Non-compact Symplectic Manifold Research Articles

Removing a Ray from a Noncompact Symplectic Manifold

Abstract We prove that any noncompact symplectic manifold that admits a properly embedded ray with a wide neighborhood is symplectomorphic to the complement of the ray by constructing an explicit symplectomorphism in the case of the standard Euclidean space. This allows us to excide a ray from a symplectic manifold and we use this excision trick to find a nowhere vanishing Liouville vector field on every cotangent bundle.

On the Vergne conjecture

Consider a Hamiltonian action by a compact Lie group on a possibly non-compact symplectic manifold. We give a short proof of a geometric formula for the decomposition into irreducible representations of the equivariant index of a \({{\mathrm{{{\mathrm{Spin}}}^c}}}\)-Dirac operator in this context. This formula was conjectured by Vergne in (Eur Math Soc Zurich I:635–664, 2007) and proved by Ma and Zhang in (Acta Math 212:11–57, 2014).

$S^1$-equivariant local index and transverse index for non-compact symplectic manifolds

We define an $S^1$-equivariant index for non-compact symplectic manifolds with Hamiltonian $S^1$-action. We use the perturbation by Dirac-type operator along the $S^1$-orbits. We give a formulation and a proof of quantization conjecture for this $S^1$-equivariant index. We also give comments on the relation between our $S^1$-equivariant index and the index of transverse elliptic operators.

Geometric quantization for proper moment maps: the Vergne conjecture

We establish an analytic interpretation for the index of certain transversally elliptic symbols on non-compact manifolds. By using this interpretation, we establish a geometric quantization formula for a Hamiltonian action of a compact Lie group acting on a non-compact symplectic manifold with proper moment map. In particular, we present a solution to a conjecture of Michèle Vergne in her ICM 2006 plenary lecture.

Geometric Quantization in the Non-compact Setting

Geometric Quantization in the Non-compact Setting

Seidel’s long exact sequence on Calabi-Yau manifolds

In this paper, we generalize construction of Seidel's long exact sequence of Lagrangian Floer cohomology to that of compact Lagrangian submanifolds with vanishing Malsov class on general Calabi-Yau manifolds. We use the framework of anchored Lagrangian submanifolds developed in \cite{fooo:anchor} and some compactness theorem of \emph{smooth} $J$-holomorphic sections of Lefschetz Hamiltonian fibration for a generic choice of $J$. The proof of the latter compactness theorem involves a study of proper pseudoholomorphic curves in the setting of noncompact symplectic manifolds with cylindrical ends.

Integrals of equivariant forms over noncompact symplectic manifolds

This article is a result of the AIM workshop on Moment Maps and Surjectivity in Vari- ous Geometries (August 9 - 13, 2004) organized by T. Holm, E. Lerman and S. Tolman. At that workshop I was introduced to the work of T. Hausel and N. Proudfoot on hyperkahler quotients (HP). One interesting feature of their article is that they consider integrals of equivariant forms over non-compact symplectic manifolds which do not converge, so they formally define these integrals as sums over the zeroes of vector fields, as in the Berline- Vergne localization formula. In this article we introduce a geometric-analytic regularization technique which makes such integrals converge and utilizes the symplectic structure of the manifold. We also prove that the Berline-Vergne localization formula holds for these inte- grals as well. The key step here is to redefine the collection of integrals R M �(X), X 2 g, as a distribution on the Lie algebra g. We expect our regularization technique to generalize to non-compact group actions extending the results of (L1, L2).

Geometric quantization for proper moment maps

We establish a geometric quantization formula for Hamiltonian actions of a compact Lie group acting on a non-compact symplectic manifold such that the associated moment map is proper. In particular, we give a solution to a conjecture of Michèle Vergne. To cite this article: X. Ma, W. Zhang, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

String equation and quantum cohomology for noncompact symplectic manifolds

We use our Gromov–Witten invariant theory in a previous Note for noncompact geometrically bounded symplectic manifolds to get solutions of the generalized string equation and dilation equation and their variants. More solutions of the WDVV equation and quantum products on cohomology groups are also obtained for the symplectic manifolds with finitely dimensional cohomology groups. To cite this article: G. Lu, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Gromov–Witten invariants and rigidity of Hamiltonian loops with compact support on noncompact symplectic manifolds

In this paper the Gromov-Witten invariants on a class of noncompact symplectic manifolds are defined by combining Ruan-Tian’s method with that of McDuff-Salamon. The main point of the arguments is to introduce a method dealing with the transversality problems in the case of noncompact manifolds. Moreover, the techniques are also used to study the topological rigidity of Hamiltonian loops with compact support on a class of noncompact symplectic manifolds.

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