Arnold Conjecture and Gromov–witten Invariant

Fukaya, Kenji; Ono, Kaoru

doi:10.1016/s0040-9383(98)00042-1

Cited by 400 publications

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“…Where the notation of [9,10] differs, for instance in whether Kuranishi neighbourhoods are (V, E, Γ, s, ψ) with V a manifold or (V, E, s, ψ) with V an orbifold, we generally follow [9].…”

Section: Background Materials On Kuranishi Spaces and Multisectionsmentioning

confidence: 99%

“…Here we follow [10,Def. 5.1] in taking E p to be a vector bundle, rather than a finite-dimensional vector space as in [9, Def.…”

Section: Background Materials On Kuranishi Spaces and Multisectionsmentioning

confidence: 99%

“…Good coordinate systems. Good coordinate systems are convenient choices of finite coverings of X by Kuranishi neighbourhoods, [10,Def. 6.1], [9, Lem.…”

Section: Orientation Conventions Suppose X Xmentioning

confidence: 99%

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Immersed Lagrangian floer theory

Akaho¹,

Joyce²

2010

J. Differential Geom.

188

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Abstract. Let (M, ω) be a compact symplectic 2n-manifold, and L a compact embedded Lagrangian submanifold in M . Fukaya, Oh, Ohta and Ono [9] construct Lagrangian Floer cohomology for such M, L, yielding groups HF * (L, b; Λnov) for one Lagrangian or HF * `( L 1 , b 1 ), (L 2 , b 2 ); Λnov´for two, where b, b 1 , b 2 are choices of bounding cochains, and exist if and only if L, L 1 , L 2 have unobstructed Floer cohomology. These are independent of choices up to canonical isomorphism, and have important invariance properties under Hamiltonian equivalence. Floer cohomology groups are the morphism groups in the derived Fukaya category of (M, ω), and so are an essential part of the Homological Mirror Symmetry Conjecture of Kontsevich.The goal of this paper is to extend [9] to immersed Lagrangians L in M with immersion ι : L → M , with transverse self-intersections. In the embedded case, Floer cohomology HF * (L, b; Λnov) is a modified, 'quantized' version of singular homology H n− * (L; Λnov) over the Novikov ring Λnov. In our immersed case, HF * (L, b; Λnov) turns out to be a quantized version ofThe theory becomes simpler and more powerful for graded Lagrangians in Calabi-Yau manifolds, when we can work over a smaller Novikov ring ΛCY. The proofs involve associating a gapped filtered A∞ algebra over Λ 0 nov or Λ 0 CY to ι : L → M , which is independent of nearly all choices up to canonical homotopy equivalence, and is built using a series of finite approximations called A N,0 algebras for N = 0, 1, 2, . . ..

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Section: Background Materials On Kuranishi Spaces and Multisectionsmentioning

confidence: 99%

“…Here we follow [10,Def. 5.1] in taking E p to be a vector bundle, rather than a finite-dimensional vector space as in [9, Def.…”

Section: Background Materials On Kuranishi Spaces and Multisectionsmentioning

confidence: 99%

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Immersed Lagrangian floer theory

Akaho¹,

Joyce²

2010

J. Differential Geom.

188

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“…As in [12], for technical reasons the closed symplectic manifold (M, ω) is assumed to be strongly semi-positive whenever we use Floer or quantum (co)homology, though in view of the recent developments (see [15], [25], [26], [27]) it is likely that this assumption can be removed.…”

Section: Commutator Length Of Elements In Ham (M ω)mentioning

confidence: 99%

“…Given a (time-dependent) Hamiltonian function H on M denote by P(H) the space of equivalence classesγ = [γ, f ] of pairs (γ, f ), where γ : S 1 → M is a contractible time-1 periodic trajectory of the Hamiltonian flow of H. The theorems proving Arnold's conjecture imply that any Hamiltonian symplectomorphism of M has a closed contractible 1-periodic orbit and therefore P(H) is always non-empty (see [15], [25]; in the semi-positive case the conjecture was first proven in [21]). …”

Section: Floer Cohomology: Basic Definitionsmentioning

confidence: 99%

Commutator length of symplectomorphisms

Entov¹

2004

Commentarii Mathematici Helvetici

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Abstract.Each element x of the commutator subgroup [G, G] of a group G can be represented as a product of commutators. The minimal number of factors in such a product is called the commutator length of x. The commutator length of G is defined as the supremum of commutator lengths of elements of [G, G].We show that for certain closed symplectic manifolds (M, ω), including complex projective spaces and Grassmannians, the universal cover Ham (M, ω) of the group of Hamiltonian symplectomorphisms of (M, ω) has infinite commutator length. In particular, we present explicit examples of elements in Ham (M, ω) that have arbitrarily large commutator length -the estimate on their commutator length depends on the multiplicative structure of the quantum cohomology of (M, ω). By a different method we also show that in the case c 1 (M ) = 0 the group Ham (M, ω) and the universal cover Symp 0 (M, ω) of the identity component of the group of symplectomorphisms of (M, ω) have infinite commutator length. Mathematics Subject Classification (2000). 53D22, 53D05, 53D40, 53D45.

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Fredholm theory and transversality for noncompact pseudoholomorphic mapsin symplectizations

Dragnev

2004

Comm Pure Appl Math

165

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We study pseudoholomorphic maps from a punctured Riemann surface into the symplectization of a contact manifold. A Fredholm theory yields the virtual dimension of the moduli spaces of such maps in terms of the Euler characteristic of the Riemann surface and the asymptotics data given by the periodic solutions of the Reeb vector field associated to the contact form. The transversality results establish the existence of additional structure for these spaces. To be more precise, we prove that these spaces are generically smooth manifolds, and therefore their virtual dimension coincides with their actual dimension.

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Arnold Conjecture and Gromov–witten Invariant

Cited by 400 publications

References 0 publications

Immersed Lagrangian floer theory

Immersed Lagrangian floer theory

Commutator length of symplectomorphisms

Fredholm theory and transversality for noncompact pseudoholomorphic mapsin symplectizations

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