scholarly journals Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds

2007 ◽  
pp. 525-570 ◽  
Author(s):  
Yong-Geun Oh
Keyword(s):  
Symplectic Manifolds ◽  
Spectral Invariants ◽  
Hamiltonian Paths

scholarly journals DUALITY IN FILTERED FLOER–NOVIKOV COMPLEXES

2010 ◽  
Vol 02 (02) ◽  
pp. 233-258 ◽  
Author(s):  
MICHAEL USHER
Keyword(s):  
Chain Complex ◽  
Bilinear Pairing ◽  
Symplectic Manifolds ◽  
Duality Relation ◽  
Spectral Invariants ◽  
Floer Theory ◽  
Intersection Pairing ◽  
Period Map ◽  
New Formula ◽  
Special Case

We prove that a certain bilinear pairing (analogous to the Poincaré–Lefschetz intersection pairing) between filtered sub- and quotient complexes of a Floer-type chain complex and of its "opposite complex" is always nondegenerate on homology. This implies a duality relation for the Oh–Schwarz-type spectral invariants of these complexes which (in Hamiltonian Floer theory) was established in the special case that the period map has discrete image by Entov and Polterovich. The duality relation served as a key lemma in Entov and Polterovich's construction of a Calabi quasimorphism on certain rational symplectic manifolds, and the result that we prove here implies that their construction remains valid when the rationality hypothesis is dropped. Apart from this, we also use the nondegeneracy of the pairing to establish a new formula for what we have previously called the boundary depth of a Floer chain complex; this formula shows that the boundary depth is unchanged under passing to the opposite complex.

scholarly journals DESCENT AND C0-RIGIDITY OF SPECTRAL INVARIANTS ON MONOTONE SYMPLECTIC MANIFOLDS

2012 ◽  
Vol 04 (04) ◽  
pp. 481-498 ◽  
Author(s):  
SOBHAN SEYFADDINI
Keyword(s):  
Symplectic Manifolds ◽  
Spectral Invariants ◽  
Open Set ◽  
Continuity Properties ◽  
Hofer Metric

We obtain estimates showing that on monotone symplectic manifolds (asymptotic) spectral invariants of Hamiltonians which vanish on a non-empty open set, U, descend from [Formula: see text] to Hamc(M\U). Furthermore, we show that these invariants are continuous with respect to the C0-topology on Hamc(M\U).We apply the above results to Hofer geometry and establish unboundedness of the Hofer diameter of Hamc(M\U) for stably displaceable U. We also answer a question of F. Le Roux about C0-continuity properties of the Hofer metric.

scholarly journals Spectral invariants of distance functions

2016 ◽  
Vol 08 (04) ◽  
pp. 655-676 ◽  
Author(s):  
Suguru Ishikawa
Keyword(s):  
Symplectic Manifold ◽  
Symplectic Form ◽  
Floer Homology ◽  
Symplectic Manifolds ◽  
Distance Functions ◽  
Spectral Invariants ◽  
Euclidian Space ◽  
Closed Riemann Surface ◽  
Genus Formula ◽  
Spectral Invariant

Calculating the spectral invariant of Floer homology of the distance function, we can find new superheavy subsets in symplectic manifolds. We show if convex open subsets in Euclidian space with the standard symplectic form are disjointly embedded in a spherically negative monotone closed symplectic manifold, their complement is superheavy. In particular, the [Formula: see text] bouquet in a closed Riemann surface with genus [Formula: see text] is superheavy. We also prove some analogous properties of a monotone closed symplectic manifold. These can be used to extend Seyfaddni’s result about lower bounds of Poisson bracket invariant.

Constructing symplectic manifolds

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Final Section ◽  
Symplectic Manifolds ◽  
Homotopy Equivalence ◽  
Codimension Two ◽  
Open Manifolds ◽  
Connected Sums ◽  
Symplectic Forms ◽  
Ambient Manifold ◽  
Blowing Up ◽  
Four Manifolds

This chapter examines various ways to construct symplectic manifolds and submanifolds. It begins by studying blowing up and down in both the complex and the symplectic contexts. The next section is devoted to a discussion of fibre connected sums and describes Gompf’s construction of symplectic four-manifolds with arbitrary fundamental group. The chapter also contains an exposition of Gromov’s telescope construction, which shows that for open manifolds the h-principle rules and the inclusion of the space of symplectic forms into the space of nondegenerate 2-forms is a homotopy equivalence. The final section outlines Donaldson’s construction of codimension two symplectic submanifolds and explains the associated decompositions of the ambient manifold.

scholarly journals Hamilton–Jacobi formalism on locally conformally symplectic manifolds

2021 ◽  
Vol 62 (3) ◽  
pp. 033506
Author(s):  
Oğul Esen ◽  
Manuel de León ◽  
Cristina Sardón ◽  
Marcin Zajşc
Keyword(s):  
Symplectic Manifolds ◽  
Jacobi Formalism ◽  
Locally Conformally Symplectic

scholarly journals Poisson structures of near-symplectic manifolds and their cohomology

2021 ◽  
Vol 62 (3) ◽  
pp. 033513
Author(s):  
Panagiotis Batakidis ◽  
Ramón Vera
Keyword(s):  
Symplectic Manifolds ◽  
Poisson Structures

scholarly journals NON-SYMPLECTIC INVOLUTIONS ON MANIFOLDS OF -TYPE

2020 ◽  
pp. 1-25
Author(s):  
CHIARA CAMERE ◽  
ALBERTO CATTANEO ◽  
ANDREA CATTANEO
Keyword(s):  
Moduli Spaces ◽  
Quadratic Forms ◽  
Symplectic Manifolds ◽  
Hilbert Schemes ◽  
Small Rank ◽  
Twisted Sheaves ◽  
Formula Type

We study irreducible holomorphic symplectic manifolds deformation equivalent to Hilbert schemes of points on a $K3$ surface and admitting a non-symplectic involution. We classify the possible discriminant quadratic forms of the invariant and coinvariant lattice for the action of the involution on cohomology and explicitly describe the lattices in the cases where the invariant lattice has small rank. We also give a modular description of all $d$ -dimensional families of manifolds of $K3^{[n]}$ -type with a non-symplectic involution for $d\geqslant 19$ and $n\leqslant 5$ and provide examples arising as moduli spaces of twisted sheaves on a $K3$ surface.

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