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Minimality and irreducibility of symplectic four-manifolds

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M. J. D. Hamilton, D. Kotschick

Symplectic Geometry · Submitted 9 Sep 2005

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

The authors' main claim We prove that all minimal symplectic four-manifolds are essentially irreducible.

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arXiv IDmath/0509204 (version 2)
Pith IDZWGGT44Y, Pith's permanent record of this paper and its review
CitationD. Kotschick, M. J. D. Hamilton. "Minimality and irreducibility of symplectic four-manifolds." arXiv:math/0509204 (2005). https://pith.science/paper/ZWGGT44Y
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The compiled record

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-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,910 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.22 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.45 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

We prove that all minimal symplectic four-manifolds are essentially irreducible. We also clarify the relationship between holomorphic and symplectic minimality of K\"ahler surfaces. This leads to a new proof of the deformation-invariance of holomorphic minimality for complex surfaces with even first Betti number which are not Hirzebruch surfaces.

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 We prove that all minimal symplectic four-manifolds are essentially irreducible. result 0.96 0.96 0.90 asserted 1.00
s1 We also clarify the relationship between holomorphic and symplectic minimality of K\"ahler surfaces. contribution 0.73 0.89 0.05 asserted 0.99
s2 This leads to a new proof of the deformation-invariance of holomorphic minimality for complex surfaces with even first Betti number which are not Hirzebruch surfaces. result 0.62 0.87 0.09 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

We prove that all minimal symplectic four-manifolds are essentially irreducible. We also clarify the relationship between holomorphic and symplectic minimality of K\"ahler surfaces. This leads to a new proof of the deformation-invariance of holomorphic minimality for complex surfaces with even first Betti number which are not Hirzebruch surfaces.

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