Hamiltonian Torus Actions Research Articles

Integrable systems, toric degenerations and Okounkov bodies

Let $$X$$ be a smooth projective variety of dimension $$n$$ over $$\mathbb {C}$$ equipped with a very ample Hermitian line bundle $$\mathcal {L}$$ . In the first part of the paper, we show that if there exists a toric degeneration of $$X$$ satisfying some natural hypotheses (which are satisfied in many settings), then there exists a surjective continuous map from $$X$$ to the special fiber $$X_0$$ which is a symplectomorphism on an open dense subset $$U$$ . From this we are then able to construct a completely integrable system on $$X$$ in the sense of symplectic geometry. More precisely, we construct a collection of real-valued functions $$\{H_1, \ldots , H_n\}$$ on $$X$$ which are continuous on all of $$X$$ , smooth on an open dense subset $$U$$ of $$X$$ , and pairwise Poisson-commute on $$U$$ . Moreover, our integrable system in fact generates a Hamiltonian torus action on $$U$$ . In the second part, we show that the toric degenerations arising in the theory of Newton-Okounkov bodies satisfy all the hypotheses of the first part of the paper. In this case the image of the ‘moment map’ $$\mu = (H_1, \ldots , H_n): X \rightarrow \mathbb {R}^n$$ is precisely the Newton-Okounkov body $$\Delta = \Delta (R, v)$$ associated to the homogeneous coordinate ring $$R$$ of $$X$$ , and an appropriate choice of a valuation $$v$$ on $$R$$ . Our main technical tools come from algebraic geometry, differential (Kähler) geometry, and analysis. Specifically, we use the gradient-Hamiltonian vector field, and a subtle generalization of the famous Łojasiewicz gradient inequality for real-valued analytic functions. Since our construction is valid for a large class of projective varieties $$X$$ , this manuscript provides a rich source of new examples of integrable systems. We discuss concrete examples, including elliptic curves, flag varieties of arbitrary connected complex reductive groups, spherical varieties, and weight varieties.

Classification of Hamiltonian torus actions with two-dimensional quotients

We construct all possible Hamiltonian torus actions for which all the non-empty reduced spaces are two dimensional (and not single points) and the manifold is connected and compact, or, more generally, the moment map is proper as a map to a convex set. This construction completes the classification of tall complexity one spaces.

Localization and specialization for Hamiltonian torus actions

We consider a Hamiltonian T action on a compact symplectic manifold (M,ω) with d isolated fixed points. For every fixed point p there exists a class ap ∈ H∗ T (M ;Q) such that the collection {ap}, for all fixed points, forms a basis for H∗ T (M ;Q) as an H∗(BT ;Q) module. The map induced by inclusion, ι∗ : H∗ T (M ;Q) → H∗ T (M ;Q) = ⊕j=1Q[x1, . . . , xn] is injective. We will use such classes {ap} to give necessary and sufficient conditions for f = (f1, . . . , fd) in ⊕j=1Q[x1, . . . , xn] to be in the image of ι∗, i.e. to represent an equiviariant cohomology class on M . We may recover the GKM Theorem when the one skeleton is 2-dimensional. Moreover, our techniques give combinatorial description when we restrict to a smaller torus, even though we are then no longer in GKM case.

A convexity theorem for three tangled Hamiltonian torus actions, and super-integrable systems

A completely integrable system on a symplectic manifold is called super-integrable when the number of independent integrals of motion is more than half the dimension of the manifold. Several important completely integrable systems are super-integrable: the harmonic oscillators, the Kepler system, the non-periodic Toda lattice, etc. Motivated by an additional property of the super-integrable system of the Toda lattice (Agrotis et al., 2006) [2], we will give a generalization of the Atiyah and Guillemin–Sternbergʼs convexity theorem.

Quasimap Floer Cohomology for Varying Symplectic Quotients

AbstractWe show that quasimap Floer cohomology for varying symplectic quotients resolves several puzzles regarding displaceability of toric moment fibers. For example, we present a compact Hamiltonian torus action containing an open subset of non-displaceable orbits and a codimension four singular set, partly answering a question of McDuff, and we determine displaceability for most of the moment fibers of a symplectic ellipsoid.

New techniques for obtaining Schubert-type formulas for Hamiltonian manifolds

In [GT], Goldin and Tolman extend some ideas from Schubert calculus to the more general setting of Hamiltonian torus actions on compact symplectic manifolds with isolated fixed points. (See also [Kn99,Kn08].) The main goal of this paper is to build on this work by finding more effective formulas. More explicitly, given a generic component of the moment map, they define a canonical class alpha(p) in the equivariant cohomology of the manifold M for each fixed point p is an element of M. When they exist, canonical classes form a natural basis of the equivariant cohomology of M. In particular, when M is a flag variety, these classes are the equivariant Schubert classes. It is a long-standing problem in combinatorics to find positive integral formulas for the equivariant structure constants associated to this basis. Since computing the restriction of the canonical classes to the fixed points determines these structure constants, it is important to find effective formulas for these restrictions. In this paper, we introduce new techniques for calculating the restrictions of a canonical class alpha(p) to a fixed point q. Our formulas are nearly always simpler, in the sense that they count the contributions over fewer paths. Moreover, our formula is manifestly positive and integral in certain important special cases.

