Betti numbers of the moduli space of rank 3 parabolic Higgs bundles.

Parabolic Higgs bundles on a Riemann surface are of interest for many reasons, one of them being their importance in the study of representations of the fundamental group of the punctured surface in the complex general linear group. In this paper we calculate the Betti numbers of the moduli space of...

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Authors: García Prada, O., Gothen, P.B., Muñoz, Vicente
Format: article
Publication Date:2007
Country:España
Institution:Universidad Complutense de Madrid (UCM)
Repository:Docta Complutense
Language:English
OAI Identifier:oai:docta.ucm.es:20.500.14352/50605
Online Access:https://hdl.handle.net/20.500.14352/50605
Access Level:Open access
Keyword:512.7
Parabolic bundles
Higgs bundles
Moduli spaces.
Geometria algebraica
1201.01 Geometría Algebraica
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spelling Betti numbers of the moduli space of rank 3 parabolic Higgs bundles.García Prada, O.Gothen, P.B.Muñoz, Vicente512.7Parabolic bundlesHiggs bundlesModuli spaces.Geometria algebraica1201.01 Geometría AlgebraicaParabolic Higgs bundles on a Riemann surface are of interest for many reasons, one of them being their importance in the study of representations of the fundamental group of the punctured surface in the complex general linear group. In this paper we calculate the Betti numbers of the moduli space of rank 3 parabolic Higgs bundles with fixed and non-fixed determinant, using Morse theory. A key point is that certain critical submanifolds of the Morse function can be identified with moduli spaces of parabolic triples. These moduli spaces come in families depending on a real parameter and we carry out a careful analysis of them by studying their variation with this parameter. Thus we obtain in particular information about the topology of the moduli spaces of parabolic triples for the value of the parameter relevant to the study of parabolic Higgs bundles. The remaining critical submanifolds are also described: one of them is the moduli space of parabolic bundles, while the remaining ones have a description in terms of symmetric products of the Riemann surface. As another consequence of our Morse theoretic analysis, we obtain a proof of the parabolic version of a theorem of Laumon, which states that the nilpotent cone (the preimage of zero under the Hitchin map) is a Lagrangian subvariety of the moduli space of parabolic Higgs bundles.American Mathematical SocietyUniversidad Complutense de Madrid20072007-01-0120072007-01-01journal articlehttp://purl.org/coar/resource_type/c_6501info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/20.500.14352/50605reponame:Docta Complutenseinstname:Universidad Complutense de Madrid (UCM)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:docta.ucm.es:20.500.14352/506052026-06-02T12:44:21Z
dc.title.none.fl_str_mv Betti numbers of the moduli space of rank 3 parabolic Higgs bundles.
title Betti numbers of the moduli space of rank 3 parabolic Higgs bundles.
spellingShingle Betti numbers of the moduli space of rank 3 parabolic Higgs bundles.
García Prada, O.
512.7
Parabolic bundles
Higgs bundles
Moduli spaces.
Geometria algebraica
1201.01 Geometría Algebraica
title_short Betti numbers of the moduli space of rank 3 parabolic Higgs bundles.
title_full Betti numbers of the moduli space of rank 3 parabolic Higgs bundles.
title_fullStr Betti numbers of the moduli space of rank 3 parabolic Higgs bundles.
title_full_unstemmed Betti numbers of the moduli space of rank 3 parabolic Higgs bundles.
title_sort Betti numbers of the moduli space of rank 3 parabolic Higgs bundles.
dc.creator.none.fl_str_mv García Prada, O.
Gothen, P.B.
Muñoz, Vicente
author García Prada, O.
author_facet García Prada, O.
Gothen, P.B.
Muñoz, Vicente
author_role author
author2 Gothen, P.B.
Muñoz, Vicente
author2_role author
author
dc.contributor.none.fl_str_mv Universidad Complutense de Madrid
dc.subject.none.fl_str_mv 512.7
Parabolic bundles
Higgs bundles
Moduli spaces.
Geometria algebraica
1201.01 Geometría Algebraica
topic 512.7
Parabolic bundles
Higgs bundles
Moduli spaces.
Geometria algebraica
1201.01 Geometría Algebraica
description Parabolic Higgs bundles on a Riemann surface are of interest for many reasons, one of them being their importance in the study of representations of the fundamental group of the punctured surface in the complex general linear group. In this paper we calculate the Betti numbers of the moduli space of rank 3 parabolic Higgs bundles with fixed and non-fixed determinant, using Morse theory. A key point is that certain critical submanifolds of the Morse function can be identified with moduli spaces of parabolic triples. These moduli spaces come in families depending on a real parameter and we carry out a careful analysis of them by studying their variation with this parameter. Thus we obtain in particular information about the topology of the moduli spaces of parabolic triples for the value of the parameter relevant to the study of parabolic Higgs bundles. The remaining critical submanifolds are also described: one of them is the moduli space of parabolic bundles, while the remaining ones have a description in terms of symmetric products of the Riemann surface. As another consequence of our Morse theoretic analysis, we obtain a proof of the parabolic version of a theorem of Laumon, which states that the nilpotent cone (the preimage of zero under the Hitchin map) is a Lagrangian subvariety of the moduli space of parabolic Higgs bundles.
publishDate 2007
dc.date.none.fl_str_mv 2007
2007-01-01
2007
2007-01-01
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/20.500.14352/50605
url https://hdl.handle.net/20.500.14352/50605
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv American Mathematical Society
publisher.none.fl_str_mv American Mathematical Society
dc.source.none.fl_str_mv reponame:Docta Complutense
instname:Universidad Complutense de Madrid (UCM)
instname_str Universidad Complutense de Madrid (UCM)
reponame_str Docta Complutense
collection Docta Complutense
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15.301629

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