Recursos de colección
Gay-Balmaz, Francois; Ratiu, Tudor S.
The Lagrangian and Hamiltonian structures for an ideal gauge-charged ﬂuid are determined. Using a Kaluza–Klein point of view, the equations of motion are obtained by Lagrangian and Poisson reductions associated to the automorphism group of a principal bundle. As a consequence of the Lagrangian approach, a Kelvin–Noether theorem is obtained. The Hamiltonian formulation determines a non-canonical Poisson bracket associated to these equations.
Spaeth, Peter W.
On any closed symplectic manifold, we construct a path-connected neighborhood of the identity in the Hamiltonian diﬀeomorphism group with the property that each Hamiltonian diﬀeomorphism in this neighborhood admits a Hofer and spectral length minimizing path to the identity. This neighborhood is open in the $C^1$-topology. The construction utilizes a continuation argument and chain level result in the Floer theory of Lagrangian intersections.
Battaglia, Fiammetta; Prato, Elisa
The purpose of this article is to view Penrose rhombus tilings from the perspective of symplectic geometry. We show that each thick rhombus in such a tiling can be naturally associated to a highly singular 4-dimensional compact symplectic space $M_R$, while each thin rhombus can be associated to another such space $M_r$; both spaces are invariant under the Hamiltonian action of a 2-dimensional quasitorus, and the images of the corresponding moment mappings give the rhombuses back. The spaces $M_R$ and $M_r$ are diﬀeomorphic but not symplectomorphic.
We prove a theorem which asserts that the Lie algebra of all holomorphic vector ﬁelds on a compact Kähler manifold with a perturbed extremal metric has the structure similar to the case of an unperturbed extremal Kähler metric proved by Calabi.
Many interesting $C∗$-algebras can be viewed as quantizations of Poisson manifolds. I propose that a Poisson manifold may be quantized by a twisted polarized convolution $C∗$-algebra of a symplectic groupoid. Toward this end, I define polarizations for Lie groupoids and sketch the construction of this algebra. A large number of examples show that this idea unifies previous geometric constructions, including geometric quantization of symplectic manifolds and the $C∗$-algebra of a Lie groupoid. I sketch a few new examples, including twisted groupoid $C∗$-algebras as quantizations of bundle affine Poisson structures.
Lee, Dan A; Lipshitz, Robert
Suppose that the compact and connected Lie group $G$ acts holomorphically on the irreducible complex projective manifold $M$, and that the action linearizes to the Hermitian ample line bundle $L$ on $M$. Assume that $0$ is a regular value of the associated moment map. The spaces of global holomorphic sections of powers of $L$ may be decomposed over the finite dimensional irreducible representations of $G$. We study how the holomorphic sections in each equivariant piece asymptotically concentrate along the zero locus of the moment map. In the special case where $G$ acts freely on the zero locus of the moment map, this relates the scaling limits of the Szego...
In this paper, defining Poisson functions on super manifolds, we show that the graphs of Poisson functions are Dirac structures, and find Poisson functions which include as special cases both quasi-Poisson structures and twisted Poisson structures.
Extending our reduction construction in (S. Hu, Hamiltonian symmetries and reduction in generalized geometry, Houston J. Math., to appear, math.DG/0509060, 2005.) to the Hamiltonian action of a Poisson Lie group, we show that generalized Kähler reduction exists even when only one generalized complex structure in the pair is preserved by the group action. We show that the constructions in string theory of the (geometrical) T-duality with H-fluxes for principle bundles naturally arise as reductions of factorizable Poisson Lie group actions. In particular, the groups involved may be non-abelian.
We develop a formalism for relative Gromov–Witten invariants following Li J. Li, Stable morphisms to singular schemes and relative stable morphisms, J. Differential Geom. 57 (3) (2001), 509–578, J. Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 (2) (2002), 199–293 that is analogous to the symplectic field theory (SFT) of Eliashberg, Givental and Hofer Y. Eliashberg, A. Givental and H. Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. (Special Volume, Part II) (2000), 560–673 GAFA 2000 (Tel Aviv, 1999). This formalism allows us to express natural degeneration formulae in terms of generating functions and re-derive the formulae of Caporaso–Harris L. Caporaso and J. Harris, Counting plane curves...
Lisca, Paolo; Stipsicz, András I.
Phong, Duong H.; Sturm, Jacob
Oh, Yong-Geun; Müller, Stefan
Karshon, Yael; Kessler, Liat; Pinsonnault, Martin
Let ($M$, \omega) be a four-dimensional compact connected symplectic manifold. We prove that ($M$, \omega) admits only finitely many inequivalent Hamiltonian effective 2-torus actions. Consequently, if $M$ is simply connected, the number of conjugacy classes of 2-tori in the symplectomorphism group Sympl($M$, \omega) is finite. Our proof is “soft”. The proof uses the fact that if a symplectic four-manifold admits a toric action, then the restriction of the period map to the set of exceptional homology classes is proper. In an appendix, we present the Gromov–McDuff properness result for a general compact symplectic four-manifold.
Hutchings, Michael; Taubes, Clifford Henry
We describe a strategy for classifying symplectic fillings of contact structures supported by planar open books. We demonstrate the efficacy of this strategy in certain cases.
An effective class in a closed symplectic four-manifold is a twodimensional homology class which is realized by a J-holomorphic cycle for every tamed almost complex structure J. We first prove that effective classes are orthogonal to Lagrangian tori with respect to the intersection form. We then deduce an invariant under birational transformations of closed symplectic four-manifolds. We finally prove using the same techniques of symplectic field theory that the unit cotangent bundle of a compact orientable hyperbolic Lagrangian surface does not embed as a hypersurface of contact type in a rational or ruled symplectic four-manifold.
From a hermitian metric on the anticanonical bundle on a Del Pezzo surface, and a holomorphic section of it, we construct a one-parameter family of bihermitian metrics (or equivalently generalized Kähler structures). The construction appears to be linked to noncommutative geometry.
Donaldson, Simon; Guillemin, Victor; Mrowka, Tomasz; Tian, Gang