Moduli of parabolic Higgs bundles and Atiyah algebroids

Marina Logares; Johan Martens

doi:10.1515/CRELLE.2010.090

Moduli of parabolic Higgs bundles and Atiyah algebroids

Marina Logares^*, Johan Martens

^*Corresponding author for this work

School of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper we study the geometry of the moduli space of (non-strongly) parabolic Higgs bundles over a Riemann surface with marked points. We show that this space possesses a Poisson structure, extending the one on the dual of an Atiyah algebroid over the moduli space of parabolic vector bundles. By considering the case of full flags, we get a Grothendieck-Springer resolution for all other flag types, in particular for the moduli spaces of twisted Higgs bundles, as studied by Markman and Bottacin and used in the recent work of Laumon-Ngo. We discuss the Hitchin system, and demonstrate that all these moduli spaces are integrable systems in the Poisson sense.

Original language	English
Pages (from-to)	89-116
Number of pages	28
Journal	Journal für die reine und angewandte Mathematik
Volume	2010
Issue number	649
DOIs	https://doi.org/10.1515/CRELLE.2010.090
Publication status	Published - Dec 2010

Keywords / Materials (for Non-textual outputs)

MANIFOLDS
INTEGRABLE SYSTEMS
VECTOR-BUNDLES
SHEAVES
RIEMANN SURFACE
PAIRS
SPACES
THEOREM
EQUATIONS
SPECTRAL CURVES

Access to Document

10.1515/CRELLE.2010.090

Cite this

@article{0c3801afab914c98911af1372ec796cc,

title = "Moduli of parabolic Higgs bundles and Atiyah algebroids",

abstract = "In this paper we study the geometry of the moduli space of (non-strongly) parabolic Higgs bundles over a Riemann surface with marked points. We show that this space possesses a Poisson structure, extending the one on the dual of an Atiyah algebroid over the moduli space of parabolic vector bundles. By considering the case of full flags, we get a Grothendieck-Springer resolution for all other flag types, in particular for the moduli spaces of twisted Higgs bundles, as studied by Markman and Bottacin and used in the recent work of Laumon-Ngo. We discuss the Hitchin system, and demonstrate that all these moduli spaces are integrable systems in the Poisson sense.",

keywords = "MANIFOLDS, INTEGRABLE SYSTEMS, VECTOR-BUNDLES, SHEAVES, RIEMANN SURFACE, PAIRS, SPACES, THEOREM, EQUATIONS, SPECTRAL CURVES",

author = "Marina Logares and Johan Martens",

year = "2010",

month = dec,

doi = "10.1515/CRELLE.2010.090",

language = "English",

volume = "2010",

pages = "89--116",

journal = "Journal f{\"u}r die reine und angewandte Mathematik",

issn = "0075-4102",

publisher = "De Gruyter",

number = "649",

}

TY - JOUR

T1 - Moduli of parabolic Higgs bundles and Atiyah algebroids

AU - Logares, Marina

AU - Martens, Johan

PY - 2010/12

Y1 - 2010/12

N2 - In this paper we study the geometry of the moduli space of (non-strongly) parabolic Higgs bundles over a Riemann surface with marked points. We show that this space possesses a Poisson structure, extending the one on the dual of an Atiyah algebroid over the moduli space of parabolic vector bundles. By considering the case of full flags, we get a Grothendieck-Springer resolution for all other flag types, in particular for the moduli spaces of twisted Higgs bundles, as studied by Markman and Bottacin and used in the recent work of Laumon-Ngo. We discuss the Hitchin system, and demonstrate that all these moduli spaces are integrable systems in the Poisson sense.

AB - In this paper we study the geometry of the moduli space of (non-strongly) parabolic Higgs bundles over a Riemann surface with marked points. We show that this space possesses a Poisson structure, extending the one on the dual of an Atiyah algebroid over the moduli space of parabolic vector bundles. By considering the case of full flags, we get a Grothendieck-Springer resolution for all other flag types, in particular for the moduli spaces of twisted Higgs bundles, as studied by Markman and Bottacin and used in the recent work of Laumon-Ngo. We discuss the Hitchin system, and demonstrate that all these moduli spaces are integrable systems in the Poisson sense.

KW - MANIFOLDS

KW - INTEGRABLE SYSTEMS

KW - VECTOR-BUNDLES

KW - SHEAVES

KW - RIEMANN SURFACE

KW - PAIRS

KW - SPACES

KW - THEOREM

KW - EQUATIONS

KW - SPECTRAL CURVES

U2 - 10.1515/CRELLE.2010.090

DO - 10.1515/CRELLE.2010.090

M3 - Article

SN - 0075-4102

VL - 2010

SP - 89

EP - 116

JO - Journal für die reine und angewandte Mathematik

JF - Journal für die reine und angewandte Mathematik

IS - 649

ER -

Moduli of parabolic Higgs bundles and Atiyah algebroids

Abstract

Keywords / Materials (for Non-textual outputs)

Access to Document

Fingerprint

Cite this