Hamiltonian Fashion Research Articles

Equivariant Lagrangian Floer homology via cotangent bundles of EGN$EG_N$

AbstractWe provide a construction of equivariant Lagrangian Floer homology , for a compact Lie group acting on a symplectic manifold in a Hamiltonian fashion, and a pair of ‐Lagrangian submanifolds . We do so by using symplectic homotopy quotients involving cotangent bundles of an approximation of . Our construction relies on Wehrheim and Woodward's theory of quilts, and the telescope construction. We show that these groups are independent of the auxiliary choices involved in their construction, and are ‐bimodules. In the case when , we show that their chain complex is homotopy equivalent to the equivariant Morse complex of . Furthermore, if zero is a regular value of the moment map and if acts freely on , we construct two ‘Kirwan morphisms’ from to (respectively, from to ). Our construction applies to the exact and monotone settings, as well as in the setting of the extended moduli space of flat ‐connections of a Riemann surface, considered in Manolescu and Woodward's work. Applied to the latter setting, our construction provides an equivariant symplectic side for the Atiyah–Floer conjecture.

A dual pair for the contact group

AbstractGeneralizing the canonical symplectization of contact manifolds, we construct an infinite dimensional non-linear Stiefel manifold of weighted embeddings into a contact manifold. This space carries a symplectic structure such that the contact group and the group of reparametrizations act in a Hamiltonian fashion with equivariant moment maps, respectively, giving rise to a dual pair, called the EPContact dual pair. Via symplectic reduction, this dual pair provides a conceptual identification of non-linear Grassmannians of weighted submanifolds with certain coadjoint orbits of the contact group. Moreover, the EPContact dual pair gives rise to singular solutions for the geodesic equation on the group of contact diffeomorphisms. For the projectivized cotangent bundle, the EPContact dual pair is closely related to the EPDiff dual pair due to Holm and Marsden.

A Splitting Result for Real Submanifolds of a Kähler Manifold

Let (Z,omega ) be a connected Kähler manifold with an holomorphic action of the complex reductive Lie group U^mathbb {C}, where U is a compact connected Lie group acting in a hamiltonian fashion. Let G be a closed compatible Lie group of U^mathbb {C} and let M be a G-invariant connected submanifold of Z. Let xin M. If G is a real form of U^mathbb {C}, we investigate conditions such that Gcdot x compact implies U^mathbb {C}cdot x is compact as well. The vice-versa is also investigated. We also characterize G-invariant real submanifolds such that the norm-square of the gradient map is constant. As an application, we prove a splitting result for real connected submanifolds of (Z,omega ) generalizing a result proved in Gori and Podestà (Ann Global Anal Geom 26: 315–318, 2004), see also Bedulli and Gori (Results Math 47: 194–198, 2005), Biliotti (Bull Belg Math Soc Simon Stevin 16: 107–116 2009).

Hamiltonian circle actions with almost minimal isolated fixed points

Let the circle act in a Hamiltonian fashion on a connected compact symplectic manifold (M,ω) of dimension 2n. Then the S1-action has at least n+1 fixed points. In a previous paper, we study the case when the fixed point set consists of precisely n+1 isolated points. In this paper, we study the case when the fixed point set consists of exactly n+2 isolated points. We show that in this case n must be even. We find equivalent conditions on the first Chern class of M and a particular weight of the S1-action. We also show that the particular weight can completely determine the integral cohomology ring and the total Chern class of M, and the sets of weights of the S1-action at all the fixed points. We will see that all these data are isomorphic to those of known examples, G˜2(Rn+2) with n≥2 even, equipped with standard circle actions.

Beyond cohomological assignments

Let a torus T act in a Hamiltonian fashion on a compact symplectic manifold (M,ω). The assignment ring AT(M) is an extension of the equivariant cohomology ring HT(M); it is modeled on the GKM description of the equivariant cohomology of a GKM space. We show that AT(M) is a finitely generated S(t⁎)-module, and give a criterion guaranteeing that a given set of assignments generates (alternatively, is a basis for) this module. We define two new types of assignments, delta classes and bridge classes, and show that if the torus T is 2-dimensional, then all assignments of sufficiently high degree are generated by cohomological, delta, and bridge classes. In particular, if M is 6-dimensional, then we can find a basis of such classes.

