scholarly journals REMARKS ON INVARIANTS OF HAMILTONIAN LOOPS

2010 ◽  
Vol 02 (03) ◽  
pp. 277-325 ◽  
Author(s):  
EGOR SHELUKHIN
Keyword(s):  
Maslov Index ◽  
Universal Cover ◽  
Symplectic Manifolds ◽  
Linear Functions ◽  
Torus Action ◽  
Hamiltonian Diffeomorphisms ◽  
Toric Manifolds ◽  
Mixed Action ◽  
Nonlinear Maslov Index ◽  
Moment Polytopes

In this paper the interrelations between several natural morphisms on the π1 of groups of Hamiltonian diffeomorphisms are investigated. As an application, the equality of the (nonlinear) Maslov index of loops of quantomorphisms of prequantizations of ℂPn and the Calabi–Weinstein invariant is shown, settling affirmatively a conjecture by Givental. We also prove, in the wake of a remark by Woodward, the proportionality of the mixed action-Maslov morphism and the Futaki invariant on loops of Hamiltonian biholomorphisms of Fano Kahler manifolds. Finally, a family of generalized action-Maslov invariants is computed for toric manifolds, on loops coming from the torus action, via barycenters of their moment polytopes, with an application to mass-linear functions recently introduced by McDuff and Tolman. In addition, we reinterpret the quasimorphism of Py on the universal cover of the group of Hamiltonian diffeomorphisms of monotone symplectic manifolds.

scholarly journals THE NONLINEAR MASLOV INDEX AND THE CALABI HOMOMORPHISM

2007 ◽  
Vol 09 (06) ◽  
pp. 769-780 ◽  
Author(s):  
GABI BEN SIMON
Keyword(s):  
Projective Space ◽  
Open Subset ◽  
Complex Projective Space ◽  
Maslov Index ◽  
Universal Cover ◽  
Hamiltonian Diffeomorphisms ◽  
Calabi Invariant ◽  
Nonlinear Maslov Index ◽  
Complex Projective ◽  
Calabi Homomorphism

In this paper, we find that the asymptotic nonlinear Maslov index defined on the universal cover of the group of all contact Hamiltonian diffeomorphisms of the standard (2n - 1)-dimensional contact sphere is a quasimorphism. Then we show our main result: Let M be standard (n - 1)-dimensional complex projective space. We prove that the value of the pullback of the asymptotic nonlinear Maslov index to the universal cover of the group of Hamiltonian diffeomorphisms of M, when evaluated on a diffeomorphism supported in a sufficiently small open subset of M, equals [Formula: see text] times the Calabi invariant of this diffeomorphism.

scholarly journals Symplectic Capacities of Toric Manifolds and Related Results

2006 ◽  
Vol 181 ◽  
pp. 149-184 ◽  
Author(s):  
Guangcun Lu
Keyword(s):  
Line Bundle ◽  
Blow Up ◽  
Ample Line Bundle ◽  
Symplectic Manifolds ◽  
Toric Manifolds ◽  
Seshadri Constant ◽  
Combinatorial Data

AbstractIn this paper we give concrete estimations for the pseudo symplectic capacities of toric manifolds in combinatorial data. Some examples are given to show that our estimates can compute their pseudo symplectic capacities. As applications we also estimate the symplectic capacities of the polygon spaces. Other related results are impacts of symplectic blow-up on symplectic capacities, symplectic packings in symplectic toric manifolds, the Seshadri constant of an ample line bundle on toric manifolds, and symplectic capacities of symplectic manifolds withS1-action.

scholarly journals Convexity of the moment map image for torus actions on b m -symplectic manifolds

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170420 ◽  
Author(s):  
Victor W. Guillemin ◽  
Eva Miranda ◽  
Jonathan Weitsman
Keyword(s):  
Symplectic Manifold ◽  
Integrable Systems ◽  
Symplectic Manifolds ◽  
Moment Map ◽  
Torus Action ◽  
Convexity Theorem ◽  
Theme Issue ◽  
Torus Actions ◽  
Finite Dimensional ◽  
The Moment

We prove a convexity theorem for the image of the moment map of a Hamiltonian torus action on a b m -symplectic manifold. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

scholarly journals Givental’s Non-linear Maslov Index on Lens Spaces

2020 ◽  
Author(s):  
Gustavo Granja ◽  
Yael Karshon ◽  
Milena Pabiniak ◽  
Sheila Sandon
Keyword(s):  
Projective Space ◽  
Maslov Index ◽  
Universal Cover ◽  
Lens Spaces ◽  
Real Projective Space ◽  
Arnold Conjecture ◽  
Identity Component ◽  
Contact Rigidity ◽  
Non Linear

Abstract Givental’s non-linear Maslov index, constructed in 1990, is a quasimorphism on the universal cover of the identity component of the contactomorphism group of real projective space. This invariant was used by several authors to prove contact rigidity phenomena such as orderability, unboundedness of the discriminant and oscillation metrics, and a contact geometric version of the Arnold conjecture. In this article, we give an analogue for lens spaces of Givental’s construction and its applications.

scholarly journals Non-contractible periodic orbits in Hamiltonian dynamics on closed symplectic manifolds

2016 ◽  
Vol 152 (9) ◽  
pp. 1777-1799 ◽  
Author(s):  
Viktor L. Ginzburg ◽  
Başak Z. Gürel
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Symplectic Manifold ◽  
Floer Homology ◽  
Hamiltonian Dynamics ◽  
Symplectic Manifolds ◽  
Homotopy Classes ◽  
Hamiltonian Diffeomorphisms ◽  
Favorable Conditions ◽  
Hamiltonian Diffeomorphism

We study Hamiltonian diffeomorphisms of closed symplectic manifolds with non-contractible periodic orbits. In a variety of settings, we show that the presence of one non-contractible periodic orbit of a Hamiltonian diffeomorphism of a closed toroidally monotone or toroidally negative monotone symplectic manifold implies the existence of infinitely many non-contractible periodic orbits in a specific collection of free homotopy classes. The main new ingredient in the proofs of these results is a filtration of Floer homology by the so-called augmented action. This action is independent of capping and, under favorable conditions, the augmented action filtration for toroidally (negative) monotone manifolds can play the same role as the ordinary action filtration for atoroidal manifolds.

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