Flat rank two vector bundles on genus two curves Viktoria Heu, Frank Loray

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Viktoria Heu, Frank Loray. Flat rank two vector bundles on genus two curves. 259 (1247), 2019, Memoirs of the American Mathematical Society, ￿10.1090/memo/1247￿. ￿hal-00927061v2￿

HAL Id: hal-00927061 https://hal.archives-ouvertes.fr/hal-00927061v2 Submitted on 15 Jul 2015

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. FLAT RANK 2 VECTOR BUNDLES ON GENUS 2 CURVES

VIKTORIA HEU AND FRANK LORAY

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 log- arithmic 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 Poincar´efam- ily 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 sl2C-connection over curves of genus 2 and prove the transversality of the induced flow with the locus of unstable bundles.

Contents Introduction 2 1. Preliminaries on connections 6 1.1. Logarithmic connections 6 1.2. Twists and trace 7 1.3. Projective connections and Riccati foliations 7 1.4. Parabolic structures 8 1.5. Elementary transformations 8 1.6. Stability and moduli spaces 9 2. Hyperelliptic correspondence 10 2.1. Topological considerations 11 2.2. A direct algebraic approach 15 3. Flat vector bundles over X 16 3.1. Flatness criterion 16 3.2. Semi-stable bundles and the Narasimhan-Ramanan theorem 17 3.3. Semi-stable decomposable bundles 18 3.4. Semi-stable indecomposable bundles 21 3.5. Unstable and indecomposable: the 6 + 10 Gunning bundles 23

Date: July 10, 2015. 1991 Mathematics Subject Classification. Primary 14H60; Secondary 34Mxx, 32G34, 14Q10. Key words and phrases. Vector Bundles, Moduli Spaces, Parabolic Connections, Higgs Bundles, Kum- mer Surface. The first author is supported by the ANR grants ANR-13-BS01-0001-01 and ANR-13-JS01-0002-01. The second author is supported by CNRS. 1 2 V.HEU AND F.LORAY

3.6. Computation of a system of coordinates 25 4. Anticanonical subbundles 30 4.1. Tyurin subbundles 31 4.2. Extensions of the canonical bundle 37 4.3. Tyurin parametrization 38 5. Flat parabolic vector bundles over the quotient X/ι 42 5.1. Flatness criterion 43 5.2. Dictionary: how special bundles on X occur as special bundles on X/ι 45 5.3. Semi-stable bundles and projective charts 52 5.4. Moving weights and wall-crossing phenomena 58 5.5. Galois and Geiser involutions 62 5.6. Summary: the moduli stack Bun(X) 63 6. The moduli stack Higgs(X) and the Hitchin fibration 64 6.1. A Poincar´efamily on the 2-fold cover Higgs(X/ι) 66 6.2. The Hitchin fibration 68 6.3. Explicit Hitchin Hamiltonians on Higgs(