Log-concavity of complexity one Hamiltonian torus actions

Let (M,ω) be a closed 2n-dimensional symplectic manifold equipped with a Hamiltonian Tn−1-action. Then Atiyah–Guillemin–Sternberg convexity theorem implies that the image of the moment map is an (n−1)-dimensional convex polytope. In this Note, we show that the density function of the Duistermaat–Heckman measure is log-concave on the image of the moment map.

Equivariant cohomology for Hamiltonian torus actions on symplectic orbifolds

We study Hamiltonian R-actions on symplectic orbifolds [M/S], where R and S are tori. We prove an injectivity theorem and generalize Tolman and Weitsman’s proof of the GKM theorem [TW] in this setting. The main example is the symplectic reduction X//S of a Hamiltonian T-manifold X by a subtorus S ⊂ T. This includes the class of symplectic toric orbifolds. We define the equivariant Chen–Ruan cohomology ring and use the above results to establish a combinatorial method of computing this equivariant Chen–Ruan cohomology in terms of orbifold fixed point data.

Quasi-States, Quasi-Morphisms, and the Moment Map

We prove that symplectic quasi-states and quasi-morphisms on a symplectic manifold descend under symplectic reduction on a superheavy level set of a Hamiltonian torus action. Using a construction due to Abreu and Macarini, in each dimension at least four we produce a closed symplectic toric manifold with infinite dimensional spaces of symplectic quasi-states and quasi-morphisms, and a one-parameter family of non-displaceable Lagrangian tori. By using McDuff's method of probes, we also show how Ostrover and Tyomkin's method for finding distinct spectral quasi-states in symplectic toric Fano manifolds can also be used to find different superheavy toric fibers.

Toric moment mappings and Riemannian structures

Coadjoint orbits for the group SO(6) parametrize Riemannian G-reductions in six dimensions, and we use this correspondence to interpret symplectic fibrations between these orbits, and to analyse moment polytopes associated to the standard Hamiltonian torus action on the coadjoint orbits. The theory is then applied to describe so-called intrinsic torsion varieties of Riemannian structures on the Iwasawa manifold.

Symplectic Origami

An origami manifold is a manifold equipped with a closed 2-form which is symplectic except on a hypersurface where it is like the pullback of a symplectic form by a folding map and its kernel fibrates with oriented circle fibers over a compact base. We can move back and forth between origami and symplectic manifolds using cutting (unfolding) and radial blow-up (folding), modulo compatibility conditions. We prove an origami convexity theorem for hamiltonian torus actions, classify toric origami manifolds by polyhedral objects resembling paper origami and discuss examples. We also prove a cobordism result and compute the cohomology of a special class of origami manifolds.

Generalized complex Hamiltonian torus actions: examples and constraints

Consider an effective Hamiltonian torus action T × M → M on a topologically twisted, generalized complex manifold M of dimension 2n. We prove that dim(T) ⩽ n − 2 and that the topological twisting survives Hamiltonian reduction. We then construct a large new class of such actions satisfying dim(T) = n − 2, using a surgery procedure on toric manifolds.

Convexity properties of generalized moment maps

In this paper, we consider generalized moment maps for Hamiltonian actions on H -twisted generalized complex manifolds introduced by Lin and Tolman [15]. The main purpose of this paper is to show convexity and connectedness properties for generalized moment maps. We study Hamiltonian torus actions on compact H -twisted generalized complex manifolds and prove that all components of the generalized moment map are Bott-Morse functions. Based on this, we shall show that the generalized moment maps have a convex image and connected fibers. Furthermore, by applying the arguments of Lerman, Meinrenken, Tolman, and Woodward [13] we extend our results to the case of Hamiltonian actions of general compact Lie groups on H -twisted generalized complex orbifolds.

On the Existence of Star Products on Quotient Spaces of Linear Hamiltonian Torus Actions

We discuss BFV deformation quantization (Bordemann et al. in A homological approach to singular reduction in deformation quantization, singularity theory, pp. 443–461. World Scientific, Hackensack, 2007) in the special case of a linear Hamiltonian torus action. In particular, we show that the Koszul complex on the moment map of an effective linear Hamiltonian torus action is acyclic. We rephrase the nonpositivity condition of Arms and Gotay (Adv Math 79(1):43–103, 1990) for linear Hamiltonian torus actions. It follows that reduced spaces of such actions admit continuous star products.

