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Morse Theory for the Space of Higgs Bundles

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Graeme Wilkin

Differential Geometry · Submitted 5 Nov 2006

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

The authors' main claim The main result is that the gradient flow with initial conditions $(A'', \phi)$ converges to a critical point of this functional, the isomorphism class of which is given by the graded object associated to the Harder-Narasimhan-Seshadri filtration of $(A'', \phi)$.

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arXiv IDmath/0611113 (version 2)
Pith IDDQGRMZRP, Pith's permanent record of this paper and its review
CitationGraeme Wilkin. "Morse Theory for the Space of Higgs Bundles." arXiv:math/0611113 (2006). https://pith.science/paper/DQGRMZRP
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,093 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.17 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.72 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

Here we prove the necessary analytic results to construct a Morse theory for the Yang-Mills-Higgs functional on the space of Higgs bundles over a compact Riemann surface. The main result is that the gradient flow with initial conditions $(A'', \phi)$ converges to a critical point of this functional, the isomorphism class of which is given by the graded object associated to the Harder-Narasimhan-Seshadri filtration of $(A'', \phi)$. In particular, the results of this paper show that the failure of hyperk\"ahler Kirwan surjectivity for rank 2 fixed determinant Higgs bundles does not occur because of a failure of the existence of a Morse theory.

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 Here we prove the necessary analytic results to construct a Morse theory for the Yang-Mills-Higgs functional on the space of Higgs bundles over a compact Riemann surface. contribution 1.00 0.94 0.09 asserted 1.00
s1 The main result is that the gradient flow with initial conditions $(A'', \phi)$ converges to a critical point of this functional, the isomorphism class of which is given by the graded object associated to the Harder-Narasimhan-Seshadri filtration of $(A'', \phi)$. result 1.00 0.91 0.96 asserted 1.00
s2 In particular, the results of this paper show that the failure of hyperk\"ahler Kirwan surjectivity for rank 2 fixed determinant Higgs bundles does not occur because of a failure of the existence of a Morse theory. interpretation 0.99 0.81 0.11 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

Here we prove the necessary analytic results to construct a Morse theory for the Yang-Mills-Higgs functional on the space of Higgs bundles over a compact Riemann surface. The main result is that the gradient flow with initial conditions $(A'', \phi)$ converges to a critical point of this functional, the isomorphism class of which is given by the graded object associated to the Harder-Narasimhan-Seshadri filtration of $(A'', \phi)$. In particular, the results of this paper show that the failure of hyperk\"ahler Kirwan surjectivity for rank 2 fixed determinant Higgs bundles does not occur because of a failure of the existence of a Morse theory.

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