Flat Rank Two Vector Bundles on Genus Two Curves.

Cover -- Title page -- Introduction -- Chapter 1. Preliminaries on connections -- 1.1. Logarithmic connections -- 1.2. Twists and trace -- 1.3. Projective connections and Riccati foliations -- 1.4. Parabolic structures -- 1.5. Elementary transformations -- 1.6. Stability and moduli spaces -- Chapter 2. Hyperelliptic correspondence -- 2.1. Topological considerations -- 2.2. A direct algebraic approach -- Chapter 3. Flat vector bundles over -- 3.1. Flatness criterion -- 3.2. Semi-stable bundles and the Narasimhan-Ramanan theorem -- 3.3. Semi-stable decomposable bundles -- 3.4. Semi-stable indecomposable bundles -- 3.5. Unstable and indecomposable: the 6+10 Gunning bundles -- 3.6. Computation of a system of coordinates -- Chapter 4. Anticanonical subbundles -- 4.1. Tyurin subbundles -- 4.2. Extensions of the canonical bundle -- 4.3. Tyurin parametrization -- Chapter 5. Flat parabolic vector bundles over the quotient / -- 5.1. Flatness criterion -- 5.2. Dictionary: how special bundles on occur as special bundles on / -- 5.3. Semi-stable bundles and projective charts -- 5.4. Moving weights and wall-crossing phenomena -- 5.5. Galois and Geiser involutions -- 5.6. Summary: the moduli stack \BUN( ) -- Chapter 6. The moduli stack ℌ ( ) and the Hitchin fibration -- 6.1. A Poincaré family on the 2-fold cover \HIGGS( / ) -- 6.2. The Hitchin fibration -- 6.3. Explicit Hitchin Hamiltonians on \HIGGS( / ) -- 6.4. Explicit Hitchin Hamiltonians on \HIGGS( ) -- 6.5. Comparison to existing formulae -- Chapter 7. The moduli stack ℭ ( ) -- 7.1. An explicit atlas -- 7.2. The apparent map on \CON( / ) -- 7.3. A Lagrangian section of \CON( )→\BUN( ) -- Chapter 8. Application to isomonodromic deformations -- 8.1. Darboux coordinates -- 8.2. Hamiltonian system -- 8.3. Transversality to the locus of Gunning bundles -- 8.4. Projective structures and Hejhal's theorem.

Bibliography -- Back Cover.

The authors 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 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 well-known moduli space of the underlying parabolic bundles with the classical approaches by Narasimhan-Ramanan, 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 Narasimhan-Ramanan 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 Geemen-Previato. 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.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.