OW, R, FK, BB, LV] in the algebraic category; [F, OR] in the smooth category. This is the first in a series of papers in which we study complexity one torus actions in arbitrary dimension. In this paper we study the basic building blocks: the preimages under the moment map of sufficiently small open subsets in t * . We provide invariants which determine these spaces up to an equivariant symplectomorphism. Our techniques apply to "large" complexity one spaces, as long as they are centered (see Definition 1.4). In this paper, because we wish to restrict to the preimages of open subsets of t * , we do not insist that our manifolds be compact. Instead, we assume that the moment map is proper as a map to an open convex set U ⊂ t * , that is, that the preimage of every compact subset of U is compact. For instance, if M is compact, Φ is proper. Definition 1.2. Let T be a torus. A proper Hamiltonian T-manifold is a connected symplectic manifold (M, ω) together with an effective action of T , an open convex subset U ⊆ t * , and a proper moment map Φ : M −→ U . Here, t is the Lie algebra of T and t * the dual space. For brevity, in this paper we call (M, ω, Φ, U) a complexity k space, where k = 1 2 dim M − dim T . An isomorphism between two such spaces over the same set U is an equivariant symplectomorphism that respects the moment maps. Example 1.3. Let (M, ω, Φ, U) be a proper Hamiltonian T -manifold. For any open convex subset V ⊆ U , the preimage Φ −1 (V ) is a proper Hamiltonian Tmanifold over V . The fact that it is connected follows from the facts that the restriction Φ : Φ −1 (V ) −→ V is proper and its image and fibers are connected (see Theorem 2.3) by easy point-set topology. Here are a few examples of complexity one spaces. The complete flags on C 3 form a six dimensional compact symplectic manifold with a two dimensional Hamiltonian torus action. The Grassmannians Gr + (2, R 5 ) and Gr + (2, R 6 ) of oriented two-planes in R 5 and R 6 are also complexity one spaces; see section 14 for more details. Any symplectic toric manifold gives rise to complexity one spaces in several ways: one can either restrict the action to a codimension one subtorus, or take the product of the manifold with a surface. Finally, the example in [To], of a symplectic manifold with a Hamiltonian torus action with isolated fixed points that is not equivariantly Kähler, is a complexity one space. We now describe invariants of a complexity one space. The Liouville measure on a 2n dimensional symplectic manifold (M, ω) is given by integration of the volume form ω n /n! with respect to the symplectic orientation. In the presence of a Hamiltonian action, the Duistermaat-Heckman measure is the push-forward of Liouville measure by the moment map. It is equal to Lebesgue measure on t * times the Duistermaat-Heckman function, which is piecewise linear. 2 Assume M is connected. For any value α ∈ Φ(M ), if the symplectic quotient Φ −1 (α)/T is not a single point, it is homeomorphic to a connected closed oriented surface (see Proposition 6.1). The genus of this surface does not depend on α (see Corollary 9.7); we call it the genus of the complexity one space.