Non-minimal scalar-flat Kaehler surfaces and parabolic stability

Details Non-minimal scalar-flat Kaehler surfaces and parabolic stability Non-minimal scalar-flat Kaehler surfaces and parabolic stability

2004-04-22

arxiv.org/abs/math/0404423v5

Inventiones Mathematicae, vol 162 (2005) p237-270

Mathematics

Differential Geometry

Final version, 30 pages, the appendix has been suppressed and will appear elsewhere, to be published in Inventiones Mathematic

Scientific paper

10.1007/s00222-004-0436-6

A new construction is presented of scalar-flat Kaehler 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 CP^2 blown up at 10 suitably chosen points, admits a scalar-flat Kaehler 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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