Stability of measures on Kähler manifolds

Let (M,ω) be a Kähler manifold and let K be a compact group that acts on M in a Hamiltonian fashion. We study the action of KC on probability measures on M. First of all we identify an abstract setting for the momentum mapping and give numerical criteria for stability, semi-stability and polystability. Next we apply this setting to the action of KC on measures. We get various stability criteria for measures on Kähler manifolds. The same circle of ideas gives a very general surjectivity result for a map originally studied by Hersch and Bourguignon–Li–Yau.

One-connectivity and finiteness of HamiltonianS1-manifolds with minimal fixed sets

Let the circle act effectively in a Hamiltonian fashion on a compact symplectic manifold $(M, \omega)$. Assume that the fixed point set $M^{S^1}$ has exactly two components, $X$ and $Y$, and that $\dim(X) + \dim(Y) +2 = \dim(M)$. We first show that $X$, $Y$ and $M$ are simply connected. Then we show that, up to $S^1$-equivariant diffeomorphism, there are finitely many such manifolds in each dimension. Moreover, we show that in low dimensions, the manifold is unique in a certain category. We use techniques from both areas of symplectic geometry and geometric topology.

NONLINEAR SYMPLECTIC GRASSMANNIANS AND HAMILTONIAN ACTIONS IN PREQUANTUM LINE BUNDLES

In this paper we extend to the Fréchet setting the following well-known fact about finite-dimensional symplectic geometry: if a Lie group G acts on a symplectic manifold in a Hamiltonian fashion with momentum map μ, given x ∈ M the isotropy group Gxacts linearly on the tangent space in a Hamiltonian fashion, with momentum map the Taylor expansion of μ up to degree 2. We use this result to give a conceptual explanation for a formula of Donaldson in [Scalar curvature and projective embeddings. I, J. Differential Geom.59(3) (2001) 479–522], which describes the momentum map of the Hamiltonian infinitesimal action of the Lie algebra of the group of Hamiltonian diffeomorphisms of a closed integral symplectic manifold, on sections of its prequantum line bundle.

Singular Reduction of Generalized Complex Manifolds

In this paper, we develop results in the direction of an analogue of Sjamaar and Lerman's singular reduction of Hamiltonian symplectic manifolds in the context of reduction of Hamiltonian generalized complex manifolds (in the sense of Lin and Tolman). Specifically, we prove that if a compact Lie group acts on a generalized complex manifold in a Hamiltonian fashion, then the partition of the global quotient by orbit types induces a partition of the Lin-Tolman quotient into generalized complex manifolds. This result holds also for reduction of Hamiltonian generalized K\"ahler manifolds.

Fixed points of symplectic periodic flows

AbstractThe study of fixed points is a classical subject in geometry and dynamics. If the circle acts in a Hamiltonian fashion on a compact symplectic manifold M, then it is classically known that there are at least $\frac {1}{2}\,{\dim M}+1$ fixed points; this follows from Morse theory for the momentum map of the action. In this paper we use Atiyah–Bott–Berline–Vergne (ABBV) localization in equivariant cohomology to prove that this conclusion also holds for symplectic circle actions with non-empty fixed sets, as long as the Chern class map is somewhere injective—the Chern class map assigns to a fixed point the sum of the action weights at the point. We complement this result with less sharp lower bounds on the number of fixed points, under no assumptions; from a dynamical systems viewpoint, our results imply that there is no symplectic periodic flow with exactly one or two equilibrium points on a compact manifold of dimension at least eight.

