Electronic ISBN:  9781470404833 
Product Code:  MEMO/187/879.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 187; 2007; 80 ppMSC: Primary 14;
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 the authors calculate the Betti numbers of the moduli space of rank 3 parabolic Higgs bundles with fixed and nonfixed 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 the authors carry out a careful analysis of them by studying their variation with this parameter. Thus the authors 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 their Morse theoretic analysis, the authors 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.

Table of Contents

Chapters

1. Introduction

2. Parabolic Higgs bundles

3. Morse theory on the moduli space

4. Parabolic triples

5. Critical values and flips

6. Parabolic triples with $r_1 = 2$ and $r_2 = 1$

7. Critical submanifolds of type (1,1,1)

8. Critical submanifolds of type (1,2)

9. Critical submanifolds of type (2,1)

10. Betti numbers of the moduli space of rank three parabolic bundles

11. Betti numbers of the moduli space of rank three parabolic Higgs bundles

12. The fixed determinant case


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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 the authors calculate the Betti numbers of the moduli space of rank 3 parabolic Higgs bundles with fixed and nonfixed 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 the authors carry out a careful analysis of them by studying their variation with this parameter. Thus the authors 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 their Morse theoretic analysis, the authors 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.

Chapters

1. Introduction

2. Parabolic Higgs bundles

3. Morse theory on the moduli space

4. Parabolic triples

5. Critical values and flips

6. Parabolic triples with $r_1 = 2$ and $r_2 = 1$

7. Critical submanifolds of type (1,1,1)

8. Critical submanifolds of type (1,2)

9. Critical submanifolds of type (2,1)

10. Betti numbers of the moduli space of rank three parabolic bundles

11. Betti numbers of the moduli space of rank three parabolic Higgs bundles

12. The fixed determinant case