Moduli of J-Holomorphic Curves with Lagrangian Boundary Conditions and Open Gromov-Witten Invariants for an $S^1$-Equivariant Pair

Chiu-Chu Melissa Liu

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Moduli of J-Holomorphic Curves with Lagrangian Boundary Conditions and Open Gromov-Witten Invariants for an $S^1$-Equivariant Pair

Chiu-Chu Melissa Liu

Symplectic Geometry · Submitted 17 Oct 2002

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim The stable compactification of the moduli space of parametrized $J$-holomorphic curves in $X$ with boundary in $L$ (with prescribed topological data) is compact and Hausdorff in Gromov's $C^\infty$-topology.

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arXiv IDmath/0210257 (version 2)

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CitationChiu-Chu Melissa Liu. "Moduli of J-Holomorphic Curves with Lagrangian Boundary Conditions and Open Gromov-Witten Invariants for an $S^1$-Equivariant Pair." arXiv:math/0210257 (2002). https://pith.science/paper/OHPRAZBG

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Let $(X,\omega)$ be a symplectic manifold, $J$ be an $\omega$-tame almost complex structure, and $L$ be a Lagrangian submanifold. The stable compactification of the moduli space of parametrized $J$-holomorphic curves in $X$ with boundary in $L$ (with prescribed topological data) is compact and Hausdorff in Gromov's $C^\infty$-topology. We construct a Kuranishi structure with corners in the sense of Fukaya and Ono. This Kuranishi structure is orientable if $L$ is spin. In the special case where the expected dimension of the moduli space is zero, and there is an $S^1$ action on the pair $(X,L)$ which preserves $J$ and acts freely on $L$, we define the Euler number for this $S^1$ equivariant pair and the prescribed topological data. We conjecture that this rational number is the one computed by localization techniques using the given $S^1$ action.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Let $(X,\omega)$ be a symplectic manifold, $J$ be an $\omega$-tame almost complex structure, and $L$ be a Lagrangian submanifold.	background 1.00	0.06	0.02	asserted 0.72
s1	The stable compactification of the moduli space of parametrized $J$-holomorphic curves in $X$ with boundary in $L$ (with prescribed topological data) is compact and Hausdorff in Gromov's $C^\infty$-topology.	result 0.96	0.45	0.36	asserted 1.00
s2	We construct a Kuranishi structure with corners in the sense of Fukaya and Ono.	contribution 0.72	0.95	0.11	asserted 1.00
s3	This Kuranishi structure is orientable if $L$ is spin.	result 0.89	0.74	0.06	asserted 1.00
s4	In the special case where the expected dimension of the moduli space is zero, and there is an $S^1$ action on the pair $(X,L)$ which preserves $J$ and acts freely on $L$, we define the Euler number for this $S^1$ equivariant pair and the prescribed topological data.	contribution 0.93	0.92	0.19	asserted 1.00
s5	We conjecture that this rational number is the one computed by localization techniques using the given $S^1$ action.	interpretation 0.58	0.17	0.23	speculative 1.00

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Original abstract

Let $(X,\omega)$ be a symplectic manifold, $J$ be an $\omega$-tame almost complex structure, and $L$ be a Lagrangian submanifold. The stable compactification of the moduli space of parametrized $J$-holomorphic curves in $X$ with boundary in $L$ (with prescribed topological data) is compact and Hausdorff in Gromov's $C^\infty$-topology. We construct a Kuranishi structure with corners in the sense of Fukaya and Ono. This Kuranishi structure is orientable if $L$ is spin. In the special case where the expected dimension of the moduli space is zero, and there is an $S^1$ action on the pair $(X,L)$ which preserves $J$ and acts freely on $L$, we define the Euler number for this $S^1$ equivariant pair and the prescribed topological data. We conjecture that this rational number is the one computed by localization techniques using the given $S^1$ action.

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How the literature uses it

29 arXiv papers cite this in 65 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 34 sentences without a confident label

Uses its method or tool 3 papers

As in [ref] , to deal with the nodes of type $E$ in sense of [ref] , we assume that $L$ is homological trivial in $H_3(M,)$ and we fix a $4$-chain $K$ such that [equation]

Open Gromov-Witten theory without Obstruction · Statement of the results · 0.89
The Lemma follows from the fact that the contribution of nontrivial (that is with at least an edge) decorated graphs is zero if we use of a perturbations invariants with respect to the inducted $S^1$-action (as in [ref] ).

Frame ambiguity in Open Gromov-Witten invariants · Katz-Liu computation · 0.90
The construction combines ideas of [ref] (the Kimura-Stasheff-Voronov compactification) and [ref] (real Gromov-Witten theory).

Disjoinable Lagrangian spheres and dilations · The two-parameter family · 0.94

Compares with it 1 paper

The invariant of Liu [ref] agrees with the Open Gromov-Witten invariants $F_g,n(A)^f$ under the correspondence ([ref]).

Frame ambiguity in Open Gromov-Witten invariants · Katz-Liu computation · 0.98

Cites it as support 1 paper

In other words, the first reason to blow up is so we can discard unwanted symmetric topologies (see [ref] ) - such configurations cannot be in the image of the map $^::_0,k+2l(β+β)^/2$.

Moduli of Open Stable Maps to a Homogeneous Space · Introduction · 0.98

Cites it as background 18 papers

In Liu [ref] , the general notion of a stable map from a Riemann surfaces with boundary to an arbitrary pair $(X,L)$, is given.

Moduli of Open Stable Maps to a Homogeneous Space · The groupoid of disc-map configurations · 0.95
Liu in [ref] defines an invariant counting open curves associated to a triple $(X,L,ρ)$, where $ρ$ is an admissible $S^1$ action on $X$ (this means that $ρ$ preserves $L$ and acts freely on $L$).

Frame ambiguity in Open Gromov-Witten invariants · Katz-Liu computation · 0.84
Foundational issues concerning the construction of $_g,k,l$ are addressed in [ref] .

Intersection theory on moduli of disks, open KdV and Virasoro · Moduli of Riemann surfaces with boundary · 0.83

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