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Non-minimal scalar-flat Kähler surfaces and parabolic stability

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A new construction is presented of scalar-flat Kähler metrics on non-minimal ruled surfaces. The method is based on the resolution of singularities of orbifold ruled surfaces which are closely related to rank-2 parabolically stable holomorphic bundles. This rather general construction is shown also to give new examples of low genus: in particular, it is shown that \(\mathbb{CP}^2\) blown up at 10 suitably chosen points, admits a scalar-flat Kähler metric; this answers a question raised by Claude LeBrun in 1986 in connection with the classification of compact self-dual 4-manifolds.

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Authors and Affiliations

  1. MIT, 77 Massachusetts Avenue, Cambridge, MA, 02139, USA

    Yann Rollin

  2. School of Mathematics, King’s buildings, Edinburgh, Scotland, UK

    Michael Singer

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  1. Yann Rollin
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  2. Michael Singer
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Correspondence to Yann Rollin or Michael Singer.

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Rollin, Y., Singer, M. Non-minimal scalar-flat Kähler surfaces and parabolic stability. Invent. math. 162, 235–270 (2005). https://doi.org/10.1007/s00222-004-0436-6

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  • DOI: https://doi.org/10.1007/s00222-004-0436-6

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