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Symplectic geometry on moduli spaces of J-holomorphic curves

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Joseph Coffey, Liat Kessler, Alvaro Pelayo

Symplectic Geometry · Submitted 19 May 2009

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

The authors' main claim Then we give conditions on a compatible almost complex structure J on (M,\omega) that ensure that the restriction of the form to the moduli space of simple immersed J-holomorphic Sigma-curves in a homology class A in H_2(M,\Z) is a symplectic form, and show applications and examples.

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arXiv ID0905.3009 (version 2)
Pith IDGJZOZQ24, Pith's permanent record of this paper and its review
CitationAlvaro Pelayo, Joseph Coffey, Liat Kessler. "Symplectic geometry on moduli spaces of J-holomorphic curves." arXiv:0905.3009 (2009). https://pith.science/paper/GJZOZQ24
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,734 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.39 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.05 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

Let (M,\omega) be a symplectic manifold, and Sigma a compact Riemann surface. We define a 2-form on the space of immersed symplectic surfaces in M, and show that the form is closed and non-degenerate, up to reparametrizations. Then we give conditions on a compatible almost complex structure J on (M,\omega) that ensure that the restriction of the form to the moduli space of simple immersed J-holomorphic Sigma-curves in a homology class A in H_2(M,\Z) is a symplectic form, and show applications and examples. In particular, we deduce sufficient conditions for the existence of J-holomorphic Sigma-curves in a given homology class for a generic J.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Let (M,\omega) be a symplectic manifold, and Sigma a compact Riemann surface. background 1.00 0.09 0.02 asserted 0.74
s1 We define a 2-form on the space of immersed symplectic surfaces in M, and show that the form is closed and non-degenerate, up to reparametrizations. contribution 0.77 0.95 0.15 asserted 1.00
s2 Then we give conditions on a compatible almost complex structure J on (M,\omega) that ensure that the restriction of the form to the moduli space of simple immersed J-holomorphic Sigma-curves in a homology class A in H_2(M,\Z) is a symplectic form, and show applications and examples. contribution 0.66 0.95 0.64 asserted 1.00
s3 In particular, we deduce sufficient conditions for the existence of J-holomorphic Sigma-curves in a given homology class for a generic J. result 0.88 0.95 0.29 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

Let (M,\omega) be a symplectic manifold, and Sigma a compact Riemann surface. We define a 2-form on the space of immersed symplectic surfaces in M, and show that the form is closed and non-degenerate, up to reparametrizations. Then we give conditions on a compatible almost complex structure J on (M,\omega) that ensure that the restriction of the form to the moduli space of simple immersed J-holomorphic Sigma-curves in a homology class A in H_2(M,\Z) is a symplectic form, and show applications and examples. In particular, we deduce sufficient conditions for the existence of J-holomorphic Sigma-curves in a given homology class for a generic J.

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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

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