Flat rank 2 vector bundles on genus 2 curves

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 under- lying 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 allow 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 vanGeemen-Previato. We explicitly describe the isomonodromic foliation in the moduli space of vector bundles with sl(2,C)-connection over curves of genus 2 and prove the transversality of the induced flow with the locus of unstable bundles.
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Pré-publication, Document de travail
This new version (with 20 pages more) contains more details, pictures, tables and results. 2014
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Dernière modification le : vendredi 22 juin 2018 - 01:19:53
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  • HAL Id : hal-00927061, version 2
  • ARXIV : 1401.2449

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Viktoria Heu, Frank Loray. Flat rank 2 vector bundles on genus 2 curves. This new version (with 20 pages more) contains more details, pictures, tables and results. 2014. 〈hal-00927061v2〉

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