Toric structures on near-symplectic 4-manifolds

A near-symplectic structure on a 4-manifold is a closed 2-form that is symplectic away from the 1-dimensional submanifold along which it vanishes and that satisfies a certain transversality condition along this vanishing locus. We investigate near-symplectic 4-manifolds equipped with singular Lagrangian torus fibrations which are locally induced by effective Hamiltonian torus actions. We show how such a structure is completely characterized by a singular integral affine structure on the base of the fibration whenever the vanishing locus is nonempty. The base equipped with this geometric structure generalizes the moment map image of a toric 4-manifold in the spirit of earlier work by the second author on almost toric symplectic 4-manifolds. We use the geometric structure on the base to investigate the problem of making given smooth torus actions on 4-manifolds symplectic or Hamiltonian with respect to near-symplectic structures and to give interesting constructions of structures which are locally given by torus actions but have nontrivial global monodromy.

Rigid subsets of symplectic manifolds

AbstractWe show that there is an hierarchy of intersection rigidity properties of sets in a closed symplectic manifold: some sets cannot be displaced by symplectomorphisms from more sets than the others. We also find new examples of rigidity of intersections involving, in particular, specific fibers of moment maps of Hamiltonian torus actions, monotone Lagrangian submanifolds (following the works of P. Albers and P. Biran-O. Cornea) as well as certain, possibly singular, sets defined in terms of Poisson-commutative subalgebras of smooth functions. In addition, we get some geometric obstructions to semi-simplicity of the quantum homology of symplectic manifolds. The proofs are based on the Floer-theoretical machinery of partial symplectic quasi-states.

Towards generalizing Schubert calculus in the symplectic category

The main purpose of this article is to extend some of the ideas from Schubert calculus to the more general setting of Hamiltonian torus actions on compact symplectic manifolds with isolated fixed points. Given a generic component of the moment map, we define a canonical class \alpha_p in the equivariant cohomology of the manifold M for each fixed point p of M. When they exist, canonical classes form a natural basis of the equivariant cohomology of M; in particular, when M is a flag variety, these classes are the equivariant Schubert classes. We show that the restriction of a canonical class \alpha_p to a fixed point q can be calculated by a rational function which depends only on the value of the moment map, and the restriction of other canonical classes to points of index exactly two higher. Therefore, the structure constants can be calculated by a similar rational function. Our restriction formula is manifestly positive in many cases, including when M is a flag manifold. Finally, we prove the existence of integral canonical classes in the case that M is a GKM manifold the moment map component is index increasing. In this case, our restriction formula specializes to an easily computable rational sum which depends only on the GKM graph.

A personal tour through symplectic topology and geometry

(i) Gromov’s Compactness Theorem for pseudo-holomorphic curves in symplectic manifolds ([23]) and the topology of symplectomorphism groups of rational ruled surfaces (sections 2 and 3, references [1, 2]). (ii) Atiyah-Guillemin-Sternberg’s Convexity Theorem for the moment map of Hamiltonian torus actions ([9, 25]) and Kahler geometry of toric orbifolds in symplectic coordinates (sections 4 and 5, references [3, 4, 5]). (iii) Donaldson’s moment map framework for the action of the symplectomorphism group on the space of compatible almost complex structures ([17]) and the topology of the space of compatible integrable complex structures of a rational ruled surface (sections 6 and 7, references [6, 7]).

Continuous families of Hamiltonian torus actions

We determine conditions under which two Hamiltonian torus actions on a symplectic manifold M are homotopic by a family of Hamiltonian torus actions, when M is a toric manifold and when M is a coadjoint orbit.

Reduced Delzant spaces and a convexity theorem

The convexity theorem of Atiyah and Guillemin–Sternberg says that any connected compact manifold with Hamiltonian torus action has a moment map whose image is the convex hull of the image of the fixed point set. Sjamaar–Lerman proved that the Marsden–Weinstein reduction of a connected Hamitonian G -manifold is a stratified symplectic space. Suppose 1 → A → G → T → 1 is an exact sequence of compact Lie groups and T is a torus. Then the reduction of a Hamiltonian G -manifold with respect to A yields a Hamiltonian T -space. We show that if the A -moment map is proper, then the convexity theorem holds for such a Hamiltonian T -space, even when it is singular. We also prove that if, furthermore, the T -space has dimension 2 dim T and T acts effectively, then the moment polytope is sufficient to essentially distinguish their homeomorphism type, though not their diffeomorphism types. This generalizes a theorem of Delzant in the smooth case. This paper is a concise version of a companion paper [B. Lian. B. Song, A convexity theorem and reduced Delzant spaces, math.DG/0509429].