Lagrangian floer Theory over integers: spherically positive symplectic manifolds

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.

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Invariance of symplectic cohomology and twisted cotangent bundles over surfaces

International Journal of Mathematics ◽

10.1142/s0129167x20500706 ◽

2020 ◽

Vol 31 (09) ◽

pp. 2050070

Author(s):

Gabriele Benedetti ◽

Alexander F. Ritter

Keyword(s):

Symplectic Form ◽

Cotangent Bundle ◽

Local System ◽

Symplectic Manifolds ◽

Existence Results ◽

Open Convex ◽

Floer Theory ◽

Cotangent Bundles ◽

Sample Application ◽

Symplectic Cohomology

We prove that symplectic cohomology for open convex symplectic manifolds is invariant when the symplectic form undergoes deformations which may be nonexact and noncompactly supported, provided one uses the correct local system of coefficients in Floer theory. As a sample application beyond the Liouville setup, we describe in detail the symplectic cohomology for disc bundles in the twisted cotangent bundle of surfaces, and we deduce existence results for periodic magnetic geodesics on surfaces. In particular, we show the existence of geometrically distinct orbits by exploiting properties of the BV-operator on symplectic cohomology.

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Morse theory for the Hofer length functional

Journal of Topology and Analysis ◽

10.1142/s1793525314500083 ◽

2014 ◽

Vol 06 (02) ◽

pp. 193-210 ◽

Cited By ~ 1

Author(s):

Yasha Savelyev

Keyword(s):

Morse Theory ◽

Symplectic Manifolds ◽

Floer Theory

Following [16], we develop here a connection between Morse theory for the (positive) Hofer length functional L : Ω Ham (M, ω) → ℝ, with Gromov–Witten/Floer theory, for monotone symplectic manifolds (M, ω). This gives some immediate restrictions on the topology of the group of Hamiltonian symplectomorphisms (possibly relative to the Hofer length functional), and a criterion for non-existence of certain higher index geodesics for the Hofer length functional. The argument is based on a certain automatic transversality phenomenon which uses Hofer geometry to conclude transversality and may be useful in other contexts. Strangely the monotone assumption seems essential for this argument, as abstract perturbations necessary for the virtual moduli cycle, decouple us from underlying Hofer geometry, causing automatic transversality to break.

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Constructing symplectic manifolds

10.1093/oso/9780198794899.003.0008 ◽

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.

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Holomorphic Lagrangian subvarieties in holomorphic symplectic manifolds with Lagrangian fibrations and special Kähler geometry

European Journal of Mathematics ◽

10.1007/s40879-021-00488-3 ◽

2021 ◽

Author(s):

Ljudmila Kamenova ◽

Misha Verbitsky

Keyword(s):

Symplectic Manifolds ◽

Kahler Geometry

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Hamilton–Jacobi formalism on locally conformally symplectic manifolds

Journal of Mathematical Physics ◽

10.1063/5.0021790 ◽

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

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Poisson structures of near-symplectic manifolds and their cohomology

Journal of Mathematical Physics ◽

10.1063/5.0041230 ◽

2021 ◽

Vol 62 (3) ◽

pp. 033513

Author(s):

Panagiotis Batakidis ◽

Ramón Vera

Keyword(s):

Symplectic Manifolds ◽

Poisson Structures

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NON-SYMPLECTIC INVOLUTIONS ON MANIFOLDS OF -TYPE

Nagoya Mathematical Journal ◽

10.1017/nmj.2019.43 ◽

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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