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Flat Rank Two Vector Bundles on Genus Two Curves
About this Title
Viktoria Heu, IRMA, 7 rue René-Descartes, 67084 Strasbourg Cedex, France and Frank Loray, University of Rennes, CNRS, IRMAR—UMR 6625, F-35000 Rennes, France
Publication: Memoirs of the American Mathematical Society
Publication Year:
2019; Volume 259, Number 1247
ISBNs: 978-1-4704-3566-0 (print); 978-1-4704-5249-0 (online)
DOI: https://doi.org/10.1090/memo/1247
Published electronically: April 16, 2019
Keywords: Vector Bundles,
moduli spaces,
parabolic connections,
Higgs bundles,
Kummer surface
MSC: Primary 14H60; Secondary 34Mxx, 32G34, 14Q10
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
Abstract
We study the moduli space of trace-free 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 we 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. We establish explicit links between the well-known moduli space of the underlying parabolic bundles with the classical approaches by Narasimhan-Ramanan, Tyurin and Bertram. This allows us to explain a certain number of geometric phenomena in the considered moduli spaces such as the classical $(16,6)$-configuration of the Kummer surface. We also recover a Poincaré family due to Bolognesi on a degree 2 cover of the Narasimhan-Ramanan moduli space. We explicitly compute the Hitchin integrable system on the moduli space of Higgs bundles and compare the Hitchin Hamiltonians with those found by van Geemen-Previato. We 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.- M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181–207. MR 86359, DOI 10.1090/S0002-9947-1957-0086359-5
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