On a symplectic generalization of Petrie’s conjecture

Motivated by the Petrie conjecture, we consider the following questions: Let a circle act in a Hamiltonian fashion on a compact symplectic manifold ( M , ω ) (M,\omega ) which satisfies H 2 i ( M ; R ) = H 2 i ( C P n , R ) H^{2i}(M;\mathbb {R}) = H^{2i}({\mathbb C}{\mathbb P}^n,\mathbb {R}) for all i i . Is H j ( M ; Z ) = H j ( C P n ; Z ) H^j(M;\mathbb {Z}) = H^j({\mathbb C}{\mathbb P}^n;\mathbb {Z}) for all j j ? Is the total Chern class of M M determined by the cohomology ring H ∗ ( M ; Z ) H^*(M;\mathbb {Z}) ? We answer these questions in the six-dimensional case by showing that H j ( M ; Z ) H^j(M;\mathbb {Z}) is equal to H j ( C P 3 ; Z ) H^j({\mathbb C}{\mathbb P}^3;\mathbb {Z}) for all j j , by proving that only four cohomology rings can arise, and by computing the total Chern class in each case. We also prove that there are no exotic actions. More precisely, if H ∗ ( M ; Z ) H^*(M;\mathbb {Z}) is isomorphic to H ∗ ( C P 3 ; Z ) H^*({\mathbb C}{\mathbb P}^3;\mathbb {Z}) or H ∗ ( G ~ 2 ( R 5 ) ; Z ) H^*(\widetilde {G}_2(\mathbb {R}^5);\mathbb {Z}) , then the representations at the fixed components are compatible with one of the standard actions; in the remaining two cases, the representation is strictly determined by the cohomology ring. Finally, our results suggest a natural question: Do the remaining two cohomology rings actually arise? This question is closely related to some interesting problems in symplectic topology, such as embeddings of ellipsoids.

Theta functions on the Kodaira–Thurston manifold

The Kodaira—Thurston manifold M is a compact, 4-dimensional nilmanifold which is symplectic and complex but not Kahler. We describe a construction of ϑ-functions associated to M, which parallels the classical theory of ϑ-function associated to the torus (from the point of view of representation theory and geometry), and which yields pseudoperiodic complex-valued functions on ℝ 4 . There exists a three-step nilpotent Lie group G which acts transitively on the Kodaira—Thurston manifold and on the universal cover of M in a Hamiltonian fashion. The ϑ-functions discussed in this paper are intimately related to the representation theory of G in much the same way that the classical ϑ-functions are related to the Heisenberg group. One aspect of our results is a connection between the representation theory of G and the existence of Lagrangian and special Lagrangian foliations and torus fibrations in M; in particular, we show that G-invariant special Lagrangian foliations can be detected by a simple algebraic condition on certain subalgebras of the Lie algebra of G. Crucial to our generalization of ϑ-functions is the spectrum of the Laplacian Δ acting on sections of certain line bundles over M. One corollary of our work is a verification of a theorem of Guillemin-Uribe describing the structure (in the semiclassical limit) of the low-lying spectrum of Δ.

On the moment map on symplectic manifolds

We consider a connected symplectic manifold $M$ acted on properly and in a Hamiltonian fashion by a connected Lie group $G$. If $G$ is compact, then we characterize the symplectic manifolds whose squared moment map is constant. We also give a sufficient condition for $G$ to admit a symplectic orbit. Then we study the case when $G$ is a non-compact Lie group proving splitting results for symplectic manifolds.

Hamiltonian actions and homogeneous Lagrangian submanifolds

We consider a connected symplectic manifold $M$ acted on properly and in a Hamiltonian fashion by a connected Lie group $G$. Inspired by recent results, we study Lagrangian orbits of Hamiltonian actions. The dimension of the moduli space of the Lagrangian orbits is given. Also, we describe under which condition a Lagrangian orbit is isolated. If $M$ is a compact Kähler manifold, we give a necessary and sufficient condition for an isometric action to admit a Lagrangian orbit. Then we investigate homogeneous Lagrangian submanifolds on the symplectic cut and on the symplectic reduction. As an application of our results, we exhibit new examples of homogeneous Lagrangian submanifolds on the blow-up at one point of the complex projective space and on the weighted projective spaces. Finally, applying our result which may be regarded as Lagrangian slice theorem for a Hamiltonian group action with a fixed point, we give new examples of homogeneous Lagrangian submanifolds on irreducible Hermitian symmetric spaces of compact or noncompact type.

A splitting result for compact symplectic manifolds

We consider compact symplectic manifolds acted on effectively by a compact connected Lie group K in a Hamiltonian fashion. We prove that the squared moment map ∥μ∥2 is constant if and only if K is semisimple and the manifold is K-equivariantly symplectomorphic to a product of a flag manifold and a compact symplectic manifold which is acted on trivially by K. In the almost-Kahler setting the symplectomorphism turns out to be an isometry.

A Note on the Moment Map on Compact Kähler Manifolds

We consider compact Kahler manifolds acted on effectively by a connected compact Lie group K of isometries in a Hamiltonian fashion. We prove that the squared moment map ||μ||2 is constant if and only if K is semisimple and the manifold is biholomorphically and K-equivariantly isometric to a product of a flag manifold and a compact Kahler manifold which is acted on trivially by K.

