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Stability of rank 2 vector bundles along isomonodromic deformations

Abstract

We are interested in the stability of holomorphic rank 2 vector bundles of degree 0 over compact Riemann surfaces, which are provided with irreducible meromophic tracefree connections. In the case of a logarithmic connection on the Riemann sphere, such a vector bundle will be trivial up to the isomonodromic deformation associated to a small move of the poles, according to a result of A. Bolibruch. In the general case of meromorphic connections over Riemann surfaces of arbitrary genus, we prove that the vector bundle will be semi-stable, up to a small isomonodromic deformation. More precisely, the vector bundle underlying the universal isomonodromic deformation is generically semi-stable along the deformation, and even maximally stable. For curves of genus g ≥ 2, this result is non-trivial even in the case of non-singular connections.

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Correspondence to Viktoria Heu.

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The author was partially supported by ANR SYMPLEXE BLAN06-3-137237.

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Heu, V. Stability of rank 2 vector bundles along isomonodromic deformations. Math. Ann. 344, 463–490 (2009). https://doi.org/10.1007/s00208-008-0316-2

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  • DOI: https://doi.org/10.1007/s00208-008-0316-2

Mathematics Subject Classification (2000)

  • 53B05
  • 14D05
  • 14H60
  • 34Mxx