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Softcover ISBN:  9781470435660 
Product Code:  MEMO/259/1247 
List Price:  $81.00 
MAA Member Price:  $72.90 
AMS Member Price:  $48.60 
eBook ISBN:  9781470452490 
Product Code:  MEMO/259/1247.E 
List Price:  $81.00 
MAA Member Price:  $72.90 
AMS Member Price:  $48.60 
Softcover ISBN:  9781470435660 
eBookISBN:  9781470452490 
Product Code:  MEMO/259/1247.B 
List Price:  $162.00$121.50 
MAA Member Price:  $145.80$109.35 
AMS Member Price:  $97.20$72.90 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 259; 2019; 103 ppMSC: Primary 14; Secondary 34; 32;
The authors study the moduli space of tracefree irreducible rank 2 connections over a curve of genus 2 and the forgetful map towards the moduli space of underlying vector bundles (including unstable bundles), for which they compute a natural Lagrangian rational section. As a particularity of the genus \(2\) case, connections as above are invariant under the hyperelliptic involution: they descend as rank \(2\) logarithmic connections over the Riemann sphere. The authors establish explicit links between the wellknown moduli space of the underlying parabolic bundles with the classical approaches by NarasimhanRamanan, Tyurin and Bertram. This allows the authors to explain a certain number of geometric phenomena in the considered moduli spaces such as the classical \((16,6)\)configuration of the Kummer surface.
The authors also recover a Poincaré family due to Bolognesi on a degree 2 cover of the NarasimhanRamanan moduli space. They explicitly compute the Hitchin integrable system on the moduli space of Higgs bundles and compare the Hitchin Hamiltonians with those found by van GeemenPreviato. They explicitly describe the isomonodromic foliation in the moduli space of vector bundles with \(\mathfrak sl_2\)connection over curves of genus 2 and prove the transversality of the induced flow with the locus of unstable bundles. 
Table of Contents

Chapters

Introduction

1. Preliminaries on connections

2. Hyperelliptic correspondence

3. Flat vector bundles over $X$

4. Anticanonical subbundles

5. Flat parabolic vector bundles over the quotient $X/\iota $

6. The moduli stack $\mathfrak {Higgs}(X)$ and the Hitchin fibration

7. The moduli stack $\mathfrak {Con} (X)$

8. Application to isomonodromic deformations


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The authors study the moduli space of tracefree irreducible rank 2 connections over a curve of genus 2 and the forgetful map towards the moduli space of underlying vector bundles (including unstable bundles), for which they compute a natural Lagrangian rational section. As a particularity of the genus \(2\) case, connections as above are invariant under the hyperelliptic involution: they descend as rank \(2\) logarithmic connections over the Riemann sphere. The authors establish explicit links between the wellknown moduli space of the underlying parabolic bundles with the classical approaches by NarasimhanRamanan, Tyurin and Bertram. This allows the authors to explain a certain number of geometric phenomena in the considered moduli spaces such as the classical \((16,6)\)configuration of the Kummer surface.
The authors also recover a Poincaré family due to Bolognesi on a degree 2 cover of the NarasimhanRamanan moduli space. They explicitly compute the Hitchin integrable system on the moduli space of Higgs bundles and compare the Hitchin Hamiltonians with those found by van GeemenPreviato. They explicitly describe the isomonodromic foliation in the moduli space of vector bundles with \(\mathfrak sl_2\)connection over curves of genus 2 and prove the transversality of the induced flow with the locus of unstable bundles.

Chapters

Introduction

1. Preliminaries on connections

2. Hyperelliptic correspondence

3. Flat vector bundles over $X$

4. Anticanonical subbundles

5. Flat parabolic vector bundles over the quotient $X/\iota $

6. The moduli stack $\mathfrak {Higgs}(X)$ and the Hitchin fibration

7. The moduli stack $\mathfrak {Con} (X)$

8. Application to isomonodromic deformations