An effective algorithm for the cohomology ring of symplectic reductions

Let G be a compact torus acting on a compact symplectic manifold M in a Hamiltonian fashion, and T a subtorus of G. We prove that the kernel of \( \kappa : H^\ast_G(M) \to H^\ast(M//G) \) is generated by a small number of classes \( \alpha \in H^\ast_G(M) \) satisfying very explicit restriction properties. Our main tool is the equivariant Kirwan map, a natural map from the G-equivariant cohomology of M to the G/T-equivariant cohomology of the symplectic reduction of M by T . We show this map is surjective. This is an equivariant version of the well-known result that the (nonequivariant) Kirwan map \( \kappa : H^\ast_G(M) \to H^\ast(M//G) \) is surjective. We also compute the kernel of the equivariant Kirwan map, generalizing the result due to Tolman and Weitsman [TW] in the case T = G and allowing us to apply their methods inductively. This result is new even in the case that dim T = 1. We close with a worked example: the cohomology ring of the product of two \( {\Bbb C}P^2{\rm s} \), quotiented by the diagonal 2-torus action.

The moment map and equivariant cohomology with generalized coefficients

Let M be a symplectic manifold acted on by a compact Lie group G in a Hamiltonian fashion, with proper moment map. In this situation we introduce a pushforward morphism P : H ∗ G(M)→ M −∞( g ∗) G , from the equivariant cohomology of M to the space of G-invariant distributions on g ∗ , which gives rise to symplectic invariants, in particular the pushforward of the Liouville measure. For the study of this pushforward morphism we make an intensive use of equivariant forms with generalized coefficients.

SINGULAR REDUCTION AND QUANTIZATION

Consider a compact prequantizable symplectic manifold M on which a compact Lie group G acts in a Hamiltonian fashion. The “quantization commutes with reduction” theorem asserts that the G-invariant part of the equivariant index of M is equal to the Riemann–Roch number of the symplectic quotient of M, provided the quotient is nonsingular. We extend this result to singular symplectic quotients, using partial desingularizations of the symplectic quotient to define its Riemann–Roch number. By similar methods we also compute multiplicities for the equivariant index of the dual of a prequantum bundle, and furthermore show that the arithmetic genus of a Hamiltonian G-manifold is invariant under symplectic reduction.

Moment maps on symplectic cones

We analyze some convexity properties of the image maps on symplectic cones, similar to the ones obtained by GuilleminSternberg and Atiyah for compact symplectic manifolds in the early 80’s. We prove the image of the moment map associated to the symplectic action of an n-torus on a symplectic cone is a polytopic convex cone in R n : Then, we generalize these results to symplectic manifolds obtained by special perturbations of the symplectic structure of a cone: we obtain sucient (and essentially necessary) conditions for the image of the moment map associated to the perturbed form to remain unchanged. Hamiltonian actions of tori and the images of their moment maps have been intensely studied in the eighties. According to the fundamental result, obtained independently by Atiyah [2] and Guillemin and Sternberg [4], the moment map of a Hamiltonian action of a torus on a compact symplectic manifold has for its image a convex polytope, spanned by the images of the xed points of the action. More recently, Prato [10] proved a convexity result concerning the image of moment maps of torus actions on non-compact symplectic manifolds. Theorem [10]. Let the torus T r act in a Hamiltonian fashion on the symplectic manifold (X;!) and denote by : X !(LieT r ) = R r the corresponding moment map. Suppose that there exists a circle S 1 =fe t 0g T r for some 02 LieT r such that 0 =h; 0iis a proper function having a minimum as its unique critical value. Then (X) is the convex hull of a nite number of rays in (LieT r ). In this paper, we prove a dierent kind of result, closer in spirit to perturbation theory: We start with a special non-compact symplectic manifold, described below, for which a similar convexity theorem holds, and consider the changes of the underlying symplectic structure which keep the image of the resulting moment maps unchanged. Let (X;!) be a symplectic cone with homothety groupft ;t 2 R + gso that t! = t!, for positive t, and compact base X=R + . Suppose that the torus T r acts symplectically on (X;!) and that this action commutes with